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[ "<title>Introduction</title>", "<p>Acute scrotal pain is a relatively common emergency presentation, both in the primary care setting and in the emergency department, comprising approximately 0.5% of all emergency visits in the United States annually [##UREF##0##1##]. Testicular torsion is a true urological emergency in such cases of acute scrotal pain. Torsion is a time-sensitive condition in which twisting of the spermatic cord occurs and testicular blood supply compromise ensues, leading to acute onset severe scrotal pain [##UREF##0##1##]. Understanding the anatomy of the testicle is important in comprehending the pathophysiology of torsion. The tunica vaginalis is usually firmly attached to the posterolateral aspect of the testicle, and within it, the spermatic cord is not mobile. In cases where the attachment of the tunica vaginalis is high, the spermatic cord can twist more easily inside, leading to intravaginal torsion [##UREF##0##1##].</p>", "<p>The incidence of testicular torsion is highest amongst prepubertal males; however, it can occur at any age [##UREF##0##1##]. Torsion typically presents with symptoms of acutely painful hemi-scrotum with a tender, elevated testis felt at a horizontal lie on clinical examination. As arterial blood supply is abruptly ceased, testicular detorsion is a race against time. Every hour that passes from the onset of symptoms has been shown to decrease the salvageability rate of the torted testis. Another significant factor that impacts testicular salvage is the degree of torsion. In most cases, 90-180 degrees of testicular rotation is capable of compromising testicular blood flow. Further degrees of torsion are rarer and significantly decrease the viability of the testes. The best salvage rates are seen within less than eight hours from the onset but become rare if more than 24 hours have elapsed [##UREF##0##1##,##UREF##1##2##].</p>", "<p>A testicular ultrasound can be invaluable when available in a timely manner; however, it must not delay quick surgical intervention. It is considered the main adjunctive diagnostic modality beyond clinical examination. A color Doppler flow ultrasound for testicular torsion is approximately 93% sensitive and 100% specific, aiding in both diagnosis of testicular torsion as well as an assessment of testicular volume [##REF##23044376##3##,##UREF##2##4##]. Once the diagnosis is made, the standard of care is immediate surgical intervention for testicular detorsion and bilateral orchidopexy if the testis is viable, or orchidectomy if necrosis has occurred [##REF##32683416##5##].</p>", "<p>The surgical management of testicular torsion depends on whether the testis is salvageable during surgical exploration. A black-colored testis is deemed necrotic, leading to orchidectomy, while a purple to whitish-pink-colored testis is considered viable, and bilateral orchidopexy is performed [##REF##32683416##5##,##REF##16330742##6##]. One major sequela following orchidopexy for torsion is the decrease in testicular volume. As testicular volumes decrease so does its ability for spermatogenesis and testosterone production. A poorly functioning testis can have long-term effects on patients in terms of fertility, as well as decreased libido, sexual dysfunction, and psychological impacts [##REF##16330742##6##].</p>", "<p>The aim of this study is to assess testicular volume loss post orchidopexy in patients who presented with testicular torsion as well as to identify the significance of the degree of rotation and duration of torsion in post-fixation volume loss.</p>" ]

[ "<title>Materials and methods</title>", "<p>All patients who underwent scrotal exploration for a primary diagnosis of testicular torsion between June 1, 2016, and January 15, 2023, were reviewed. All data were recorded from the hospital’s electronic database. Patients were excluded if they underwent an orchidectomy, had a diagnosis other than testicular torsion once scrotal exploration was done, or did not perform a follow-up scrotal ultrasound. Additionally, patients who were referred from other centers and had preoperative ultrasounds done outside our institute or who underwent an orchidopexy for undescended testis earlier in life were excluded.</p>", "<p>The information obtained from the electronic files included the patients’ demographics such as age, duration of symptoms, and laterality. Images were reviewed for preoperative ultrasound findings, which included confirmation of testicular torsion as well as testicular volume measurements. Routine postoperative scrotal ultrasound is not done in our center unless patients have postoperative concerns that necessitate it. However, patients with at least six months of follow-up were contacted by phone and testicular volumes were measured by scrotal ultrasound. Testicular measurement was done using the formula of length (mm) × width (mm) × weight (mm) × 0.72. All scrotal ultrasounds were done using a GE LOGIQ E9 ultrasound machine (General Electric, Boston, Massachusetts, United States) using a linear 9 Hz transducer probe. All radiographic reporting was done by a senior radiology resident.</p>", "<p>The local protocol in our center for a patient who presents with acute scrotal pain is simultaneous immediate shifting to ultrasound assessment and urological consultation. Once testicular torsion is suspected, patients are booked and shifted for surgery. The standard operative procedure practiced in our center is vertical scrotal incision starting on the affected side followed by delivery of the torted testis and assessment for viability and color followed by prompt detorsion while recording the degree of torsion. Once detorted, warm compressors are kept, and contralateral orchidopexy is performed. If the torted testis regains color and is visually viable, a dartos pouch is fashioned and 3-point fixation at 3, 6, and 9 o’clock is done using 3-0 Vicryl (Ethicon Inc., Raritan, New Jersey, United States). The dartos layer is also closed using 3-0 Vicryl, whereas skin closure is done using Rapide Vicryl. All surgeries were done under spinal anesthesia and performed by a senior urology resident.</p>", "<p>For statistical analysis purposes, degrees of testicular torsion were classified into mild (90-180 degrees), moderate (180-360 degrees), and severe (&gt;360 degrees). Furthermore, time for surgery was recorded in hours from the onset of symptoms until surgery start time and classified into mild (Less than four hours), moderate (four to six hours), and severe (more than six hours). A linear regression model was used to predict the relationship between testicular volume loss and the independent variables of degree of torsion and time to surgery. The equation used for the regression model was “Volume = β0 + β1 * Independent variable” in which β1 is the regression coefficient for the degree of torsion and β0 is the intercept. </p>", "<p>Additionally, given that time is an ordinal value, Spearman correlation coefficients were utilized to assess the relationship between the time of surgery and postoperative testicular volume loss. All statistical analysis was conducted using IBM SPSS Statistics for Windows, Version 29.0 (Released 2022; IBM Corp., Armonk, New York, United States), and 95% confidence intervals were calculated for the treatment’s success rates with p-values of &lt; 0.05 considered statistically significant.</p>" ]

[ "<title>Results</title>", "<p>A total of 109 patient records were reviewed within the specific time frame. Forty-seven patients were excluded as per the exclusion criteria mentioned, which gave us a sample size of 62 patients. The patient and surgical parameters are given in Table ##TAB##0##1##.</p>", "<p>Our data showed that 29 (46.7%) patients presented with right-sided testicular torsion and 33 (53.3%) patients presented with left-sided testicular torsion. In terms of degrees of torsion, 19 patients (30.6%) had a mild degree whereas 28 (45.1%) patients and 15 (24.1%) patients had moderate and severe torsion respectively. The mean preoperative testicular volume on the unaffected side was 17.9 ml + 1.7 and the postoperative mean volume was 17.5 ml + 1.9. Comparatively, the mean preoperative volume of the affected testis was 18.5 ml + 2.1 whereas the mean postoperative volume was calculated for the different degrees of torsion and was as follows: mild 18.0 ml + 0.7, moderate 16.5 ml + 0.4, and severe 13.6 ml + 0.6.</p>", "<p>In terms of time to surgery, 14 (22.5%) patients were considered within the mild group (&lt; four hours), 31 (50%) patients and 17 (27.4%) patients were considered moderate (four to six hours), and severe (&gt; six hours) respectively. The mean preoperative testicular volume in both the unaffected and affected side are identical to the previously mentioned volumes. However, the mean preoperative volume and the mean postoperative volume were calculated for the different times to surgery and were as follows: mild 17.8 ml + 0.5, moderate 16.2 ml + 0.3, and severe 12.9 ml + 0.8.</p>", "<p>Figure ##FIG##0##1## illustrates how the mean testicular volume loss in ml increases as the severity of the degree of torsion and time to surgery increases. However, it can be noted that time to surgery (orange curve) has a more pronounced effect on the mean volume loss than the degree of torsion. </p>", "<p>Table ##TAB##1##2## and Table ##TAB##2##3## demonstrate the results of the linear regression models as they pertain to the severity of the degree of torsion and time to surgery with the postoperative testicular volume loss. Furthermore, the following results describe the relationship between the severity of the time to surgery and the postoperative volume loss in the affected testis as seen in Table ##TAB##2##3##. Increasing severity of the degree of torsion as well as the time for surgery have statistically significant (p-value &lt;0.05) effects on postoperative testicular volume loss in ml.</p>", "<p>Spearman correlation coefficients quantifying the relationship between postoperative testicular volume loss and severity of time to surgery were calculated and showed mild torsion: ρ = 0.65 (p &lt; 0.05) moderate torsion: ρ = 0.52 (p &lt; 0.05), severe torsion: ρ = 0.40 (p &lt; 0.05). This is a positive correlation and signifies that as time to surgery increases, postoperative testicular volume loss tends to be higher. Moreover, the analysis showed that on average with every additional hour from the onset of symptoms to surgery, the approximate volume loss will be 0.15 ml; however, once time exceeds the 4.5-hour mark, the mean volume loss is 0.4 ml for every additional hour.</p>" ]

[ "<title>Discussion</title>", "<p>The management of testicular torsion is immediate surgical intervention with the aim of untwisting the spermatic cord and restoring blood supply to the affected testis as soon as possible [##UREF##3##7##]. Testicular salvageability is directly correlated to the time it takes to undergo surgical correction [##REF##25671040##8##]. This has been best described in the systematic review done by Mellick et al., which concluded that if the surgical correction is conducted within less than six hours, the testis salvageability rate is around 97.2%, whereas this number decreases to 7.4% in patients who present after 48 hours [##REF##28953100##9##]. Moreover, testicular spermatogenesis and hormonal production are proportionately affected by the total testicular volume; hence, testicular volume following repair is an important factor for determining postoperative testis function. Mellick et al. also noted a permanent effect on testicular spermatogenesis and endocrine function, which occurs once the period of torsion exceeds eight hours [##REF##28953100##9##]. These findings align with what we observed in our study, which showed that the longer the time increases, the higher the volume loss. However, when comparing our findings to the systematic review, we notice that significant postoperative volume loss can occur in patients who underwent orchidopexy as early as four hours after the onset of symptoms.</p>", "<p>Furthermore, we demonstrated that time is clearly the more important determinant of postoperative testicular volume loss, as seen in Figure ##FIG##0##1##. However, the effect of the degree of torsion cannot be understated. This is clear in the comparative graph in Figure ##FIG##0##1##, which demonstrates the different mean volume losses between the different severity grades of time and degree of torsion. Additionally, our regression model values show a significant statistical correlation between higher degrees of testicular torsion and increased postoperative volume loss. Howe et al., in their study conducted in 2017, also looked into the degree of twisting and its clinical significance on testicular torsion outcomes [##REF##29354505##10##]. They concluded that when the spermatic cord undergoes more than 360 degrees of twisting, there is up to a 25% chance of orchiectomy. However, in the current study, we were more concerned with the salvageable testicular volume, and to add to their findings, we conclude that a severe degree of torsion (360 degrees and more) was statistically (p-value &lt;0.05) associated with the highest amount of volume loss where we saw a mean volume loss of around 4.9 ml.</p>", "<p>Additionally, in a study conducted in 2016 by Dias Filho et al., they reviewed the spermatic cord rotation effect on the outcomes of intravaginal testicular torsion [##REF##27619663##11##], and demonstrated similar findings to the current study. They concluded that presentation delay is the major factor in determining surgical outcomes. However, the degree of spermatic cord rotation exerts a multiplicative effect on time to surgery and increases the chances of orchiectomy. Concurrently, they also found that both presentation delay, as well as degree of torsion, were inversely proportional to chances of orchidopexy [##REF##27619663##11##]. However, they did not study the exact effect that these variables have on post-orchidopexy volumes as was done in the present study. </p>", "<p>When looking at the time to surgery as an independent factor for testicular volume loss, it can be seen from Figure ##FIG##0##1## that there appears to be a directly proportionate relationship in which more time leads to higher volume loss. Although the relationship is directly proportionate [##UREF##4##12##], it is not particularly linear. This was noted when our statistical analysis showed that the mean testicular volume loss per hour appears to be significantly higher once the time to surgery exceeds 4.5 hours. Comparatively, the total time average volume loss from the onset of symptoms to surgical correction is only 0.15 ml. This signifies within four to five hours from the onset of symptoms, significant volume loss should be expected even if orchidopexy is done, and the patient should be counseled accordingly to manage the post-operative expectations.</p>", "<p>Furthermore, another study published in 2015, which investigated the factors influencing testicular atrophy following torsion, showed that if the time to surgery exceeds 24 hours, then 91% of patients are expected to develop significant testicular atrophy postoperatively [##REF##26509312##13##]. Our findings are consistent with this; however, we observed that the onset of significant atrophy actually occurs around four hours from the onset of symptoms. This underlines the importance of immediate diagnosis to prevent long-term permanent damage.</p>", "<p>The psychological and clinical impacts of post-orchidopexy testicular volume loss cannot be understated. In addition to concerns that arise regarding future fertility, the masculine perception of the individual can be affected by the physical size of the testis, leading to feelings of inadequacy and even low self-esteem [##REF##30774733##14##,##REF##10782149##15##]. It is well established that libido is considerably affected by stress and can lead to performance anxiety [##REF##10782149##15##]. Patients who suffer from abnormally small testicular sizes or particularly from an uneven-appearing scrotum might experience an added psychological effect leading to decreased ability to perform sexually and even a tendency to avoid sexual intercourse due to fear of being stigmatized [##REF##25269643##16##].</p>", "<p>Limitations of our study include the fact that ultrasound imaging was not performed by the same radiologist, which can lead to differences in calculating testicular volumes. This was difficult to address given that the presentation of torsion is an acute emergency and imaging needs to be done immediately by the on-call senior radiologist without the possibility of delay. Furthermore, the postoperative testicular volume was measured at approximately six-month intervals due to resource and schedule limitations. Further imaging at longer intervals, such as one and two years postoperatively, can help further assess the effects torsion has on testicular volume.</p>" ]

[ "<title>Conclusions</title>", "<p>Our study indicates that earlier surgical intervention and correction of torsion are associated with enhanced preservation of postoperative testicular volume. Both the degree of torsion and time to surgery influence mean volume loss; however, time to surgery shows a greater effect on mean volume loss. These results highlight the importance of early diagnosis and intervention in cases of testicular torsion to minimize the risk of long-term testicular volume loss.</p>" ]

[ "<p>Introduction</p>", "<p>Testicular torsion is an urological emergency. It is a time-sensitive condition in which twisting of the spermatic cord and testicular blood supply occurs, causing acute onset severe scrotal pain. The incidence of testicular torsion is highest amongst prepubertal males; however, it can occur at any age. Every hour that passes from the onset of symptoms has been shown to decrease the salvageability rate of the torted testis. Another significant factor that impacts testicular salvage is the degree of torsion. Prompt surgical exploration of the scrotum and orchidopexy, if the testis is salvageable, is the mainstay of treatment. A major sequela following orchidopexy for torsion is the decrease in testicular volume. The aim of this study is to assess testicular volume loss post orchidopexy in patients who presented with testicular torsion, as well as to identify the significance of the degree of rotation and duration of torsion in post-fixation volume loss.</p>", "<p>Methods</p>", "<p>This is a retrospective study in which all patients who underwent scrotal exploration for a primary diagnosis of testicular torsion between June 1, 2016, to January 15, 2023, were reviewed. The information obtained included the patients’ demographics such as age, duration of symptoms, and laterality. Ultrasound images were reviewed for pre- and postoperative findings which included confirmation of testicular torsion as well as testicular volume measurements. Patients were excluded if they underwent an orchidectomy, had a diagnosis other than testicular torsion once scrotal exploration was done, or did not perform a follow-up scrotal ultrasound. Additionally, patients who underwent an orchidopexy for undescended testis earlier in life were also excluded. For statistical analysis purposes, degrees of testicular torsion and time to surgery were classified into mild, moderate, and severe.</p>", "<p>Results</p>", "<p>A total of 109 patient records were reviewed within the specific time frame. Of these, 47 patients were excluded as per the exclusion criteria mentioned previously, which gave us a sample size of 62 patients. Our findings showed that increasing severity of the degree of torsion as well as the time for surgery have statistically significant (p-value &lt;0.05) effects on postoperative testicular volume loss. However, it was noted that time to surgery has a more pronounced effect on the mean volume loss than the degree of torsion. Moreover, the analysis also showed that, on average, with every additional hour from the onset of symptoms to surgery, the approximate volume loss is 0.15 ml. However, once time exceeds the 4.5-hour mark, the mean volume loss is 0.4 ml for each additional hour.</p>", "<p>Conclusion</p>", "<p>The current study indicates that earlier surgical intervention and correction of torsion are associated with enhanced preservation of postoperative testicular volume. Both the degree of torsion and time to surgery influence mean volume loss; however, time to surgery has a greater impact on the mean volume loss. These results highlight the importance of early diagnosis and intervention in cases of testicular torsion to minimize the risk of long-term testicular volume loss.</p>" ]

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[ "<fig position=\"anchor\" fig-type=\"figure\" id=\"FIG1\"><label>Figure 1</label><caption><title>Mean postoperative volume loss in ml in affected testis with regard to severity of degree of torsion (blue curve) compared to that of time to surgery severity (orange curve)</title></caption></fig>" ]

[ "<table-wrap position=\"float\" id=\"TAB1\"><label>Table 1</label><caption><title>Patient and surgical parameters</title></caption><table frame=\"hsides\" rules=\"groups\"><tbody><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\"> </td><td rowspan=\"1\" colspan=\"1\">Mean + SD</td><td rowspan=\"1\" colspan=\"1\">Maximum Value</td><td rowspan=\"1\" colspan=\"1\">Minimum Value</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Age (in years)</td><td rowspan=\"1\" colspan=\"1\">17.4 + 1.5 (32-14)</td><td rowspan=\"1\" colspan=\"1\">32</td><td rowspan=\"1\" colspan=\"1\">14</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Duration (in hours)</td><td rowspan=\"1\" colspan=\"1\">5.8 + 1.2 (30-2)</td><td rowspan=\"1\" colspan=\"1\">30</td><td rowspan=\"1\" colspan=\"1\">2</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Degree of Torsion</td><td rowspan=\"1\" colspan=\"1\">180 + 90 (90-540)</td><td rowspan=\"1\" colspan=\"1\">90</td><td rowspan=\"1\" colspan=\"1\">540</td></tr></tbody></table></table-wrap>", "<table-wrap position=\"float\" id=\"TAB2\"><label>Table 2</label><caption><title>Severity of degree of torsion with postoperative testicular volume loss in affected testis</title></caption><table frame=\"hsides\" rules=\"groups\"><tbody><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">\nDegree of Torsion\n</td><td rowspan=\"1\" colspan=\"1\">\nRegression Coefficient\n</td><td rowspan=\"1\" colspan=\"1\">\nP-Value\n</td></tr><tr><td rowspan=\"1\" colspan=\"1\">\nMild (90-180 degrees)\n</td><td rowspan=\"1\" colspan=\"1\">\n0.3\n</td><td rowspan=\"1\" colspan=\"1\">\n&lt; 0.01\n</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">\nModerate (180-360 degrees)\n</td><td rowspan=\"1\" colspan=\"1\">\n0.2\n</td><td rowspan=\"1\" colspan=\"1\">\n&lt; 0.01\n</td></tr><tr><td rowspan=\"1\" colspan=\"1\">\nSevere (&gt; 360 degrees)\n</td><td rowspan=\"1\" colspan=\"1\">\n0.1\n</td><td rowspan=\"1\" colspan=\"1\">\n&lt; 0.05\n</td></tr></tbody></table></table-wrap>", "<table-wrap position=\"float\" id=\"TAB3\"><label>Table 3</label><caption><title>Severity of time to surgery with postoperative testicular volume loss in affected testis</title></caption><table frame=\"hsides\" rules=\"groups\"><tbody><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">\nTime to surgery\n</td><td rowspan=\"1\" colspan=\"1\">\nRegression Coefficient\n</td><td rowspan=\"1\" colspan=\"1\">\nP-Value\n</td></tr><tr><td rowspan=\"1\" colspan=\"1\">\nMild (&lt; 4 hours)\n</td><td rowspan=\"1\" colspan=\"1\">\n0.2\n</td><td rowspan=\"1\" colspan=\"1\">\n&lt; 0.05\n</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">\nModerate (4 -6 hours)\n</td><td rowspan=\"1\" colspan=\"1\">\n0.1\n</td><td rowspan=\"1\" colspan=\"1\">\n&lt; 0.05\n</td></tr><tr><td rowspan=\"1\" colspan=\"1\">\nSevere (&gt; 6hours)\n</td><td rowspan=\"1\" colspan=\"1\">\n0.1\n</td><td rowspan=\"1\" colspan=\"1\">\n&lt; 0.01\n</td></tr></tbody></table></table-wrap>" ]

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[ "<fn-group content-type=\"other\"><title>Author Contributions</title><fn fn-type=\"other\"><p><bold>Concept and design:</bold>  Ahmed A. Al Rashed, Nader Awad, Khalid Abdulaziz, Basma Malalla, Ali H. Al Aradi</p><p><bold>Acquisition, analysis, or interpretation of data:</bold>  Ahmed A. Al Rashed, Nader Awad, Khalid Abdulaziz, Basma Malalla, Ali H. Al Aradi</p><p><bold>Drafting of the manuscript:</bold>  Ahmed A. Al Rashed, Nader Awad, Khalid Abdulaziz, Basma Malalla, Ali H. Al Aradi</p><p><bold>Critical review of the manuscript for important intellectual content:</bold>  Ahmed A. Al Rashed, Nader Awad, Khalid Abdulaziz, Basma Malalla, Ali H. Al Aradi</p><p><bold>Supervision:</bold>  Ahmed A. Al Rashed, Nader Awad, Khalid Abdulaziz, Basma Malalla, Ali H. Al Aradi</p></fn></fn-group>", "<fn-group content-type=\"other\"><title>Human Ethics</title><fn fn-type=\"other\"><p>Consent was obtained or waived by all participants in this study</p></fn></fn-group>", "<fn-group content-type=\"other\"><title>Animal Ethics</title><fn fn-type=\"other\"><p><bold>Animal subjects:</bold> All authors have confirmed that this study did not involve animal subjects or tissue.</p></fn></fn-group>", "<fn-group content-type=\"competing-interests\"><fn fn-type=\"COI-statement\"><p>The authors have declared that no competing interests exist.</p></fn></fn-group>" ]

[ "<graphic xlink:href=\"cureus-0015-00000050543-i01\" position=\"float\"/>" ]

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[ "<title>Introduction</title>", "<p>Obesity, which is considered a multifactorial and complex disease that negatively affects health, is one of the most important causes of preventable deaths today. It contributes to the development of many health problems, such as type 2 diabetes mellitus (DM), cardiovascular disease (CVD), hypertension (HT), hyperlipidemia (HL), cerebrovascular disease, various cancers, obstructive sleep apnea syndrome (OSAS), fatty liver, gastroesophageal reflux, polycystic ovary syndrome (PCOS), osteoarthrosis, and depression [##REF##25590212##1##,##REF##15609891##2##]. Therefore, it creates a significant burden on the health budgets of societies. The prevalence of obesity is increasing in our country and reaching epidemic proportions as it is all over the World. In the Turkey Diabetes Epidemiology Studies (TURDEP) conducted in 1998 and 2010, it was seen that the prevalence of obesity in our country increased from 22.3% to 31.2% [##REF##12196426##3##,##REF##23407904##4##]. In 2016, the WHO reported that the country where obesity is most common in Europe is Turkey, with a prevalence of 29.5% [##UREF##0##5##].</p>", "<p>Three methods are used in obesity treatment: lifestyle change, pharmacotherapy, and bariatric surgery. Clinical studies have shown the effectiveness of lifestyle change and behavioral interventions in obesity. Drugs that provide 5% weight loss in three to six months have been accepted as effective in the treatment and have been approved by pharmaceutical institutions worldwide for managing chronic obesity. Adding pharmacotherapy to lifestyle changes helps with further weight loss. It facilitates patients' compliance with treatment and helps improve obesity-related health risks, thus contributing to increased quality of life.</p>", "<p>Liraglutide (LG), one of the limited number of obesity drugs in our country, is a long-acting GLP-1 receptor agonist (GLP-1 RA) that is resistant to metabolism by the dipeptidyl peptidase (DPP)-IV enzyme [##REF##31790314##6##]. GLP-1 analogs induce weight loss through many central and peripheral mechanisms. It stimulates glucose-dependent insulin release, reduces the glucagon response, and reduces appetite by slowing gastric emptying [##REF##21563859##7##]. In vitro studies have shown that liraglutide has a central effect and directly stimulates the \"cocaine- and amphetamine-regulated transcript\" and \"pro-opiomelanocortin\" neurons and indirect inhibition in neurons expressing \"Agouti-related peptide\" and \"neuropeptide Y\" in the arcuate nucleus of the hypothalamus [##REF##28031776##8##]. Thus, appetite is suppressed, energy intake is reduced, and weight loss occurs with these mechanisms. The positive effects of LG on weight loss and metabolic parameters have been emphasized in various studies [##REF##26132939##9##, ####REF##32594453##10##, ##REF##31062937##11##, ##REF##36478514##12##, ##REF##33473176##13####33473176##13##].</p>", "<p>In Turkey, LG 3 mg (Saxenda®; Novo Nordisk, Bagsvaerd, Denmark) was approved for treating obesity in May 2018. In this study, we aimed to evaluate the effects of LG treatment on weight loss, glycemic and lipid parameters, and the side effects (SE) of this drug as a contribution to the limited number of studies conducted in our country.</p>" ]

[ "<title>Materials and methods</title>", "<p>Study design and participants</p>", "<p>This single-center and retrospective study was approved by the Ethics Committee of Bursa City Hospital (approval number 2023-19/2) and was performed per the Declaration of Helsinki. </p>", "<p>The 67 participants between 18 and 65 years old who used liraglutide for at least 16 weeks due to obesity treatment between July 2020 and September 2022 in Bursa City Hospital were included in the study. Individuals who had undergone bariatric surgery, previously used glucagon-like peptide-1 receptor agonists (GLP-1RA) or another drug that affects weight, and had diseases that could cause weight loss, such as cancer, physiatric disease, eating disorders, and chronic kidney disease, were not included in the study. Pregnant or breastfeeding women are also excluded. LG was given to obese people (BMI&gt;30kg/m2) who could not achieve adequate weight loss despite complying with lifestyle changes or to people with BMI&gt;27 kg/m2 and at least one comorbidity (uncontrolled diabetes mellitus (DM), hypertension (HT), obstructive sleep apnea syndrome (OSAS), hyperlipidemia (HL), etc.)</p>", "<p>Treatment was started with 0.6 mg daily, and the dose was titrated weekly and increased to 3 mg/day, according to side effects (SEs). All patients were also given a personalized low-calorie-restricted diet and at least 150 minutes of weekly physical activity.</p>", "<p>The participants' diagnoses, medications, prescriptions, demographic characteristics, and laboratory results were accessed in the hospital computer database. The body weight (BW), body mass index (BMI), comorbidities of the patients, and follow-up laboratory results at four and 16 weeks were recorded. The patients were questioned about possible drug-related SEs at each follow-up examination.</p>", "<p>The BW was detected on a scale without shoes and extra clothing. BMI was calculated as the BW (kilograms) divided by the squared height (in meters). All biochemical parameters were analyzed from serum samples after eight hours of fasting. Plasma values of fasting glucose (FG), fasting insulin (FI), glycosylated hemoglobin (HbA1c), and lipid profile [triglycerides (TG) and low-density lipoprotein (LDL)] were recorded. Homeostatic Model Assessment-Insulin Resistance Index (HOMA-IR): FG (mg/l)' FI (mU/l) /405 was used to measure insulin resistance for all individuals. Type 2 DM, prediabetes, HT, HL, OSAS, PCOS, and a history of CVD were obtained from the patient's records. Prediabetes and type 2 DM were diagnosed according to the American Diabetes Association's diabetes diagnostic criteria [##UREF##1##14##].</p>", "<p>Statistical analysis </p>", "<p>We used the IBM SPSS 21.0 Statistic version 23 package program (IBM Inc., Armonk, New York) and performed statistical analyses. Continuous variables were expressed as mean ± standard deviation for descriptive statistics, and categorical variables were expressed as frequency and percentages. The Shapiro-Wilk test was used to test the normal distribution. The mean values of variables with normal distribution were compared using Student's t-test or analysis of variance (ANOVA) and those without a normal distribution by the Mann-Whitney U test. The significance level was considered as p-values &lt;0.05. </p>" ]

[ "<title>Results</title>", "<p>Seventy-one patients were evaluated in the study. Due to the higher cost of LG, 10 participants could not continue the treatment for 16 weeks. Therefore, it was not included in the statistical analysis. Sixty (89.5%) women and seven (10.5%) men with a mean age of 42.8 ± 4.4 years met the inclusion criteria. At the beginning of the study in patients, mean BW and BMI were 103.8±18.7 kg and 35.2±7.21kg/m2. The participants' baseline characteristics are presented in Table ##TAB##0##1##. Patients were classified as overweight, class 1, class 2, and class 3 obese according to their BMI. It was determined that most patients were in the class 2 obese (n=17, 25.3%). Of the study patients, 19 (28.4%) were prediabetic, 45 (67.1%) were normoglycemic, and three (4.5%) were diabetic. There was no concomitant disease in 38 (56.7%) patients. Other comorbidities are shown in Table ##TAB##0##1##. </p>", "<p>In this study, all patients reached the LG 3 mg/day target dose and were followed up. The mean BW levels decreased from 103.8±18.7 kg at the beginning of the therapy to 97.6± 17.5 at four weeks and 92.1± 16.4 at 16 weeks. Comparative analyses were found between baseline and four weeks (p=0.023), baseline and 16 weeks (p&lt;0.001), and four and 16 weeks (p=0.019). The mean BMI decreased from 35.2 ± 7.21kg/m2 at baseline to 33.72±7.22 kg/m2 at four weeks and 29.61± 7.14 kg/m2 at 16 weeks. Comparative analyses were conducted between baseline and four weeks (p=0.045), baseline and 16 weeks (p&lt;0.001), and four and 16 weeks (p=0.034). At the end of the 16 weeks, the percentage of body weight loss (BWL) change was found to be comparable between obesity classes 1, 2 and 3 (-9.81±1.93 %, -11.02±2.11 %, and -12.94±2.94respectively; p=0.954), and similar rates of ≥5% BWL were achieved between the three groups (72.6 %, 74.8 % and 78.5 %, respectively; p=0.623).</p>", "<p>When evaluated according to their average BWL, the mean BWL and BMI loss of patients using LG in the first four and 16 weeks after treatment initiation were -6.17 ± 1.34 kg, -1.51 ± 1.25 kg/m2, and -11.71±2.21 kg, -5.56±1.88 kg/m2, respectively (Table ##TAB##1##2##). After the four and 16 weeks of beginning the LG use, the patients who lost more than 5% of initial BW were 38.8% vs. 76.1%, respectively (p=0.034). At four weeks, 14.9% of participants had ≥10% BWL, and this rate increased to 59.7% at 16 weeks (Table ##TAB##1##2##).</p>", "<p>Changes in metabolic parameters such as fasting glucose (FG), fasting insulin (FI), homeostatic model assessment-insulin resistance (HOMA-IR), glycosylated hemoglobin (HbA1c), low-density lipoprotein (LDL), and triglycerides (TG) values before starting treatment and at the four and 16 weeks are summarized in Table ##TAB##2##3##. A statistically significant difference was observed between the baseline, week four, and week 16 mean HOMA-IR values (p&lt;0.001). The baseline HOMA-IR levels were statistically significantly higher than week four and week 16 HOMA-IR levels (p&lt;0.001). A statistically significant difference was observed in HbA1c levels between baseline, four weeks, and 16 weeks (p&lt;0.001). The 16-week mean HbA1clevels were statistically significantly lower than the baseline and 4-week mean HbA1c levels (p&lt;0.001). Similar findings were shown for mean FG and FI levels, with the 16-week levels significantly lower than baseline and 4-week (p&lt;0.001, p&lt;0.001, respectively). There was a significant decrease in baseline LDL and TG concentrations at the end of the four and 16 weeks (p&lt;0.001, p&lt;0.001, respectively). In contrast, the difference between the four and 16 weeks was insignificant (p=0.234, p=0.089, respectively).</p>", "<p>While 45 (67.2%) patients did not experience any SEs after starting LG treatment, the most common SEs were nausea (29.4%), abdominal pain (11.8%), vomiting (10.3%), diarrhea (7.2%), and others (15.9%) (headache, dyspepsia, influenza-like symptoms, constipation). Despite these digestive SEs, none of the patients discontinued their treatment.</p>" ]

[ "<title>Discussion</title>", "<p>This study aimed to demonstrate the effectiveness and SE profile of overweight and obese patients with LG treatment evaluated in clinical practice. The findings of this retrospective study showed that mean BW, BMI, FG, FI, HOMA-IR, HbA1c, LDL, and TG levels were significantly reduced in obese or overweight patients at the 16-week follow-up. </p>", "<p>Obesity is a chronic disease associated with high morbidity and mortality risks and limited quality of life that require long-term medical attention. In addition, the increase in health expenditures brings heavy burdens to the country's economies. Treatment options include diet and exercise, medication, or surgery. Many drugs with different mechanisms of action can be used to treat obesity. Although pharmacotherapy is an effective method in the treatment of obesity, drug costs are often a limiting factor. Obesity treatment options worldwide include phentermine, phentermine/topiramate, lorcaserin, naltrexone/bupropion, diethylpropion, orlistat, and LG. In Turkey, only orlistat and LG are approved for use in treating obesity. Pancreatic lipase inhibitor orlistat has serious gastrointestinal SEs, and it is difficult to achieve tolerability. In a European study, GLP-1 analogs are more effective in weight loss than orlistat and glimepirid [##REF##21844879##15##]. In another retrospective study from Spain, BWL with LG (-7.7 kg) was significantly greater than that observed with orlistat (-3.3 kg), and approximately two and a half times more patients lost at least 5% of their initial BW with LG than with orlistat [##UREF##2##16##]. Opioid antagonists with antidepressant effects Naltrexone-bupropion, sympathomimetic phentermine+ topiramate, and pramlintide were found to be as effective as LG in weight loss, but they were not recommended due to SEs; therefore, the use of LG comes to the fore [##REF##37062832##17##].</p>", "<p>The SCALE randomized controlled clinical trial followed 3731 participants with obesity receiving LG; over 13 months, 63.2% and 33.1% of all participants significantly lost at least 5% and 10% of their BW, respectively [##REF##26132939##9##]. In a meta-analysis, Konwar et al. included approximately 6000 patients who did not have DM but were obese and using LG and observed 2.8-11.8 kg of BWL in their follow-up at 12-56 weeks [##REF##35936066##18##]. Recently, Cetiner et al. from Turkey evaluated 201 patients using LG for 12 months and showed significant BWL. After three months from the LG treatment, 72.14% of the patients (n=145/201), and at the end of six months, almost all (n=96/106) had more than 5% weight loss observed. Additionally, the mean weight loss was 17.79 ± 8.93 kg for those who continued treatment for 12 months [##REF##37782186##19##]. Our investigation determined that LG 3.0 mg for patients with obesity or overweight led to significant BWL of 6.1 to 11.7 kg (4.9%-10.9%) at four and 16 weeks of treatment, respectively. The results revealed that over 75 % and 55% of the participants who used the LG for the initial 16 weeks achieved ≥ 5% and ≥10%BWL, respectively. These findings are similar to or mostly higher than those reported in previous randomized controlled clinical trials [##REF##26132939##9##, ##REF##26284720##20##, ####REF##27005405##21##, ##REF##23812094##22####23812094##22##]. Additionally, our results are superior to those of Italian [##REF##32594453##10##], Canadian [##REF##31062937##11##], and Spanish [##UREF##2##16##]; real-life studies demonstrated that 64% to 68% of patients exhibited &gt;5% BWL and 20% to 35% of patients exhibited &gt;10% BWL at four to seven months of treatment. In contrast, our study compared to a smaller cohort from Switzerland (n=54) showed that the percentage of patients reaching ≥5% weight loss at 16 weeks was lower [##REF##36478514##12##]. In that study with a four-month follow-up, 87% of subjects showed ≥5% BWL, and this percentage increased to 96% at 10 months [##REF##36478514##12##]. This difference in results can be attributed to different nutritional habits between populations. In Turkey, where dietary habits consume extremely high amounts of carbohydrates, transitioning to a low-calorie diet combined with the appetite-suppressing effects of LG may have resulted in more significant weight loss in a short period.</p>", "<p>In patients using LG, diabetes regulation is impaired due to its effect on decreased appetite and gastrointestinal intolerance. In addition, significant weight losses were noted. A recently published randomized placebo-controlled trial of LG found that patients who experienced nausea achieved a more significant absolute weight loss [##REF##35894080##23##]. Previous studies found a direct correlation between drug dose and weight loss [##REF##32594453##10##,##REF##26284720##20##]. The BWL increases, especially when the dose is increased to 2.4-3.0 mg/day [##UREF##3##24##]. In this study, all patients started with LG 0.6 mg and reached the 3mg maximum dose with dose titration within four weeks. Therefore, the relationship between weight loss and drug dose could not be evaluated.</p>", "<p>In another study conducted in Canada, the effectiveness of LG was determined according to the degree of obesity, and no difference in the effectiveness of LG was found in stage 1, stage 2, and stage 3 obese patients [##REF##31062937##11##]. In our study, the percentage of BW change and ≥5% BWL were similar between obesity classes. </p>", "<p>Our study, consistent with a recent meta-analysis, showed a significant decrease in glycemic control variables such as HbA1c, FG, FI, HOMA-IR, and fasting lipid parameters [##REF##35813188##25##]. However, it is unclear whether GLP-1RAs ameliorate metabolic parameters to the same extent in obese patients with and without diabetes. Santini et al. found considerable improvement in triglycerides, glucose profile, and insulin resistance but no significant changes in total cholesterol, LDL, or high-density lipoprotein (HDL) levels [##REF##36478514##12##]. </p>", "<p>The SEs of LG were consistent with findings in previous reports. Our evaluation of LG's SEs showed that 67.1% did not experience any SEs, while 32.9% reported SEs. The most common SE was nausea, observed in approximately one in three participants. Other reported SEs included abdominal pain, vomiting, diarrhea, headache, dyspepsia, influenza-like symptoms, and constipation. However, these SEs are mild to moderate and transient with symptomatic treatment or dose reduction [##REF##20949699##26##]. In our study, patients had nausea during the increase in dose, especially in the transition to 1.2-1.8 mg/day, and many of them were given only symptomatic treatment. Also, LG is not recommended for those with a personal or family history of pancreatitis and multiple endocrine neoplasia (MEN) 2A and 2B [##REF##23481614##27##].</p>", "<p>Despite the positive effectiveness in metabolic control and weight loss of LG, it continues to be an expensive treatment in our country. We think that GLP-1 analogs should be included in the scope of health insurance, considering that they will provide severe benefits in the fight against obesity in the future.</p>", "<p>This study has several limitations. The first is that the study was single-center, retrospective, and small sample size. Secondly, it cannot provide long-term BW changes because the follow-up lasted only 16 weeks. Due to the cost of the medicine, the duration of use of patients is shortened, and long-term results cannot be evaluated. Finally, most of our study participants were women; thus, studies with a greater number of both genders are needed.</p>" ]

[ "<title>Conclusions</title>", "<p>In recent years, medical treatments for obesity, in addition to lifelong adequate and balanced nutrition, physical activity, and behavioral therapies, have come forth. This study showed a clinically significant decrease in BW and improved cardiometabolic parameters in four and 16 weeks of treatment with LG. It stands as a safe and effective medical treatment modality for addressing the issue of obesity. Unfortunately, financial limitations in drug use remain a significant obstacle.</p>" ]

[ "<p>Background</p>", "<p>One of the essential chronic diseases is obesity, which negatively affects the individual and society. Liraglutide (LG) is an effective treatment for both obesity treatment and metabolic control. This study aims to show the effect of a 3.0 mg dose of LG, injected subcutaneously once a day, on weight loss and metabolic parameters.</p>", "<p>Methods</p>", "<p>This retrospective single-center study included 67 patients (60 women and seven men) with a BMI of at least 27 kg/m² with comorbidities or a BMI of at least 30 kg/m². Demographic characteristics, anthropometric measurements, and biochemical data of the participants were evaluated at the end of the four and 16 weeks.</p>", "<p>Results</p>", "<p>The mean body weight (BW) loss of patients using LG at 16 weeks was -11.71±2.21 kg. After the four and 16 weeks of beginning the LG use, the patients who lost more than 5% of initial BW were 38.8% vs. 76.1%, respectively (p=0.034). The mean baseline Homeostatic Model Assessment-Insulin Resistance Index, hemoglobin A1c, low-density lipoprotein, and triglycerides values were significantly higher than the 4 and 16 weeks (p&lt;0.001). Twenty-two (32.8 %) patients experienced side effects (SE) after starting LG treatment, and the most common SE was found to be nausea (29.4%).</p>", "<p>Conclusion</p>", "<p>The use of LG, which is not covered by insurance, together with diet and exercise, has been shown to have clinically significant weight loss and a positive effect on glycemic values ​​and lipid profile.</p>" ]

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[ "<table-wrap position=\"float\" id=\"TAB1\"><label>Table 1</label><caption><title>Characteristics of the participants at baseline</title><p>BMI - body mass index; CVD - cardiovascular disease; OSAS - obstructive sleep apnea syndrome; PCOS - polycystic ovary syndrome</p></caption><table frame=\"hsides\" rules=\"groups\"><tbody><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Variables</td><td rowspan=\"1\" colspan=\"1\">(n = 67)</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Demographic parameters</td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Age, years</td><td rowspan=\"1\" colspan=\"1\">42.8 ± 4.4</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Gender (female), n (%)</td><td rowspan=\"1\" colspan=\"1\">60 (89.5)</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Anthropometric parameters</td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr><td rowspan=\"1\" colspan=\"1\">Weight, kg</td><td rowspan=\"1\" colspan=\"1\">103.8 ± 18.7</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">BMI, kg/m²</td><td rowspan=\"1\" colspan=\"1\">35.2 ± 7.21</td></tr><tr><td rowspan=\"1\" colspan=\"1\">    Overweight (27 to 29.9 kg/m²), n (%)</td><td rowspan=\"1\" colspan=\"1\">5 (7.5)</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">    Obese class 1 (30 to 34.9 kg/m²), n (%)</td><td rowspan=\"1\" colspan=\"1\">29 (43.3)</td></tr><tr><td rowspan=\"1\" colspan=\"1\">    Obese class 2 (35 to 39.9 kg/m²), n (%)</td><td rowspan=\"1\" colspan=\"1\">17 (25.3)</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">    Obese class 3 (&gt;40 kg/m²), n (%)</td><td rowspan=\"1\" colspan=\"1\">16(23.9)</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Diabetes status</td><td rowspan=\"1\" colspan=\"1\"> </td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">     Normoglycemic, n (%)</td><td rowspan=\"1\" colspan=\"1\">45 (67.1)</td></tr><tr><td rowspan=\"1\" colspan=\"1\">     Prediabetic, n (%)</td><td rowspan=\"1\" colspan=\"1\">19 (28.4)</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">     Diabetic, n (%)</td><td rowspan=\"1\" colspan=\"1\">3 (4.5)</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Hypothyroidism, n (%)</td><td rowspan=\"1\" colspan=\"1\">9 (13.4)</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Hypertension, n (%)</td><td rowspan=\"1\" colspan=\"1\">14 (20.9)</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Hyperlipidemia, n (%)</td><td rowspan=\"1\" colspan=\"1\">26 (38.8)</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">CVD</td><td rowspan=\"1\" colspan=\"1\">2 (2.9)</td></tr><tr><td rowspan=\"1\" colspan=\"1\">OSAS</td><td rowspan=\"1\" colspan=\"1\">5 (7.4)</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">PCOS</td><td rowspan=\"1\" colspan=\"1\">7 (10.4)</td></tr></tbody></table></table-wrap>", "<table-wrap position=\"float\" id=\"TAB2\"><label>Table 2</label><caption><title>The weight loss achieved with the use of liraglutide for four and 16 weeks</title><p>Data are expressed as mean ± SD. Statistically significant values (p&lt;0.05) are shown in bold.</p><p>BWL - body weight loss; BMI - body mass index</p></caption><table frame=\"hsides\" rules=\"groups\"><tbody><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Variables (n=67)</td><td rowspan=\"1\" colspan=\"1\">4 weeks</td><td rowspan=\"1\" colspan=\"1\">16 weeks</td><td rowspan=\"1\" colspan=\"1\">p-value</td></tr><tr><td rowspan=\"1\" colspan=\"1\">BWL, kg</td><td rowspan=\"1\" colspan=\"1\">-6.17 ± 1.34</td><td rowspan=\"1\" colspan=\"1\">-11.71±2.21</td><td rowspan=\"1\" colspan=\"1\">&lt;0.001</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">BWL change %</td><td rowspan=\"1\" colspan=\"1\">-4.97± 1.09</td><td rowspan=\"1\" colspan=\"1\">-10.94±2.79</td><td rowspan=\"1\" colspan=\"1\">&lt;0.001</td></tr><tr><td rowspan=\"1\" colspan=\"1\">BWL ≥5% of baseline, n (%)</td><td rowspan=\"1\" colspan=\"1\">26 (38.8)</td><td rowspan=\"1\" colspan=\"1\">51 (76.1)</td><td rowspan=\"1\" colspan=\"1\">0.034</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">BWL ≥10% of baseline, n (%)</td><td rowspan=\"1\" colspan=\"1\">10(14.9)</td><td rowspan=\"1\" colspan=\"1\">40(59.7)</td><td rowspan=\"1\" colspan=\"1\">0.011</td></tr><tr><td rowspan=\"1\" colspan=\"1\">BMI loss, kg/m²\n</td><td rowspan=\"1\" colspan=\"1\">-1.51±1.25</td><td rowspan=\"1\" colspan=\"1\">-5.56±1.88</td><td rowspan=\"1\" colspan=\"1\">&lt;0.001</td></tr></tbody></table></table-wrap>", "<table-wrap position=\"float\" id=\"TAB3\"><label>Table 3</label><caption><title>Metabolic parameters of participants at baseline and follow-up </title><p>Data are expressed as mean ± SD; *p - comparisons between baseline and 4 weeks; †p - comparisons between 4 and 16 weeks; §p - comparisons between baseline and 16 weeks; p&lt;0.05 signifies statistically significant values</p><p>FG - fasting glucose; FI - fasting insulin; HOMA-IR - homeostatic model assessment-insulin resistance; HbA1c - glycosylated hemoglobin; LDL - low-density lipoprotein; TG - triglycerides</p></caption><table frame=\"hsides\" rules=\"groups\"><tbody><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Variables n=67</td><td rowspan=\"1\" colspan=\"1\">Baseline</td><td rowspan=\"1\" colspan=\"1\">4 weeks</td><td rowspan=\"1\" colspan=\"1\">16 weeks</td><td rowspan=\"1\" colspan=\"1\">p*</td><td rowspan=\"1\" colspan=\"1\">p^†\n</td><td rowspan=\"1\" colspan=\"1\">p^§\n</td></tr><tr><td rowspan=\"1\" colspan=\"1\">FG (mg/dl)</td><td rowspan=\"1\" colspan=\"1\">108.27 ± 29.76</td><td rowspan=\"1\" colspan=\"1\">101.18 ± 25.43</td><td rowspan=\"1\" colspan=\"1\">97.68 ± 21.27</td><td rowspan=\"1\" colspan=\"1\">&lt;0.001</td><td rowspan=\"1\" colspan=\"1\">&lt;0.001</td><td rowspan=\"1\" colspan=\"1\">&lt;0.001</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">FI (IU/ml)</td><td rowspan=\"1\" colspan=\"1\">24.45 ± 19.78</td><td rowspan=\"1\" colspan=\"1\">22.17 ± 16.78</td><td rowspan=\"1\" colspan=\"1\">20.45 ± 14.55</td><td rowspan=\"1\" colspan=\"1\">&lt;0.001</td><td rowspan=\"1\" colspan=\"1\">&lt;0.001</td><td rowspan=\"1\" colspan=\"1\">&lt;0.001</td></tr><tr><td rowspan=\"1\" colspan=\"1\">HOMA-IR</td><td rowspan=\"1\" colspan=\"1\">5.25±3.91</td><td rowspan=\"1\" colspan=\"1\">3.98 ± 2.45</td><td rowspan=\"1\" colspan=\"1\">3.06 ± 1.97</td><td rowspan=\"1\" colspan=\"1\">&lt;0.001</td><td rowspan=\"1\" colspan=\"1\">&lt;0.001</td><td rowspan=\"1\" colspan=\"1\">&lt;0.001</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">HbA1c (%)</td><td rowspan=\"1\" colspan=\"1\">5.93 ± 0.88</td><td rowspan=\"1\" colspan=\"1\">5.85 ± 0.79</td><td rowspan=\"1\" colspan=\"1\">5.76 ± 0.71</td><td rowspan=\"1\" colspan=\"1\">&lt;0.001</td><td rowspan=\"1\" colspan=\"1\">&lt;0.001</td><td rowspan=\"1\" colspan=\"1\">&lt;0.001</td></tr><tr><td rowspan=\"1\" colspan=\"1\">LDL (mg/dl)</td><td rowspan=\"1\" colspan=\"1\">146.9 ± 37.9</td><td rowspan=\"1\" colspan=\"1\">129.7 ± 24.5</td><td rowspan=\"1\" colspan=\"1\">118.3 ± 21.7</td><td rowspan=\"1\" colspan=\"1\">&lt;0.001</td><td rowspan=\"1\" colspan=\"1\">0.234</td><td rowspan=\"1\" colspan=\"1\">&lt;0.001</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">TG (mg/dl)</td><td rowspan=\"1\" colspan=\"1\">201.2 ± 64.3</td><td rowspan=\"1\" colspan=\"1\">187.9 ± 45.7</td><td rowspan=\"1\" colspan=\"1\">176.2 ± 39.3</td><td rowspan=\"1\" colspan=\"1\">&lt;0.001</td><td rowspan=\"1\" colspan=\"1\">0.089</td><td rowspan=\"1\" colspan=\"1\">&lt;0.001</td></tr></tbody></table></table-wrap>" ]

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[ "<fn-group content-type=\"other\"><title>Author Contributions</title><fn fn-type=\"other\"><p><bold>Concept and design:</bold>  Ayşen Akkurt Kocaeli</p><p><bold>Acquisition, analysis, or interpretation of data:</bold>  Ayşen Akkurt Kocaeli</p><p><bold>Drafting of the manuscript:</bold>  Ayşen Akkurt Kocaeli</p><p><bold>Critical review of the manuscript for important intellectual content:</bold>  Ayşen Akkurt Kocaeli</p><p><bold>Supervision:</bold>  Ayşen Akkurt Kocaeli</p></fn></fn-group>", "<fn-group content-type=\"other\"><title>Human Ethics</title><fn fn-type=\"other\"><p>Consent was obtained or waived by all participants in this study. Ethics Committee of the Health Sciences University, Bursa City Hospital issued approval E-13012450-514.05.99-248899121. Consent was obtained or waived by all participants in this study.</p></fn></fn-group>", "<fn-group content-type=\"other\"><title>Animal Ethics</title><fn fn-type=\"other\"><p><bold>Animal subjects:</bold> All authors have confirmed that this study did not involve animal subjects or tissue.</p></fn></fn-group>", "<fn-group content-type=\"competing-interests\"><fn fn-type=\"COI-statement\"><p>The authors have declared that no competing interests exist.</p></fn></fn-group>" ]

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[ "<title>Introduction</title>", "<p>One of the prime reasons for patients seeking orthodontic treatment is improvement in their aesthetics or appearance. With a greater number of adult patients now opting for orthodontic treatment, the demand for aesthetic orthodontic materials has increased [##REF##25709410##1##]. The patients desire to undergo orthodontic treatment without compromising their appearance during the treatment period. The increasing demand for more aesthetic orthodontic appliances has elicited an aesthetic revolution marked by the emergence of invisible appliances such as aesthetic brackets, lingual appliances, and clear aligners [##REF##23767109##2##]. Ceramic brackets, clear aligners, and tooth-colored archwires are the new yardsticks for aesthetic orthodontic appliances. They are promising alternatives to conventional metallic materials that contain nickel for patients with nickel sensitivity [##REF##12835436##3##]. Orthodontic archwires have significantly evolved from their original conception. Previously made of gold, they are currently made of various alloys such as stainless steel, nickel-titanium, copper-titanium, and other such alloys [##REF##9188964##4##]. With the advent of ceramic and composite brackets, it was obvious for the archwires to change from their conventional metallic look to a more contemporary aesthetic look to enhance the patient’s appearance during the treatment period. The first esthetic transparent nonmetallic orthodontic wire known as Optiflex was made of a silica core, a silicone resin middle layer, and a stain-resistant nylon outer layer and was marketed by Ormco [##REF##1452728##5##]. An archwire is usually replaced after four to eight weeks as per the schedule followed by orthodontists [##UREF##0##6##]. Thus, the wires need to sustain their aesthetic coating for at least 8 weeks before they are changed with the successive wire.</p>", "<p>Based on preliminary research, only a few studies have been conducted (specifically in the American context). With the background of the recent coronavirus disease 2019 (COVID-19) pandemic and the popular role of strongly pigmented beverages that play an immunity-boosting role, studies exploring the effect of such beverages on orthodontic appliances may improve the decision-making process of selecting such aesthetic appliances [##UREF##1##7##].</p>" ]

[ "<title>Materials and methods</title>", "<p>Four brands of wires were included in this study. The wires were Teflon-, epoxy-, or ceramic-coated. Convenience sampling was done, and five samples of each brand were prepared to be tested in each solution. The samples needed to be in a tile form of 10 x 10 mm dimension, as the minimum size required for spectrophotometry is 8 mm diameter. Archwires of one brand were marked and cut into equal pieces that were 10 mm long. The ends of these pieces were approximated such that light could not pass through them. The approximated pieces were kept on a glass slab coated with petroleum jelly to prevent them from sticking to the glass slab. The ends of the wires were glued together using light cure composite and glue as shown in Figure ##FIG##0##1##.</p>", "<p>Three hundred ml of distilled water was used to prepare different solutions. After mixing and boiling, the solutions were cooled to room temperature and strained. A coffee solution was prepared using commercially available coffee powder (Nescafe) sachets. One teaspoon of coffee powder was added to 300 ml of boiling distilled water and stirred for uniform mixing as per the manufacturer’s instructions. Tea was prepared by adding commercially available tea powder. Two tablets of commercially available AYUSH kadha (Dhootapapeshwar dispersible tablet; Shree Dhootapapeshwar Limited, Punjab, India) were mixed as per instructions given by the manufacturer. Two vitamin C tablets (Limcee) were dissolved in 300 ml water and stirred well. A tablespoon of Chyavanprash was mixed in water at room temperature and stirred well. For making turmeric milk, one teaspoon of commercially available turmeric powder was added to boiling milk and allowed to cool.</p>", "<p>All solutions were divided into four parts of 75 ml solution each using a measuring cylinder for staining four brands of archwire.</p>", "<p>Before the specimens were immersed into the solution, the color of each sample was measured using the spectrophotometer and recorded as color at T0. The samples were immersed in their respective solutions for two weeks, four weeks, and eight weeks for 30 minutes each. Fresh solutions were supplemented every day.</p>", "<p>Color measurement of the samples was done as follows.</p>", "<p>Samples were tested at two, four, and eight weeks after immersing them in various solutions such as turmeric milk, AYUSH kadha, vitamin C tablets’ solution, coffee, Chyavanprash solution, and tea.</p>", "<p>After the first measurement (T0), the samples were placed in a container with the prepared staining solution. Color measurements were repeated after two weeks(T1), four weeks (T2), and eight weeks (T3) of immersion in the solution. Before each measurement, samples were removed from the solution and rinsed with water. Excess water on the surfaces was blotted with tissue papers, and the samples were allowed to dry. Thereafter, the samples were subjected to spectrophotometric analysis. The spectrophotometer model used in this study was VITA Zahnfrabik H. Rauter GmbH &amp; Co. KG, Germany, Sr No. H57127.</p>", "<p>The samples were placed on a flat surface with a green background. The nose of the spectrophotometer was placed perpendicular to the center of the sample (Figure ##FIG##1##2##). The spectrophotometer automatically generated three measurements from which it calculated a mean color measurement which was seen on the spectrophotometer’s screen. Color changes were characterized using the Commission Internationale de I’Eclairage L*a*b* color space (CIE L*a*b*).</p>", "<p>The ΔE value of each sample was thus calculated.</p>", "<p>Color differences (ΔE*) were determined using the following equation:</p>", "<p>ΔE= [(ΔL)² + (Δa)² + (Δb)²]^1/2</p>", "<p>Where ΔE = color difference between the respective samples before and after the intervention.</p>", "<p>ΔL = differences in the 'L.' value [darkness (0) or lightness (100)]</p>", "<p>Δa = differences in the \"a' value [redness (positive a*) or greenness (negative a*)]</p>", "<p>Δb = differences in the b value [yellowness (positive b*) or blueness (negative b*)]</p>", "<p>L*, a*, and b* values before (T0) and after immersion at each time interval (T1, T2, T3).</p>", "<p>To relate the amount of color change (ΔE *) to a clinical environment, the data were converted to National Bureau of Standards (NBS) units as follows: NBS units =ΔE* x 0.92.</p>", "<p>The definitions of color changes quantified by NBS units were used. These values were suggested by Koksal and Dikbas as shown in Table ##TAB##0##1## [##REF##18309623##8##].</p>", "<p>Statistics</p>", "<p>A comparison of aesthetic degradation due to color changes among four brands of archwires was done by applying the one-way analysis of variance (ANOVA) test. The p values were calculated for all samples to determine whether the color change that occurred in the samples was statistically significant or not. Descriptive statistics were used to test the degree of color change using ΔE based on NBS units.</p>" ]

[ "<title>Results</title>", "<p>Table ##TAB##1##2## provides the descriptive statistics like mean and standard deviation for color change, i.e., ΔE obtained for four brands of wires dipped in six solutions for two weeks. At two weeks, the highest ΔE value of 26.92 (0.35) was noticed in the U Orthodontics (New Delhi, India) archwire after immersion in Chyavanprash solution (Table ##TAB##1##2## and Figure ##FIG##2##3##). The least ΔE value of 1.87 (0.39) was observed in the Libral Traders (New Delhi, India) archwire group in a vitamin C solution (Table ##TAB##1##2## and Figure ##FIG##2##3##).</p>", "<p>Table ##TAB##2##3## provides descriptive statistics like the mean and standard deviation for color change, i.e., ΔE, obtained for four brands of wire dipped in six solutions for four weeks. At four weeks, the color change intensified for all the archwires with a significant increase in ΔE.</p>", "<p>JJ Orthodontics (Kerala, India) showed the ΔE 3.09 (0.27) in a vitamin C solution. Koden (Kerala, India) archwires showed more color change in a tea solution as compared to other archwires with ΔE 12.16 (0.23). Overall, the color change was less intense in the vitamin C solution and with Libral Traders archwires, whereas color change increased in the Chyavanprash solution and the U Orthodontics archwires (Figure ##FIG##3##4##).</p>", "<p>The pairwise intergroup comparison at T2, i.e., four weeks, suggested that the difference in color change among various brands of archwires was statistically significant for most of the solutions. The result was statistically highly significant for all intergroup comparisons for AYUSH kadha, turmeric milk, Chyavanprash, and tea. There was no difference in color degradation between JJ Orthodontics and U Orthodontics archwires in the coffee solution. Libral and Koden had a similar amount of color change in the vitamin C solution as the p value was &gt;0.05.</p>", "<p>Table ##TAB##3##4## provides the descriptive statistics like mean and standard deviation for color change, i.e., ΔE, obtained for four brands of wire dipped in six solutions for eight weeks (T3). At eight weeks, the color change intensified for all the archwires with a significant increase in ΔE (Figure ##FIG##4##5##). The color change was maximum in U Orthodontics archwires and Chyavanprash solution (Figure ##FIG##4##5##). The difference was statistically significant for all archwires in all solutions.</p>", "<p>NBS values at the end of eight weeks suggested that almost all archwires showed ‘much’ difference in color. The vitamin C solution caused only appreciable color changes in the archwires as compared to the Chyavanprash solution, which led to ‘very much’ change.</p>", "<p>All intergroup comparisons at the end of eight weeks (T3) indicated that changes produced by the vitamin C solution are not statistically significant for archwires. P value was &lt;0.001 for all brand groups in the Chyavanprash solution except in JJ Orthodontics versus the Libral Traders group (Figure ##FIG##4##5##). Also, the color change among most groups of brands in the vitamin C solution was almost similar and thus statistically insignificant.</p>", "<p>Overall results showed that none of the archwires resisted color change after being immersed in staining solutions after two, four, and eight weeks, respectively.</p>" ]

[ "<title>Discussion</title>", "<p>Multiple studies have been conducted on the color stability of various archwires in different staining solutions such as coffee, tea, cola, wine, etc. The consumption of beverages in the Indian context is quite different and has considerably changed after the COVID pandemic. As per the guidelines given by the AYUSH Department of the Government of India, it was recommended to drink AYUSH kadha and golden milk (turmeric milk) once or twice daily to boost immunity [##UREF##1##7##]. Chyavanprash, which is composed of a highly concentrated mixture of nutrient-rich plants and minerals, was also suggested by the same guidelines, as it intends to boost immunity [##REF##31035513##9##]. As the ingredients of these beverages tend to have a staining effect, our study aimed whether the aesthetic archwires maintained their color on consistently encountering these staining solutions.</p>", "<p>Usually, the duration between two appointments to change archwires is four to six weeks [##UREF##0##6##]. Previous studies by da Silvaa et al. (2013), Deepika S et al. (2016), and Anand A (2020) measured the color change after three weeks [##REF##22591261##10##, ####UREF##2##11##, ##UREF##3##12####3##12##]. This is the minimum duration that wires should resist color change before they are replaced. Hence, the time intervals of two, four, and eight weeks (T1, T2, and T3, respectively) were selected for our study.</p>", "<p>Of the four brands of wires used in this study, two had Teflon coating (JJ Orthodontics and U Orthodontics) while one had ceramic coating (Koden) and one had epoxy coating (Libral Traders). Results showed that irrespective of the brand and coating, all archwires displayed a staining effect when immersed in different solutions. The finding that epoxy-coated archwires were more color-stable than Teflon-coated wires is consistent with the findings of the study conducted by Anand A (2018) who used red wine, orange juice, and mouthwash as staining solutions [##UREF##3##12##]. JJ Orthodontics showed the minimum color change as per the ΔE values in vitamin C as compared to the other wires. U Orthodontics showed less staining (6.86) than Libral Trades (7.39) in turmeric milk. U Orthodontics wires, which were Teflon-coated, showed maximum staining among all the archwires followed by JJ Orthodontics, which were also Teflon-coated. Libral Traders’ archwires resisted staining the most followed by Koden archwires. Thus, epoxy coating and ceramic coating seemed to have better stain resistance as compared to other coatings. Studies conducted by Anand A et al., Alsanea et al., and Ismail N et al. had similar conclusions [##UREF##3##12##, ####REF##30923691##13##, ##UREF##4##14####4##14##]. Teflon-coated wires were more stained even when immersed in fluoridated and non-fluoridated mouthwashes as per the study by Hussein L et al. [##UREF##5##15##]. As compared to rhodium coating, epoxy-coated archwires had an almost equal color change value. Whereas rhodium archwires were superior to Teflon-coated archwires in maintaining their color stability [##UREF##4##14##]. According to all these experiments, the Teflon-coated aesthetic archwires were more prone to color changes when dipped in various dietary staining solutions. Teflon-coated archwires' higher propensity for a color shift may be caused by the production process [##REF##28301016##16##].</p>", "<p>With respect to the solutions (i.e. Chyavanprash, tea, coffee, turmeric milk, vitamin C solution, and AYUSH Kadha) included in this study, ΔE was observed over two, four, and eight weeks. Some studies conducted on ceramic brackets concluded that wine was the most staining solution in comparison with other staining substances such as mouthwash and cola drinks [##UREF##3##12##]. Studies by Mutlu-Sagesen L et al. and Ertaş E et al. concluded that coffee was the most staining solution in comparison with other staining substances such as tea and cola drinks [##REF##16279728##17##, ####REF##16916243##18####16916243##18##]. Ismail N et al. suggested that adding milk to the preparation reduced the staining effect of coffee and tea and may reduce the concentration of staining pigments present in these solutions [##UREF##4##14##]. The studies mentioned above support the use of such solutions to test the color stability of archwires.</p>", "<p>Although the present study did not statistically test the highest staining agent, vitamin C tablets dissolved in distilled water were observed to have a lower staining effect followed by AYUSH kadha tablet solution according to the ΔE findings. The reason for this observation could be that both these tablets were completely dispersible and thus showed minimum remnants on the wire surface and could be easily cleaned with tap water.</p>", "<p>Since none of the previous studies included Chyavanprash solution, AYUSH kadha, turmeric milk, or vitamin C solution, their effect on archwires was a novel finding in the study. The staining of archwires was visible to the naked eye at all time intervals for all solutions, suggesting that none of the archwires could be the yardsticks for good aesthetic materials.</p>", "<p>Limitations</p>", "<p>This study explored the color stability of four brands of aesthetic archwires in six beverages. However, it also has some limitations that provide scope for future research. For example, more brands of wires can be incorporated in such a study. The in vivo evaluation of the color stability of archwires should be evaluated as the environment of the oral cavity may have a different effect on the staining of archwires. However, it may not be possible to monitor the effect of a single solution on an archwire if the study is conducted intraorally as other factors such as the patient’s oral hygiene and salivary flow can change the results.</p>", "<p>Clinical significance</p>", "<p>Based on the current research, it can be concluded that currently, epoxy-coated archwires could be preferred for a patient undergoing fixed orthodontic therapy with aesthetic brackets. With the background of the COVID-19 pandemic, vitamin C and Ayush kadha are suitable for the aesthetic preservation of archwires.</p>" ]

[ "<title>Conclusions</title>", "<p>Our study tested the staining effect of six solutions on four different brands of archwires at two, four, and eight weeks. At the end of all time intervals, none of the archwires resisted a color change irrespective of the brand or coating of archwires. With respect to the solutions, all solutions, i.e. Chyavanprash, tea, coffee, vitamin C, turmeric milk, and AYUSH kadha, displayed a staining effect on all the aesthetic archwires. ΔE values suggest that there could be a difference in the degree of color change in the various staining solutions, the statistical significance of which can be investigated in future studies.</p>", "<p>Since the consumption of beverages apart from tea and coffee used in this study is becoming popular worldwide, the results of this study can be implicated not only in the Indian context but also globally. Overall, this study provides a basis for further research, which includes more solutions and archwires to statistically determine the most aesthetically stable archwires. This can help clinicians guide their patients better in maintaining the aesthetics of their appliances throughout the treatment.</p>" ]

[ "<p>Introduction</p>", "<p>One of the prime reasons for patients seeking orthodontic treatment is improvement in their aesthetics or appearance. With a greater number of adult patients now opting for orthodontic treatment, the demand for aesthetic orthodontic materials has increased. With the background of the recent coronavirus disease 2019 (COVID-19) pandemic and the popular role of strongly pigmented beverages that play an immunity-boosting role, studies exploring the effect of such beverages on orthodontic appliances may improve the decision-making process of selecting such aesthetic appliances.</p>", "<p>Materials and methods</p>", "<p>Four brands of wires and six beverages were included in this study. The wires were Teflon-, epoxy-, or ceramic-coated. Convenience sampling was done, and five samples of each brand were prepared to be tested in each solution. Samples were tested under a spectrophotometer after immersing them in various solutions for two, four, and eight weeks. A comparison of aesthetic degradation due to color changes amongst four brands of archwires was done by applying the one-way analysis of variance (ANOVA) test. P values were calculated for all samples to determine whether the color change that occurred in the samples was statistically significant or not.</p>", "<p>Results</p>", "<p>Overall results showed that none of the archwires resisted color change after being immersed in staining solutions after two, four, and eight weeks, respectively, which was found to be statistically significant. </p>", "<p>Conclusion</p>", "<p>At the end of all time intervals, none of the archwires resisted a color change irrespective of the brand or coating of archwires. This result was found to be statistically significant. With respect to the solutions, all solutions from Chyavanprash, tea, coffee, vitamin C, turmeric milk, and AYUSH kadha displayed a staining effect on all the aesthetic archwires.</p>" ]

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[ "<fig position=\"anchor\" fig-type=\"figure\" id=\"FIG1\"><label>Figure 1</label><caption><title>Preparation of samples by juxtaposing 10 mm long pieces of archwires</title></caption></fig>", "<fig position=\"anchor\" fig-type=\"figure\" id=\"FIG2\"><label>Figure 2</label><caption><title>Measurement of color change using the spectrophotometer</title></caption></fig>", "<fig position=\"anchor\" fig-type=\"figure\" id=\"FIG3\"><label>Figure 3</label><caption><title>Bar graph showing mean color change (ΔE) values at T1 for four brands of wires in six solutions</title><p>The data has been represented as the mean color change, i.e. (ΔE), at the first interval</p></caption></fig>", "<fig position=\"anchor\" fig-type=\"figure\" id=\"FIG4\"><label>Figure 4</label><caption><title>Bar graph showing the mean color change (ΔE) values at T2 for four brands of wires in six solutions</title><p>The data have been represented as the mean color change, i.e., (ΔE), at the second interval.</p></caption></fig>", "<fig position=\"anchor\" fig-type=\"figure\" id=\"FIG5\"><label>Figure 5</label><caption><title>Bar graph showing mean color change (ΔE) values at T3 for four brands of wires in six solutions</title><p>The data have been represented as the mean color change, i.e., (ΔE), at the third interval.</p></caption></fig>" ]

[ "<table-wrap position=\"float\" id=\"TAB1\"><label>Table 1</label><caption><title>Critical marks of color change according to National Bureau Standards (NBS)</title></caption><table frame=\"hsides\" rules=\"groups\"><tbody><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">NBS Unit</td><td colspan=\"2\" rowspan=\"1\">Definitions of Color Differences</td></tr><tr><td rowspan=\"1\" colspan=\"1\">0.0-0.5</td><td rowspan=\"1\" colspan=\"1\">Trace</td><td rowspan=\"1\" colspan=\"1\">Extremely slight change</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">0.5-1.5</td><td rowspan=\"1\" colspan=\"1\">Slight</td><td rowspan=\"1\" colspan=\"1\">Slight change</td></tr><tr><td rowspan=\"1\" colspan=\"1\">1.5-3</td><td rowspan=\"1\" colspan=\"1\">Noticeable</td><td rowspan=\"1\" colspan=\"1\">Perceivable change</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">3.0-6.0</td><td rowspan=\"1\" colspan=\"1\">Appreciable</td><td rowspan=\"1\" colspan=\"1\">Marked change</td></tr><tr><td rowspan=\"1\" colspan=\"1\">6.0-12.0</td><td rowspan=\"1\" colspan=\"1\">Much</td><td rowspan=\"1\" colspan=\"1\">Extremely marked change</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">12.0+</td><td rowspan=\"1\" colspan=\"1\">Very much</td><td rowspan=\"1\" colspan=\"1\">Change to other color</td></tr></tbody></table></table-wrap>", "<table-wrap position=\"float\" id=\"TAB2\"><label>Table 2</label><caption><title>The overall intergroup comparison of color change (ΔE) of different brands of aesthetic archwires in various staining solutions at T1, i.e., two weeks</title><p>The data is represented as change in color (ΔE) as Mean (SD = standard deviation), F = ratio of variance, p= p value such that p&gt; 0.05 means no significant difference, *p&lt; 0.05 means significant change and **p&lt; 0.001 means a highly significant change in color.</p></caption><table frame=\"hsides\" rules=\"groups\"><tbody><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">T1</td><td rowspan=\"1\" colspan=\"1\">Coffee Mean (SD)</td><td rowspan=\"1\" colspan=\"1\">Tea Mean (SD)</td><td rowspan=\"1\" colspan=\"1\">Vit C Mean (SD)</td><td rowspan=\"1\" colspan=\"1\">Chyavanprash Mean (SD)</td><td rowspan=\"1\" colspan=\"1\">Ayush Kadha Mean (SD)</td><td rowspan=\"1\" colspan=\"1\">Turmeric Milk Mean (SD)</td></tr><tr><td rowspan=\"1\" colspan=\"1\">JJ Orthodontics</td><td rowspan=\"1\" colspan=\"1\">5.47 (1.12)</td><td rowspan=\"1\" colspan=\"1\">5.4 (0.42)</td><td rowspan=\"1\" colspan=\"1\">1.94 (0.57)</td><td rowspan=\"1\" colspan=\"1\">8.33 (0.76)</td><td rowspan=\"1\" colspan=\"1\">6.71 (0.5)</td><td rowspan=\"1\" colspan=\"1\">6.09 (0.44)</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Libral Traders</td><td rowspan=\"1\" colspan=\"1\">3.16 (1.22)</td><td rowspan=\"1\" colspan=\"1\">2.19 (0.4)</td><td rowspan=\"1\" colspan=\"1\">1.87 (0.39)</td><td rowspan=\"1\" colspan=\"1\">9.03 (0.33)</td><td rowspan=\"1\" colspan=\"1\">3.6 (0.4)</td><td rowspan=\"1\" colspan=\"1\">3.6 (0.4)</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Koden</td><td rowspan=\"1\" colspan=\"1\">1.92 (0.46)</td><td rowspan=\"1\" colspan=\"1\">11.09(0.49)</td><td rowspan=\"1\" colspan=\"1\">2.17 (0.4)</td><td rowspan=\"1\" colspan=\"1\">6.84 (0.23)</td><td rowspan=\"1\" colspan=\"1\">4.99 (0.15)</td><td rowspan=\"1\" colspan=\"1\">4.99 (0.15)</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">U Orthodontics</td><td rowspan=\"1\" colspan=\"1\">9.36 (0.25)</td><td rowspan=\"1\" colspan=\"1\">5.4 (0.16)</td><td rowspan=\"1\" colspan=\"1\">2.11 (0.28)</td><td rowspan=\"1\" colspan=\"1\">26.92 (0.35)</td><td rowspan=\"1\" colspan=\"1\">5.01 (0.15)</td><td rowspan=\"1\" colspan=\"1\">5.01 (0.15)</td></tr><tr><td rowspan=\"1\" colspan=\"1\">One-way ANOVA F test</td><td rowspan=\"1\" colspan=\"1\">F = 70.44</td><td rowspan=\"1\" colspan=\"1\">F = 441.19</td><td rowspan=\"1\" colspan=\"1\">F = 0.55</td><td rowspan=\"1\" colspan=\"1\">F = 2056.0</td><td rowspan=\"1\" colspan=\"1\">F =196.2</td><td rowspan=\"1\" colspan=\"1\">F =50.466</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">P value</td><td rowspan=\"1\" colspan=\"1\">p&lt; 0.001**</td><td rowspan=\"1\" colspan=\"1\">p&lt; 0.001**</td><td rowspan=\"1\" colspan=\"1\">p&lt; 0.001**</td><td rowspan=\"1\" colspan=\"1\">p&lt; 0.001**</td><td rowspan=\"1\" colspan=\"1\">p&lt; 0.001**</td><td rowspan=\"1\" colspan=\"1\">p&lt; 0.001**</td></tr></tbody></table></table-wrap>", "<table-wrap position=\"float\" id=\"TAB3\"><label>Table 3</label><caption><title>The table represents the overall intergroup comparison of the color change (ΔE) of different brands of aesthetic archwires in various staining solutions at T2</title><p>The data are represented as change in color (ΔE) as mean (SD = standard deviation), F = ratio of variance, p = p value such that p&gt; 0.05 means no significant difference, *p&lt; 0.05 means significant change, and **p&lt; 0.001 means highly significant change in color.</p></caption><table frame=\"hsides\" rules=\"groups\"><tbody><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">T2</td><td rowspan=\"1\" colspan=\"1\">Coffee Mean (SD)</td><td rowspan=\"1\" colspan=\"1\">Tea Mean (SD)</td><td rowspan=\"1\" colspan=\"1\">Vit C Mean (SD)</td><td rowspan=\"1\" colspan=\"1\">Chyavanprash Mean (SD)</td><td rowspan=\"1\" colspan=\"1\">Ayush Kadha Mean (SD)</td><td rowspan=\"1\" colspan=\"1\">Turmeric Milk Mean (SD)</td></tr><tr><td rowspan=\"1\" colspan=\"1\">JJ Orthodontics</td><td rowspan=\"1\" colspan=\"1\">8.03 (0.63)</td><td rowspan=\"1\" colspan=\"1\">8.15 (0.36)</td><td rowspan=\"1\" colspan=\"1\">3.09 (0.27)</td><td rowspan=\"1\" colspan=\"1\">11.06 (0.26)</td><td rowspan=\"1\" colspan=\"1\">7.88 (0.34)</td><td rowspan=\"1\" colspan=\"1\">6.09 (0.26)</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Libral Traders</td><td rowspan=\"1\" colspan=\"1\">4.96(0.31)</td><td rowspan=\"1\" colspan=\"1\">3.69 (0.4)</td><td rowspan=\"1\" colspan=\"1\">3.82 (0.32)</td><td rowspan=\"1\" colspan=\"1\">11.71 (0.23)</td><td rowspan=\"1\" colspan=\"1\">3.23 (0.07)</td><td rowspan=\"1\" colspan=\"1\">3.37 (0.18)</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Koden</td><td rowspan=\"1\" colspan=\"1\">4.61 (0.32)</td><td rowspan=\"1\" colspan=\"1\">12.16 (0.23)</td><td rowspan=\"1\" colspan=\"1\">3.89 (0.27)</td><td rowspan=\"1\" colspan=\"1\">10.46 (0.31)</td><td rowspan=\"1\" colspan=\"1\">4.11 (0.15)</td><td rowspan=\"1\" colspan=\"1\">4.93 (0.19)</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">U Orthodontics</td><td rowspan=\"1\" colspan=\"1\">8.18 (0.43)</td><td rowspan=\"1\" colspan=\"1\">7.32 (0.25)</td><td rowspan=\"1\" colspan=\"1\">4.85 (0.21)</td><td rowspan=\"1\" colspan=\"1\">14.31 (0.26)</td><td rowspan=\"1\" colspan=\"1\">6.98 (0.23)</td><td rowspan=\"1\" colspan=\"1\">7.76 (0.12)</td></tr><tr><td rowspan=\"1\" colspan=\"1\">One-way Anova F test</td><td rowspan=\"1\" colspan=\"1\">F = 93.66</td><td rowspan=\"1\" colspan=\"1\">F = 577.68</td><td rowspan=\"1\" colspan=\"1\">F = 34.80</td><td rowspan=\"1\" colspan=\"1\">F = 195.96</td><td rowspan=\"1\" colspan=\"1\">F = 499.32</td><td rowspan=\"1\" colspan=\"1\">F = 437.24</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">P value</td><td rowspan=\"1\" colspan=\"1\">p&lt; 0.001**</td><td rowspan=\"1\" colspan=\"1\">p&lt; 0.001**</td><td rowspan=\"1\" colspan=\"1\">p&lt; 0.001**</td><td rowspan=\"1\" colspan=\"1\">p&lt; 0.001**</td><td rowspan=\"1\" colspan=\"1\">p&lt; 0.001**</td><td rowspan=\"1\" colspan=\"1\">p&lt; 0.001**</td></tr></tbody></table></table-wrap>", "<table-wrap position=\"float\" id=\"TAB4\"><label>Table 4</label><caption><title>Overall intergroup comparison of the color change of different brands of aesthetic archwires in various staining solutions at T3</title><p>The data are represented as the change in color (ΔE) as mean (SD = standard deviation), F = ratio of variance, p = p value such that p&gt; 0.05 means no significant difference, *p&lt; 0.05 means significant change, and **p&lt; 0.001 means a highly significant change in color.</p></caption><table frame=\"hsides\" rules=\"groups\"><tbody><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">T3</td><td rowspan=\"1\" colspan=\"1\">Coffee Mean (SD)</td><td rowspan=\"1\" colspan=\"1\">Tea Mean (SD)</td><td rowspan=\"1\" colspan=\"1\">Vit C Mean (SD)</td><td rowspan=\"1\" colspan=\"1\">Chyavanprash Mean (SD)</td><td rowspan=\"1\" colspan=\"1\">Ayush Kadha Mean (SD)</td><td rowspan=\"1\" colspan=\"1\">Turmeric Milk Mean (SD)</td></tr><tr><td rowspan=\"1\" colspan=\"1\">JJ Orthodontics</td><td rowspan=\"1\" colspan=\"1\">10.28 (1.2)</td><td rowspan=\"1\" colspan=\"1\">9.83 (0.19)</td><td rowspan=\"1\" colspan=\"1\">4.95 (0.37)</td><td rowspan=\"1\" colspan=\"1\">15.45 (0.27)</td><td rowspan=\"1\" colspan=\"1\">9.48 (0.2)</td><td rowspan=\"1\" colspan=\"1\">10.32 (0.28)</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Libral Traders</td><td rowspan=\"1\" colspan=\"1\">7.81 (1.81)</td><td rowspan=\"1\" colspan=\"1\">7.03 (0.38)</td><td rowspan=\"1\" colspan=\"1\">5.69 (0.29)</td><td rowspan=\"1\" colspan=\"1\">15.48 (0.23)</td><td rowspan=\"1\" colspan=\"1\">4.53 (0.19)</td><td rowspan=\"1\" colspan=\"1\">7.39 (0.43)</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Koden</td><td rowspan=\"1\" colspan=\"1\">5.46 (0.86)</td><td rowspan=\"1\" colspan=\"1\">12.64 (0.32)</td><td rowspan=\"1\" colspan=\"1\">5.34 (0.23)</td><td rowspan=\"1\" colspan=\"1\">11.21 (0.4)</td><td rowspan=\"1\" colspan=\"1\">5.07 (0.2)</td><td rowspan=\"1\" colspan=\"1\">5.29 (0.22)</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">U Orthodontics</td><td rowspan=\"1\" colspan=\"1\">11.63 (4.03)</td><td rowspan=\"1\" colspan=\"1\">10.17 (0.09)</td><td rowspan=\"1\" colspan=\"1\">5.72 (0.37)</td><td rowspan=\"1\" colspan=\"1\">17.24 (0.33)</td><td rowspan=\"1\" colspan=\"1\">8.49 (0.33)</td><td rowspan=\"1\" colspan=\"1\">6.86 (0.2)</td></tr><tr><td rowspan=\"1\" colspan=\"1\">One-way ANOVA F test</td><td rowspan=\"1\" colspan=\"1\">F = 6.838</td><td rowspan=\"1\" colspan=\"1\">F=355.206</td><td rowspan=\"1\" colspan=\"1\">F = 6.044</td><td rowspan=\"1\" colspan=\"1\">F = 326.209</td><td rowspan=\"1\" colspan=\"1\">F=524.21</td><td rowspan=\"1\" colspan=\"1\">F= 242.29</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">P value</td><td rowspan=\"1\" colspan=\"1\">P=0.004*</td><td rowspan=\"1\" colspan=\"1\">p&lt; 0.001**</td><td rowspan=\"1\" colspan=\"1\">P=0.006*</td><td rowspan=\"1\" colspan=\"1\">p&lt; 0.001**</td><td rowspan=\"1\" colspan=\"1\">p&lt; 0.001**</td><td rowspan=\"1\" colspan=\"1\">p&lt; 0.001**</td></tr></tbody></table></table-wrap>" ]

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[ "<fn-group content-type=\"other\"><title>Author Contributions</title><fn fn-type=\"other\"><p><bold>Concept and design:</bold>  Sharvari S. Khedkar, Usha Shenoy, Ananya Hazare, Himija Karia, Pritam Khorgade, Nivedita Nandeshwar, Sangeeta Bhattacharya</p><p><bold>Acquisition, analysis, or interpretation of data:</bold>  Sharvari S. Khedkar, Usha Shenoy</p><p><bold>Drafting of the manuscript:</bold>  Sharvari S. Khedkar</p><p><bold>Critical review of the manuscript for important intellectual content:</bold>  Sharvari S. Khedkar, Usha Shenoy, Ananya Hazare, Himija Karia, Pritam Khorgade, Nivedita Nandeshwar, Sangeeta Bhattacharya</p><p><bold>Supervision:</bold>  Usha Shenoy, Ananya Hazare, Himija Karia, Pritam Khorgade, Nivedita Nandeshwar, Sangeeta Bhattacharya</p></fn></fn-group>", "<fn-group content-type=\"other\"><title>Human Ethics</title><fn fn-type=\"other\"><p>Consent was obtained or waived by all participants in this study</p></fn></fn-group>", "<fn-group content-type=\"other\"><title>Animal Ethics</title><fn fn-type=\"other\"><p><bold>Animal subjects:</bold> All authors have confirmed that this study did not involve animal subjects or tissue.</p></fn></fn-group>", "<fn-group content-type=\"competing-interests\"><fn fn-type=\"COI-statement\"><p>The authors have declared that no competing interests exist.</p></fn></fn-group>" ]

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{ "acronym": [], "definition": [] }

[ "<title>Introduction</title>", "<p id=\"Par2\">Fluid simulations enable the investigation of blood flow distribution in the cardiovascular system to better understand disease progression, inform surgical procedures and evaluate responses to internal and external conditions affecting the body. These simulations can also be used to reduce the risks associated with extreme environments, such as the microgravity experienced by astronauts during long-duration spaceflight where cardiovascular and muscular deconditioning can occur along with the development of a condition known as space associated neuro-ocular syndrome (SANS)^{##UREF##0##1##}, impairing vision.</p>", "<p id=\"Par3\">There are a number of cardiovascular-related changes that can arise during, or as a result of, long-duration spaceflight including large fluid shifts and stroke volume changes, variations in blood pressure, vascular tissue changes and orthostatic intolerance^{##REF##33391848##2##}. Terrestrially based research can be performed to emulate phenomena associated with spaceflight and to investigate the long-term implications on the body through methods such as long-duration head-down tilt (HDT) and water immersion experiments or parabolic flights. However, these experiments generally require extensive planning and are often associated with high costs due to duration and equipment requirements^{##REF##30253272##3##}. Furthermore, HDT experiments may not necessarily accurately emulate microgravity as they induce artificial pressure gradients, whilst parabolic flights can only provide short exposure windows of 20–30 s at a time^{##REF##33083524##4##}. Alternatively, computational fluid dynamics (CFD) simulations offer a relatively low-cost approach to model fluid changes associated with spaceflight, in either human or animal models. In addition, simulations are also advantageous in that they can use retrospective data and account for the varying sizes and scales of cardiovascular networks throughout the body.</p>", "<p id=\"Par4\">Zero-dimensional (0D) lumped parameter modelling is often employed to model large-scale arterial networks across a wide range of conditions with relatively low computational cost. However, a drawback of low dimensional models is the inability to capture localised haemodynamics such as wall shear stress (WSS) distributions, or non-uniform flow through vessels due to geometric factors such as stenosis, bifurcations, tortuosity or high degrees of vessel curvature. Three-dimensional (3D) CFD simulations of the vasculature enable the evaluation of localised flow to a high degree of spatial and temporal resolution^{##REF##24115509##5##,##REF##26512019##6##}. However, 3D CFD simulations often omit vascular networks upstream and downstream of a domain of interest due to increased computational cost. In place of an upstream geometry, inlet boundary conditions can be specified using measured data, existing literature or from 0D modelling. Downstream of a domain of interest outlet boundaries can be prescribed using zero pressure or specified flow split outlets. Alternatively, resistive elements, fractal trees or Windkessel modelling can be employed to emulate the downstream resistance and compliance of peripheral arteries and venous networks.</p>", "<p id=\"Par5\">Lower dimensional fluid mechanics studies (i.e., 0D, 1D and 2D) have previously been used to model the arterial tree leading to the cerebrovasculature, or within the eye, in attempts to understand disease development such as glaucoma^{##REF##27417037##7##}, diabetic retinopathy^{##REF##29407525##8##}, hyper- and hypotension^{##REF##24876284##9##} as well as the effects of spaceflight or ground-based HDT experiments^{##REF##33083524##4##,##REF##31412033##10##}. Studies have simulated blood flow within large arterial networks for the purposes of understanding pulse wave velocity propagation and age-related arterial stiffening^{##UREF##1##11##}, arterial particle and embolism transport^{##REF##28088305##12##,##UREF##2##13##}, calculation of the ankle-brachial index^{##REF##30385003##14##}, effect of venoarterial extracorporeal membrane oxygenation^{##REF##33333365##15##} and demonstration of meshing strategies and computational optimisation^{##UREF##3##16##}. Previous large-scale simulations have also encountered challenges in accurately evaluating localised haemodynamic metrics such as WSS due to computational limitations^{##UREF##1##11##}. Development of a large-scale human artery haemodynamics framework enables verification with existing models of blood flow under conditions such as simulated microgravity^{##REF##33083524##4##,##REF##31412033##10##,##UREF##4##17##}.</p>", "<p id=\"Par6\">In this study, we aimed to construct a physiologically possible 3D model of human blood vessels ranging from the aortic root through to the retina, by combining existing subject-specific 3D models of arterial vessels from different sources. With this geometry, we developed fluid simulations that accommodated a single continuum physics model for modelling blood across a large arterial network, and gravitational effects to investigate the distributions of blood flow to distal small arteries, specifically in the eye. We used this framework to compare haemodynamic metrics at arterial regions of interest within this vessel network in response to Earth’s gravitational conditions and simulated microgravity.</p>" ]

[ "<title>Methods</title>", "<title>Imaging methods and 3D reconstruction</title>", "<p id=\"Par7\">This study was approved by the University of Western Australia (2022/ET000688). We sourced 3D arterial vasculature models from numerous past studies^{##REF##31140054##18##–##UREF##5##22##}. We imported the associated stereolithography files into STAR-CCM+ (v15, Siemens, Munich, Germany), where we used the surface repair tool to manually perform iterative rigid transformations to closely align each adjacent model using general imaging landmarks (e.g., aortic arch), combine selected overlapping vertices and then iteratively smooth the overlapping regions to achieve an average lumen (Fig. ##FIG##0##1##).</p>", "<p id=\"Par8\">The superficial retinal arteriole section of the 3D model used data from a previous study^{##REF##31140054##18##}, which was reconstructed from a publicly available retinal fundus image of a healthy eye (CANON CF-60UVi) from the High-Resolution Fundus Image Database^{##REF##24416040##23##}. Briefly, the images were filtered and converted to binary processed images before manual segmentation of the arterioles and their diameter using open-source graphics editing software (GIMP, GIMP Team, California, United States). The resulting image was then segmented again in Mimics (v18, Materialise, Belgium) to create a 3D geometry and the centerline was extracted using 3-Matic (v10, Materialise, Belgium), before creating lofted cylinders along these centerlines and transforming the 3D model to a spherical curvature with a radius of 11.824 mm using a coordinate transformation in MATLAB (2016b, Mathworks, Massachusetts, United States). This model was duplicated and applied for both the left and right eyes, correcting for nasal and temporal orientation on either side. More detail on the image analysis and geometry creation is provided in Rebhan et al.^{##REF##31140054##18##}.</p>", "<p id=\"Par9\">The 3D model of the cerebrovascular and neck arteries was obtained as part of a previous study^{##REF##32881618##19##}, where participants were imaged using 3 T time-of-flight magnetic resonance angiography (3 T TOF MRA) (Siemens Magnetom, Skyra) with a corresponding pixel size of 0.31 mm and a slice thickness of 0.75 mm. These images were reconstructed using in-house software to create a 3D isosurface, which was subsequently smoothed to within 5% of its starting volume and reconstruction artefacts were removed. While segmenting the cerebrovascular and neck arteries, a manual mask segmentation was created that followed the centre of the optic nerve sheath from the retina to the ophthalmic artery, representing the branching central retinal artery (CRA). This segmented cylindrical line was initially at the resolution of the 3 T TOF MRA images at approximately 0.3 mm in diameter. Upon importation into STAR-CCM+ for surface repair, we reduced the thickness of this line to 163 μm using local mesh smoothing techniques to match the average measured diameter of the CRA in healthy individuals^{##REF##12789540##24##–##UREF##6##26##}.</p>", "<p id=\"Par10\">The 3D model of the aortic root and coronary arteries was obtained from previously reconstructed computed tomography (CT) coronary angiogram images from a recent study^{##REF##33358367##20##,##REF##34274284##21##} which had a pixel size of 0.45 mm and slice thickness of 0.8 mm in Mimics. The aorta, iliac and femoral artery 3D model network was obtained from reconstructed and modified imaging data from a previous study^{##UREF##5##22##}, which used continuous arterial phase CT imaging with a slice thickness and increment size of 2.5 mm and 1 mm, respectively.</p>", "<title>Computational fluid dynamics</title>", "<p id=\"Par11\">Simulations were developed in the commercial CFD package STAR-CCM+. In this study, we created two investigation cases; Earth gravity and simulated microgravity which both used the same 3D arterial geometry.</p>", "<p id=\"Par12\">We used a combination of a trimmer cell mesh in the core of the fluid geometry and prescribed anisotropic prism layer cells to capture the behaviour of the fluid at the near-wall boundary. These prism layer cells were distributed in both the small and large arterial vessels with variable thickness to the regions of the retina, cerebrovasculature, neck and coronary arteries using volumetric controls. To ensure mesh independence we used the non-uniform refinement ratio formulation of the grid convergence index (GCI)^{##UREF##7##27##,##UREF##8##28##} across a range of different haemodynamic parameters—with calculated GCI values for mass flow rate and WSS metrics falling below 2-3% indicating sufficient mesh discretisation^{##REF##31140054##18##,##REF##32881618##19##,##UREF##9##29##}. Mesh sizes can be found in Supplementary Table ##SUPPL##0##1##, settings in Supplementary Table ##SUPPL##0##2## and GCI results in Supplementary Table ##SUPPL##0##3##. The final mesh consisted of ~43 million elements.</p>", "<p id=\"Par13\">Blood was modelled as an incompressible fluid with a density of 1050 kg m^-3\n^{##REF##19958111##30##}. We assumed rigid walls with a no-slip boundary condition and a laminar flow regime, as this is expected in the majority of the fluid domain under normal healthy conditions. To capture the variation in blood viscosity due to both the non-Newtonian shear thinning nature as well as a reduction in viscosity due to the Fåhraeus-Lindqvist (FL) effect, a blended viscosity model was implemented. Using wall distance, we determined vessel diameter and prescribed the FL viscosity model^{##REF##19958111##30##,##UREF##10##31##} below vessel diameters of 0.6 mm^{##REF##3742742##32##}, the Carreau-Yasuda model as described by Karimi et al.^{##UREF##11##33##} (<italic>η</italic>_∞ = 0.0035 Pa s; <italic>η</italic>₀ = 0.16 Pa s; <italic>λ</italic> = 8.2 s; <italic>a</italic> = 0.64; <italic>n</italic> = 0.2128) in vessels greater than 1.2 mm, and linearly interpolated between these two models within this diameter range (0.6–1.2 mm).</p>", "<p id=\"Par14\">For the gravity case, we used the mass flow waveform from Brown et al.^{##REF##22189248##34##}, which was prescribed in terms of a parabolic velocity profile at the aortic root. For each of the retinal arteriole outlets, we calculated outlet resistances using a structured asymmetrical fractal tree specific for retinal arteries as described by Malek et al.^{##UREF##10##31##}, which is an extension of methods developed by those such as Olufsen^{##UREF##12##35##}. Briefly, this method describes the branching of the daughter vessel radius from the parent vessel radius in terms of an exponent law and an asymmetry index, which allows for asymmetrical weighting of vessel branching. To calculate the resistances associated with a fractal tree network, a length ratio was assumed that varied depending on the branching vessel diameter. Vessel outlets that branched to a diameter below an assumed retinal capillary bed diameter of 4 μm^{##REF##21245397##36##} were assumed to have a pressure of 0 mmHg. The non-Newtonian behaviour of blood viscosity within small arterioles due to the FL effect is known to substantially affect upstream haemodynamics^{##REF##25510364##37##}. Consequently, we used an implementation of the FL viscosity model described by Liu et al.^{##REF##19958111##30##}, assuming a haematocrit value of 0.45, a plasma viscosity of 1.2 mPa s and blood density of 1050 kg m^-3\n^{##REF##19958111##30##}. Resistance values calculated for each retinal arteriole outlet were then converted to a corresponding effective viscosity value using the Hagen-Poiseuille equation for incompressible laminar fluid flow within cylindrical pipes (Eq. ##FORMU##0##1##).Where, for outlet <italic>i</italic>, <italic>µ</italic>_{<italic>i</italic>} is the effective viscosity, <italic>R</italic>_{<italic>i</italic>} is the fractal tree calculated resistance, and <italic>r</italic>_{<italic>i</italic>} and <italic>L</italic>_{<italic>i</italic>} are the radius and length respectively.</p>", "<p id=\"Par15\">We then implemented each representative resistance effective viscosity value within a corresponding extruded outlet region specific for each arteriole. The extrusion length was set as double the outlet diameter and the distal surface was prescribed a zero-pressure condition, as per the assumption of prescribing a pressure of 0 mmHg at the capillary bed.</p>", "<p id=\"Par16\">For each of the remaining arterial outlets outside of the retinal arterioles, we calculated the desired resistance for each outlet, which we then converted into a viscosity term using the same Hagen-Poiseuille Eq. (##FORMU##0##1##) and applied this within an extruded outlet region. To do this, we assumed a pressure drop from systolic pressure to zero across each extruded outlet region (representing the pressure drop towards distal capillary beds) and used the corresponding systolic volume flow rate from the pressure and inlet waveforms respectively from Brown et al.^{##REF##22189248##34##}. The volumetric flow to each outlet was then scaled primarily by the percentage distribution of cardiac output to an overarching arterial region, and then secondarily using the corresponding outlet radius relative to the other outlet radii within the same arterial region using Murray’s law (Eq. ##FORMU##1##2##). For the percentage of cardiac output to each region, we obtained values from literature of the estimated percentage of cardiac output to different arterial regions, which are summarised in Table ##TAB##0##1##. Flow to the subclavian arteries was calculated from the residual of total cardiac output and paired arterial regions were assumed to have symmetrical distribution of flow to each.Where, for outlet <italic>i</italic>, <italic>R</italic>_{<italic>i</italic>} is the calculated resistance, <italic>P</italic>_sys and <italic>Q</italic>_sys are the systolic pressure and flow values from Brown et al.^{##REF##22189248##34##}, CO_split is the estimated percentage split of cardiac output to an arterial region as summarised in Table ##SUPPL##0##1##, <italic>r</italic>_{<italic>i</italic>} is the outlet radius and <italic>N</italic> is the number of outlets in a corresponding arterial region.</p>", "<p id=\"Par17\">Each resistance outlet viscosity term was then implemented within a corresponding extruded outlet region, which was prescribed a length twice the outlet diameter and a zero-pressure boundary condition was imposed at the distal extrusion surface. For the gravity case, we prescribed the typical Earth gravitational acceleration of 9.81 m s^–2\n^{##UREF##13##38##} (1 g) acting inferiorly to emulate an upright position.</p>", "<p id=\"Par18\">For the simulated microgravity case, we modified the inlet waveform from Brown et al.^{##REF##22189248##34##} to account for the effects observed during spaceflight. Cardiac output is generally reported to increase in response to microgravity, with documented increases of 10%^{##REF##30975860##39##}, 20%^{##REF##28546443##40##,##REF##16301338##41##} as well as up to 30-40%^{##REF##25774397##42##}, and even in excess of 50%^{##UREF##0##1##,##REF##28798205##43##}. Heart rate is known to be relatively constant during spaceflight^{##REF##9688754##44##}, if not slightly decreased^{##UREF##14##45##}, indicating that there is an increase in stroke volume. To model this change, we vertically scaled the waveform from Brown et al. until the stroke volume (and hence cardiac output) was increased by an assumed 20%^{##REF##30137190##46##} with the aim to emulate a moderate increase in pre-load from increased venous return as observed during spaceflight^{##REF##25774397##42##,##REF##30137190##46##}. Furthermore, arterial resistance of the external iliac artery outlets (from Table ##TAB##0##1##) was increased by 93% to emulate the increased lower limb vascular resistance observed during spaceflight^{##REF##11247959##47##} relative to an upright position, while all other outlets remained at the corresponding Earth gravity resistances, as few other arterial networks have observed changes in resistance in response to microgravity^{##REF##32973543##48##}. Finally, we set the gravitational acceleration to 0 m s^-2.</p>", "<title>Simulation execution</title>", "<p id=\"Par19\">All simulations were solved using the finite-volume method within STAR-CCM+. We used the segregated flow solver and the implicit unsteady model with second-order temporal discretization, which uses the Semi-Implicit Method for Pressure-Linked Equations (SIMPLE) algorithm for coupling pressure and velocity. We used a time step of 0.001 s and inner iterations were terminated if normalised momentum and continuity residuals fell below 10^-4. Simulations were run for 3 cardiac cycles, whereupon data was extracted over a final fourth cycle, which was sufficient for the difference in cycle averaged metrics (i.e., global WSS, ICA and VA flow, etc.) to remain below 2–3% between the final two cycles for both the control and simulated microgravity cases. Simulations were run on Magnus, a Cray XC40 supercomputer (Pawsey Supercomputing Centre, Perth, Australia) using 600 cores across 25 compute nodes, each providing 24 cores per node. Simulations required approximately 20,500 core hours to complete, equating to 34 h of run time.</p>", "<title>Data extraction</title>", "<p id=\"Par20\">For each case, we extracted mass flow rates leading to the cerebrovasculature and retina, maximal velocity waveforms within the CRA and M1 segments of the middle cerebral artery (MCA) and surface averaged time-averaged WSS (TAWSS), and oscillatory shear index (OSI) within the retina, Circle of Willis (CoW), carotid bifurcations, coronary and iliac arteries as well as within the ascending and descending aorta. Surface averaged data is presented as surface average ± surface standard deviation.</p>" ]

[ "<title>Results</title>", "<title>Haemodynamic responses to simulated microgravity</title>", "<p id=\"Par21\">Qualitative distributions of the relative change in TAWSS and OSI between control and simulated microgravity conditions across the continuous arterial geometry as well as detailed views of regions of interest can be seen in Fig. ##FIG##1##2##. We extracted the surface averaged haemodynamic metrics across regions of interest throughout the entire 3D geometry (Fig. ##FIG##2##3##). Across both cases, absolute TAWSS was greatest in the CoW. The most substantial differences in TAWSS between gravity and simulated microgravity cases were found in the coronary arteries with increases of 41% (2.54 ± 2.74 Pa vs. 1.80 ± 1.97 Pa), in the left and right carotid bifurcations with increases of 36–37% (left, 3.72 ± 3.11 Pa vs. 2.74 ± 2.21 Pa; right, 4.14 ± 3.93 Pa vs. 3.02 ± 2.72 Pa), in the CoW with increases of 37% (6.04 ± 5.66 Pa vs. 4.40 ± 3.91 Pa) and within the left and right retinal arterioles with increases of 29-31% (left, 0.76 ± 2.27 Pa vs. 0.58 ± 1.92 Pa; right, 0.65 ± 2.37 Pa vs. 0.50 ± 1.99 Pa). Less substantial increases of 23% in the ascending (2.39 ± 1.08 Pa vs. 1.95 ± 0.91 Pa) and 17% in the descending (2.96 ± 5.17 Pa vs. 2.52 ± 4.05 Pa) aorta were found. In comparison, the TAWSS in the iliac arteries decreased by 4% (3.58 ± 2.30 Pa vs. 3.73 ± 2.35 Pa). Across both cases, absolute OSI was greatest in the ascending and descending aorta. In general, we observed a decrease in surface averaged distributions of OSI between gravity and simulated microgravity cases, with the largest decreases of –19% and –14%, respectively, in the left and right retinal arterioles (left, 0.010 ± 0.133 vs. 0.013 ± 0.137; right, 0.012 ± 0.132 vs. 0.014 ± 0.136), –16% and –10%, respectively, in the left and right carotid bifurcations (left, 0.127 ± 0.097 vs. 0.151 ± 0.107; right, 0.125 ± 0.107 vs. 0.138 ± 0.110) and –14% in the coronary arteries (0.109 ± 0.096 vs. 0.126 ± 0.096). Conversely, OSI in the descending aorta and CoW remained unchanged (0–1%), while a 7% increase was observed in the iliac arteries (0.066 ± 0.103 vs. 0.062 ± 0.110).</p>", "<title>Head and neck artery response to simulated microgravity</title>", "<p id=\"Par22\">In response to a 20% increase in cardiac output at the aortic root in the case of simulated microgravity, the computed velocities and flow rates within the cerebrovasculature were observed to increase (Fig. ##FIG##3##4##). A summary of waveform metrics are presented in Table ##TAB##1##2##. We found increases in systolic and average mass flow rates under simulated microgravity conditions compared to gravity conditions. Within the M1 segments of the middle cerebral arteries, peak and average maximal velocity were observed to increase in response to simulated microgravity, preferentially on the left side. Average maximal velocity was 34% and 42% greater in the left compared to the right M1 segment in the gravity and simulated microgravity cases respectively. The CRA observed similar increases in peak and average maximal velocity in response to simulated microgravity. In general, the average velocity leading to the retinal arterioles was 9–10% greater in the left compared to the right eye across both gravity and simulated microgravity cases. Given the fixed CRA cross-section, we extracted average volumetric flow rates (average ± standard deviation) leading to the eyes, where we found the peak and average retinal blood flow increased by 32% (57.7 ± 4.1 μl min^-1 vs. 43.8 ± 2.8 μl min^-1) and 31% (9.9 ± 0.4 μl min^–1 vs 7.6 ± 0.3 μl min^–1), respectively, in the simulated microgravity case compared to the gravity case.</p>", "<title>Retinal vasculature response to simulated microgravity</title>", "<p id=\"Par23\">We calculated the mean of surface averaged haemodynamic metrics across the left and right retinal arterioles, which were distributed by corresponding vessel diameter (Fig. ##FIG##4##5##). In general, TAWSS was found to be greatest in the smallest arterioles (g: 0.71–1.12 Pa and μg: 0.92–1.44 Pa across 10–20 μm diameters), followed by in the larger diameter vessels (g: 0.60–0.66 Pa and μg: 0.78–0.86 Pa across 90–110 μm diameters). Relative to the gravity case, TAWSS increased uniformly across all diameter bands in response to simulated microgravity (29-30%). OSI was almost uniformly distributed across small to large arterioles (g: 0.012–0.013 and μg: 0.010–0.011 across 10–130 μm diameters). The oscillatory shear decreased with increasing diameter in the larger arteriole vessels above 140 μm, irrespective of exposure condition. OSI also decreased in response to simulated microgravity across all diameter bands (–12–19%).</p>" ]

[ "<title>Discussion</title>", "<p id=\"Par24\">Since the beginning of spaceflight, there has been interest in understanding the effects of space travel on the human body. While the implications of microgravity on muscle mass and strength have been investigated in both animal models^{##REF##33911096##49##} and humans^{##REF##7649906##50##}, the effect on blood flow and arterial biomechanics is less understood. Some studies have examined blood flow stasis and thrombosis using ultrasound and revealed important flow abnormalities during microgravity^{##REF##31722025##51##}. Others have used computational modelling to replicate the haemodynamic effects of pressure changes and weightlessness^{##REF##33633331##52##}. In this study, we investigated the effects of simulated microgravity on vascular biomechanics in a large three-dimensional model of the arterial system, contiguous from the heart through to the eye.</p>", "<p id=\"Par25\">To achieve this, we combined 3D models from different imaging modality data to develop a large 3D model indicative of an arterial blood flow network, similar to methods used previously for 3D geometries spanning from the lower limbs to the CoW^{##UREF##1##11##,##REF##33333365##15##}. We developed a simulation framework for CFD analysis with continuous physical fluid characteristics and fractal tree and resistance outlets which were applied to this large 3D geometry. We then implemented the rudimentary effects of simulated microgravity in the arterial system. This work serves as an interesting proof of concept for future research that may seek to investigate the effects of physiological stimuli on large interconnected arterial networks.</p>", "<p id=\"Par26\">There is considerable interest in the reported vision loss associated with SANS due to prolonged spaceflight. Computer simulations may help reveal some of the biomechanics that may be contributing to the development of SANS^{##UREF##15##53##}. Salerni et al.^{##REF##31412033##10##} constructed a 0D model of the cerebrovasculature, retinal and choroidal vessels, incorporating the effects of changes in aqueous humour and cerebrospinal fluid flow, compression of the lamina cribrosa and the osmosis of fluid at the blood-brain barrier (BBB) in response to simulated microgravity. Although the focus of their study was the investigation of different oncotic pressures and their influence on intraocular and transmural pressures, they found that the parallel configuration of the retina and choroid chambers resulted in increases in normalised retinal flow of approximately 5% for the simulated microgravity case with a weakened BBB, while parallel flow in the choroid and the ciliary body gradually decreased. Although our simulation did not incorporate the effects of intraocular, oncotic or intracranial pressures, we did find that the increase in cardiac output and change in gravitational field in simulated microgravity conditions increased retinal blood flow relative to upright Earth gravity conditions. Within the eye, similar changes to those calculated in our simulations have also been observed during spaceflight. Using colour Doppler ultrasound, Sirek et al.^{##REF##24479252##54##} measured changes in peak systolic velocity in the CRA before, during and after spaceflight. From a database of 14 astronauts, they found an average increase in velocity of 36.1% combined across the left and right eyes from pre-flight values to inflight values, similar to the increases in CRA peak velocity calculated in our study (30-31%). The slightly lower relative changes in velocities in our study may be explained by the assumption of rigid geometry, as Sirek et al. also observed an 11% increase in optic nerve sheath diameter, which may compress the CRA, decreasing its diameter resulting in increases in velocity^{##REF##33242219##55##}; this was not accounted for in our model. Interestingly, ground-based experiments have found even greater increases in retinal blood flow, with Laurie et al.^{##REF##28611153##56##} measuring CRA velocity increases of 43–48% in HDT and HDT with hypercapnia compared to seated measures.</p>", "<p id=\"Par27\">An interesting question that arises from these findings is what elevated flow, and therefore shear stress, may mean in the context of the retina and conditions such as SANS. Elevation of shear stress to 2 Pa in bovine retinal endothelial cells has been previously shown to significantly increase retinal endothelium permeability by up to a factor of 14^{##REF##11262618##57##}. Higher shear stress conditions (&gt; 1 Pa), as opposed to low-moderate shear (0.1–0.5 Pa), have also been associated with pro-inflammatory responses and barrier dysfunction in human retinal endothelial cells^{##REF##31394104##58##}. Higher vascular permeability may not necessarily pose a risk to osmotic balance at the vessel wall provided albumin transport is matched^{##REF##20543206##59##}, however, albumin concentration has been found to be significantly lower in astronauts^{##UREF##16##60##,##REF##35389755##61##}. Evidence of retinal endothelial cell dysfunction has been observed previously in mice flown on the international space station (ISS), which exhibited significantly higher retinal endothelial cell apoptosis compared to both Earth controls, as well as to mice that also flew on the ISS while in a centrifuged habitat that produced an effective 1 g of artificial gravity^{##UREF##17##62##}. Our results show that an assumed increase in cardiac output of 20% due to emulating the increase in pre-load from increased venous return^{##REF##25774397##42##,##REF##30137190##46##} during simulated microgravity may result in up to a 30% increase in WSS in the retina, and where higher shear stress is primarily distributed in the smaller arterioles. Although an assumed value of 20% was used in this study^{##REF##28546443##40##,##REF##16301338##41##,##REF##30137190##46##}, long-duration spaceflight between 3–6 months has reported the possibility of greater increases in cardiac outputs between 35-41%^{##REF##25774397##42##}, with some estimates as high or in excess of 50%^{##UREF##0##1##,##REF##28798205##43##}. As discussed in later sections, if the assumed cardiac waveform in our study is lower than what may be the case for an individual, a higher baseline cardiac output coupled with the corresponding increase in shear stress in the retina attributed to the microgravity environment (which may be a potentially greater increase in cardiac output than the assumed 20%) may predispose these vessels to possible endothelial dysfunction and leakiness, subsequently contributing to the development of oedema in and around the retina. Recent research has postulated a multi-hit hypothesis to the progression of SANS, whereby any oedema caused by endothelial dysfunction may impact the outflow of cerebral spinal fluid which is already impaired in space^{##REF##35412862##63##}—in turn contributing to the pressure accumulation on the posterior eye^{##REF##28546443##40##}. Nonetheless, given the heterogeneous and non-individual-specific nature of the underlying 3D models used in this study, and the mirroring of retinal vasculature about the left and right sides, these results provide only the initial trends of the shear stress related responses to simulated microgravity in the eye. As such, the remaining pathophysiology of this condition remains highly complex and is likely contributed to by a multitude of factors, each requiring subsequent investigation.</p>", "<p id=\"Par28\">Simulated blood flow to the brain has also been reported by Gallo et al.^{##REF##33083524##4##}. In comparison to our findings where blood flow to the cerebrovasculature increased in response to simulated microgravity, they reported general decreases in blood flow in regions throughout the body, such as in the vertebral (–17%) and internal carotid (–19%) arteries. However, their study purposely compared the simulated microgravity results to a reference supine condition. Hence, many of their findings are the inverse to the results in our study, where we observed increases in vertebral and internal carotid artery flows between 21–28%, due to our gravity condition being in an upright reference position. Nevertheless, as suggested by the authors, results of increased flow upon exposure to simulated microgravity would be expected relative to an upright condition^{##REF##33083524##4##,##REF##29167331##64##}. Ground-based emulated microgravity research of neck artery flow by Ogoh et al.^{##REF##31691384##65##} measured flow leading to the cerebral vasculature after 57 days of –6 degrees HDT rest as an analogue for prolonged spaceflight. Relative to pre-HDT rest in a supine condition, they observed an average reduction in blood flow in the ICA after 30 (–23%) and 57 days (–15%), while the vertebral arteries remained unchanged, resulting in the vertebral arteries carrying an increased proportion of the cerebrovascular flow relative to pre-HDT. Although we report results relative to an upright condition (and as indicated, our results appear inverted as relative to upright increases compared to relative to supine reductions) we found greater changes in average flow in the vertebral compared to internal carotid arteries, reflecting an increase in proportional cerebrovascular flow in the vertebral arteries between the simulated microgravity and gravity cases.</p>", "<p id=\"Par29\">Interestingly, in general, we also observed higher flows in the left arteries leading to the cerebrovasculature compared to the right, which is also reflected in the flows leading to the eye in the central retinal artery. Although possibly anatomically specific to an individual, this may be due to the left carotid and vertebral arteries originating from branches either closer or directly from the aortic arch, compared to the right side branching from the brachiocephalic artery. Interestingly, naturally higher left-side flow has previously been observed, particularly between vertebral artery sides^{##UREF##18##66##}, which is consistent with the findings of our study in that left vertebral artery flow was substantially higher than that of right vertebral artery flow. Interestingly, though warranting further investigation, additional preferential arterial flow to the left side of the cerebrovasculature may potentially contribute to the findings of additional flow stasis observed in the left jugular vein during spaceflight, which is less pronounced on the right side^{##REF##35502899##67##–##REF##11822475##69##}.</p>", "<p id=\"Par30\">Blood flow simulations of isolated 3D geometries leading to, and within the cerebrovasculature, under different gravitational loadings have also been performed previously. Kim et al.^{##UREF##4##17##} simulated blood flow through compliant carotid bifurcation and CoW 3D geometries, as well as incorporating an autoregulatory mechanism at the arterial outlets, to investigate the changes observed in response to spaceflight. In the carotid artery bifurcation, they found that in order to maintain consistent blood flow to the outlets as per their autoregulation algorithm, the carotid diameter in the simulated microgravity case increased by 6.2% relative to the upright gravity case. As a result, distributions of TAWSS between the cases were observed to decrease almost uniformly under simulated microgravity relative to upright. Similar changes were also observed leading to and within the CoW, with diameter increases in the ICAs (3%), basilar (4.4%) and MCAs (6.9%) under the simulated microgravity case relative to upright. Similar changes in TAWSS were observed with almost uniform decreases in all regions of the CoW and proximal arteries. In comparison, our simulations used a rigid wall boundary condition preventing vessel wall change, and consequently yielded the inverse result, with almost uniform increases in surface averaged TAWSS across the upper body regions of the 3D geometry.</p>", "<p id=\"Par31\">Within the brain itself, MCA velocity changes have also been observed in response to spaceflight microgravity or terrestrial microgravity emulation. In response to HDT and HDT with induced hypercapnia, Laurie et al.^{##REF##28611153##56##} measured increases in average MCA velocity of approximately 20%, an increase similar to the results found in our study (20–28%). In comparison, cerebral blood flow measured in four astronauts after 1 and 2 weeks in space^{##REF##17185344##70##} was found to elucidate non-significant changes in MCA average velocity relative to the pre-flight measurement. However, 1 astronaut did observe a substantial increase in average MCA velocity at the 2-week mark, an increase of approximately 28% relative to pre-flight measurements. The lack of change in MCA velocity could indicate cerebral autoregulation acting for this duration of spaceflight, though this may not necessarily occur across all individuals. Similar non-significant changes in MCA average velocity have also been observed during parabolic flights (15 bouts of 20 s of parabolic freefall), where small average increases (4%) were observed across 16 participants^{##REF##30741333##71##}. In comparison to these findings, Iwasaki et al.^{##REF##33103234##72##} found that MCA blood flow velocity in 11 astronauts pre and post-spaceflight (between 3–6 months prior to the flight and within 3 days of landing), for either supine or sitting measurements significantly increased by between 10–13%. The blood flow velocity was then observed to reduce to pre-flight levels after a recovery window of between 1–6 months of landing. This increase in MCA velocity is less substantial than the changes observed in our study, however, this could be attributed to the study subjects returning to Earth for imaging, where the reintroduction of Earth’s gravitational vector may have influenced cerebral blood flow within the first 3 days of landing.</p>", "<p id=\"Par32\">The question remains, however, what these changes in cerebral flows as per the findings of our study, and similarly to others, mean for individuals in microgravity. Although postural changes may result in brain blood flow increasing or decreasing throughout a day within autoregulatory bounds, our results show that simulated microgravity results in a constant increase of 20–30% in brain blood flow, and close to 40% increases in shear stress within the brain. This may have consequences in that increased perfusion of the brain may lead to exacerbated autoregulatory responses resulting in prolonged vessel dilation, reduced myogenic response, which is consistent with mouse spaceflight models^{##REF##23457215##73##}, decreased cerebrovascular resistance and consequently any increased pressure acting on the cerebral endothelium could result in oedema^{##UREF##19##74##}. Typical time-averaged shear acting across the sensitive BBB within the cerebrovasculature is in the range of 0.3–3 Pa^{##REF##24009582##75##}, and where moderate shear stress within this range has been found to be beneficial to the barrier function of cerebral endothelial cells^{##REF##21569296##76##}. However, severely elevated pulsatile shear stress (&gt; 4 Pa) has been associated with the downregulation of BBB tight junction markers, impeding endothelial cell contact^{##REF##27702879##77##}. Our findings show that simulated microgravity serves to substantially increase the shear stresses acting both within the cerebrovasculature and retinal vessels. Coupled with additional blood throughput potentially leading to venous stasis and congestion^{##UREF##0##1##,##REF##34705011##78##}, our findings are consistent with causes of fluid oedema associated with exposure to microgravity, which is often observed as a key contributor to the pathogenesis associated with the development of SANS. Nonetheless, future work is required to improve the understanding of the development of SANS as well as the clinical implications of constant elevated flow to the brain.</p>", "<p id=\"Par33\">There is limited data on the effects of microgravity on the coronary arteries, in particular shear stress. We found that TAWSS in the coronary arteries increased in response to simulated microgravity, although both gravity and simulated microgravity case values fell within the normal and atheroprotective shear stress range of 1–7 Pa^{##REF##31566246##79##}. Although anecdotal, this finding may support NASA data reporting that, when compared to healthy terrestrially based control populations, astronauts following spaceflight do not have increased differences in cardiovascular and coronary artery disease or standardised mortality^{##REF##28784652##80##,##REF##29855508##81##}.</p>", "<p id=\"Par34\">Various in vitro cellular model and in vivo animal model studies have been used to investigate the functional effects of emulated microgravity on endothelial cells and arteries. Hindlimb unloading (HU) is an animal model technique involving the suspension of rodents to create a downwards head tilt and pressure gradient across the body, similar to head-down tilt (HDT) in humans. Despite minimal morphological changes^{##REF##10601157##82##}, functional changes such as vasoconstriction and relaxation responses in young HU rat abdominal aorta samples have been found to be reduced relative to control rats^{##REF##7665402##83##}. Similar diminished vasoconstriction responses have also been observed in the mesenteric arteries of HU rats^{##REF##18218919##84##} as well as in mice that have flown in space^{##REF##23099650##85##}. Alternatively, Shi et al.^{##REF##22808143##86##} found that cultured human umbilical vein endothelial cells experiencing 24 h of emulated microgravity conditions using a clinostat upregulated endothelial nitric oxide synthase, increased cell migration and promoted angiogenic pathways. Similar findings of increased endothelial cell migration and nitric oxide production have been observed by Siamwala et al.^{##REF##20174953##87##} after 2 h of similarly emulated microgravity. Increases in endothelial nitric oxide synthase have also been observed in the aortas of HU mice^{##REF##23511048##88##}. In our study, surface averaged TAWSS (1.95–2.52 Pa) across the aorta increased by 17–23% in the simulated microgravity case. Given the mechanoactivation of endothelial nitric oxide synthase is associated with higher shear stresses^{##REF##34073212##89##}, haemodynamic responses to simulated or emulated microgravity may induce, on average, somewhat favourable endothelial conditions in larger arteries, such as the aorta, and contribute to any reductions in vasoconstriction.</p>", "<p id=\"Par35\">Blood flow changes in the lower limbs have also been investigated previously from spaceflight data, simulations and ground analogue experiments. Gallo et al.^{##REF##33083524##4##} implemented a large 0D-1D model, combining a 1D arterial tree with 0D representations of circulatory regions and baroreceptor mechanisms to understand the deconditioning of the cardiovascular system during long-duration spaceflight. Compared to upper body flow, they calculated smaller decreases in flow to the lower limb regions of the inner iliac (-2.27%) and femoral (–4.87%) arteries in response to simulated microgravity. Although again, the changes in flow are inverted compared to ours due to using a supine position as their relative condition. Despite this, we also observed a greater proportion of flow distributed to the upper body compared to the lower limbs. After 5 weeks of HDT, a study by Palombo et al.^{##REF##25654096##90##} found that the diameters of the femoral artery were significantly reduced, while non-significant reductions in wall shear rates were measured with ultrasound at the near (–2%) and far (–9%) walls. In our study, we calculated similar small decreases in TAWSS across the iliac arteries (–4%) in response to simulated microgravity, while interestingly the absolute values of TAWSS remained higher compared to the upstream regions such as those in the aorta. Nonetheless, this decrease in shear stress reflects a reduction in blood flow towards the lower limbs in response to simulated microgravity, which is consistent with decreases (though reversible after one month of Earth gravity) in superficial blood flow that has been observed and measured pre and post flight in the lower limbs of astronauts^{##UREF##20##91##}. Prolonged reductions in flow to lower limbs may have implications for the metabolic health in these regions, particularly given the documented musculoskeletal wasting that occurs in space with the reduction in gravitational loading^{##REF##7649906##50##,##REF##10336885##92##}. Furthermore, prolonged reductions in perfusion to the lower limbs may present additional risks in the context of the development of peripheral artery conditions or disease, which are generally characterised by reductions in perfusion and ischaemia in these regions^{##UREF##21##93##}. Promising potential countermeasures include lower body negative pressure devices, which serve to counteract upward fluid shift and redistribution by introducing negative pressure about the lower limbs^{##REF##35207555##94##}. These devices may also serve as a potential countermeasure for the pathophysiological development of SANS, which is suspected to be caused, at least in part, by this fluid gradient and redistribution of fluid throughout the body^{##REF##35207555##94##}.</p>", "<p id=\"Par36\">Despite differences in exact geometry, dimensional representation or environmental conditions, and variation in demographics associated with imaging sources, we observed some similarities between our results and existing large-scale simulation networks. Blanco et al.^{##REF##25347874##95##} developed a 1D anatomically detailed arterial network model consisting of over 2,000 arterial vessels using a 3D circulatory representation as a geometrical substrate. Although the inlet flow rate in their model was greater than the inlet flow rate used in our gravity case, the average flow calculated across the VAs matched our results to within 2%, though the average flow in the ICAs was 25% greater in their study compared to ours. Xiao et al.^{##UREF##1##11##} developed a simulation of a 3D deformable full-body arterial network consisting of arterial vessels ranging from the tibial artery to the CoW. Using diameters and flow data provided in their work, the average velocity in the left middle cerebral artery was found to be 54% greater than the same artery in our gravity simulation. However, their inlet flow waveform was substantially higher, with a systolic flow rate approximately double that used in our gravity case simulation. Xiao et al. also calculated and presented the shear stress throughout the entire geometry but highlighted that the mesh used was insufficient to ensure grid independence in the WSS fields and that the results were only to provide an indication of capability. Our simulation framework demonstrates that this resolution is achievable to capture the WSS and associated haemodynamic metrics.</p>", "<p id=\"Par37\">Nonetheless, despite serving as an initial proof of concept study for investigating continuously connected arterial networks in response to environments, such as simulated microgravity, the methods proposed in this work are not without limitations and need for future development.</p>", "<p id=\"Par38\">Firstly, as we used a mixture of 3D model data from different imaging modalities and sources across different subjects, the 3D model developed does not represent a single individual. Consequently, the underlying 3D model heterogeneity may influence the absolute data reported, such as the distributions of surface averaged shear stress or the amplitude of velocity waveforms. As such, although absolute data is reported for reference, the findings from this study aimed to focus on the relative changes and trends offered by rudimentary simulation of microgravity, whereby any systematic heterogeneity effects may be nullified through cancellation given the use of the same 3D model in both Earth gravity and simulated microgravity cases. Nonetheless, by incorporating real imaging data, the model does at least represent a continuous human arterial network that is physiologically possible, albeit not singularly subject specific and formed from sources with varying demographics. Future work using the methods and approaches described in this study would ultimately use individual-specific imaging data for the construction of the continuous arterial network 3D model. One key benefit of this approach, however, is that (as demonstrated in our study) retrospective imaging data can be combined to form the 3D arterial network. Consequently, future work with individual astronaut data would potentially be feasible—enabling greater insights into the haemodynamics occurring throughout a large proportion of the arterial cardiovasculature in the spaceflight environment. Alternatively, subject-specific data could be combined to understand how individuals with pre-existing cardiovascular risk factors may be predisposed in environments of varying gravitational load, such as during spaceflight, on Martian or lunar surfaces, or during the acute hyper-gravity associated with planetary exit or re-entry.</p>", "<p id=\"Par39\">Secondly, we used a rigid wall model that did not account for the movement of the arterial wall. Although astronauts undergoing 6-month duration spaceflight have been observed to experience the equivalent of 10–20 years of arterial aging and stiffening^{##UREF##22##96##}, this remains a limitation given vessels are inherently compliant in healthy populations, of which would be the case in astronauts currently undertaking spaceflight missions. Alternatively, wall movement could be achieved in future work using fluid-structure interaction (FSI) modelling. This was not performed, however, in this initial study given the significant additional computational load required for FSI simulation, computational stability and interfacing requirements across such a large arterial domain, and as the arterial wall thicknesses and tissue material properties were either unable to be resolved or known, respectively. Furthermore, in future work that would ideally use individual-specific imaging, obtaining wall thickness and tissue material properties would either be impossible, severely invasive or limited to only the larger arteries. Additionally, these methods would need to account for vessel pretension embedded in the 3D reconstructions from vessel imaging being captured, which represents vessels already at arterial pressure. Despite rigid wall modelling not accounting for the deformation fluctuations experienced throughout the cardiac cycle and instead representing a snapshot in time, comparable distributions of WSS between rigid and FSI simulations have previously been observed, although rigid wall simulations generally overestimate instantaneous WSS compared to FSI^{##REF##22189248##34##,##REF##22981220##97##}. Surface and time-averaged metrics, such as those used in this study, have also been observed to be similar between rigid wall and FSI methods^{##REF##22153749##98##}.</p>", "<p id=\"Par40\">Thirdly, for all arterial outlets except for the retinal arterioles, we used a mixture of known estimated flow splits to arterial networks and then Murray’s law to estimate the distribution of flow throughout numerous arterial outlet regions. This approach was adopted due to the simplicity of implementation, but it limited the simulations in terms of accounting for any changes in vessel compliance or autoregulatory constrictions/dilations. Alternative outlet modelling approaches that should be considered include using varying power law exponents based on literature or modelling distal resistance and compliance using multi-element Windkessel models, which may enable the incorporation of autoregulatory mechanisms in the cerebrovasculature as well as account for the effects of venous stasis and congestion that are generally associated with the spaceflight environment^{##UREF##0##1##,##REF##34705011##78##}. Additionally, MRI or ultrasound methods could be employed to measure subject-specific regional flow distributions, as opposed to the assumed values as described in Table ##TAB##0##1##.</p>", "<p id=\"Par41\">Fourthly, the cardiac output of the inlet flow condition adapted from Brown et al.^{##REF##22189248##34##} may be inadequate for the geometry developed, which is reflected in the variation in results for the gravity condition with other large arterial simulations. Future work should aim to use subject-specific measured cardiac output, which could be obtained using MRI or duplex ultrasound methods. Alternatively, in the absence of cardiac output data, a parametrically swept range of different cardiac outputs could be investigated. However, as the goal of this study was to provide an initial framework for comparing relative changes in response to simulated microgravity in a large 3D continuous arterial network, the relative changes were found to be somewhat consistent with emerging microgravity research and measured data. Additionally, while we incorporated the aortic root as part of the geometry, we implemented the flow at the aortic valve surface as a simplified parabolic velocity flow profile with a fixed orifice area, neglecting the natural helical and three-dimensional nature of blood flow ejected from the aortic valve.</p>", "<p id=\"Par42\">Finally, we assumed the flow to be within the laminar flow regime, which is a common approach in arteries outside of the aorta, however, turbulence is likely induced within the ascending aorta due to the high ejection velocities at the aortic valve as well as during the deceleration phase of the cardiac cycle. As a key region of interest in this study were the vessels leading to and within the eye, which are known to exhibit mostly laminar flow, this regime was considered appropriate and any turbulence generated at the aortic root was assumed to have minimal effect on reported haemodynamics in these regions. Modelling using large eddy simulation (LES) may be more appropriate in future studies, particularly to investigate changes in haemodynamics within the aorta and nearby larger arteries, though this was not performed due to the increasingly high computational cost associated with this modelling approach.</p>", "<p id=\"Par43\">In this study, we aimed to demonstrate that large-scale 3D arterial networks can be constructed across a wide range of vessel calibres from 3D models derived from numerous image datasets and that the resulting geometry can be used to understand the change in haemodynamics in response to simulated microgravity. From our simulations, we found similarities with existing spaceflight simulation models and measured data—specifically that blood flow and shear stress decrease towards the lower limbs and increase towards the cerebrovasculature and within the eyes in response to simulated microgravity exposure relative to an upright position in Earth gravity. This framework may also prove useful to simulate the changes in haemodynamics in other equally challenging environments influencing the cardiovascular system.</p>" ]

[]

[ "<p id=\"Par1\">We investigated variations in haemodynamics in response to simulated microgravity across a semi-subject-specific three-dimensional (3D) continuous arterial network connecting the heart to the eye using computational fluid dynamics (CFD) simulations. Using this model we simulated pulsatile blood flow in an upright Earth gravity case and a simulated microgravity case. Under simulated microgravity, regional time-averaged wall shear stress (TAWSS) increased and oscillatory shear index (OSI) decreased in upper body arteries, whilst the opposite was observed in the lower body. Between cases, uniform changes in TAWSS and OSI were found in the retina across diameters. This work demonstrates that 3D CFD simulations can be performed across continuously connected networks of small and large arteries. Simulated results exhibited similarities to low dimensional spaceflight simulations and measured data—specifically that blood flow and shear stress decrease towards the lower limbs and increase towards the cerebrovasculature and eyes in response to simulated microgravity, relative to an upright position in Earth gravity.</p>", "<title>Subject terms</title>" ]

[ "<title>Supplementary information</title>", "<p>\n\n</p>" ]

[ "<title>Supplementary information</title>", "<p>The online version contains supplementary material available at 10.1038/s41526-024-00348-w.</p>", "<title>Acknowledgements</title>", "<p>This work was supported by resources provided by the Pawsey Supercomputing Centre with funding from the Australian Government and the Government of Western Australia. HC is supported by a Forrest Research Foundation Scholarship and an Australian Government Research Training Programme Scholarship at The University of Western Australia. DG is supported by a National Health and Medical Research Council Principal Research Fellowship (APP1080914). We would like to acknowledge Prof Natzi Sakalihasan (University of Liege) and Prof Carl Schultz (University of Western Australia) as we used 3D models derived from medical images acquired as part of their independent research.</p>", "<title>Author contributions</title>", "<p>All authors contributed to the study conception and design. 3D file generation and simulation preparation: H.C., L.K. and L.P. Simulation execution and data extraction: H.C. Data analysis: H.C., L.K., L.P., D.G. and B.D. Manuscript drafting: H.C. All authors reviewed, commented and edited each iteration of the manuscript. Supervision: D.G. and B.D. All authors read and approved the final manuscript.</p>", "<title>Data availability</title>", "<p>Data not directly presented in the article, such as geometries and simulation files, can be made available on reasonable request to the authors.</p>", "<title>Code availability</title>", "<p>The fluid simulation code used in this study is commercially available.</p>", "<title>Competing interests</title>", "<p id=\"Par44\">The authors declare no competing interests.</p>" ]

[ "<fig id=\"Fig1\"><label>Fig. 1</label><caption><title>Continuous arterial network model.</title><p>Examples of types of corresponding image data are provided alongside detailed views of the 3D geometry. This model was comprised of four 3D reconstructions associated with separate image datasets, consisting of the retinal arterioles (<bold>a</bold>) with corresponding image adapted from^{##REF##24416040##23##} (CC BY 3.0), cerebrovasculature and neck arteries (<bold>b</bold>), aorta and iliac arteries (<bold>c</bold>) and the aortic root and coronary arteries (<bold>d</bold>). The 3D reconstructions of these regions were combined within STAR-CCM+.</p></caption></fig>", "<fig id=\"Fig2\"><label>Fig. 2</label><caption><title>Entire arterial network surface distributions of haemodynamic metrics.</title><p>Distributions show the relative change (%) in time-averaged wall shear stress (TAWSS) (<bold>a</bold>) and oscillatory shear index (OSI) (<bold>b</bold>) from Earth gravity to simulated microgravity (g→µg). Detail views (with varying colour bar scales for TAWSS) of the retinal, cerebrovascular, carotid bifurcation, coronary, branching visceral and renal, and iliac arteries are presented.</p></caption></fig>", "<fig id=\"Fig3\"><label>Fig. 3</label><caption><title>Surface averaged haemodynamic metrics.</title><p>Data is presented for the Earth gravity (g; white) and simulated microgravity (µg; grey) cases. Measures of time-averaged wall shear stress (TAWSS) (<bold>a</bold>) and oscillatory shear index (OSI) (<bold>b</bold>) are presented as a surface average with error bars representing the corresponding regional surface standard deviation.</p></caption></fig>", "<fig id=\"Fig4\"><label>Fig. 4</label><caption><title>Blood flow waveform changes throughout the cerebrovasculature and the eye.</title><p>Waveforms are presented for Earth gravity (g; solid black line) and simulated microgravity (µg; dashed red line). Maximal velocity (<italic>V</italic>_max) in the left (<bold>a</bold>) and right (<bold>e</bold>) central retinal arteries (CRA) and left (<bold>b</bold>) and right (<bold>f</bold>) M1 segment of the middle cerebral artery, as well as mass flow rate waveforms (MFR) in the left (<bold>c</bold>) and right (<bold>g</bold>) internal carotid (ICA_L and ICA_R) and left (<bold>d</bold>) and right (<bold>h</bold>) vertebral (VA_L and VA_R) arteries over the final cardiac cycle are presented.</p></caption></fig>", "<fig id=\"Fig5\"><label>Fig. 5</label><caption><title>Haemodynamic metrics within the retinal arterioles.</title><p>Diameter distributed surface averaged haemodynamic metrics for the Earth gravity (g; white) and simulated microgravity (µg; grey) cases. Measures of surface averaged time-averaged wall shear stress (TAWSS) (<bold>a</bold>) and oscillatory shear index (OSI) (<bold>b</bold>) are presented as a combined mean of the left and right eye diameter bins, with error bars representing the corresponding standard deviation of each bin average.</p></caption></fig>" ]

[ "<table-wrap id=\"Tab1\"><label>Table 1</label><caption><p>Assumed flow distribution to different arterial networks expressed as a percentage of cardiac output.</p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th>Artery region (<italic>N</italic>)</th><th>Cardiac output</th><th>Refs.</th></tr></thead><tbody><tr><td>Celiac artery (1)</td><td>14.00%</td><td>^{##UREF##23##99##}</td></tr><tr><td>Cerebral arteries (32)</td><td>12.78%</td><td>^{##REF##21486813##100##}</td></tr><tr><td>Coronary arteries (26)</td><td>5.00%</td><td>^{##UREF##24##101##}</td></tr><tr><td>External carotid artery (2)</td><td>9.45%</td><td>^{##REF##21486813##100##}</td></tr><tr><td>Internal iliac artery (2)</td><td>4.00%</td><td>^{##UREF##23##99##}</td></tr><tr><td>External iliac artery (2)</td><td>9.00%</td><td>^{##UREF##23##99##}</td></tr><tr><td>Ophthalmic artery (2)</td><td>0.68%</td><td>^{##REF##23518769##102##}</td></tr><tr><td>Mesenteric artery (1)</td><td>16.00%</td><td>^{##UREF##23##99##}</td></tr><tr><td>Renal artery (2)</td><td>23.00%</td><td>^{##UREF##23##99##}</td></tr><tr><td>Subclavian artery (2)</td><td>6.09%</td><td>-</td></tr></tbody></table></table-wrap>", "<table-wrap id=\"Tab2\"><label>Table 2</label><caption><p>Different calculated flow and velocity metrics extracted from the gravity (g) and simulated microgravity (µg) cases, as well as the relative change (g → µg) from gravity to simulated microgravity.</p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th>Waveform metric</th><th>g</th><th>µg</th><th>% Change (g → µg)</th></tr></thead><tbody><tr><td>ICA_L MFR_peak (kg s^-1)</td><td>0.0164</td><td>0.0210</td><td>28%</td></tr><tr><td>ICA_R MFR_peak (kg s^-1)</td><td>0.0159</td><td>0.0217</td><td>36%</td></tr><tr><td>VA_L MFR_peak (kg s^-1)</td><td>0.0113</td><td>0.0142</td><td>26%</td></tr><tr><td>VA_R MFR_peak (kg s^-1)</td><td>0.0056</td><td>0.0070</td><td>25%</td></tr><tr><td>ICA_L MFR_ave (kg s^-1)</td><td>0.0037</td><td>0.0045</td><td>21%</td></tr><tr><td>ICA_R MFR_ave (kg s^-1)</td><td>0.0035</td><td>0.0042</td><td>21%</td></tr><tr><td>VA_L MFR_ave (kg s^-1)</td><td>0.0021</td><td>0.0027</td><td>25%</td></tr><tr><td>VA_R MFR_ave (kg s^-1)</td><td>0.0010</td><td>0.0013</td><td>28%</td></tr><tr><td>M1_L\n<italic>V</italic>_{max, peak} (cm s^-1)</td><td>235.49</td><td>301.57</td><td>28%</td></tr><tr><td>M1_R\n<italic>V</italic>_{max, peak} (cm s^-1)</td><td>207.03</td><td>250.24</td><td>21%</td></tr><tr><td>CRA_L\n<italic>V</italic>_{max, peak} (cm s^-1)</td><td>5.96</td><td>7.88</td><td>32%</td></tr><tr><td>CRA_R\n<italic>V</italic>_{max, peak} (cm s^-1)</td><td>5.24</td><td>6.86</td><td>31%</td></tr><tr><td>M1_L\n<italic>V</italic>_{max, ave} (cm s^-1)</td><td>41.97</td><td>53.58</td><td>28%</td></tr><tr><td>M1_R\n<italic>V</italic>_{max, ave} (cm s^-1)</td><td>31.33</td><td>37.64</td><td>20%</td></tr><tr><td>CRA_L\n<italic>V</italic>_{max, ave} (cm s^-1)</td><td>1.00</td><td>1.31</td><td>31%</td></tr><tr><td>CRA_R\n<italic>V</italic>_{max, ave} (cm s^-1)</td><td>0.92</td><td>1.19</td><td>30%</td></tr><tr><td>CRA_L\n<italic>Q</italic>_peak (µl min^-1)</td><td>45.78</td><td>60.60</td><td>32%</td></tr><tr><td>CRA_R\n<italic>Q</italic>_peak (µl min^-1)</td><td>41.83</td><td>54.76</td><td>31%</td></tr><tr><td>CRA_L\n<italic>Q</italic>_ave (µl min^-1)</td><td>7.77</td><td>10.20</td><td>31%</td></tr><tr><td>CRA_R\n<italic>Q</italic>_ave (µl min^-1)</td><td>7.41</td><td>9.61</td><td>30%</td></tr></tbody></table></table-wrap>" ]

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[ "<supplementary-material content-type=\"local-data\" id=\"MOESM1\"></supplementary-material>" ]

[ "<table-wrap-foot><p>The number of vessels associated with each arterial network is provided in brackets (<italic>N</italic>). Paired arteries are assumed to have symmetrical flow distribution to each side. Flow to the subclavian arteries was assumed to be the residual of cardiac output.</p></table-wrap-foot>", "<table-wrap-foot><p>Peak (MFR_peak) and average (MFR_ave) mass flow in the left (ICA_L) and right (ICA_R) internal carotid and left (VA_L) and right (VA_R) vertebral arteries as well as peak (<italic>V</italic>_{max, peak}) and average (<italic>V</italic>_{max, ave}) velocity in the M1 segment of the left (M1_L) and right (M1_R) middle cerebral artery and left (CRA_L) and right (CRA_R) central retinal arteries are presented. Peak (<italic>Q</italic>_peak) and average (<italic>Q</italic>_ave) volume flow rates are also provided for the left and right central retinal arteries.</p></table-wrap-foot>", "<fn-group><fn><p><bold>Publisher’s note</bold> Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.</p></fn></fn-group>" ]

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{ "acronym": [], "definition": [] }

[ "<title>Introduction</title>", "<p id=\"Par2\">Millimeter wave (mmWave) has attracted huge interest in wireless technologies, in particular for the fifth generation of mobile communication (5G) applications in recent years. The demand for massive data rates and low latency in 5G systems can be addressed with the large available bandwidth in mmWave frequencies. The Advanced Antenna System (AAS) is a recommended necessity for 5G applications in mmWave frequencies. AAS employs a large antenna array with beamforming capability to improve the performance and spectral efficiency of 5G systems^{##UREF##0##1##}. AAS requirements for 5G applications from the latest standards can be found in Ref.^{##UREF##0##1##} in terms of frequency bands, antenna element properties, array configurations, and beamforming characteristics. The 3rd Generation Partnership Project (3GPP) unites well-known standard organizations to develop protocols for mobile telecommunication systems^{##UREF##2##3##}. Some standard mmWave frequency bands for 5G systems are^{##UREF##0##1##–##UREF##2##3##}:<list list-type=\"bullet\"><list-item><p id=\"Par3\"><italic>n257</italic>: 26.5–29.5 GHz;</p></list-item><list-item><p id=\"Par4\"><italic>n258</italic>: 24.25–27.5 GHz;</p></list-item><list-item><p id=\"Par5\"><italic>n261</italic>: 27.5–28.35 GHz (subset of <italic>n257</italic>).</p></list-item></list></p>", "<p id=\"Par6\">The single antenna element is recommended to be dual-polarized with the half power beam width of 65°^{##UREF##3##4##}. The 5G antenna array known as AAS utilizes adaptive beam forming, multiple input multiple output (MIMO), and Spatial Division Multiple Access (SDMA)^{##UREF##0##1##,##UREF##1##2##}. The main requirements of AAS for 5G mmWave applications are presented in Table ##TAB##0##1##.</p>", "<p id=\"Par7\">Beamforming is a key technique in AAS that concentrates the power toward the desired direction and nulls the undesired directions. Beamforming can be implemented as analog, digital, and hybrid configurations. Analog beamforming is easy to implement but has a limited number and characteristics of fixed beams. Digital beamforming employs a separate RF chain for each array element, leading to a complicated structure, but provides very flexible and efficient beamforming. Hybrid beamforming utilizes both analog and digital beamforming as each RF chain is associated with multiple antenna elements^{##UREF##4##5##,##UREF##5##6##}. The hybrid beamforming seems to be more efficient since the beamforming is performed in the analog domain using fewer RF chains.</p>", "<p id=\"Par8\">It is clear that analog beamformers represent an important component of hybrid beamforming systems. Several different analog RF beamforming networks have been proposed mostly following the Butler matrix and Rotman lens topologies. The Butler matrix is a microwave network as the analog implementation of the fast Fourier transform including couplers and phase shifters. The phase shifts at the output ports can be determined by a combination of the phase shifts of all the signal paths. The Rotman lens is a scanning system that can be used in various systems as a fundamental multiple-beam antenna. In a Rotman lens, the required phase distribution on the antenna ports is achieved by true time-delay (TTD) through a shaping path and maintains a constant time delay over a broadband frequency range of operation^{##UREF##6##7##}.</p>", "<p id=\"Par9\">The Butler matrix is developed by integrating the couplers, phase shifters, and crossovers exhibiting significantly narrower bandwidths than most wideband antenna arrays. Rotman lenses are passive beamforming networks by implement true time-delay with relatively wideband performance^{##UREF##7##8##}.</p>", "<p id=\"Par10\">The main advantages of Rotman lenses compared to Butler and Blass Matrices are lower weight and hardware cost while having wider bandwidth and beam steering. Therefore, a Rotman lens is suitable for applications that require both a large scan of the radiation pattern and wide frequency range coverage^{##UREF##8##9##}.</p>", "<p id=\"Par11\">In Ref.^{##UREF##9##10##}, a 5 × 8 Butler matrix operating at the frequency range of 27.8–30.8 GHz is presented where output signals with equal power divisions and five differential phases can be obtained. However, this beamforming network does not provide continuous beams.</p>", "<p id=\"Par12\">A 16-element antenna array with a Butler matrix covering the band 26–31.4 GHz and ± 42° beam switching is presented in Ref.^{##UREF##10##11##}. The maximum gain is 12 dBi with − 19 dB SLL, and 9 dBi with − 8 dB SLL for the beams with ± 13° and ± 42°, respectively.</p>", "<p id=\"Par13\">In Ref.^{##UREF##11##12##} a wide angular Rotman lens operating in the 28 GHz band is proposed. The 6-port Rotman lens is connected to eight linear antenna arrays consisting of five series-fed rectangular patches. The angular scan is from − 60° to 60° with a variation of less than 8 dB. However, the bandwidth is low and the gain variation is high.</p>", "<p id=\"Par14\">In Ref.^{##UREF##12##13##} a PCB-based Rotman lens consisting of an eight-element Yagi–Uda antenna array at 28 GHz is demonstrated. The proposed beamformer operates across 25.5–28.5 GHz with 7 switchable beams covering ± 30° with a realized gain of up to 9.4 dBi. However, the scan loss for the side angle (± 30°) is relatively high (4.5 dB).</p>", "<p id=\"Par15\">With the extensive review of the literature on the mmWave Rotman lens topic, the following major problems in the existing designs can be recognized:<list list-type=\"bullet\"><list-item><p id=\"Par16\">Lack of Rotman lens design with wide-angle scanning around ± 60° over the wide frequency bandwidth.</p></list-item><list-item><p id=\"Par17\">High scan loss for wide-angle beam compared to the central beam.</p></list-item><list-item><p id=\"Par18\">Low SLL for wide-angle beam less than 10 dB.</p></list-item><list-item><p id=\"Par19\">Lack of integration of the antenna with the beamforming network.</p></list-item></list></p>", "<p id=\"Par20\">On the other hand, the Rotman lens beamforming network suitable for 5G AAS needs to be in line with the requirements indicated in Table ##TAB##0##1##. Therefore, the design of a Rotman lens with a minimum of 8 antenna elements with possible dual-polarization capability in a wide operational bandwidth over 5G frequency bands (24–30 GHz) for the scanning coverage of ± 60° and SLL &gt; 10 dB is demanded by industry^{##UREF##0##1##,##UREF##1##2##}.</p>", "<p id=\"Par21\">In this research work, a novel wide-angle Rotman lens beamformer is developed to meet the AAS requirements for 5G applications. An exhaustive design methodology for the Rotman lens is extracted comprising different components of the Rotman lens including parallel-plate contour, beam ports, array ports, and dummy ports. For the antenna elements, an end-fire Vivaldi antenna in which the direction of radiation is along the line of the antenna is suggested for easy integration with the beamformer and facilitates dual polarization implementation as well as possible stacking to obtain 2-D beamforming. The beam ports and non-uniform array ports are designed in detail to accommodate good matching and enhanced SLL. A novel integrated matched load is also introduced for dummy ports. The designed beamformer is fabricated and tested to verify the results. A comparison between the proposed Rotman lens relative to the recent works is also provided.</p>", "<p id=\"Par22\">The following merits can be summarized for the proposed Rotman lens beamformer taking into account addressing the mentioned problems with the existing designs.<list list-type=\"bullet\"><list-item><p id=\"Par23\">Optimized Rotman lens design methodology for wide scanning angle ± 53° covering ± 60° with 3 dB beamwidth.</p></list-item><list-item><p id=\"Par24\">Low scan loss/dropping gain for the side beam (± 53°) less than 1.9 dB.</p></list-item><list-item><p id=\"Par25\">Non-uniform antenna ports to satisfy the SLL &gt; 10 dB for the side-beam (± 53°).</p></list-item><list-item><p id=\"Par26\">End-fire antenna element integrated with the beamformer to eliminate the connector loss, possible dual-polarization, and stacking 2-D beamforming capabilities.</p></list-item><list-item><p id=\"Par27\">PCB-based low-cost and high-performance beamformer covering multi 5G <italic>n257</italic>, <italic>n258</italic>, and <italic>n261</italic> frequency bands.</p></list-item></list></p>" ]

[]

[ "<title>Fabrication and measurement results</title>", "<p id=\"Par95\">The designed Rotman lens beamformer including 8 beam ports and integrated 8 Vivaldi antennas is fabricated and measured as shown in Fig. ##FIG##10##11## to validate the design and simulation results.</p>", "<p id=\"Par96\">The SMPM cables are used for measurement to be compatible with the input connectors. One port is excited at each stage and the other ports are terminated with SMPM terminators as designed in Ref.^{##UREF##25##26##}.</p>", "<p id=\"Par97\">Due to the suitable matching of the tapered beam ports and a large number of S-parameters configurations, the only measured S-parameters are shown in Fig. ##FIG##11##12##. Reflection coefficient and mutual coupling of the different ports are demonstrated with solid lines and dashed lines respectively.</p>", "<p id=\"Par98\">The measured results show a good matching of the input ports () and mutual coupling () of better than 28 dB for the band 24–40 GHz.</p>", "<p id=\"Par99\">To test the beamformer performance, the radiation patterns are measured and plotted in comparison with the simulated patterns at 24 GHz, 27 GHz, and 30 GHz in Fig. ##FIG##12##13##. The simulation and measurement results are depicted in the form of solid lines and dashed lines respectively. The simulated and measured peak gain and simulated radiation efficiency of the Rotman lens beamforming network are also presented for beam ports 1, 2, 3, and 4 in Figs. ##FIG##13##14## and ##FIG##14##15## due to achieving identical results for the symmetric ports.</p>", "<p id=\"Par100\">The radiation pattern results show a good agreement between simulations and measurements. The beam steering from − 53° to 53° in 15° increments with a slight beam-pointing error of less than 2° is obtained. The radiation patterns show a scan loss at a maximum scanning angle of less than 1.6 dB, 1.8 dB, and 1.9 dB at 24 GHz, 27 GHz, and 30 GHz respectively due to Rotman amplitude error and single element radiation pattern. Also, the scanning directions remain unchanged across the bandwidth as the Rotman lens is a true-time-delay beamformer. The radiation patterns exhibit some ripples for the wide angle which is due to the imperfection and multipath reflections of the anechoic chamber. The proposed Rotman lens offers the SLL better than 10 dB for whole ports that meet the minimum AAS requirements.</p>", "<p id=\"Par101\">The measured peak gain shows the average 10 dBi gain for the center ports (Port 4 and Port 5) and the scan loss of less than 1.9 dB for the wide-angle ports (Port 1 and Port 8). It can be observed that the average radiation efficiency is 62% under every input feed port of the Rotman lens across the bandwidth.</p>", "<p id=\"Par102\">As a result, the proposed Rotman lens provides proper beamforming capability in the wide range of ± 53° to cover ± 60° with the 3dB beamwidth in wide target frequency bands.</p>", "<p id=\"Par103\">The proposed Rotman lens beamformer is compared with some recently recognized Rotman lenses for 5G mmWave applications as summarized in Table ##TAB##5##6##.</p>", "<p id=\"Par104\">As can be seen, most of the beamformer designs are restricted to the limited bandwidth and scan angle while the SLL is below 10 dB and the scan loss is almost high for the wide-angle beam. The proposed Rotman lens beamformer in this work offers a wide bandwidth (24–30 GHz) and wide scanning range of ± 60° with SLL &gt; 10 dB and scan loss &lt; 1.9 dB which meets the 5G AAS requirements and exhibits better performances compared to most of the other literary works.</p>" ]

[]

[ "<title>Conclusion</title>", "<p id=\"Par109\">Beamforming is an important part of mmWave technologies to improve the link budget and spectral efficiency and abilities of MIMO and SDMA. The beamforming requirements for 5G mmWave applications indicate the wide-angular coverage of ± 60° and SLL &gt; 10 dB. A wide-angle Rotman lens with a detailed improved design methodology to satisfy the 5G mmWave beamforming is presented. The enhanced SLL is obtained by imposing an optimized distribution on the aperture of the antenna ports. The proposed seamless beamformer and antenna array is fabricated using low-cost PCB technology on Rogers 4350B () substrate with a thickness of 0.254 mm and successfully tested to verify the feasibility of the design methodology. The overall results demonstrate that the proposed beamformer exhibits wideband impedance and radiation characteristics over a bandwidth of 24 –30 GHz and beam-scanning capability over a scan range of ± 60° with SLL &gt; 10 dB and scan loss &lt; 1.9 dB. Dual-polarization and 2-D beamforming configurations of the beamformer are also proposed.</p>", "<p id=\"Par110\">The resulting beamformer features unique characteristics such as wide angular coverage with acceptable SLL and low scan loss over a wide frequency bandwidth of 24–30 GHz covering three standard 5G <italic>n257</italic>, <italic>n258</italic>, and <italic>n261</italic> bands.</p>", "<p id=\"Par111\">The proposed beamforming network is qualified for various mmWave and 5G applications such as AAS, massive MIMO systems, hybrid beamforming systems, remote sensing, and automotive radars.</p>" ]

[ "<p id=\"Par1\">The attachment of 5G with millimeter wave (mmWave) frequencies offers massive capacity and low latency to reveal the full 5G experiences. High directive gain and beamforming are considered essential for mmWave 5G systems. The main requirements of the beamforming network for 5G mmWave applications are the scanning coverage of ± 60° and SLL &gt; 10 dB in a wide operational bandwidth over the standard 5G frequency bands. In this paper, a novel PCB-based wide-angle Rotman lens beamformer is designed, simulated, and successfully measured to meet the mentioned requirements for 5G mmWave applications. A comprehensive improved design methodology is provided for all components of the Rotman lens to reach a wide scanning angle, enhanced sidelobe level, and low scan loss. The end-fire Vivaldi antenna is selected as an array element for easy integration to the beamforming network as well as its capability to use in dual-polarization configuration. The proposed Rotman lens is operational in the 24–30 GHz frequency band covering 5G <italic>n257</italic>, <italic>n258</italic>, and <italic>n261</italic> frequency bands. The results show a nearly constant 8 beams across the whole bandwidth steering from − 53° to 53° in 15° increments to provide ± 60° coverage with the SLL &gt; 10 dB and scan loss &lt; 1.9 dB. The retrieved novelties from this work contain an effective design methodology for an optimized Rotman lens with wide-scan angle and low phase and amplitude error, non-uniform distribution based array ports, and integration with end-fire antenna for possible dual polarization and 2-D beamforming capabilities. The comparison of the proposed beamformer with the most recent works shows several advantages in terms of integrated structure and performances including bandwidth, wide scanning angle, SLL, and scan loss. With such performances, this beamformer can be used for various mmWave and 5G applications such as advanced antenna systems, massive MIMO systems, and hybrid beamforming systems.</p>", "<title>Subject terms</title>" ]

[ "<title>Rotman lens beamformer design</title>", "<p id=\"Par28\">The Rotman lens is a wide-angle lens that can be utilized as a wideband beamformer. The schematic of a conventional Rotman lens is shown in Fig. ##FIG##0##1##, which consists of a parallel-plate contour surrounded by <italic>M</italic>-beam ports and <italic>N</italic>-array ports. Each beam port steers the beam in a certain angular direction coming up with <italic>M</italic>-discrete beams. The array ports are connected via transmission lines to the radiating elements of a linear antenna array. The loaded dummy ports are connected to the parallel-plate region to provide an appropriate termination^{##UREF##13##14##}.</p>", "<p id=\"Par29\">The design of the Rotman lens starts with defining the general requirements of the beamformer such as the operating frequency range, the number of beam ports (<italic>M</italic>), the desired beam steering angle (± <italic>θ</italic>), the number of radiating elements (<italic>N</italic>) for specific gain performance, and the spacing between array elements (<italic>d</italic>)^{##UREF##14##15##}.</p>", "<p id=\"Par30\">The circular arc on the left side of the Rotman lens as the beam contour has the on-axis focal length located at 0° angle and the off-axis focal length located at angles α°. The general shape of the parallel-plate contour is determined based on the four basic Rotman lens parameters as shown in Fig. ##FIG##0##1##.</p>", "<p id=\"Par31\">Also, the length of transmission lines connecting each array-port to the lens (<italic>W</italic>) is essential for Rotman length design^{##UREF##13##14##}.</p>", "<p id=\"Par32\">Initial geometrical condition of the on-axis focal length for appropriate amplitude performance and also physical arrangments of the input and output ports, considering maximum scanning angle <italic>θ</italic> and array length (<italic>N</italic> − 1)<italic>d</italic>, can be defined as^{##UREF##15##16##,##UREF##16##17##}:</p>", "<p id=\"Par33\">The angle between the on-axis focal length and the off-axis focal length is the focal angle () and the ratio between them is parameter </p>", "<p id=\"Par34\">The expansion factor is the ratio between the focal angle and array beam angle as:</p>", "<p id=\"Par35\">The indirect factor of utility controls the amplitude and phase error and corresponds to the distance of any point on the array from the axis () to the on-axis focal length () as expressed by:</p>", "<p id=\"Par36\">The maximum distance of gives the maximum of the indirect factor of utility :</p>", "<p id=\"Par37\">The upper limit of the indirect factor of utility appears when the transmission line <italic>W</italic> = 0 as:</p>", "<p id=\"Par38\">The limiting value of the indirect factor of utility versus is depicted for some focal angle values in Fig. ##FIG##1##2##. Due to the fact that the useful range of is between 0.5 and 0.8, Fig. ##FIG##1##2## can be considered for choosing an appropriate range of for a given ^{##UREF##14##15##}.</p>", "<p id=\"Par39\">In the case of fabricating the Rotman lens on a dielectric substrate with permittivity (), all dimensions of the lens are divided by a factor of .</p>", "<p id=\"Par40\">The transmission line length that connects the element port to the array antenna can be calculated as^{##UREF##13##14##}:</p>", "<p id=\"Par41\">The parameters and have a comparable effect on Rotman lens geometry and significantly influence the gain performance and phase error. The parameters and need to be selected in conjunction with other parameters to reach the optimized gain performance and phase error reduction.</p>", "<p id=\"Par42\">The phase error is as a result of the path length difference between an arbitrary central ray through the center of array ports and any other arbitrary ray that can be evaluated as a function of scanning angle <italic>θ</italic> and indirect factor of utility . Thus, the normalized path length error can be calculated as^{##UREF##13##14##,##UREF##16##17##}:where is the normalized distance from a point on the beam arc to the origin, is the normalized distance of any other point, is the normalized transmission line length, is the dielectric constant, and is the effective dielectric constant of the transmission line as shown in Fig. ##FIG##0##1##. The total path difference stemming from all ports can be expressed as^{##UREF##13##14##,##UREF##16##17##}:</p>", "<p id=\"Par43\">In order to estimate the amplitude performance, the approximation of the coupling between a beam port width and array port width can be presented as^{##UREF##16##17##}:where is the phase constant, , is the separation between ports <italic>i</italic> and <italic>j</italic>, and and are the angles between the boresight direction of ports and the line connecting the port phase centers.</p>", "<p id=\"Par44\">The results of the Rotman lens design show that the minimization of the only path length error does not lead to the acceptable amplitude performance^{##UREF##16##17##}. Therefore, the optimum performance of the Rotman lens can be achieved by considering both path length error and amplitude performance. Various optimization methods in particular numerical-based methods can be used for choosing and optimally to reach reasonable phase error and amplitude performance simultaneously.</p>", "<p id=\"Par45\">After designing the shape of the parallel-plate contour, the beam ports, array ports, and dummy ports should be designed. Firstly, the phase center of the corresponding ports should be determined and then the ports need to be matched with transmission lines.</p>", "<p id=\"Par46\">The coordinate of the array port phase center can be calculated as^{##UREF##17##18##}:</p>", "<p id=\"Par47\">The phase center location for beam ports can be also expressed as:</p>", "<p id=\"Par48\">When the location of beam ports and array ports are determined, a horn-type tapering transmission can be used for connecting the transmission lines to the lens body aiming to provide appropriate matching. Also, the wall side of the lens body is connected to the number of matched dummy ports to make a reflection-less parallel-plate contour. There is no specific requirement for the number of dummy ports. Some designers implement multiple dummy ports while others utilize a single dummy port for each side of the lens body. However, some studies indicate that the number of dummy ports does not change the main beam performance and only may affect the side lobe levels (SLLs)^{##UREF##18##19##,##UREF##19##20##}. Therefore, the main intention of the beam port, array port, and dummy port design is to provide appropriate reflection and transmission coefficients and SLLs.</p>", "<title>Novel wide-angle Rotman lens beamformer design</title>", "<p id=\"Par49\">The proposed beamforming network suitable for 5G mmWave Advanced Antenna System (AAS) should be at least 8 × 8 array supporting mmWave frequency bands allocated to 5G as defined in Section I steering ± 60° and ± 15° in horizontal and elevation planes respectively. Thus, designing an 8-element array, wideband, and wide-angle (± 60) beamformer with dual-polarization capability is a basic demanding requirement for 5G AAS. A novel Rotman lens is designed to meet the basic beamforming requirements for 5G application. The Rotman lens is modeled and optimized with Matlab. The modeled beamformer is simulated and optimized using the Ansys HFSS electromagnetic simulator package. The design procedure consists of the following steps:<list list-type=\"bullet\"><list-item><p id=\"Par50\">End-fire single antenna element design for dual-polarization capability.</p></list-item><list-item><p id=\"Par51\">Lens body design and optimization for wide-angle steering (± 60).</p></list-item><list-item><p id=\"Par52\">Beam and array ports and connected transmission line design.</p></list-item><list-item><p id=\"Par53\">Dummy port design.</p></list-item><list-item><p id=\"Par54\">Non-uniform array port design for SLL reduction.</p></list-item></list></p>", "<title>Single element design</title>", "<p id=\"Par55\">The single antenna element suitable for 5G AAS is recommended to be dual-polarized with a half-power beam width of at least 65° and in line with 3GPP mmWave frequency bands. In addition, the single element is preferred to be integrated into the beamforming network to avoid using large numbers of connectors and associated losses. Thus, an end-fire type antenna element is selected for easy integration to the beamforming network as well as its capability to use in dual-polarization configuration. As a consequence, a novel Vivaldi antenna is designed to be utilized in the AAS beamformer. The proposed antenna is fabricated on Rogers 4350B () substrate with a thickness of 0.254 mm. The design methodology, fabrication, and results are comprehensively discussed in Ref.^{##UREF##20##21##}.</p>", "<p id=\"Par56\">The structure of the proposed Vivaldi antenna element is presented in Fig. ##FIG##2##3##. The antenna has a compact size of 12 × 5.5 × 0.254 mm³. Therefore, the space element is d = 5.5 mm if the antenna is used in the array. The proposed antenna can operate in 23–45 GHz covering 5G n257, n258, n259, n260, and n261 frequency bands and exhibit a nearly constant end-fire radiation pattern with a measured gain of more than 5dBi across the whole bandwidth^{##UREF##20##21##}.</p>", "<title>Lens body design</title>", "<p id=\"Par57\">It is intended to design a basic 8-element Rotman lens beamformer between 24 and 30 GHz covering 5G <italic>n257</italic>, <italic>n258</italic>, and <italic>n261</italic> frequency bands for AAS applications. The Rotman lens is designed for fabrication on a dielectric substrate. The dielectric substrate used is Rogers 4350B () with a thickness of 0.254 mm. The design methodology can be presented as a step-by-step process based on the approach detailed in “<xref rid=\"Sec2\" ref-type=\"sec\">Rotman lens beamformer design</xref>” section.</p>", "<title>Determine the requirements</title>", "<p id=\"Par58\">As indicated, the operational frequency band is between 24 and 30 GHz. The number of array elements is <italic>N</italic> = 8 and the number of beam ports is assumed to be <italic>M</italic> = 8. The steering angle to meet the AAS requirement is which is considered to be a wide angle. The 3dB beam width of the <italic>N</italic>-element array is:</p>", "<p id=\"Par59\">Considering the array beam width , the maximum coverage angle can be moderated as . Thus, assuming the steering angle , the 3 dB beam width of the resultant array can cover the AAS coverage requirement .</p>", "<p id=\"Par60\">The element spacing equal to the width of a single antenna element is <italic>d</italic> = 5.5 mm. The substrate parameters are and <italic>h</italic> = 0.254 mm.<list list-type=\"order\"><list-item><p id=\"Par61\">Calculate the minimum on-axis focal length using Eq. (##FORMU##4##1##).</p></list-item><list-item><p id=\"Par62\">Set the and and calculate the initial parameters using (##FORMU##18##5##), and (##FORMU##11##3##) respectively.</p><p id=\"Par63\">The initial can be selected from Fig. ##FIG##1##2## as .</p></list-item><list-item><p id=\"Par64\">Optimize the value of using (##FORMU##49##9##) and (##FORMU##52##10##) to obtain optimum phase and amplitude performance. In this research work, the genetic algorithm (GA) is employed for numerical optimization using Matlab. The optimized parameters are indicated as:</p></list-item><list-item><p id=\"Par65\">Specify the difference in transmission line length for the side array port compared to the center array port (<italic>W</italic>) using (##FORMU##32##7##) as:</p><p id=\"Par66\">Divide the dimensions by a factor of . The parameters are divided to as the designed dielectric constant is recommended 3.66 for Rogers 4350B^{##UREF##21##22##}.</p></list-item><list-item><p id=\"Par67\">The modeled body lens parameters are summarized in Table ##TAB##1##2##.</p></list-item></list></p>", "<title>Beam and array ports and transmission line design</title>", "<p id=\"Par68\">The phase center location of beam ports can be calculated using (##FORMU##64##12a##) and (##FORMU##65##12b##) as:</p>", "<p id=\"Par69\">Also, the phase center location of array ports can be calculated using (##FORMU##61##11a##) and (##FORMU##62##11b##) as:where the origin is the center of the array port arc. Considering the center location of the ports, the width of the lens port aperture is roughly .</p>", "<p id=\"Par70\">The microstrip width () of the feed line can be calculated as^{##UREF##22##23##}:where is the characteristic impedance, <italic>h,</italic> and <italic>t</italic> are the substrate and track thicknesses respectively. Considering the 50 Ω characteristic impedance and proposed substrate parameters, . However, the microstrip line width is slightly optimized as in the simulation process.</p>", "<p id=\"Par71\">After determining the phase center location, lens port width, and feed line width, a horn with the appropriate length is tapered toward the feed to overcome the impedance discontinuity problem due to connecting the large lens port width to the small feed line width.</p>", "<p id=\"Par72\">According to Ref.^{##UREF##23##24##}, the length of the triangular transition is suggested to be as follows:where is the width of the lens port aperture.</p>", "<p id=\"Par73\">In this work, the tapering length is optimized based on a model and extracted results as shown in Fig. ##FIG##3##4## to obtain optimum matching and insertion loss. As a result, the optimized horn length is where the number 4.57 in Ref.^{##UREF##23##24##} is adjusted as . The schematic of the tapering transition is depicted in Fig. ##FIG##4##5##.</p>", "<p id=\"Par74\">Due to the very small distance between the ports and easy interconnection purpose, SMPM connectors are used. The impedance matching between the SMPM connector and the microstrip line is very sensitive for mmWave bands. The SMPM connectors are used for the beam ports using the transition procedure as detailed in Ref.^{##UREF##24##25##}.</p>", "<title>Dummy port design</title>", "<p id=\"Par75\">In this design, we employ only a single dummy port on each side of the parallel plate contour with a wide aperture width in order to convene multiple dummy ports and simplify the structure.</p>", "<p id=\"Par76\">The dummy ports are matched using a novel absorber sheet based termination load as described extensively in Ref.^{##UREF##25##26##}.</p>", "<p id=\"Par77\">The proposed high-performance and cost-effective microstrip termination load is based on the combination of a printed monopole antenna and an absorber sheet as shown in Fig. ##FIG##5##6## that can be easily integrated with a microstrip line or used with a connector as a termination load for test measurement. The results show a good impedance matching between 20 and 67 GHz that can be effectively used as loaded dummy ports for mmWave applications.</p>", "<p id=\"Par78\">In this design, this termination load is integrated into the dummy ports to provide good termination and reflection-less side walls of the Rotman lens.</p>", "<p id=\"Par79\">Once the whole Rotman lens parameters are specified, a mathematic-based geometry of the Rotman lens can be generated by Matlab as shown in Fig. ##FIG##6##7##.</p>", "<p id=\"Par80\">The generated geometry can be imported to full wave simulation packages. Thus, the modeled Rotman lens geometry is imported to Ansys HFSS for simulation and further optimization.</p>", "<title>Non-uniform array port design</title>", "<p id=\"Par81\">The side lobe level (SLL) is a challenging parameter in the Rotman lens due to the unwanted reflections from side walls, beam and array ports, and also dummy ports^{##UREF##18##19##}.</p>", "<p id=\"Par82\">Chebyshev is a famous tapered distribution that can be used to set SLL to a specified value (<italic>s</italic>). In an <italic>N-</italic>element array, the peak value of the Chebyshev polynomial of the order <italic>N − 1</italic> can be expressed as^{##UREF##26##27##}:where <italic>s</italic> is SLL in dB and is the main lobe position that can be calculated as:</p>", "<p id=\"Par83\">The half power beam width (HPBW) of the scanning array can be obtained using:where <italic>L</italic> is the array length, <italic>d</italic> is the inter-element space, and is the scanning angle.</p>", "<p id=\"Par84\">In a non-uniform array design, the SLL can be controlled by the amplitude distribution among the elements and there is a tradeoff between SLL and HPBW where by decreasing the SLL, the HPBW is decreased^{##UREF##26##27##–##UREF##28##29##}.</p>", "<p id=\"Par85\">In this research work, firstly an improved distribution scheme for the target array is achieved based on a Matlab code aiming to enhance the SLL while the HPBW is almost constant. The resultant as depicted in Fig. ##FIG##7##8## and the distribution coefficient is presented in Table ##TAB##2##3##.</p>", "<p id=\"Par86\">The improved non-uniform amplitude distribution is applied to the array port by altering the port width. Assuming the relation of power and impedance^{##UREF##21##22##}:</p>", "<p id=\"Par87\">The impedance of microstrip as a function of microstrip width to substrate height is^{##UREF##25##26##}:</p>", "<p id=\"Par88\">The following relation can be extracted for the width of the microstrip line:</p>", "<p id=\"Par89\">The width of the array port based on the amplitude coefficient using (21), when the center port width is 4.65 mm can be calculated as indicated in Table ##TAB##2##3##.</p>", "<p id=\"Par90\">The new Rotman geometry including the non-uniform array ports as indicated in Table ##TAB##2##3## is generated for further optimization by the full wave simulation. To this effect, the Rotman lens parameters, and amplitude weighting together with the position of the array ports are optimized using genetic algorithm (GA) by HFSS in terms of phase and amplitude errors and SLLs in the widest angle beam () as the worst-case for 24 GHz and 30 GHz as the start and stop operating frequencies. The optimized array ports width and corresponding distribution coefficient are shown in Table ##TAB##3##4##. The symmetrically oriented array port numbers can be found in Fig. ##FIG##6##7##.</p>", "<p id=\"Par91\">The simulated resultant radiation patterns by exciting different beam ports for uniform and improved non-uniform distribution of array ports are presented in Fig. ##FIG##8##9##.</p>", "<p id=\"Par92\">It is clear that applying an optimized non-uniform distribution coefficient improves the SLL as well as the amplitude and phase performances. The minimum SLL for uniform distribution is around 9 dB while it is increased to around 12 dB for optimized distribution.</p>", "<p id=\"Par93\">The structure of the final Rotman lens beamforming network with optimized parameters is shown in Fig. ##FIG##9##10##. Also, Table ##TAB##4##5## summarizes the designed and optimized Rotman lens parameters.</p>", "<p id=\"Par94\">It can be concluded that the optimized values are very close to the designed values confirming a good convergence between the optimized simulation parameters and the proposed design procedure.</p>", "<title>Possible dual-polarized 2-D configuration</title>", "<p id=\"Par105\">The end-fire antenna element integrated with the beamforming network facilitates dual polarization implementation and possible stacking to obtain 2-D beamforming.</p>", "<p id=\"Par106\">To design the 8 × 8 dual-polarized 2-D AAS, the proposed Rotman lens beamformers are crossed and vertical to each other as shown in Fig. ##FIG##15##16##.</p>", "<p id=\"Par107\">To realize the crossing, the extended length of the transmission lines is equal to the self-length of the Rotman lens. 8 proposed Rotman lens beamformers are staked along the <italic>x</italic>-direction and <italic>y</italic>-direction with stacking spacing <italic>d</italic> (Fig. ##FIG##15##16##).</p>", "<p id=\"Par108\">The first stage of the beamformer can be connected to the second stage of the beamformers via the SMPM connectors to construct the 2-D beamforming network. The recommended steering angle of the second stage of the beamformer is ± 15° (Table ##TAB##0##1##) which can be generated using a 3-beam port Rotman lens with a step angle of 10°.</p>" ]

[ "<title>Author contributions</title>", "<p>A.A.; Formal analysis, Resources, Writing-original draft, Writing-review &amp; editing, A.S.; Supervision, H.A.; Supervision.</p>", "<title>Data availability</title>", "<p>All the data required to evaluate the findings of this work is available in the manuscript. Any other additional data related to this work may be requested from the corresponding author.</p>", "<title>Competing interests</title>", "<p id=\"Par112\">The authors declare no competing interests.</p>" ]

[ "<fig id=\"Fig1\"><label>Figure 1</label><caption><p>Schematic of a conventional Rotman-lens parameters.</p></caption></fig>", "<fig id=\"Fig2\"><label>Figure 2</label><caption><p>Limiting values of versus for some values.</p></caption></fig>", "<fig id=\"Fig3\"><label>Figure 3</label><caption><p>Fabricated Vivaldi antenna array.</p></caption></fig>", "<fig id=\"Fig4\"><label>Figure 4</label><caption><p>Tapering transition modeling and simulation.</p></caption></fig>", "<fig id=\"Fig5\"><label>Figure 5</label><caption><p>Schematic of the tapering transition.</p></caption></fig>", "<fig id=\"Fig6\"><label>Figure 6</label><caption><p>Fabricated SMPM terminator.</p></caption></fig>", "<fig id=\"Fig7\"><label>Figure 7</label><caption><p>Rotman lens modeling and geometry.</p></caption></fig>", "<fig id=\"Fig8\"><label>Figure 8</label><caption><p>Comparison of SLL for uniform and non-uniform array ports.</p></caption></fig>", "<fig id=\"Fig9\"><label>Figure 9</label><caption><p>Comparison of SLL for uniform and improved non-uniform array ports.</p></caption></fig>", "<fig id=\"Fig10\"><label>Figure 10</label><caption><p>Proposed Rotman lens beamformer structure.</p></caption></fig>", "<fig id=\"Fig11\"><label>Figure 11</label><caption><p>Fabrication and measurement of the Proposed Rotman lens beamformer.</p></caption></fig>", "<fig id=\"Fig12\"><label>Figure 12</label><caption><p>Measured S-parameters of beam ports.</p></caption></fig>", "<fig id=\"Fig13\"><label>Figure 13</label><caption><p>Simulated and measured radiation patterns at 24, 27, and 30 GHz.</p></caption></fig>", "<fig id=\"Fig14\"><label>Figure 14</label><caption><p>Simulated and measured peak gain versus frequency for different ports.</p></caption></fig>", "<fig id=\"Fig15\"><label>Figure 15</label><caption><p>Simulated radiation efficiency versus frequency for different ports.</p></caption></fig>", "<fig id=\"Fig16\"><label>Figure 16</label><caption><p>The model of 8 × 8 dual-polarized 2-D AAS.</p></caption></fig>" ]

[ "<table-wrap id=\"Tab1\"><label>Table 1</label><caption><p>Main requirements of AAS^{##UREF##0##1##}.</p></caption><table frame=\"hsides\" rules=\"groups\"><tbody><tr><td align=\"left\">Operating frequency range</td><td align=\"left\"><p>26.5–29.5 GHz</p><p>24.25–27.5 GHz</p><p>27.5–28.35 GHz</p></td></tr><tr><td align=\"left\">Array size</td><td align=\"left\">8 × 8</td></tr><tr><td align=\"left\">Polarization</td><td align=\"left\">Dual-polarized</td></tr><tr><td align=\"left\">Beamwidth of a single element (°)</td><td align=\"left\">65° for both H/V</td></tr><tr><td align=\"left\">Side lobe level (SLL)</td><td align=\"left\"> &gt; 10 dB</td></tr><tr><td align=\"left\">Scan angles in the horizontal plane</td><td align=\"left\"> ± 60°</td></tr><tr><td align=\"left\">Scan angles in the vertical plane</td><td align=\"left\"> ± 15°</td></tr></tbody></table></table-wrap>", "<table-wrap id=\"Tab2\"><label>Table 2</label><caption><p>Designed Rotman lens parameters.</p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th align=\"left\">Parameter</th><th align=\"left\">F0 (mm)</th><th align=\"left\">F1 (mm)</th><th align=\"left\">α (deg)</th><th align=\"left\">β</th><th align=\"left\">ζ</th><th align=\"left\">W (mm)</th></tr></thead><tbody><tr><td align=\"left\">Value</td><td char=\".\" align=\"char\">31.93</td><td char=\".\" align=\"char\">29.05</td><td char=\".\" align=\"char\">29.7°</td><td char=\".\" align=\"char\">0.91</td><td char=\".\" align=\"char\">0.7</td><td char=\".\" align=\"char\">0.76</td></tr></tbody></table></table-wrap>", "<table-wrap id=\"Tab3\"><label>Table 3</label><caption><p>Improved Chebyshev distribution by Matlab to enhance the SLL while the HPBW is almost constant.</p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th align=\"left\">Array port</th><th align=\"left\">1 and 8</th><th align=\"left\">2 and 7</th><th align=\"left\">3 and 6</th><th align=\"left\">4 and 5</th></tr></thead><tbody><tr><td align=\"left\">Amplitude weighting</td><td char=\".\" align=\"char\">0.65</td><td char=\".\" align=\"char\">0.85</td><td char=\".\" align=\"char\">0.89</td><td align=\"left\">1</td></tr><tr><td align=\"left\">Width of port (mm)</td><td char=\".\" align=\"char\">2.9</td><td char=\".\" align=\"char\">3.85</td><td char=\".\" align=\"char\">4.1</td><td align=\"left\">4.65</td></tr></tbody></table></table-wrap>", "<table-wrap id=\"Tab4\"><label>Table 4</label><caption><p>Optimized distribution by full-wave simulation.</p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th align=\"left\">Array port</th><th align=\"left\">1 and 8</th><th align=\"left\">2 and 7</th><th align=\"left\">3 and 6</th><th align=\"left\">4 and 5</th></tr></thead><tbody><tr><td align=\"left\">Amplitude weighting</td><td align=\"left\">0.66</td><td align=\"left\">0.87</td><td align=\"left\">0.92</td><td align=\"left\">1</td></tr><tr><td align=\"left\">Array port width (mm)</td><td align=\"left\"> 2.88</td><td align=\"left\"> 3.94</td><td align=\"left\"> 4.17</td><td align=\"left\"> 4.61</td></tr></tbody></table></table-wrap>", "<table-wrap id=\"Tab5\"><label>Table 5</label><caption><p>Designed and optimized Rotman lens parameters.</p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th align=\"left\">Parameters</th><th align=\"left\">Designed value</th><th align=\"left\">Final optimized value</th></tr></thead><tbody><tr><td align=\"left\"></td><td align=\"left\">31.93 mm</td><td char=\".\" align=\"char\">32.66 mm</td></tr><tr><td align=\"left\"></td><td align=\"left\">29.05 mm</td><td char=\".\" align=\"char\">29.85 mm</td></tr><tr><td align=\"left\"></td><td align=\"left\">29.7°</td><td char=\".\" align=\"char\">29.81°</td></tr><tr><td align=\"left\"></td><td align=\"left\">0.91</td><td char=\".\" align=\"char\">0.913</td></tr><tr><td align=\"left\"></td><td align=\"left\">0.55 mm</td><td char=\".\" align=\"char\">0.53 mm</td></tr><tr><td align=\"left\"></td><td align=\"left\">4.65 mm</td><td char=\".\" align=\"char\">4.73 mm</td></tr><tr><td align=\"left\"></td><td align=\"left\">22 mm</td><td char=\".\" align=\"char\">23.7 mm</td></tr><tr><td align=\"left\"></td><td align=\"left\">2.9 mm</td><td char=\".\" align=\"char\">2.88 mm</td></tr><tr><td align=\"left\"></td><td align=\"left\">3.85 mm</td><td char=\".\" align=\"char\">3.94 mm</td></tr><tr><td align=\"left\"></td><td align=\"left\">4.1 mm</td><td char=\".\" align=\"char\">4.17 mm</td></tr><tr><td align=\"left\"></td><td align=\"left\">4.65 mm</td><td char=\".\" align=\"char\">4.61 mm</td></tr><tr><td align=\"left\">W (Difference)</td><td align=\"left\">0.76 mm</td><td char=\".\" align=\"char\">0.45 mm</td></tr></tbody></table></table-wrap>", "<table-wrap id=\"Tab6\"><label>Table 6</label><caption><p>Performance comparison of the proposed Rotman lens.</p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th align=\"left\">Ref.</th><th align=\"left\">Type</th><th align=\"left\">Possible dual-pol</th><th align=\"left\">Freq. band (GHz)</th><th align=\"left\">Scan angle (°)</th><th align=\"left\">SLL (dB)</th><th align=\"left\">Scan loss (dB)</th></tr></thead><tbody><tr><td align=\"left\">^{##UREF##9##10##}</td><td align=\"left\">Butler</td><td align=\"left\">No</td><td align=\"left\">27.8–30.8</td><td char=\".\" align=\"char\"> ± 45</td><td align=\"left\">11</td><td align=\"left\">1</td></tr><tr><td align=\"left\">^{##UREF##10##11##}</td><td align=\"left\">Butler</td><td align=\"left\">Yes</td><td align=\"left\">26–31.4</td><td char=\".\" align=\"char\"> ± 42</td><td align=\"left\">8</td><td align=\"left\">3</td></tr><tr><td align=\"left\">^{##UREF##11##12##}</td><td align=\"left\">Rotman</td><td align=\"left\">No</td><td align=\"left\">28</td><td char=\".\" align=\"char\"> ± 60</td><td align=\"left\">8</td><td align=\"left\">3</td></tr><tr><td align=\"left\">^{##UREF##12##13##}</td><td align=\"left\">Rotman</td><td align=\"left\">No</td><td align=\"left\">25.5–28.5</td><td char=\".\" align=\"char\"> ± 30</td><td align=\"left\">8.5</td><td align=\"left\">4.5</td></tr><tr><td align=\"left\">This work</td><td align=\"left\">Rotman</td><td align=\"left\">Yes</td><td align=\"left\">24–30</td><td char=\".\" align=\"char\"> ± 60</td><td align=\"left\">10</td><td align=\"left\">1.9</td></tr></tbody></table></table-wrap>" ]

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id=\"M29\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${F}_{0}$$\\end{document}</tex-math><mml:math id=\"M30\"><mml:msub><mml:mi>F</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ4\"><label>4</label><alternatives><tex-math id=\"M31\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\zeta ={y}_{3}\\gamma /{F}_{0}.$$\\end{document}</tex-math><mml:math id=\"M32\" display=\"block\"><mml:mrow><mml:mi>ζ</mml:mi><mml:mo>=</mml:mo><mml:msub><mml:mi>y</mml:mi><mml:mn>3</mml:mn></mml:msub><mml:mi>γ</mml:mi><mml:mo stretchy=\"false\">/</mml:mo><mml:msub><mml:mi>F</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo>.</mml:mo></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq13\"><alternatives><tex-math id=\"M33\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${y}_{3}$$\\end{document}</tex-math><mml:math id=\"M34\"><mml:msub><mml:mi>y</mml:mi><mml:mn>3</mml:mn></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq14\"><alternatives><tex-math id=\"M35\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\zeta }_{max}$$\\end{document}</tex-math><mml:math id=\"M36\"><mml:msub><mml:mi>ζ</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">max</mml:mi></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ5\"><label>5</label><alternatives><tex-math id=\"M37\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${y}_{3({\\text{max}})}=\\left(N-1\\right)d/2\\to {\\zeta }_{\\text{max}}=(N-1)d\\gamma /2{F}_{0}.$$\\end{document}</tex-math><mml:math id=\"M38\" display=\"block\"><mml:mrow><mml:msub><mml:mi>y</mml:mi><mml:mrow><mml:mn>3</mml:mn><mml:mo stretchy=\"false\">(</mml:mo><mml:mtext>max</mml:mtext><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mfenced close=\")\" open=\"(\"><mml:mi>N</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mfenced><mml:mi>d</mml:mi><mml:mo stretchy=\"false\">/</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">→</mml:mo><mml:msub><mml:mi>ζ</mml:mi><mml:mtext>max</mml:mtext></mml:msub><mml:mo>=</mml:mo><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>N</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mi>d</mml:mi><mml:mi>γ</mml:mi><mml:mo stretchy=\"false\">/</mml:mo><mml:mn>2</mml:mn><mml:msub><mml:mi>F</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo>.</mml:mo></mml:mrow></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ6\"><label>6</label><alternatives><tex-math id=\"M39\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\zeta }_{W=0}=\\frac{2\\sqrt{1-\\beta C}}{S}\\sqrt{1-\\frac{1-\\beta C}{{S}^{2}}},$$\\end{document}</tex-math><mml:math id=\"M40\" display=\"block\"><mml:mrow><mml:msub><mml:mi>ζ</mml:mi><mml:mrow><mml:mi>W</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mfrac><mml:mrow><mml:mn>2</mml:mn><mml:msqrt><mml:mrow><mml:mn>1</mml:mn><mml:mo>-</mml:mo><mml:mi>β</mml:mi><mml:mi>C</mml:mi></mml:mrow></mml:msqrt></mml:mrow><mml:mi>S</mml:mi></mml:mfrac><mml:msqrt><mml:mrow><mml:mn>1</mml:mn><mml:mo>-</mml:mo><mml:mfrac><mml:mrow><mml:mn>1</mml:mn><mml:mo>-</mml:mo><mml:mi>β</mml:mi><mml:mi>C</mml:mi></mml:mrow><mml:msup><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mn>2</mml:mn></mml:msup></mml:mfrac></mml:mrow></mml:msqrt><mml:mo>,</mml:mo></mml:mrow></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equa\"><alternatives><tex-math id=\"M41\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$C={\\text{cos}}\\alpha \\cdot S={\\text{sin}}\\alpha .$$\\end{document}</tex-math><mml:math id=\"M42\" display=\"block\"><mml:mrow><mml:mi>C</mml:mi><mml:mo>=</mml:mo><mml:mtext>cos</mml:mtext><mml:mi>α</mml:mi><mml:mo>·</mml:mo><mml:mi>S</mml:mi><mml:mo>=</mml:mo><mml:mtext>sin</mml:mtext><mml:mi>α</mml:mi><mml:mo>.</mml:mo></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq15\"><alternatives><tex-math id=\"M43\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\zeta $$\\end{document}</tex-math><mml:math id=\"M44\"><mml:mi>ζ</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq16\"><alternatives><tex-math id=\"M45\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\beta $$\\end{document}</tex-math><mml:math id=\"M46\"><mml:mi>β</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq17\"><alternatives><tex-math id=\"M47\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\alpha $$\\end{document}</tex-math><mml:math id=\"M48\"><mml:mi>α</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq18\"><alternatives><tex-math id=\"M49\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\zeta $$\\end{document}</tex-math><mml:math id=\"M50\"><mml:mi>ζ</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq19\"><alternatives><tex-math id=\"M51\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\beta $$\\end{document}</tex-math><mml:math id=\"M52\"><mml:mi>β</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq20\"><alternatives><tex-math id=\"M53\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\alpha $$\\end{document}</tex-math><mml:math id=\"M54\"><mml:mi>α</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq21\"><alternatives><tex-math id=\"M55\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\zeta $$\\end{document}</tex-math><mml:math id=\"M56\"><mml:mi>ζ</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq22\"><alternatives><tex-math id=\"M57\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\beta $$\\end{document}</tex-math><mml:math id=\"M58\"><mml:mi>β</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq23\"><alternatives><tex-math id=\"M59\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\alpha $$\\end{document}</tex-math><mml:math id=\"M60\"><mml:mi>α</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq24\"><alternatives><tex-math id=\"M61\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\varepsilon }_{r}$$\\end{document}</tex-math><mml:math id=\"M62\"><mml:msub><mml:mi>ε</mml:mi><mml:mi>r</mml:mi></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq25\"><alternatives><tex-math id=\"M63\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\sqrt{{\\varepsilon }_{r}}$$\\end{document}</tex-math><mml:math id=\"M64\"><mml:msqrt><mml:msub><mml:mi>ε</mml:mi><mml:mi>r</mml:mi></mml:msub></mml:msqrt></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ7\"><label>7</label><alternatives><tex-math id=\"M65\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$a{(W/{F}_{0})}^{2}+bW/{F}_{0}+c=0,$$\\end{document}</tex-math><mml:math id=\"M66\" display=\"block\"><mml:mrow><mml:mi>a</mml:mi><mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>W</mml:mi><mml:mo stretchy=\"false\">/</mml:mo><mml:msub><mml:mi>F</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:msup><mml:mo>+</mml:mo><mml:mi>b</mml:mi><mml:mi>W</mml:mi><mml:mo stretchy=\"false\">/</mml:mo><mml:msub><mml:mi>F</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo>+</mml:mo><mml:mi>c</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo></mml:mrow></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equb\"><alternatives><tex-math id=\"M67\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$a=1-\\frac{{\\left(1-\\beta \\right)}^{2}}{{\\left(1-\\beta C\\right)}^{2}}-\\frac{{\\zeta }^{2}}{{\\beta }^{2}},$$\\end{document}</tex-math><mml:math id=\"M68\" display=\"block\"><mml:mrow><mml:mi>a</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn><mml:mo>-</mml:mo><mml:mfrac><mml:msup><mml:mrow><mml:mfenced close=\")\" open=\"(\"><mml:mn>1</mml:mn><mml:mo>-</mml:mo><mml:mi>β</mml:mi></mml:mfenced></mml:mrow><mml:mn>2</mml:mn></mml:msup><mml:msup><mml:mrow><mml:mfenced close=\")\" open=\"(\"><mml:mn>1</mml:mn><mml:mo>-</mml:mo><mml:mi>β</mml:mi><mml:mi>C</mml:mi></mml:mfenced></mml:mrow><mml:mn>2</mml:mn></mml:msup></mml:mfrac><mml:mo>-</mml:mo><mml:mfrac><mml:msup><mml:mrow><mml:mi>ζ</mml:mi></mml:mrow><mml:mn>2</mml:mn></mml:msup><mml:msup><mml:mrow><mml:mi>β</mml:mi></mml:mrow><mml:mn>2</mml:mn></mml:msup></mml:mfrac><mml:mo>,</mml:mo></mml:mrow></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equc\"><alternatives><tex-math id=\"M69\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$b=-2+\\frac{{2\\zeta }^{2}}{\\beta }+\\frac{2\\left(1-\\beta \\right)}{1-\\beta C}-\\frac{{\\zeta }^{2}{S}^{2}\\left(1-\\beta \\right)}{{\\left(1-\\beta C\\right)}^{2}},$$\\end{document}</tex-math><mml:math id=\"M70\" display=\"block\"><mml:mrow><mml:mi>b</mml:mi><mml:mo>=</mml:mo><mml:mo>-</mml:mo><mml:mn>2</mml:mn><mml:mo>+</mml:mo><mml:mfrac><mml:msup><mml:mrow><mml:mn>2</mml:mn><mml:mi>ζ</mml:mi></mml:mrow><mml:mn>2</mml:mn></mml:msup><mml:mi>β</mml:mi></mml:mfrac><mml:mo>+</mml:mo><mml:mfrac><mml:mrow><mml:mn>2</mml:mn><mml:mfenced close=\")\" open=\"(\"><mml:mn>1</mml:mn><mml:mo>-</mml:mo><mml:mi>β</mml:mi></mml:mfenced></mml:mrow><mml:mrow><mml:mn>1</mml:mn><mml:mo>-</mml:mo><mml:mi>β</mml:mi><mml:mi>C</mml:mi></mml:mrow></mml:mfrac><mml:mo>-</mml:mo><mml:mfrac><mml:mrow><mml:msup><mml:mrow><mml:mi>ζ</mml:mi></mml:mrow><mml:mn>2</mml:mn></mml:msup><mml:msup><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mn>2</mml:mn></mml:msup><mml:mfenced close=\")\" open=\"(\"><mml:mn>1</mml:mn><mml:mo>-</mml:mo><mml:mi>β</mml:mi></mml:mfenced></mml:mrow><mml:msup><mml:mrow><mml:mfenced close=\")\" open=\"(\"><mml:mn>1</mml:mn><mml:mo>-</mml:mo><mml:mi>β</mml:mi><mml:mi>C</mml:mi></mml:mfenced></mml:mrow><mml:mn>2</mml:mn></mml:msup></mml:mfrac><mml:mo>,</mml:mo></mml:mrow></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equd\"><alternatives><tex-math id=\"M71\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$c={-\\zeta }^{2}+\\frac{{\\zeta }^{2}{S}^{2}}{1-\\beta C}-\\frac{{\\zeta }^{4}{S}^{4}}{4{\\left(1-\\beta C\\right)}^{2}},$$\\end{document}</tex-math><mml:math id=\"M72\" display=\"block\"><mml:mrow><mml:mi>c</mml:mi><mml:mo>=</mml:mo><mml:msup><mml:mrow><mml:mo>-</mml:mo><mml:mi>ζ</mml:mi></mml:mrow><mml:mn>2</mml:mn></mml:msup><mml:mo>+</mml:mo><mml:mfrac><mml:mrow><mml:msup><mml:mrow><mml:mi>ζ</mml:mi></mml:mrow><mml:mn>2</mml:mn></mml:msup><mml:msup><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mn>2</mml:mn></mml:msup></mml:mrow><mml:mrow><mml:mn>1</mml:mn><mml:mo>-</mml:mo><mml:mi>β</mml:mi><mml:mi>C</mml:mi></mml:mrow></mml:mfrac><mml:mo>-</mml:mo><mml:mfrac><mml:mrow><mml:msup><mml:mrow><mml:mi>ζ</mml:mi></mml:mrow><mml:mn>4</mml:mn></mml:msup><mml:msup><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mn>4</mml:mn></mml:msup></mml:mrow><mml:mrow><mml:mn>4</mml:mn><mml:msup><mml:mrow><mml:mfenced close=\")\" open=\"(\"><mml:mn>1</mml:mn><mml:mo>-</mml:mo><mml:mi>β</mml:mi><mml:mi>C</mml:mi></mml:mfenced></mml:mrow><mml:mn>2</mml:mn></mml:msup></mml:mrow></mml:mfrac><mml:mo>,</mml:mo></mml:mrow></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Eque\"><alternatives><tex-math id=\"M73\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$C={\\text{cos}}\\alpha \\cdot S={\\text{sin}}\\alpha .$$\\end{document}</tex-math><mml:math id=\"M74\" display=\"block\"><mml:mrow><mml:mi>C</mml:mi><mml:mo>=</mml:mo><mml:mtext>cos</mml:mtext><mml:mi>α</mml:mi><mml:mo>·</mml:mo><mml:mi>S</mml:mi><mml:mo>=</mml:mo><mml:mtext>sin</mml:mtext><mml:mi>α</mml:mi><mml:mo>.</mml:mo></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq26\"><alternatives><tex-math id=\"M75\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\alpha $$\\end{document}</tex-math><mml:math id=\"M76\"><mml:mi>α</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq27\"><alternatives><tex-math id=\"M77\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\beta $$\\end{document}</tex-math><mml:math id=\"M78\"><mml:mi>β</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq28\"><alternatives><tex-math id=\"M79\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\alpha $$\\end{document}</tex-math><mml:math id=\"M80\"><mml:mi>α</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq29\"><alternatives><tex-math id=\"M81\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\beta $$\\end{document}</tex-math><mml:math id=\"M82\"><mml:mi>β</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq30\"><alternatives><tex-math id=\"M83\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\zeta $$\\end{document}</tex-math><mml:math id=\"M84\"><mml:mi>ζ</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq31\"><alternatives><tex-math id=\"M85\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$(\\Delta l=\\Delta L/{F}_{0})$$\\end{document}</tex-math><mml:math id=\"M86\"><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi mathvariant=\"normal\">Δ</mml:mi><mml:mi>l</mml:mi><mml:mo>=</mml:mo><mml:mi mathvariant=\"normal\">Δ</mml:mi><mml:mi>L</mml:mi><mml:mo stretchy=\"false\">/</mml:mo><mml:msub><mml:mi>F</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ8\"><label>8</label><alternatives><tex-math id=\"M87\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\Delta l=\\sqrt{{\\varepsilon }_{r}}\\left(r-h\\right)+\\sqrt{{\\varepsilon }_{eff}}w+{y}_{3}{\\text{sin}}\\theta ,$$\\end{document}</tex-math><mml:math id=\"M88\" display=\"block\"><mml:mrow><mml:mi mathvariant=\"normal\">Δ</mml:mi><mml:mi>l</mml:mi><mml:mo>=</mml:mo><mml:msqrt><mml:msub><mml:mi>ε</mml:mi><mml:mi>r</mml:mi></mml:msub></mml:msqrt><mml:mfenced close=\")\" open=\"(\"><mml:mi>r</mml:mi><mml:mo>-</mml:mo><mml:mi>h</mml:mi></mml:mfenced><mml:mo>+</mml:mo><mml:msqrt><mml:msub><mml:mi>ε</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">eff</mml:mi></mml:mrow></mml:msub></mml:msqrt><mml:mi>w</mml:mi><mml:mo>+</mml:mo><mml:msub><mml:mi>y</mml:mi><mml:mn>3</mml:mn></mml:msub><mml:mtext>sin</mml:mtext><mml:mi>θ</mml:mi><mml:mo>,</mml:mo></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq32\"><alternatives><tex-math id=\"M89\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$h=H/{F}_{0}$$\\end{document}</tex-math><mml:math id=\"M90\"><mml:mrow><mml:mi>h</mml:mi><mml:mo>=</mml:mo><mml:mi>H</mml:mi><mml:mo stretchy=\"false\">/</mml:mo><mml:msub><mml:mi>F</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq33\"><alternatives><tex-math id=\"M91\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$r=R/{F}_{0}$$\\end{document}</tex-math><mml:math id=\"M92\"><mml:mrow><mml:mi>r</mml:mi><mml:mo>=</mml:mo><mml:mi>R</mml:mi><mml:mo stretchy=\"false\">/</mml:mo><mml:msub><mml:mi>F</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq34\"><alternatives><tex-math id=\"M93\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$w=W/{F}_{0}$$\\end{document}</tex-math><mml:math id=\"M94\"><mml:mrow><mml:mi>w</mml:mi><mml:mo>=</mml:mo><mml:mi>W</mml:mi><mml:mo stretchy=\"false\">/</mml:mo><mml:msub><mml:mi>F</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq35\"><alternatives><tex-math id=\"M95\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\varepsilon }_{r}$$\\end{document}</tex-math><mml:math id=\"M96\"><mml:msub><mml:mi>ε</mml:mi><mml:mi>r</mml:mi></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq36\"><alternatives><tex-math id=\"M97\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\varepsilon }_{eff}$$\\end{document}</tex-math><mml:math id=\"M98\"><mml:msub><mml:mi>ε</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">eff</mml:mi></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ9\"><label>9</label><alternatives><tex-math id=\"M99\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\left|\\Delta l\\right|=\\sum_{1}^{M}\\sum_{1}^{N}\\left|\\sqrt{{\\varepsilon }_{r}}\\left(r-h\\right)+\\sqrt{{\\varepsilon }_{eff}}w+{y}_{3}{\\text{sin}}\\theta \\right|.$$\\end{document}</tex-math><mml:math id=\"M100\" display=\"block\"><mml:mrow><mml:mfenced close=\"|\" open=\"|\"><mml:mi mathvariant=\"normal\">Δ</mml:mi><mml:mi>l</mml:mi></mml:mfenced><mml:mo>=</mml:mo><mml:munderover><mml:mo>∑</mml:mo><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mi>M</mml:mi></mml:munderover><mml:munderover><mml:mo>∑</mml:mo><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mi>N</mml:mi></mml:munderover><mml:mfenced close=\"|\" open=\"|\"><mml:msqrt><mml:msub><mml:mi>ε</mml:mi><mml:mi>r</mml:mi></mml:msub></mml:msqrt><mml:mfenced close=\")\" open=\"(\"><mml:mi>r</mml:mi><mml:mo>-</mml:mo><mml:mi>h</mml:mi></mml:mfenced><mml:mo>+</mml:mo><mml:msqrt><mml:msub><mml:mi>ε</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">eff</mml:mi></mml:mrow></mml:msub></mml:msqrt><mml:mi>w</mml:mi><mml:mo>+</mml:mo><mml:msub><mml:mi>y</mml:mi><mml:mn>3</mml:mn></mml:msub><mml:mtext>sin</mml:mtext><mml:mi>θ</mml:mi></mml:mfenced><mml:mo>.</mml:mo></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq37\"><alternatives><tex-math id=\"M101\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${WB}_{i}$$\\end{document}</tex-math><mml:math id=\"M102\"><mml:msub><mml:mrow><mml:mi mathvariant=\"italic\">WB</mml:mi></mml:mrow><mml:mi>i</mml:mi></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq38\"><alternatives><tex-math id=\"M103\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${WA}_{j}$$\\end{document}</tex-math><mml:math id=\"M104\"><mml:msub><mml:mrow><mml:mi mathvariant=\"italic\">WA</mml:mi></mml:mrow><mml:mi>j</mml:mi></mml:msub></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ10\"><label>10</label><alternatives><tex-math id=\"M105\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${S}_{{WB}_{i}{WA}_{j}}={J}_{0}\\left(\\frac{K{WB}_{i}}{2}{\\text{sin}}{\\phi B}_{i}\\right){J}_{0}\\left(\\frac{K{WA}_{j}}{2}{\\text{sin}}{\\phi A}_{j}\\right)\\sqrt{\\frac{{WB}_{i}{WA}_{j}}{\\lambda d}}{e}^{-j\\left(k{d}_{ij}+\\pi /4\\right)},$$\\end{document}</tex-math><mml:math id=\"M106\" display=\"block\"><mml:mrow><mml:msub><mml:mi>S</mml:mi><mml:mrow><mml:msub><mml:mrow><mml:mi mathvariant=\"italic\">WB</mml:mi></mml:mrow><mml:mi>i</mml:mi></mml:msub><mml:msub><mml:mrow><mml:mi mathvariant=\"italic\">WA</mml:mi></mml:mrow><mml:mi>j</mml:mi></mml:msub></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:msub><mml:mi>J</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mfenced close=\")\" open=\"(\"><mml:mfrac><mml:mrow><mml:mi>K</mml:mi><mml:msub><mml:mrow><mml:mi mathvariant=\"italic\">WB</mml:mi></mml:mrow><mml:mi>i</mml:mi></mml:msub></mml:mrow><mml:mn>2</mml:mn></mml:mfrac><mml:mtext>sin</mml:mtext><mml:msub><mml:mrow><mml:mi>ϕ</mml:mi><mml:mi>B</mml:mi></mml:mrow><mml:mi>i</mml:mi></mml:msub></mml:mfenced><mml:msub><mml:mi>J</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mfenced close=\")\" open=\"(\"><mml:mfrac><mml:mrow><mml:mi>K</mml:mi><mml:msub><mml:mrow><mml:mi mathvariant=\"italic\">WA</mml:mi></mml:mrow><mml:mi>j</mml:mi></mml:msub></mml:mrow><mml:mn>2</mml:mn></mml:mfrac><mml:mtext>sin</mml:mtext><mml:msub><mml:mrow><mml:mi>ϕ</mml:mi><mml:mi>A</mml:mi></mml:mrow><mml:mi>j</mml:mi></mml:msub></mml:mfenced><mml:msqrt><mml:mfrac><mml:mrow><mml:msub><mml:mrow><mml:mi mathvariant=\"italic\">WB</mml:mi></mml:mrow><mml:mi>i</mml:mi></mml:msub><mml:msub><mml:mrow><mml:mi mathvariant=\"italic\">WA</mml:mi></mml:mrow><mml:mi>j</mml:mi></mml:msub></mml:mrow><mml:mrow><mml:mi>λ</mml:mi><mml:mi>d</mml:mi></mml:mrow></mml:mfrac></mml:msqrt><mml:msup><mml:mrow><mml:mi>e</mml:mi></mml:mrow><mml:mrow><mml:mo>-</mml:mo><mml:mi>j</mml:mi><mml:mfenced close=\")\" open=\"(\"><mml:mi>k</mml:mi><mml:msub><mml:mi>d</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">ij</mml:mi></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:mi>π</mml:mi><mml:mo stretchy=\"false\">/</mml:mo><mml:mn>4</mml:mn></mml:mfenced></mml:mrow></mml:msup><mml:mo>,</mml:mo></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq39\"><alternatives><tex-math id=\"M107\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$k=2\\pi /\\lambda $$\\end{document}</tex-math><mml:math id=\"M108\"><mml:mrow><mml:mi>k</mml:mi><mml:mo>=</mml:mo><mml:mn>2</mml:mn><mml:mi>π</mml:mi><mml:mo stretchy=\"false\">/</mml:mo><mml:mi>λ</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq40\"><alternatives><tex-math id=\"M109\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${J}_{0}\\left(x\\right)=(sinx)/x$$\\end{document}</tex-math><mml:math id=\"M110\"><mml:mrow><mml:msub><mml:mi>J</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mfenced close=\")\" open=\"(\"><mml:mi>x</mml:mi></mml:mfenced><mml:mo>=</mml:mo><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>s</mml:mi><mml:mi>i</mml:mi><mml:mi>n</mml:mi><mml:mi>x</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo stretchy=\"false\">/</mml:mo><mml:mi>x</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq41\"><alternatives><tex-math id=\"M111\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${d}_{ij}$$\\end{document}</tex-math><mml:math id=\"M112\"><mml:msub><mml:mi>d</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">ij</mml:mi></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq42\"><alternatives><tex-math id=\"M113\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\phi B}_{i}$$\\end{document}</tex-math><mml:math id=\"M114\"><mml:msub><mml:mrow><mml:mi>ϕ</mml:mi><mml:mi>B</mml:mi></mml:mrow><mml:mi>i</mml:mi></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq43\"><alternatives><tex-math id=\"M115\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\phi A}_{j}$$\\end{document}</tex-math><mml:math id=\"M116\"><mml:msub><mml:mrow><mml:mi>ϕ</mml:mi><mml:mi>A</mml:mi></mml:mrow><mml:mi>j</mml:mi></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq44\"><alternatives><tex-math id=\"M117\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\alpha $$\\end{document}</tex-math><mml:math id=\"M118\"><mml:mi>α</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq45\"><alternatives><tex-math id=\"M119\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\beta $$\\end{document}</tex-math><mml:math id=\"M120\"><mml:mi>β</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq46\"><alternatives><tex-math id=\"M121\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$({x}_{2}\\cdot {y}_{2})$$\\end{document}</tex-math><mml:math id=\"M122\"><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>x</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mo>·</mml:mo><mml:msub><mml:mi>y</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ11\"><label>11a</label><alternatives><tex-math id=\"M123\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${X}_{2}=\\frac{{x}_{2}}{{F}_{0}}=1-\\frac{\\frac{1}{2}{\\zeta }^{2}{\\text{sin}}^{2}\\alpha +\\left(1-\\beta \\right)W}{1-\\beta {\\text{cos}}\\alpha },$$\\end{document}</tex-math><mml:math id=\"M124\" display=\"block\"><mml:mrow><mml:msub><mml:mi>X</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mo>=</mml:mo><mml:mfrac><mml:msub><mml:mi>x</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:msub><mml:mi>F</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:mfrac><mml:mo>=</mml:mo><mml:mn>1</mml:mn><mml:mo>-</mml:mo><mml:mfrac><mml:mrow><mml:mfrac><mml:mn>1</mml:mn><mml:mn>2</mml:mn></mml:mfrac><mml:msup><mml:mrow><mml:mi>ζ</mml:mi></mml:mrow><mml:mn>2</mml:mn></mml:msup><mml:msup><mml:mrow><mml:mtext>sin</mml:mtext></mml:mrow><mml:mn>2</mml:mn></mml:msup><mml:mi>α</mml:mi><mml:mo>+</mml:mo><mml:mfenced close=\")\" open=\"(\"><mml:mn>1</mml:mn><mml:mo>-</mml:mo><mml:mi>β</mml:mi></mml:mfenced><mml:mi>W</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn><mml:mo>-</mml:mo><mml:mi>β</mml:mi><mml:mtext>cos</mml:mtext><mml:mi>α</mml:mi></mml:mrow></mml:mfrac><mml:mo>,</mml:mo></mml:mrow></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ12\"><label>11b</label><alternatives><tex-math id=\"M125\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${Y}_{2}=\\frac{{y}_{2}}{{F}_{0}}=\\zeta \\left(1-\\frac{W}{\\beta }\\right).$$\\end{document}</tex-math><mml:math id=\"M126\" display=\"block\"><mml:mrow><mml:msub><mml:mi>Y</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mo>=</mml:mo><mml:mfrac><mml:msub><mml:mi>y</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:msub><mml:mi>F</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:mfrac><mml:mo>=</mml:mo><mml:mi>ζ</mml:mi><mml:mfenced close=\")\" open=\"(\"><mml:mn>1</mml:mn><mml:mo>-</mml:mo><mml:mfrac><mml:mi>W</mml:mi><mml:mi>β</mml:mi></mml:mfrac></mml:mfenced><mml:mo>.</mml:mo></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq47\"><alternatives><tex-math id=\"M127\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$({x}_{1}.{y}_{1})$$\\end{document}</tex-math><mml:math id=\"M128\"><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>x</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>.</mml:mo><mml:msub><mml:mi>y</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ13\"><label>12a</label><alternatives><tex-math id=\"M129\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${X}_{1}=\\frac{{x}_{1}}{{F}_{0}}={\\rho }_{0\\left[1-{\\text{cos}}({\\alpha }{^\\prime}+\\psi )\\right]},$$\\end{document}</tex-math><mml:math id=\"M130\" display=\"block\"><mml:mrow><mml:msub><mml:mi>X</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>=</mml:mo><mml:mfrac><mml:msub><mml:mi>x</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:msub><mml:mi>F</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:mfrac><mml:mo>=</mml:mo><mml:msub><mml:mi>ρ</mml:mi><mml:mrow><mml:mn>0</mml:mn><mml:mfenced close=\"]\" open=\"[\"><mml:mn>1</mml:mn><mml:mo>-</mml:mo><mml:mtext>cos</mml:mtext><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>α</mml:mi><mml:msup><mml:mrow/><mml:mo>′</mml:mo></mml:msup><mml:mo>+</mml:mo><mml:mi>ψ</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mfenced></mml:mrow></mml:msub><mml:mo>,</mml:mo></mml:mrow></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ14\"><label>12b</label><alternatives><tex-math id=\"M131\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${Y}_{1}=\\frac{{y}_{1}}{{F}_{0}}={\\rho }_{0}{\\text{sin}}\\left({\\alpha }{^\\prime}+\\psi \\right),$$\\end{document}</tex-math><mml:math id=\"M132\" display=\"block\"><mml:mrow><mml:msub><mml:mi>Y</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>=</mml:mo><mml:mfrac><mml:msub><mml:mi>y</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:msub><mml:mi>F</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:mfrac><mml:mo>=</mml:mo><mml:msub><mml:mi>ρ</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mtext>sin</mml:mtext><mml:mfenced close=\")\" open=\"(\"><mml:mi>α</mml:mi><mml:msup><mml:mrow/><mml:mo>′</mml:mo></mml:msup><mml:mo>+</mml:mo><mml:mi>ψ</mml:mi></mml:mfenced><mml:mo>,</mml:mo></mml:mrow></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equf\"><alternatives><tex-math id=\"M133\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\rho }_{0}=1-\\frac{1-{\\beta }^{2}}{2(1-\\beta {\\text{cos}}\\alpha )},$$\\end{document}</tex-math><mml:math id=\"M134\" display=\"block\"><mml:mrow><mml:msub><mml:mi>ρ</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo>=</mml:mo><mml:mn>1</mml:mn><mml:mo>-</mml:mo><mml:mfrac><mml:mrow><mml:mn>1</mml:mn><mml:mo>-</mml:mo><mml:msup><mml:mrow><mml:mi>β</mml:mi></mml:mrow><mml:mn>2</mml:mn></mml:msup></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>1</mml:mn><mml:mo>-</mml:mo><mml:mi>β</mml:mi><mml:mtext>cos</mml:mtext><mml:mi>α</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mfrac><mml:mo>,</mml:mo></mml:mrow></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equg\"><alternatives><tex-math id=\"M135\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\alpha }{^\\prime}={\\text{sin}}^{-1}\\left(\\frac{{\\text{sin}}\\theta }{\\gamma }\\right),$$\\end{document}</tex-math><mml:math id=\"M136\" display=\"block\"><mml:mrow><mml:mi>α</mml:mi><mml:msup><mml:mrow/><mml:mo>′</mml:mo></mml:msup><mml:mo>=</mml:mo><mml:msup><mml:mrow><mml:mtext>sin</mml:mtext></mml:mrow><mml:mrow><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mfenced close=\")\" open=\"(\"><mml:mfrac><mml:mrow><mml:mtext>sin</mml:mtext><mml:mi>θ</mml:mi></mml:mrow><mml:mi>γ</mml:mi></mml:mfrac></mml:mfenced><mml:mo>,</mml:mo></mml:mrow></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equh\"><alternatives><tex-math id=\"M137\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\psi ={\\text{sin}}^{-1}\\left(\\frac{1-{\\rho }_{0}}{{\\rho }_{0}}{\\text{sin}}{\\alpha }{^\\prime}\\right).$$\\end{document}</tex-math><mml:math id=\"M138\" display=\"block\"><mml:mrow><mml:mi>ψ</mml:mi><mml:mo>=</mml:mo><mml:msup><mml:mrow><mml:mtext>sin</mml:mtext></mml:mrow><mml:mrow><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mfenced close=\")\" open=\"(\"><mml:mfrac><mml:mrow><mml:mn>1</mml:mn><mml:mo>-</mml:mo><mml:msub><mml:mi>ρ</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:mrow><mml:msub><mml:mi>ρ</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:mfrac><mml:mtext>sin</mml:mtext><mml:mi>α</mml:mi><mml:msup><mml:mrow/><mml:mo>′</mml:mo></mml:msup></mml:mfenced><mml:mo>.</mml:mo></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq48\"><alternatives><tex-math id=\"M139\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\varepsilon }_{r}=3.48$$\\end{document}</tex-math><mml:math id=\"M140\"><mml:mrow><mml:msub><mml:mi>ε</mml:mi><mml:mi>r</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mn>3.48</mml:mn></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq49\"><alternatives><tex-math id=\"M141\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\varepsilon }_{r}=3.48$$\\end{document}</tex-math><mml:math id=\"M142\"><mml:mrow><mml:msub><mml:mi>ε</mml:mi><mml:mi>r</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mn>3.48</mml:mn></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq50\"><alternatives><tex-math id=\"M143\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\theta }_{0}={60}^{^\\circ }$$\\end{document}</tex-math><mml:math id=\"M144\"><mml:mrow><mml:msub><mml:mi>θ</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo>=</mml:mo><mml:msup><mml:mrow><mml:mn>60</mml:mn></mml:mrow><mml:msup><mml:mrow/><mml:mo>∘</mml:mo></mml:msup></mml:msup></mml:mrow></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ15\"><label>13</label><alternatives><tex-math id=\"M145\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$M=\\left[\\frac{{2\\theta }_{0}}{{BW}_{array}}\\right]\\to {BW}_{array}=\\frac{120}{8}={15}^{^\\circ }.$$\\end{document}</tex-math><mml:math id=\"M146\" display=\"block\"><mml:mrow><mml:mi>M</mml:mi><mml:mo>=</mml:mo><mml:mfenced close=\"]\" open=\"[\"><mml:mfrac><mml:msub><mml:mrow><mml:mn>2</mml:mn><mml:mi>θ</mml:mi></mml:mrow><mml:mn>0</mml:mn></mml:msub><mml:msub><mml:mrow><mml:mi mathvariant=\"italic\">BW</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"italic\">array</mml:mi></mml:mrow></mml:msub></mml:mfrac></mml:mfenced><mml:mo stretchy=\"false\">→</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant=\"italic\">BW</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"italic\">array</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mfrac><mml:mn>120</mml:mn><mml:mn>8</mml:mn></mml:mfrac><mml:mo>=</mml:mo><mml:msup><mml:mrow><mml:mn>15</mml:mn></mml:mrow><mml:msup><mml:mrow/><mml:mo>∘</mml:mo></mml:msup></mml:msup><mml:mo>.</mml:mo></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq51\"><alternatives><tex-math id=\"M147\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$({BW}_{array}={15}^{^\\circ })$$\\end{document}</tex-math><mml:math id=\"M148\"><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant=\"italic\">BW</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"italic\">array</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:msup><mml:mrow><mml:mn>15</mml:mn></mml:mrow><mml:msup><mml:mrow/><mml:mo>∘</mml:mo></mml:msup></mml:msup><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq52\"><alternatives><tex-math id=\"M149\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\theta }_{0}={60}^{^\\circ }$$\\end{document}</tex-math><mml:math id=\"M150\"><mml:mrow><mml:msub><mml:mi>θ</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo>=</mml:mo><mml:msup><mml:mrow><mml:mn>60</mml:mn></mml:mrow><mml:msup><mml:mrow/><mml:mo>∘</mml:mo></mml:msup></mml:msup></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq53\"><alternatives><tex-math id=\"M151\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\theta =60-15/2={52.5}^{^\\circ }$$\\end{document}</tex-math><mml:math id=\"M152\"><mml:mrow><mml:mi>θ</mml:mi><mml:mo>=</mml:mo><mml:mn>60</mml:mn><mml:mo>-</mml:mo><mml:mn>15</mml:mn><mml:mo stretchy=\"false\">/</mml:mo><mml:mn>2</mml:mn><mml:mo>=</mml:mo><mml:msup><mml:mrow><mml:mn>52.5</mml:mn></mml:mrow><mml:msup><mml:mrow/><mml:mo>∘</mml:mo></mml:msup></mml:msup></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq54\"><alternatives><tex-math id=\"M153\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\theta ={52.5}^{^\\circ }$$\\end{document}</tex-math><mml:math id=\"M154\"><mml:mrow><mml:mi>θ</mml:mi><mml:mo>=</mml:mo><mml:msup><mml:mrow><mml:mn>52.5</mml:mn></mml:mrow><mml:msup><mml:mrow/><mml:mo>∘</mml:mo></mml:msup></mml:msup></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq55\"><alternatives><tex-math id=\"M155\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${(\\pm 60}^{^\\circ })$$\\end{document}</tex-math><mml:math id=\"M156\"><mml:mrow><mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mo>±</mml:mo><mml:mn>60</mml:mn></mml:mrow><mml:msup><mml:mrow/><mml:mo>∘</mml:mo></mml:msup></mml:msup><mml:mrow><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq56\"><alternatives><tex-math id=\"M157\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\varepsilon }_{r}=3.48$$\\end{document}</tex-math><mml:math id=\"M158\"><mml:mrow><mml:msub><mml:mi>ε</mml:mi><mml:mi>r</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mn>3.48</mml:mn></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq57\"><alternatives><tex-math id=\"M159\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${F}_{0}$$\\end{document}</tex-math><mml:math id=\"M160\"><mml:msub><mml:mi>F</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equi\"><alternatives><tex-math id=\"M161\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${F}_{0}\\ge 2 \\left(8-1\\right)5.5 {\\text{ sin}}{52.5}^{^\\circ }=61.08\\text{ mm.}$$\\end{document}</tex-math><mml:math id=\"M162\" display=\"block\"><mml:mrow><mml:msub><mml:mi>F</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo>≥</mml:mo><mml:mn>2</mml:mn><mml:mfenced close=\")\" open=\"(\"><mml:mn>8</mml:mn><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mfenced><mml:mn>5.5</mml:mn><mml:mrow><mml:mspace width=\"0.333333em\"/><mml:mtext>sin</mml:mtext></mml:mrow><mml:msup><mml:mrow><mml:mn>52.5</mml:mn></mml:mrow><mml:msup><mml:mrow/><mml:mo>∘</mml:mo></mml:msup></mml:msup><mml:mo>=</mml:mo><mml:mn>61.08</mml:mn><mml:mspace width=\"0.333333em\"/><mml:mtext>mm.</mml:mtext></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq58\"><alternatives><tex-math id=\"M163\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${y}_{3}={y}_{3({\\text{max}})}$$\\end{document}</tex-math><mml:math id=\"M164\"><mml:mrow><mml:msub><mml:mi>y</mml:mi><mml:mn>3</mml:mn></mml:msub><mml:mo>=</mml:mo><mml:msub><mml:mi>y</mml:mi><mml:mrow><mml:mn>3</mml:mn><mml:mo stretchy=\"false\">(</mml:mo><mml:mtext>max</mml:mtext><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq59\"><alternatives><tex-math id=\"M165\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\zeta }_{\\text{max}}=0.5$$\\end{document}</tex-math><mml:math id=\"M166\"><mml:mrow><mml:msub><mml:mi>ζ</mml:mi><mml:mtext>max</mml:mtext></mml:msub><mml:mo>=</mml:mo><mml:mn>0.5</mml:mn></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq60\"><alternatives><tex-math id=\"M167\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\gamma .\\alpha $$\\end{document}</tex-math><mml:math id=\"M168\"><mml:mrow><mml:mi>γ</mml:mi><mml:mo>.</mml:mo><mml:mi>α</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equj\"><alternatives><tex-math id=\"M169\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${y}_{3({\\text{max}})}=19.25\\text{ mm}\\stackrel{{\\zeta }_{\\text{max}}=0.5}{\\to } \\gamma =1.58\\to \\alpha ={30}^{^\\circ }.$$\\end{document}</tex-math><mml:math id=\"M170\" display=\"block\"><mml:mrow><mml:msub><mml:mi>y</mml:mi><mml:mrow><mml:mn>3</mml:mn><mml:mo stretchy=\"false\">(</mml:mo><mml:mtext>max</mml:mtext><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mn>19.25</mml:mn><mml:mspace width=\"0.333333em\"/><mml:mtext>mm</mml:mtext><mml:mover><mml:mo stretchy=\"false\">→</mml:mo><mml:mrow><mml:msub><mml:mi>ζ</mml:mi><mml:mtext>max</mml:mtext></mml:msub><mml:mo>=</mml:mo><mml:mn>0.5</mml:mn></mml:mrow></mml:mover><mml:mi>γ</mml:mi><mml:mo>=</mml:mo><mml:mn>1.58</mml:mn><mml:mo stretchy=\"false\">→</mml:mo><mml:mi>α</mml:mi><mml:mo>=</mml:mo><mml:msup><mml:mrow><mml:mn>30</mml:mn></mml:mrow><mml:msup><mml:mrow/><mml:mo>∘</mml:mo></mml:msup></mml:msup><mml:mo>.</mml:mo></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq61\"><alternatives><tex-math id=\"M171\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\beta $$\\end{document}</tex-math><mml:math id=\"M172\"><mml:mi>β</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq62\"><alternatives><tex-math id=\"M173\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\beta =0.88$$\\end{document}</tex-math><mml:math id=\"M174\"><mml:mrow><mml:mi>β</mml:mi><mml:mo>=</mml:mo><mml:mn>0.88</mml:mn></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq63\"><alternatives><tex-math id=\"M175\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\alpha .\\beta $$\\end{document}</tex-math><mml:math id=\"M176\"><mml:mrow><mml:mi>α</mml:mi><mml:mo>.</mml:mo><mml:mi>β</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equk\"><alternatives><tex-math id=\"M177\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\alpha ={29.7}^{^\\circ } , \\beta =0.91.$$\\end{document}</tex-math><mml:math id=\"M178\" display=\"block\"><mml:mrow><mml:mi>α</mml:mi><mml:mo>=</mml:mo><mml:msup><mml:mrow><mml:mn>29.7</mml:mn></mml:mrow><mml:msup><mml:mrow/><mml:mo>∘</mml:mo></mml:msup></mml:msup><mml:mo>,</mml:mo><mml:mi>β</mml:mi><mml:mo>=</mml:mo><mml:mn>0.91</mml:mn><mml:mo>.</mml:mo></mml:mrow></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equl\"><alternatives><tex-math id=\"M179\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$W=1.45\\text{ mm}$$\\end{document}</tex-math><mml:math id=\"M180\" display=\"block\"><mml:mrow><mml:mi>W</mml:mi><mml:mo>=</mml:mo><mml:mn>1.45</mml:mn><mml:mspace width=\"0.333333em\"/><mml:mtext>mm</mml:mtext></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq64\"><alternatives><tex-math id=\"M181\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\sqrt{{\\varepsilon }_{r}}$$\\end{document}</tex-math><mml:math id=\"M182\"><mml:msqrt><mml:msub><mml:mi>ε</mml:mi><mml:mi>r</mml:mi></mml:msub></mml:msqrt></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq65\"><alternatives><tex-math id=\"M183\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${F}_{0}. {F}_{1}. W$$\\end{document}</tex-math><mml:math id=\"M184\"><mml:mrow><mml:msub><mml:mi>F</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo>.</mml:mo><mml:msub><mml:mi>F</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>.</mml:mo><mml:mi>W</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq66\"><alternatives><tex-math id=\"M185\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\sqrt{3.66}$$\\end{document}</tex-math><mml:math id=\"M186\"><mml:msqrt><mml:mrow><mml:mn>3.66</mml:mn></mml:mrow></mml:msqrt></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq67\"><alternatives><tex-math id=\"M187\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$({x}_{1}\\cdot {y}_{1})$$\\end{document}</tex-math><mml:math id=\"M188\"><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>x</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>·</mml:mo><mml:msub><mml:mi>y</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equm\"><alternatives><tex-math id=\"M189\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${x}_{1}=25.23\\text{ mm }\\quad {y}_{1}=14.4\\text{ mm.}$$\\end{document}</tex-math><mml:math id=\"M190\" display=\"block\"><mml:mrow><mml:msub><mml:mi>x</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>=</mml:mo><mml:mn>25.23</mml:mn><mml:mspace width=\"0.333333em\"/><mml:mtext>mm</mml:mtext><mml:mspace width=\"0.333333em\"/><mml:mspace width=\"1em\"/><mml:msub><mml:mi>y</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>=</mml:mo><mml:mn>14.4</mml:mn><mml:mspace width=\"0.333333em\"/><mml:mtext>mm.</mml:mtext></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq68\"><alternatives><tex-math id=\"M191\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$({x}_{2}.{y}_{2})$$\\end{document}</tex-math><mml:math id=\"M192\"><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>x</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mo>.</mml:mo><mml:msub><mml:mi>y</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equn\"><alternatives><tex-math id=\"M193\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${x}_{2}=0.68\\text{ mm } \\quad {y}_{2}=14.9\\text{ mm,}$$\\end{document}</tex-math><mml:math id=\"M194\" display=\"block\"><mml:mrow><mml:msub><mml:mi>x</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mo>=</mml:mo><mml:mn>0.68</mml:mn><mml:mspace width=\"0.333333em\"/><mml:mtext>mm</mml:mtext><mml:mspace width=\"0.333333em\"/><mml:mspace width=\"1em\"/><mml:msub><mml:mi>y</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mo>=</mml:mo><mml:mn>14.9</mml:mn><mml:mspace width=\"0.333333em\"/><mml:mtext>mm,</mml:mtext></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq69\"><alternatives><tex-math id=\"M195\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${W}_{p}=4.65\\text{ mm}$$\\end{document}</tex-math><mml:math id=\"M196\"><mml:mrow><mml:msub><mml:mi>W</mml:mi><mml:mi>p</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mn>4.65</mml:mn><mml:mspace width=\"0.333333em\"/><mml:mtext>mm</mml:mtext></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq70\"><alternatives><tex-math id=\"M197\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${w}_{1}$$\\end{document}</tex-math><mml:math id=\"M198\"><mml:msub><mml:mi>w</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ16\"><label>14</label><alternatives><tex-math id=\"M199\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${w}_{1}=\\frac{7.48\\times h}{{e}^{\\left({Z}_{0}\\frac{\\sqrt{{\\varepsilon }_{r}+1.41}}{87}\\right)}}-1.25\\times t\\approx 0.55\\text{ mm,}$$\\end{document}</tex-math><mml:math id=\"M200\" display=\"block\"><mml:mrow><mml:msub><mml:mi>w</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>=</mml:mo><mml:mfrac><mml:mrow><mml:mn>7.48</mml:mn><mml:mo>×</mml:mo><mml:mi>h</mml:mi></mml:mrow><mml:msup><mml:mrow><mml:mi>e</mml:mi></mml:mrow><mml:mfenced close=\")\" open=\"(\"><mml:msub><mml:mi>Z</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mfrac><mml:msqrt><mml:mrow><mml:msub><mml:mi>ε</mml:mi><mml:mi>r</mml:mi></mml:msub><mml:mo>+</mml:mo><mml:mn>1.41</mml:mn></mml:mrow></mml:msqrt><mml:mn>87</mml:mn></mml:mfrac></mml:mfenced></mml:msup></mml:mfrac><mml:mo>-</mml:mo><mml:mn>1.25</mml:mn><mml:mo>×</mml:mo><mml:mi>t</mml:mi><mml:mo>≈</mml:mo><mml:mn>0.55</mml:mn><mml:mspace width=\"0.333333em\"/><mml:mtext>mm,</mml:mtext></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq71\"><alternatives><tex-math id=\"M201\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${Z}_{0}$$\\end{document}</tex-math><mml:math id=\"M202\"><mml:msub><mml:mi>Z</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq72\"><alternatives><tex-math id=\"M203\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${w}_{1}\\approx 0.55\\text{ mm}$$\\end{document}</tex-math><mml:math id=\"M204\"><mml:mrow><mml:msub><mml:mi>w</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>≈</mml:mo><mml:mn>0.55</mml:mn><mml:mspace width=\"0.333333em\"/><mml:mtext>mm</mml:mtext></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq73\"><alternatives><tex-math id=\"M205\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${w}_{1}=0.53\\text{ mm}$$\\end{document}</tex-math><mml:math id=\"M206\"><mml:mrow><mml:msub><mml:mi>w</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>=</mml:mo><mml:mn>0.53</mml:mn><mml:mspace width=\"0.333333em\"/><mml:mtext>mm</mml:mtext></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq74\"><alternatives><tex-math id=\"M207\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${(L}_{p})$$\\end{document}</tex-math><mml:math id=\"M208\"><mml:mrow><mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>L</mml:mi></mml:mrow><mml:mi>p</mml:mi></mml:msub><mml:mrow><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq75\"><alternatives><tex-math id=\"M209\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${L}_{p}$$\\end{document}</tex-math><mml:math id=\"M210\"><mml:msub><mml:mi>L</mml:mi><mml:mi>p</mml:mi></mml:msub></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ17\"><label>15</label><alternatives><tex-math id=\"M211\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${L}_{p}\\approx 4.57{W}_{p},$$\\end{document}</tex-math><mml:math id=\"M212\" display=\"block\"><mml:mrow><mml:msub><mml:mi>L</mml:mi><mml:mi>p</mml:mi></mml:msub><mml:mo>≈</mml:mo><mml:mn>4.57</mml:mn><mml:msub><mml:mi>W</mml:mi><mml:mi>p</mml:mi></mml:msub><mml:mo>,</mml:mo></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq76\"><alternatives><tex-math id=\"M213\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${W}_{p}$$\\end{document}</tex-math><mml:math id=\"M214\"><mml:msub><mml:mi>W</mml:mi><mml:mi>p</mml:mi></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq77\"><alternatives><tex-math id=\"M215\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${L}_{p}=22\\text{ mm}$$\\end{document}</tex-math><mml:math id=\"M216\"><mml:mrow><mml:msub><mml:mi>L</mml:mi><mml:mi>p</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mn>22</mml:mn><mml:mspace width=\"0.333333em\"/><mml:mtext>mm</mml:mtext></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq78\"><alternatives><tex-math id=\"M217\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${h}_{p}=4.73$$\\end{document}</tex-math><mml:math id=\"M218\"><mml:mrow><mml:msub><mml:mi>h</mml:mi><mml:mi>p</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mn>4.73</mml:mn></mml:mrow></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ18\"><label>16</label><alternatives><tex-math id=\"M219\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${T}_{N-1}\\left({z}_{0}\\right)={10}^{s/20},$$\\end{document}</tex-math><mml:math id=\"M220\" display=\"block\"><mml:mrow><mml:msub><mml:mi>T</mml:mi><mml:mrow><mml:mi>N</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mfenced close=\")\" open=\"(\"><mml:msub><mml:mi>z</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:mfenced><mml:mo>=</mml:mo><mml:msup><mml:mrow><mml:mn>10</mml:mn></mml:mrow><mml:mrow><mml:mi>s</mml:mi><mml:mo stretchy=\"false\">/</mml:mo><mml:mn>20</mml:mn></mml:mrow></mml:msup><mml:mo>,</mml:mo></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq79\"><alternatives><tex-math id=\"M221\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${z}_{0}$$\\end{document}</tex-math><mml:math id=\"M222\"><mml:msub><mml:mi>z</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ19\"><label>17</label><alternatives><tex-math id=\"M223\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${z}_{0}={\\text{cos}}h\\left[\\frac{{{\\text{cos}}h}^{-1}({10}^{s/10})}{N-1}\\right].$$\\end{document}</tex-math><mml:math id=\"M224\" display=\"block\"><mml:mrow><mml:msub><mml:mi>z</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo>=</mml:mo><mml:mtext>cos</mml:mtext><mml:mi>h</mml:mi><mml:mfenced close=\"]\" open=\"[\"><mml:mfrac><mml:mrow><mml:msup><mml:mrow><mml:mtext>cos</mml:mtext><mml:mi>h</mml:mi></mml:mrow><mml:mrow><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msup><mml:mrow><mml:mn>10</mml:mn></mml:mrow><mml:mrow><mml:mi>s</mml:mi><mml:mo stretchy=\"false\">/</mml:mo><mml:mn>10</mml:mn></mml:mrow></mml:msup><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow><mml:mrow><mml:mi>N</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:mfrac></mml:mfenced><mml:mo>.</mml:mo></mml:mrow></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ20\"><label>18</label><alternatives><tex-math id=\"M225\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$HPBW={\\text{cos}}^{-1}\\left[{\\text{cos}}\\theta -0.443\\frac{\\lambda }{L+d}\\right],$$\\end{document}</tex-math><mml:math id=\"M226\" display=\"block\"><mml:mrow><mml:mi>H</mml:mi><mml:mi>P</mml:mi><mml:mi>B</mml:mi><mml:mi>W</mml:mi><mml:mo>=</mml:mo><mml:msup><mml:mrow><mml:mtext>cos</mml:mtext></mml:mrow><mml:mrow><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mfenced close=\"]\" open=\"[\"><mml:mtext>cos</mml:mtext><mml:mi>θ</mml:mi><mml:mo>-</mml:mo><mml:mn>0.443</mml:mn><mml:mfrac><mml:mi>λ</mml:mi><mml:mrow><mml:mi>L</mml:mi><mml:mo>+</mml:mo><mml:mi>d</mml:mi></mml:mrow></mml:mfrac></mml:mfenced><mml:mo>,</mml:mo></mml:mrow></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equo\"><alternatives><tex-math id=\"M227\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$-{\\text{cos}}^{-1}\\left[{\\text{cos}}\\theta +0.443\\frac{\\lambda }{L+d}\\right],$$\\end{document}</tex-math><mml:math id=\"M228\" display=\"block\"><mml:mrow><mml:mo>-</mml:mo><mml:msup><mml:mrow><mml:mtext>cos</mml:mtext></mml:mrow><mml:mrow><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mfenced close=\"]\" open=\"[\"><mml:mtext>cos</mml:mtext><mml:mi>θ</mml:mi><mml:mo>+</mml:mo><mml:mn>0.443</mml:mn><mml:mfrac><mml:mi>λ</mml:mi><mml:mrow><mml:mi>L</mml:mi><mml:mo>+</mml:mo><mml:mi>d</mml:mi></mml:mrow></mml:mfrac></mml:mfenced><mml:mo>,</mml:mo></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq80\"><alternatives><tex-math id=\"M229\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\theta $$\\end{document}</tex-math><mml:math id=\"M230\"><mml:mi>θ</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq81\"><alternatives><tex-math id=\"M231\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$SLL\\approx 16\\text{ dB}$$\\end{document}</tex-math><mml:math id=\"M232\"><mml:mrow><mml:mi>S</mml:mi><mml:mi>L</mml:mi><mml:mi>L</mml:mi><mml:mo>≈</mml:mo><mml:mn>16</mml:mn><mml:mspace width=\"0.333333em\"/><mml:mtext>dB</mml:mtext></mml:mrow></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ21\"><label>19</label><alternatives><tex-math id=\"M233\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$P=\\frac{{V}^{2}}{2Z}.$$\\end{document}</tex-math><mml:math id=\"M234\" display=\"block\"><mml:mrow><mml:mi>P</mml:mi><mml:mo>=</mml:mo><mml:mfrac><mml:msup><mml:mrow><mml:mi>V</mml:mi></mml:mrow><mml:mn>2</mml:mn></mml:msup><mml:mrow><mml:mn>2</mml:mn><mml:mi>Z</mml:mi></mml:mrow></mml:mfrac><mml:mo>.</mml:mo></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq82\"><alternatives><tex-math id=\"M235\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$W/h$$\\end{document}</tex-math><mml:math id=\"M236\"><mml:mrow><mml:mi>W</mml:mi><mml:mo stretchy=\"false\">/</mml:mo><mml:mi>h</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ22\"><label>20</label><alternatives><tex-math id=\"M237\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$Z=\\frac{120\\pi }{\\sqrt{{\\varepsilon }_{eff}}\\times \\left[\\frac{W}{h}+1.393+\\frac{2}{3}{\\text{ln}}\\left(\\frac{W}{h}+1.444\\right)\\right]} \\frac{W}{h}&gt;1.$$\\end{document}</tex-math><mml:math id=\"M238\" display=\"block\"><mml:mrow><mml:mi>Z</mml:mi><mml:mo>=</mml:mo><mml:mfrac><mml:mrow><mml:mn>120</mml:mn><mml:mi>π</mml:mi></mml:mrow><mml:mrow><mml:msqrt><mml:msub><mml:mi>ε</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">eff</mml:mi></mml:mrow></mml:msub></mml:msqrt><mml:mo>×</mml:mo><mml:mfenced close=\"]\" open=\"[\"><mml:mfrac><mml:mi>W</mml:mi><mml:mi>h</mml:mi></mml:mfrac><mml:mo>+</mml:mo><mml:mn>1.393</mml:mn><mml:mo>+</mml:mo><mml:mfrac><mml:mn>2</mml:mn><mml:mn>3</mml:mn></mml:mfrac><mml:mtext>ln</mml:mtext><mml:mfenced close=\")\" open=\"(\"><mml:mfrac><mml:mi>W</mml:mi><mml:mi>h</mml:mi></mml:mfrac><mml:mo>+</mml:mo><mml:mn>1.444</mml:mn></mml:mfenced></mml:mfenced></mml:mrow></mml:mfrac><mml:mfrac><mml:mi>W</mml:mi><mml:mi>h</mml:mi></mml:mfrac><mml:mo>&gt;</mml:mo><mml:mn>1</mml:mn><mml:mo>.</mml:mo></mml:mrow></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ23\"><label>21</label><alternatives><tex-math id=\"M239\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\frac{{P}_{1}}{{P}_{2}}\\approx \\frac{{Z}_{2}}{{Z}_{1}}\\approx \\frac{\\frac{{W}_{1}}{h}+1.393+\\frac{2}{3}{\\text{ln}}\\left(\\frac{{W}_{1}}{h}+1.444\\right)}{\\frac{{W}_{2}}{h}+1.393+\\frac{2}{3}{\\text{ln}}\\left(\\frac{{W}_{2}}{h}+1.444\\right)}\\approx \\frac{{\\text{ln}}\\left(1+4\\left(\\frac{h}{{W}_{2}}\\right)\\right)}{{\\text{ln}}\\left(1+4\\left(\\frac{h}{{W}_{1}}\\right)\\right)}.$$\\end{document}</tex-math><mml:math id=\"M240\" display=\"block\"><mml:mrow><mml:mfrac><mml:msub><mml:mi>P</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:msub><mml:mi>P</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:mfrac><mml:mo>≈</mml:mo><mml:mfrac><mml:msub><mml:mi>Z</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:msub><mml:mi>Z</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:mfrac><mml:mo>≈</mml:mo><mml:mfrac><mml:mrow><mml:mfrac><mml:msub><mml:mi>W</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mi>h</mml:mi></mml:mfrac><mml:mo>+</mml:mo><mml:mn>1.393</mml:mn><mml:mo>+</mml:mo><mml:mfrac><mml:mn>2</mml:mn><mml:mn>3</mml:mn></mml:mfrac><mml:mtext>ln</mml:mtext><mml:mfenced close=\")\" open=\"(\"><mml:mfrac><mml:msub><mml:mi>W</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mi>h</mml:mi></mml:mfrac><mml:mo>+</mml:mo><mml:mn>1.444</mml:mn></mml:mfenced></mml:mrow><mml:mrow><mml:mfrac><mml:msub><mml:mi>W</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mi>h</mml:mi></mml:mfrac><mml:mo>+</mml:mo><mml:mn>1.393</mml:mn><mml:mo>+</mml:mo><mml:mfrac><mml:mn>2</mml:mn><mml:mn>3</mml:mn></mml:mfrac><mml:mtext>ln</mml:mtext><mml:mfenced close=\")\" open=\"(\"><mml:mfrac><mml:msub><mml:mi>W</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mi>h</mml:mi></mml:mfrac><mml:mo>+</mml:mo><mml:mn>1.444</mml:mn></mml:mfenced></mml:mrow></mml:mfrac><mml:mo>≈</mml:mo><mml:mfrac><mml:mrow><mml:mtext>ln</mml:mtext><mml:mfenced close=\")\" open=\"(\"><mml:mn>1</mml:mn><mml:mo>+</mml:mo><mml:mn>4</mml:mn><mml:mfenced close=\")\" open=\"(\"><mml:mfrac><mml:mi>h</mml:mi><mml:msub><mml:mi>W</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:mfrac></mml:mfenced></mml:mfenced></mml:mrow><mml:mrow><mml:mtext>ln</mml:mtext><mml:mfenced close=\")\" open=\"(\"><mml:mn>1</mml:mn><mml:mo>+</mml:mo><mml:mn>4</mml:mn><mml:mfenced close=\")\" open=\"(\"><mml:mfrac><mml:mi>h</mml:mi><mml:msub><mml:mi>W</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:mfrac></mml:mfenced></mml:mfenced></mml:mrow></mml:mfrac><mml:mo>.</mml:mo></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq83\"><alternatives><tex-math id=\"M241\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${52.5}^{^\\circ }$$\\end{document}</tex-math><mml:math id=\"M242\"><mml:msup><mml:mrow><mml:mn>52.5</mml:mn></mml:mrow><mml:msup><mml:mrow/><mml:mo>∘</mml:mo></mml:msup></mml:msup></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq84\"><alternatives><tex-math id=\"M243\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${W}_{a1.8}=$$\\end{document}</tex-math><mml:math id=\"M244\"><mml:mrow><mml:msub><mml:mi>W</mml:mi><mml:mrow><mml:mi>a</mml:mi><mml:mn>1.8</mml:mn></mml:mrow></mml:msub><mml:mo>=</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq85\"><alternatives><tex-math id=\"M245\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${W}_{a2.7}=$$\\end{document}</tex-math><mml:math id=\"M246\"><mml:mrow><mml:msub><mml:mi>W</mml:mi><mml:mrow><mml:mi>a</mml:mi><mml:mn>2.7</mml:mn></mml:mrow></mml:msub><mml:mo>=</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq86\"><alternatives><tex-math id=\"M247\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${W}_{a3.6}=$$\\end{document}</tex-math><mml:math id=\"M248\"><mml:mrow><mml:msub><mml:mi>W</mml:mi><mml:mrow><mml:mi>a</mml:mi><mml:mn>3.6</mml:mn></mml:mrow></mml:msub><mml:mo>=</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq87\"><alternatives><tex-math id=\"M249\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${W}_{a4.5}=$$\\end{document}</tex-math><mml:math id=\"M250\"><mml:mrow><mml:msub><mml:mi>W</mml:mi><mml:mrow><mml:mi>a</mml:mi><mml:mn>4.5</mml:mn></mml:mrow></mml:msub><mml:mo>=</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq88\"><alternatives><tex-math id=\"M251\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${F}_{0}$$\\end{document}</tex-math><mml:math id=\"M252\"><mml:msub><mml:mi>F</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq89\"><alternatives><tex-math id=\"M253\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${F}_{1}$$\\end{document}</tex-math><mml:math id=\"M254\"><mml:msub><mml:mi>F</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq90\"><alternatives><tex-math id=\"M255\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\alpha $$\\end{document}</tex-math><mml:math id=\"M256\"><mml:mi>α</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq91\"><alternatives><tex-math id=\"M257\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\beta $$\\end{document}</tex-math><mml:math id=\"M258\"><mml:mi>β</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq92\"><alternatives><tex-math id=\"M259\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${w}_{1}$$\\end{document}</tex-math><mml:math id=\"M260\"><mml:msub><mml:mi>w</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq93\"><alternatives><tex-math id=\"M261\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${w}_{p}$$\\end{document}</tex-math><mml:math id=\"M262\"><mml:msub><mml:mi>w</mml:mi><mml:mi>p</mml:mi></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq94\"><alternatives><tex-math id=\"M263\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${L}_{p}$$\\end{document}</tex-math><mml:math id=\"M264\"><mml:msub><mml:mi>L</mml:mi><mml:mi>p</mml:mi></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq95\"><alternatives><tex-math id=\"M265\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${w}_{a1}$$\\end{document}</tex-math><mml:math id=\"M266\"><mml:msub><mml:mi>w</mml:mi><mml:mrow><mml:mi>a</mml:mi><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq96\"><alternatives><tex-math id=\"M267\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${w}_{a2}$$\\end{document}</tex-math><mml:math id=\"M268\"><mml:msub><mml:mi>w</mml:mi><mml:mrow><mml:mi>a</mml:mi><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq97\"><alternatives><tex-math id=\"M269\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${w}_{a3}$$\\end{document}</tex-math><mml:math id=\"M270\"><mml:msub><mml:mi>w</mml:mi><mml:mrow><mml:mi>a</mml:mi><mml:mn>3</mml:mn></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq98\"><alternatives><tex-math id=\"M271\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${w}_{a4}$$\\end{document}</tex-math><mml:math id=\"M272\"><mml:msub><mml:mi>w</mml:mi><mml:mrow><mml:mi>a</mml:mi><mml:mn>4</mml:mn></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq99\"><alternatives><tex-math id=\"M273\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${S}_{ii}&lt;-15\\text{ dB}$$\\end{document}</tex-math><mml:math id=\"M274\"><mml:mrow><mml:msub><mml:mi>S</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">ii</mml:mi></mml:mrow></mml:msub><mml:mo>&lt;</mml:mo><mml:mo>-</mml:mo><mml:mn>15</mml:mn><mml:mspace width=\"0.333333em\"/><mml:mtext>dB</mml:mtext></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq100\"><alternatives><tex-math id=\"M275\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${S}_{ij}$$\\end{document}</tex-math><mml:math id=\"M276\"><mml:msub><mml:mi>S</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">ij</mml:mi></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq101\"><alternatives><tex-math id=\"M277\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\varepsilon }_{r}=3.48$$\\end{document}</tex-math><mml:math id=\"M278\"><mml:mrow><mml:msub><mml:mi>ε</mml:mi><mml:mi>r</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mn>3.48</mml:mn></mml:mrow></mml:math></alternatives></inline-formula>" ]

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{ "acronym": [], "definition": [] }

[ "<title>Introduction</title>", "<p id=\"Par12\">The cilium, a highly conserved organelle, extends from the cell surface and serves a variety of functions. Structurally, it consists of the ciliary membrane, axoneme, and basal body [##REF##30601682##1##, ##REF##32989303##2##]. As an essential organelle, the cilium is involved in several cellular process, including sensory perception, cellular motility, signaling and communication, cell division and differentiation, and cell-to-cell communication [##REF##30733609##3##]. Additionally, the cilium also contributes to tissue homeostasis and developmental signaling [##REF##37072495##4##–##REF##37642636##7##]. Consequently, aberrations in the structural integrity or functional capacity of cilia are implicated in a spectrum of genetic disorders collectively termed ciliopathies [##REF##37072495##4##, ##REF##32943623##8##]. These conditions present a diverse array of pathologies. Polycystic kidney disease (PKD), for instance, emerges from genetic mutations that trigger the formation of multiple cysts in kidney tissues [##REF##30523303##9##]. Similarly, Bardet-Biedl syndrome (BBS) originates from genetic anomalies and is characterized by a multi-systemic impact. Individuals with BBS often exhibit a combination of symptoms such as progressive vision loss, obesity, polydactyly, and kidney irregularities [##REF##33039432##10##–##REF##37466224##12##].</p>", "<p id=\"Par13\">Ciliary dysfunction and ciliopathies occur due to the absence or malfunctioning of proteins essential for ciliogenesis [##REF##30523303##9##, ##REF##28698599##13##]. These proteins required for ciliogenesis are almost synthesized in the cytoplasm and subsequently transported to the cilium through a specialized process known as intraflagellar transport (IFT) [##REF##34137439##14##]. IFT is a complex and highly regulated microtubule-based transport system that facilitates the movement of proteins along the ciliary axoneme [##REF##26498262##15##]. This system is anchored by two principal protein complexes: the retrograde IFT-A complex and the antegrade IFT-B complex. The bidirectional movement of IFT complexes within cilia relies on distinct motor proteins. Kinesin-2 is responsible for anterograde transport, moving the IFT-B complex toward the ciliary tip, while dynein-2 facilitates retrograde transport, returning the IFT-A complex to the base. This arrangement ensures coordinated bidirectional trafficking along the cilium [##REF##35767872##16##]. The core IFT machinery, together with the motor proteins, mediate the trafficking of cilia structural and signaling proteins.</p>", "<p id=\"Par14\">Furthermore, Recent studies have demonstrated that IFT-independent kinesins, also termed as non-IFT kinesins, which do not directly transport cargos in conjunction with the IFT system, also play important roles in ciliogenesis. For example, mutations in <italic>Kif7</italic>, <italic>Kif9</italic>, <italic>Kif11</italic>, or <italic>Kif19A</italic> causes abnormality in ciliary length as well as ciliopathy-related phenotypes [##REF##36654295##17##–##REF##27690357##22##]. Hence, in this review, we mainly focus on the role and mechanisms of non-IFT kinesins in ciliary formation and highlight their unique features compared to IFT kinesins. By gaining a deeper understanding of these mechanisms, insights can be gained into the modulation of ciliogenesis and can inform the development of new therapeutic strategies for ciliopathies.</p>" ]

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[ "<p id=\"Par1\">Cilia are highly conserved eukaryotic organelles that protrude from the cell surface and are involved in sensory perception, motility, and signaling. Their proper assembly and function rely on the bidirectional intraflagellar transport (IFT) system, which involves motor proteins, including antegrade kinesins and retrograde dynein. Although the role of IFT-mediated transport in cilia has been extensively studied, recent research has highlighted the contribution of IFT-independent kinesins in ciliary processes. The coordinated activities and interplay between IFT kinesins and IFT-independent kinesins are crucial for maintaining ciliary homeostasis. In this comprehensive review, we aim to delve into the specific contributions and mechanisms of action of the IFT-independent kinesins in cilia. By shedding light on their involvement, we hope to gain a more holistic perspective on ciliogenesis and ciliopathies.</p>", "<title>Subject terms</title>" ]

[ "<title>Facts</title>", "<p id=\"Par2\">\n<list list-type=\"bullet\"><list-item><p id=\"Par3\">Cilium assembly involves a specialized protein transport mechanism known as intraflagellar transport (IFT), which is characterized by the bidirectional trafficking of a large protein complex along the microtubules within cilia.</p></list-item><list-item><p id=\"Par4\">The anterograde movement of the IFT is facilitated by members of the kinesin-2 family, typically referred to as IFT-dependent kinesins.</p></list-item><list-item><p id=\"Par5\">IFT-independent kinesins, also termed non-IFT kinesins, refer to a broad category of motor proteins that do not directly transport cargos in conjunction with the IFT system.</p></list-item><list-item><p id=\"Par6\">Non-IFT kinesins have been found located at the basal body or axoneme of cilia and contribute to the maintenance of cilia and ciliary signaling pathways.</p></list-item><list-item><p id=\"Par7\">Mutants of numerous non-IFT kinesins are intricately linked with a spectrum of ciliopathies.</p></list-item></list>\n</p>", "<title>Open questions</title>", "<p id=\"Par8\">\n<list list-type=\"bullet\"><list-item><p id=\"Par9\">What are the specific mechanisms by which non-IFT kinesins coordinate their actions during various stages of ciliogenesis, and how do they contribute to this complex cellular process?</p></list-item><list-item><p id=\"Par10\">What is the physiological and pathological significance of the non-IFT kinesin-mediated ciliary homeostasis in tissue development and human disease?</p></list-item><list-item><p id=\"Par11\">While certain correlations between non-IFT kinesin and ciliopathies have been established, the underlying mechanisms remain elusive. Can non-IFT kinesins be therapeutically targeted for the treatment of ciliopathies?</p></list-item></list>\n</p>", "<title>Cilia: conserved and multifunctional organelles</title>", "<p id=\"Par15\">Cilia are microtubule-based organelles prevalent in a myriad of cell types, playing vital roles in various cellular activities. These organelles can be bifurcated into two main categories: motile cilia and primary cilia. Both types feature an axoneme composed of microtubules. Motile cilia are characterized by the “9 + 2” axoneme arrangement, which consists of nine pairs of doublet microtubules surrounding a central pair. The outer doublets of a motile cilium are linked to dynein arms and radial spokes, which are pivotal in controlling the direction and force of ciliary beating. On the other hand, primary cilia exhibit a “9 + 0” axoneme configuration, lacking the central microtubule pair, dynein arms, and radial spokes, thereby rendering them immotile [##REF##35767872##16##] (Fig. ##FIG##0##1##).</p>", "<p id=\"Par16\">Other components of the cilium include the ciliary membrane, basal body, and transition zone. The ciliary membrane, which is connected to the plasma membrane, envelops the entire axoneme of the cilium. This membrane is enriched with various signaling receptors and ion channels, including those involved in the Hedgehog pathway and Ca²⁺ channels, enabling the cilium to function as an important signaling hub [##REF##37072495##4##, ##REF##37490910##23##]. The basal body, derived from the mother centriole, reverts back to a centriole during ciliary disassembly preceding cell division [##REF##30601682##1##, ##REF##34264337##24##]. The transition zone, located between the basal body and axoneme, regulates the influx and efflux of liquids and proteins, thus establishing the composition within the cilia [##REF##36408840##25##] (Fig. ##FIG##0##1##). The collective contribution of these intricate structures and components determines the architecture and performance of cilia in cellular processes.</p>", "<p id=\"Par17\">Motile cilia are designed for dynamic movement, facilitating the generation of directed fluid flows through coordinated activity. In contrast, primary cilia act as sensitive probes, capturing various signals from the environment, and triggering responses that are crucial for regulating cell division, development, gene activity, migration, and overall cell and tissue morphology. Owing to their extensive presence in mammalian organisms and critical role in signaling pathways, the same ciliary gene mutations or abnormal expression has the potential to cause varying manifestations of ciliary abnormalities and inconsistent symptoms of ciliopathies [##REF##32943623##8##, ##REF##37963851##26##, ##REF##36796536##27##]. The variability in symptoms can result from factors such as genetic background, environmental influences, the extent of gene mutation or dysregulation, and the specific cell types or tissues affected. Nonetheless, the specific mechanisms underlying ciliopathies remain elusive, leaving ample scope for discovery in this field.</p>", "<title>IFT: protein translocation machinery in cilia</title>", "<p id=\"Par18\">During the growth of the cilium, the axoneme is assembled by the addition of new axonemal subunits to its distal tip. However, cilia lack the machinery that is necessary for protein synthesis, the site of assembly of the axoneme is far removed from the cell body, where the building materials are synthesized. The cell has solved this problem for the delivery of new axonemal building blocks to the site of axonemal assembly by means of IFT [##REF##8516294##28##, ##REF##36654447##29##]. During IFT, the non-membrane-bound particles are moved along the axonemal doublet microtubules, and beneath the ciliary membrane. The anterograde IFT-B particles moving from the ciliary base to the tip for the proper assembly and maintenance of ciliary axoneme and membrane. At the ciliary tip, the building blocks are released, and IFT-B particles are then transported back by IFT-A to the ciliary base [##REF##35767872##16##]. This IFT process is well conserved and required for the assembly of most cilia and eukaryotic flagella. The movement of these IFT particles are driven by motor proteins, including the anterograde kinesin and the retrograde dynein proteins, to move up and down the cilium [##REF##35901159##30##].</p>", "<title>Classification and characterization of kinesins</title>", "<p id=\"Par19\">Kinesins constitute a superfamily of proteins with 15 members, classified into 14 subclasses (kinesin 1 to kinesin 14B) through phylogenetic analysis [##REF##11416179##31##] (Fig. ##FIG##1##2A##). Each member of the kinesin superfamily (KIF) possesses a common motor domain, which utilizes the chemical energy from ATP hydrolysis to initiate movement along microtubules. These kinesins are generally divided into three categories based on the location of the motor domain: N-kinesins carry a motor domain in the amino-terminal region, M-kinesins have their motor domain in the middle, and C-kinesins contain it in the carboxy-terminal region. Typically, N-kinesins show directed motility towards the plus (rapidly growing) end of the microtubule, while C-kinesins move towards the minus (slowly growing) end. In contrast, M-kinesins destabilize microtubules instead of migrating along them (Fig. ##FIG##1##2B##). However, some kinesin-8 (N-kinesin) and kinesin-14 (C-kinesin) motors can both traverse and depolymerize microtubules [##REF##17346968##32##–##REF##16906148##34##]. Furthermore, certain kinesin-5 and kinesin-14 families can cross-link and slide adjacent microtubules, adding complexity to these generalizations [##REF##18984586##35##, ##REF##16892050##36##].</p>", "<p id=\"Par20\">In addition to the motor domain, many kinesins encompass a neck linker region, a stalk region, and a tail domain [##REF##2522351##37##]. The neck linker region, connected to the motor domain, acts as a flexible hinge and assists in transmitting conformational changes during the ATP hydrolysis cycle. The stalk region ensures stability, connecting the motor domain to the cargo-binding tail domain. This tail domain interacts with specific cargo molecules, enabling kinesins to transport various cargoes within cells. Further, coiled-coil segments for oligomerization are present in many kinesins, making most kinesin motors homodimers. For instance, kinesin-1 motors are heterotetramers comprising two subunits: kinesin heavy chain and kinesin light chain; kinesin-2 motors split into two subfamilies, either heterotrimers (KIF3A-KIF3B-kinesin associated protein) or homodimers (KIF17); kinesin-3 motors may exist as monomers or homodimers; and the kinesin-5 family consists of homotetrameric motors (Fig. ##FIG##1##2A##) [##REF##28284467##38##, ##REF##19773780##39##]. These kinesins, which all belong to the N-kinesins, actively transport cargoes directionally towards the plus-end of microtubules that form cylindrical polymers of 13 protofilaments. Despite the highly conserved nature of their motor domains, their cellular functions vary due to differences in these structural components.</p>", "<p id=\"Par21\">Kinesins play a plethora of roles in a microtubule-dependent manner. One of their primary functions is vesicle transport, as kinesins assist in the movement of vesicles containing important molecules and organelles to specific locations within cells [##REF##19773780##39##] (Fig. ##FIG##2##3A##). This transport process is crucial for maintaining cellular functionality and assuring proper distribution of essential components. Another pivotal role of kinesins is macromolecule transport; these proteins aid in the movement of large molecules, such as proteins and nucleic acids, within the cell [##REF##19268344##40##, ##UREF##1##41##]. By facilitating the transport of these macromolecules, kinesins contribute to key cellular processes like gene expression and cellular signaling. Kinesins also participate in cell division processes tied to mitosis and meiosis, participating in chromosome segregation to ensure accurate distribution of genetic material to daughter cells [##REF##30300593##42##].</p>", "<p id=\"Par22\">It’s noteworthy that kinesins, specifically the known IFT kinesin (kinesin-2), contributes to the dynamic nature of cilia and ensure their proper functioning [##REF##9281580##43##]. By transporting cargoes and signaling molecules and receptors to and from the ciliary membrane, kinesin-2 influences the extension of cilia and the modulation of ciliary signaling pathways [##REF##21642982##44##] (Fig. ##FIG##2##3B##). Accumulated evidence suggests that a distinctively longer neck linker region in the kinesin-2, which includes an additional three amino acid residues (Asp-Ala-Leu, DAL) at the C-terminus prior to helix α7 [##REF##29444824##45##]. This characteristic underpins the mechanistic foundation for its shorter run lengths, a trait that seems to be adapted to its specific role of transporting ciliary proteins along the axoneme of cilia.</p>", "<p id=\"Par23\">Moreover, recent works have underscored the significant involvement of non-IFT kinesins in the assembly and maintenance of cilia. While the mechanisms through which non-kinesins contribute to ciliary homeostasis remain an active area of research, preliminary findings suggest that these kinesins may be involved in various ciliary processes, such as the regulation of ciliary length, the transport of specific cargoes, or the modulation of ciliary signaling pathways. Therefore, further research into the roles of these non-IFT kinesins may yield new insights into the molecular mechanisms underlying ciliary function and dysfunction, and ultimately lead to the development of novel therapeutic strategies for ciliopathies.</p>", "<title>IFT kinesins</title>", "<p id=\"Par24\">There are two types of anterograde IFT motors, namely the heterotrimeric kinesin-2 and the homodimeric OSM-3 or KIF17. These kinesins belong to the kinesin-2 family, also known as IFT kinesins. The heterotrimeric kinesin-2 complex consists of KIF3A/KIF3B/KIFAP3 and can move at a rate of 0.2-2.4 μm/s, depending on the species and ciliary type [##REF##9487132##46##]. On the other hand, OSM-3 or KIF17 acts as a homodimer and moves approximately 1.3 μm/s along the ciliary axoneme [##REF##18304522##47##, ##REF##8232586##48##] (Fig. ##FIG##2##3B##). The biogenesis of cilia requires the anterograde IFT driven by kinesin-2, as it is responsible for transporting IFT trains. These trains are believed to deliver axoneme precursors to the tip of the axoneme, where they are incorporated, and to organize and move ciliary membrane-associated signaling complexes. For example, in the green alga <italic>Chlamydomonas</italic>, inactivation of the FLA10 subunit of heterotrimeric kinesin-2 using conditional mutants leads to a gradual halt in IFT and defects in the assembly or maintenance of motile cilia [##REF##12692152##49##]. This observation supports the hypothesis that heterotrimeric kinesin-2 drives the anterograde transport of IFT trains.</p>", "<p id=\"Par25\">The role of kinesin-2 motors in the assembly of sensory cilia in <italic>Caenorhabditis elegans</italic> amphid channels differs and presents a more intricate process [##REF##15489852##50##]. The axonemes of these cilia possess a bipartite structure characterized by a core comprising nine doublet microtubules known as the middle segment. From this middle segment, nine singlet microtubules extend to form the distal segment, which plays a critical role in certain forms of chemosensory signaling. The assembly of these axonemes involves a unique and unexpected collaboration between the heterotrimeric kinesin-2, kinesin-II, and the homodimeric kinesin-2, OSM-3. In this collaboration, the middle-segment assembly involves both motors transporting IFT trains along the middle segment, while the distal-segment assembly depends only on OSM-3 transporting IFT trains along the distal segment. Therefore, in wild-type animals, kinesin-II and OSM-3 both contribute redundantly to the assembly of the middle segment, while OSM-3 alone is responsible for constructing the distal segment.</p>", "<p id=\"Par26\">In contrast, the cilia found on olfactory receptor neurons in <italic>Drosophila</italic> also exhibit a bipartite organization and develop through a different two-step pathway [##REF##21233284##51##]. However, in this case, heterotrimeric kinesin-2 alone appears to be sufficient for the assembly of the entire axoneme. In mice, heterotrimeric kinesin-2 may have additional ciliogenic functions beyond driving IFT that cannot be compensated for KIF17, as it is required for the proper organization of centrioles, which form the basal body of the cilium [##REF##23386061##52##]. Additionally, in zebrafish, the absence of KIF17 results in a loss or disorganization of outer segments in retinal photoreceptors, while it does not affect the formation of motile cilia in the pronephros [##REF##22308397##53##]. These observations indicate that diverse mechanisms for employing kinesin-2 motors have evolved to facilitate cilium assembly.</p>", "<title>Non-IFT kinesins</title>", "<p id=\"Par27\">Beyond the well-known IFT kinesins, recent works have unveiled the involvement of non-IFT kinesins in ciliary homeostasis maintenance. These kinesins were found located at the basal body or axoneme of cilia and contribute to regulate ciliary length and the ciliary signaling pathways (Fig. ##FIG##3##4##). Such roles could be expected for the members of the kinesin-13 and kinesin-4 subfamily, known to have microtubule depolymerizing activities, therefore, negatively controlling the length of axonemal microtubules and the ciliary dependent Hedgehog signaling pathway [##REF##19933103##20##, ##REF##19666503##21##, ##REF##21620453##54##]. In addition to the depolymerizing kinesins, knockout of certain kinesin genes has identified several new kinesin members involved in diverse function at cilia.</p>", "<title>Kinesin-1 (KIF5B)</title>", "<p id=\"Par28\">Kinesin-1, the first identified plus-end-directed microtubule motor, is involved in various cellular processes through its interactions with different cargoes such as vesicles, organelles, mRNAs, and multiprotein complexes [##REF##3926325##55##, ##REF##1689058##56##]. Kinesin-1 is a heterotetramer composed of two heavy chains and two light chains. The microtubule binding motor region is found in the N-terminus of the heavy chain, which can be encoded by three different genes (<italic>Kif5A, Kif5B, Kif5C</italic>). KIF5A and KIF5C are expressed exclusively in neurons, while KIF5B is ubiquitous. Each heavy chain dimer associates with two copies of KLC1 or KLC2, which are expressed in most cell types [##REF##19773780##39##].</p>", "<p id=\"Par29\">Studies have indicated that KIF5B and KLC1 localized to the basal body and play an inhibitory role in ciliary extension, as depletion of these proteins leads to abnormally elongated cilia. Knockdown of KIF5C alone does not significantly affect ciliary length, and KIF5A is not highly expressed in hTERT-RPE cells, a cell line known to induce ciliary formation in vitro. Furthermore, genetic interaction studies suggest that the nuclear/cytoplasmic distribution of CCDC28B, a protein associated with Bardet-Biedl syndrome, is influenced by KIF5B, as targeting KIF5B leads to nuclear accumulation of CCDC28B [##REF##29445114##57##].</p>", "<title>Kinesin-3 (KIF13B/KLP-6)</title>", "<p id=\"Par30\">Kinesin-3 family members are plus-end directed motors involved in vesicle transport and endocytosis. Among them, KIF13B (also known as guanylate kinase-associated kinesin or GAKIN) is implicated in the regulation of neuronal polarity, axon formation and myelination, Golgi to plasma membrane trafficking, germ cell migration, and planar cell polarity signaling [##REF##19773780##39##].</p>", "<p id=\"Par31\">Recent studies have shown that KIF13B undergoes bursts of IFT-like bidirectional movement within primary cilia, and its depletion leads to ciliary accumulation of the cholesterol-binding membrane protein CAV1 and impaired Hedgehog signaling [##REF##28134340##58##, ##REF##35403186##59##]. Additionally, the velocities of anterograde and retrograde intraciliary movement of KIF13B are similar to those of IFT, but its movement within the cilium requires its own motor domain. Interestingly, the homolog of KIF13B, KLP-6, has been observed to move in cilia of <italic>Caenorhabditis elegans</italic> and modulate the velocities of IFT and kinesin-2 motors. KLP-6 acts as a positive regulator of ciliary length extension, as its accumulation in the cephalic male cilia promotes elongation of cilia [##REF##21757353##60##]. This demonstrates the modulation of general kinesin-2-driven IFT processes by kinesin-3 in the cilia of <italic>Caenorhabditis elegans</italic> male neurons.</p>", "<title>Kinesin-4 (KIF7/KIF27)</title>", "<p id=\"Par32\">Kinesin-4 is a remarkable motor protein due to its unique ability to depolymerize microtubules [##REF##15001780##61##]. It plays critical roles in cell division, microtubule organization, and signal transduction. Among its members, KIF7 serves as a conserved regulator of the Hedgehog signaling pathway. This kinesin facilitates the transmission of signals from the membrane protein Smoothened to the Gli/Gi transcription factors. A recent finding suggests that KIF7 regulates the length of the microtubule plus end and promotes the precise localization and proper regulation of Gli and the inhibitory factor Sufu at the tip of primary cilia. Furthermore, KIF7 mutations cause primary cilia abnormalities, including excessive length, twisting, and instability. These defects lead to the formation of ectopic tip-like compartments where Gli-Sufu complexes become localized and inappropriately activated in the absence of the sonic hedgehog ligand [##REF##19666503##21##].</p>", "<p id=\"Par33\">Another member of the kinesin-4 family, KIF27, also plays a role in cilia-related processes. KIF27, the closest mammalian homologue of KIF7, is found in motile cilia and share the ability of KIF7 to regulate axonemal microtubule dynamics. Specifically, KIF27 contributes to the assembly of the central pair of microtubules in “9 + 2” motile cilia through its interaction with Fused [##REF##19305393##62##]. Mice with defective KIF27 exhibit suppurative inflammatory responses in the nasal passages and middle ear, as well as hydrocephalus [##REF##21746835##63##].</p>", "<title>Kinesin-5 (KIF11)</title>", "<p id=\"Par34\">Kinesin-5, also known as kinesin family member 11 (KIF11) or Eg5, plays crucial roles in the formation and maintenance of bipolar spindle orientation during cell division. These activities are facilitated by its unique antiparallel tetrameric structure, which enables the motor protein to crosslink and slide adjacent microtubules [##REF##30220581##64##]. Apart from its mitotic functions, KIF11 has also been found to have non-mitotic roles, including protein transport from the Golgi complex to the cell surface, regulation of axonal growth and branching, and ciliary formation [##REF##36654295##17##, ##REF##32811879##18##, ##REF##17846176##65##, ##REF##23857769##66##].</p>", "<p id=\"Par35\">Our previous research has shown that KIF11 localizes to the basal body of primary cilia in various cell types. Knockdown of KIF11 expression in RPE1 cells leads to a decrease in ciliary length and number and perturbs Hedgehog signaling [##REF##36654295##17##]. Another study further supports the non-mitotic role of KIF11 in cilia, demonstrating that KIF11 plays a critical role in regulating ciliary behavior [##REF##32811879##18##]. Moreover, KIF11 expression is significantly higher in glioblastoma cells compared to normal cells, and there is also an overexpression of Hedgehog signaling in glioblastoma [##REF##26355032##67##]. These suggest that KIF11-mediated ciliogenesis may contribute to the overactivation of Hedgehog signaling in glioblastoma cancer cells, which holds potential implications for future cancer treatment strategies.</p>", "<title>Kinesin-8 (KIF19A)</title>", "<p id=\"Par36\">Kinesin-8 members possess remarkable capabilities of both walking towards the plus-ends of microtubules and depolymerizing these ends upon arrival, thereby exerting control over microtubule length [##REF##16906145##33##]. These motor proteins are observed on cytoplasmic microtubules during interphase and near kinetochores during cell division. Disruption of their function during mitosis leads to the formation of excessively long spindle microtubules, resulting in aberrant chromosomal segregation. This observation strongly supports the notion that precise regulation of microtubule length by kinesin-8 motors is crucial for accurate cell division.</p>", "<p id=\"Par37\">Among these motors, KIF19A has been extensively studied for its role in regulating ciliary length by depolymerizing microtubules at the tips of cilia. Depletion of KIF19A in mice results in the manifestation of ciliopathy phenotypes, including hydrocephalus and female infertility, caused by the presence of abnormally elongated cilia that are unable to generate proper fluid flow [##REF##23168168##68##]. Recent research has indirectly demonstrated that KIF19A plays a pivotal role in mediating ciliary length in mammals. For instance, depletion of adenylate cyclase 6 in mice leads to elongated cilia in airway epithelial cells, primarily due to decreased KIF19A protein levels in the cilia resulting from its degradation through autophagy [##REF##32683324##69##]. These studies shed light not only on the genetic regulation of cilia by KIF19A but also on the mechanisms underlying the regulation or control of KIF19A itself.</p>", "<title>Kinesin-9 (KIF9A/KIF9B)</title>", "<p id=\"Par38\">Kinesin-9 members are motor proteins that are exclusively expressed in tissues containing motile cilia or flagella, such as the testis, brain, and lung, as well as in flagellated microorganisms like <italic>Giardia, Leishmania, and Chlamydomonas</italic>. These kinesin motors primarily move towards the plus end of microtubules. The kinesin-9 family consists of two subfamilies: KIF9A, which includes <italic>Chlamydomonas reinhardtii</italic> KLP1, and KIF9B, which includes human KIF6. KLP1 is localized to the central pair microtubules of the axoneme and plays a role in influencing flagellar motility [##REF##8207060##70##]. Disruption in KLP1 function leads to flagella that beat slowly or become paralyzed.</p>", "<p id=\"Par39\">Recent studies have highlighted the importance of KIF9 in ciliary motility. KIF9 is highly conserved across evolutionary species and is considered the vertebrate ortholog of KLP1. It has been reported that KIF9 localizes to the axoneme of sperm flagella and cilia in multiciliated cells, such as those found in <italic>Xenopus</italic> and human airways. KIF9 is responsible for maintaining proper ciliary motility and the integrity of the distal end of the axoneme [##REF##35531639##19##]. In contrast, KIF6 is localized to both the axoneme and basal body of multiciliated cells. It is not only essential for ciliary motility but also plays a specific role in the formation of cilia in ependymal cells. Studies have shown that mutations in <italic>Kif6</italic> can lead to neurodevelopmental defects and intellectual disability in humans [##UREF##2##71##].</p>", "<title>Kinesin-13 (KIF24/KIF2A)</title>", "<p id=\"Par40\">The kinesin-13 family specifically contains M-kinesins. Unlike conventional kinesins, kinesin-13 proteins do not walk along microtubules but instead depolymerize them using ATP. This depolymerizing activity of kinesin-13 proteins operates in a range of physiological contexts such as spindle assembly, chromosome segregation, and axonal growth.</p>", "<p id=\"Par41\">Early studies have shown that kinesin-13 members in <italic>Giardia, Leishmania, and Chlamydomonas</italic> are localized to axonemes and play a role in regulating the length of flagellar [##REF##17433682##72##–##REF##31855176##74##]. However, in the mammalians, the kinesin-13 members consist of KIF2A, KIF2B, KIF2C/MACK, and KIF24. KIF24 has been reported to block ciliogenesis by recruiting CP110 at the mother centrioles and remodeling centriole microtubules through its microtubule-depolymerizing activity [##REF##34264337##24##, ##REF##21620453##54##]. Moreover, research has demonstrated that even in cycling cells, knockdown of KIF24 by small interfering RNA leads to inappropriate ciliogenesis. Another kinesin-13 member, KIF2A, has been shown to have the ability to disassemble primary cilia by depolymerizing microtubules in response to growth signals, with its activity controlled by the PLK1 [##REF##25660017##75##].</p>", "<title>Emerging roles of non-IFT kinesins in ciliopathies</title>", "<p id=\"Par42\">Considering the pivotal contribution of non-IFT kinesins to the maintenance of ciliary homeostasis, it is unsurprising that these kinesin motors are intricately linked with a spectrum of ciliopathies. Microcephaly, a neurological malformation that characterized by an abnormal small head circumference, is one of the most frequently associated clinical signs [##REF##29799801##76##]. Notably, mutations in the genes encoding kinesin motors-KIF1B, KIF14, KIF16B, KIF11, KIF10, KIF15, and KIF2A-have been identified in numerous patients with microcephaly [##REF##28892560##77##–##REF##25115524##79##].</p>", "<p id=\"Par43\">Non-IFT kinesins are also involved in other neuronal disorders related to ciliopathies. For instance, <italic>KIF4A</italic>, <italic>KIF6</italic> and <italic>KIF7</italic> has ascended to prominence as a putative gene of interest in the etiology of hydrocephalus [##REF##30679815##80##–##REF##30475797##82##]. Investigations into the developmental biology of KIF26A underscore its potential role in neural system development, as knockout mice models reveal critical deficits such as enteric nerve hypoplasia [##REF##19914172##83##, ##REF##30208315##84##]. The proteins KIF1A and KIF5 are of paramount importance for higher-order brain functionalities, namely learning and memory, exerting influence through the modulation of synaptic transmission [##REF##25724902##85##]. Peripheral neuropathies represent yet another sphere in which KIF1A and KIF1B demonstrate a genetic association [##REF##14595441##86##].</p>", "<p id=\"Par44\">Transgenic models, particularly mice with targeted deletions of KIF genes, have surfaced with a spectrum of ciliopathy syndromes. These include kidney disorders resultant from KIF26B mutations [##REF##25168025##87##], and KIF19A depletion leading to female infertility [##REF##27690357##22##]. Complementing these insights, recent discoveries have delineated biallelic variants of <italic>KIF24</italic> as pathogenic factors in skeletal ciliopathies, encompassing variants such as acromesomelic skeletal dysplasia and spondylometaphyseal dysplasia [##UREF##3##88##]. Furthermore, genetic variants in <italic>KIF1B</italic>, <italic>KIF21B</italic>, and <italic>KIF5A</italic> have been associated with increased vulnerability to multiple sclerosis [##REF##18997785##89##–##REF##29342275##91##]. Collectively, these evidences reinforce the notion that non-IFT kinesins are crucial to ciliary function and, when impaired, to the pathogenesis of a multitude of abnormalities related to ciliopathies.</p>", "<title>Concluding remarks</title>", "<p id=\"Par45\">The study of kinesins and their roles in cilia biology has undergone significant advancements over recent years, revealing the intricate mechanisms by which these motor proteins contribute to ciliary assembly, maintenance, and function. In this review, we have discussed the emerging roles of non-IFT kinesins in cilia-related processes, providing insights into their diverse functions and their implications for cellular homeostasis and human health. While IFT kinesins have long been recognized as central players in cilia assembly and maintenance, the discovery of non-IFT kinesins’ involvement adds a layer of complexity to our understanding of ciliary activities. Emerging evidence compellingly indicates that the various kinesin families are interdependent, collaboratively maintaining ciliary homeostasis. The observed interplay between IFT-associated and non-IFT kinesin proteins poses fascinating questions regarding their mechanisms of communication and cooperation. This teamwork is crucial for the modulation of the ciliary length, the precision of cargo transport, and the nuanced modulation of signaling pathways. Future research aimed at deciphering the crosstalk between these kinesin families will provide deeper insights into the mechanisms governing cilia biology.</p>", "<p id=\"Par46\">The identification of non-IFT kinesins as critical players in cilia-related processes has important implications for our understanding of ciliopathies. Mutations in various ciliary components, including kinesins, have been linked to the development of ciliopathies, underscoring the significance of these motor proteins in maintaining cellular homeostasis [##REF##32943623##8##, ##REF##28698599##13##]. For example, the presence of KIF11 in the connecting cilium of photoreceptors, and the identification of <italic>KIF11</italic> mutations in patients with retinal diseases such as MLCRD (microcephaly, lymphedema, and chorioretinal dysplasia), CDMMR (chorioretinal dysplasia, microcephaly, and mental retardation), and FEVR (familial exudative vitreoretinopathy) suggest that KIF11 may play a vital role in the pathological processes of these conditions by mediating photoreceptor ciliary homeostasis [##REF##25115524##79##, ##REF##22284827##92##, ##REF##25124931##93##]. Elucidating the roles of non-IFT kinesins in cilia biology may offer valuable insights into the molecular mechanisms underlying ciliopathy pathogenesis. Furthermore, as multiple members of the kinesin family are continuously being identified as potential targets for treating various diseases, including cancer [##REF##22825217##94##], exploring cilia and ciliary proteins as a strategy for addressing ciliopathies holds great promise [##REF##32275885##95##, ##REF##35619548##96##].</p>", "<p id=\"Par47\">As the field of kinesin research continues to advance, several intriguing questions and avenues for future investigation arise. For example, ciliary homeostasis represents a complex and finely tuned regulatory process encompassing assembly, disassembly, and maintenance phases [##REF##37833188##97##], yet the specific contributions of different kinesin proteins within this balance are not well understood. Moreover, the mechanisms by which multiple members of this large kinesin family work in concert remain elusive. Most importantly, the physiological and pathological significance of kinesin-mediated ciliary homeostasis in development and human disease remains unclear. These knowledge gaps present a compelling case for future research to unravel the intricate orchestra of kinesin activities that maintain ciliary homeostasis and to decipher their broader implications in health and disease.</p>", "<p id=\"Par48\">The core IFT-dependent machinery is crucial for the transport of ciliary and signaling proteins. However, in certain ciliated protists that are devoid of genes encoding for IFT components, and in conjunction with some metazoan spermatozoa, they use IFT-independent mechanisms for assembling axonemes that exposed to the cytosol [##REF##26654377##98##, ##REF##23891117##99##]. During this process, all or portion of this axoneme, at least temporarily, is not enveloped by plasma membrane but is instead exposed to the cytoplasm. This distinct IFT-independent ciliogenesis pathway permits a robust exchange with cytosolic proteins, and consequently, IFT is presumably excluded from playing any direct part in such cytosolic ciliogenesis events. This unconventional pathway delivers profound insights into the molecular machinations that govern non-IFT kinesins in maintaining ciliary homeostasis. For example, the basal body-localized KIF11, a newly-identified pivotal protein in ciliogenesis, which strikingly lacks inherent motor activity but is vitally influential in ciliary length. To date, the molecular mechanisms underpinning the role of KIF11 are unclear. With this context, we postulate that KIF11 may harness a mechanism resonant with the IFT-independent ciliogenesis pathway. Such a role would likely entail exploiting the cytoplasm’s microtubule framework to effectuate the translocation and assembly of requisite constituents for ciliary construction, providing a greater understanding of this kinesin protein’s involvement in the complex narrative of cilia formation and maintenance.</p>", "<p id=\"Par49\">The development of advanced imaging techniques will enable researchers to visualize kinesin behavior within cilia with unprecedented detail, offering new insights into their functions. Continued functional studies in model organisms and human genetics will provide valuable information about the roles of non-IFT kinesins in various biological contexts. In conclusion, the emerging roles of non-IFT kinesins in cilia biology have broadened our understanding of ciliary dynamics and cellular function. These motor proteins contribute to a range of processes within cilia, including assembly, length regulation, cargo transport, and signaling. By shedding light on the specific ciliary activities of non-IFT kinesins, their implications for ciliopathies, and their diverse functions beyond cilia, this review emphasizes the intricate and multifaceted nature of kinesin-mediated regulation in cellular processes. As research in this field progresses, we anticipate that further insights into the roles and mechanisms of non-IFT kinesins will continue to shape our understanding of cellular biology and human health.</p>" ]

[ "<title>Author contributions</title>", "<p>JR and LL drafted the original manuscript and prepared the figures, JR revised the manuscript. All authors have read and agreed the final version of the manuscript.</p>", "<title>Funding</title>", "<p id=\"Par50\">This work was supported by grants from the National Natural Science Foundation of China (32241014 and 32170687).</p>", "<title>Data availability</title>", "<p>Data sharing is not applicable, as no datasets were generated or analyzed during this study.</p>", "<title>Competing interests</title>", "<p id=\"Par51\">The authors declare no competing interests.</p>" ]

[ "<fig id=\"Fig1\"><label>Fig. 1</label><caption><title>Structures of motile and primary cilia.</title><p>Cilia typically have three main components: the ciliary membrane, the axoneme, and the basal body. The ciliary membrane contains numerous signaling receptors, endowing cilia with signaling functions. Axonemes are composed of doublets arranged in a “9 + 0” pattern in primary cilia. In motile cilia, the axonemes are arranged in a “9 + 2” pattern and typically include additional structures such as the central pair, radial spokes, and axonemal dynein arms, which are essential for ciliary motility. Situated between the axoneme and the basal body is the transition zone, which are connected to the ciliary membrane via Y-shaped structures.</p></caption></fig>", "<fig id=\"Fig2\"><label>Fig. 2</label><caption><title>The kinesin superfamily.</title><p><bold>A</bold> All kinesins have a motor domain head (see the figure; dark blue), which contributes to ATP hydrolysis for powering the movement along microtubules. Kinesins also contain the neck linker region, tail region and stalk region with coiled-coil segments for oligomerization. Kinesin-1 motors are heterotetramers consisting of the heavy chain KHC and light chain KLC. Kinesin-2 motors have two types: heterotrimers and homodimers. Kinesin-3 motors can also exist as homodimers or monomers. Kinesin-5 motors are heterotetramers. Except for these four kinesins, all other kinesins are homodimers. <bold>B</bold> Kinesins are mainly divided into three types based on the location of their motor domain: C-kinesins, N-kinesins, and M-kinesins. Generally, C-kinesins move towards the minus end of microtubules, N-kinesins move towards the plus end and M-kinesins destabilize microtubules.</p></caption></fig>", "<fig id=\"Fig3\"><label>Fig. 3</label><caption><title>Intracellular transport mediated by kinesins.</title><p><bold>A</bold> The process of intracellular transport begins when kinesins bind to vesicles containing organelles or protein complexes, such as endosomes and lysosomes. Kinesins move along microtubules to transport vesicles from the Golgi to the endoplasmic reticulum (ER), as well as from the Golgi network to the plasma membrane. They also facilitate the transportation of lysosomes and endosomes. <bold>B</bold> Kinesin-2 members KIF3A/KIF3B and KIF17 bind IFT-B to transport cargos from the base to the tip of cilia. And then dynein-2 is activated by IFT-A for the retrograde transport.</p></caption></fig>", "<fig id=\"Fig4\"><label>Fig. 4</label><caption><title>Non-IFT kinesins in cilia.</title><p>Non-IFT kinesins are distributed at various strategic locations within cilia, such as the axoneme and basal body. The distinct compartments of non-IFT kinesins highlight their diverse roles in the regulation of ciliary structure and function.</p></caption></fig>" ]

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[ "<fn-group><fn><p>Edited by Professor Anastasis Stephanou</p></fn><fn><p><bold>Publisher’s note</bold> Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.</p></fn></fn-group>" ]

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[ "<title>Introduction</title>", "<p id=\"Par2\">Understanding the changing characteristics of floods^{##UREF##0##1##} and their relationship with the physical causative mechanisms is a prerequisite for developing effective flood management strategies^{##UREF##1##2##,##UREF##2##3##}. Physical causes of short term variability and long term changes in extreme floods vary between the catchments^{##REF##36204013##4##,##UREF##3##5##}. Investigating the relative importance of key drivers of floods is therefore critical for improving the scientific understanding of catchment dynamics. Changing characteristics and causes of floods are well documented across many catchments in Europe^{##UREF##2##3##,##UREF##4##6##–##UREF##7##10##} and United States^{##UREF##1##2##} with extreme rainfall, soil moisture excess and snowmelt as potential drivers. Influence of rainfall and soil moisture extremes on flood peaks is evaluated in Australian^{##UREF##8##11##,##UREF##9##12##} and African^{##UREF##10##13##} catchments. However, there is no systematic study on identifying the importance of flood generating mechanisms in Indian catchments.</p>", "<p id=\"Par3\">There is significant evidence that rainfall extremes are intensifying in response to warming^{##REF##18687921##14##,##REF##26312211##15##}, whereas the evidence for increase in floods remains elusive^{##UREF##11##16##}. Increasing trends in rainfall extremes^{##REF##17138899##17##–##UREF##13##20##} and increase in the flood risk^{##UREF##14##21##,##REF##31913322##22##} are reported in India. However, studies on understanding the physical causes of such trends remain limited^{##UREF##15##23##–##REF##28442746##25##}. A recent study identifies multiday rainfall as a prominent driver of floods in India by examining the soil moisture conditions and rainfall before high flow events simulated using the Variable Infiltration Capacity (VIC) model^{##UREF##17##26##}. Authors adopt an event-based approach to identify the flood drivers but the analysis does not consider the role of groundwater in triggering floods. Floods have serious impacts on agriculture, infrastructure, water resources systems and reservoir operations. Therefore, a detailed assessment is required to classify the flood generating mechanisms in Indian catchments. Identifying the hydrological processes which trigger floods will not only improve our understanding of flood mechanisms in Indian catchments but it will also provide a foundation for robust flood risk assessment.</p>", "<p id=\"Par4\">Rainfall and subsurface antecedent wetness conditions prior to the flood event are the primary drivers of floods in India^{##UREF##16##24##,##UREF##17##26##} as snowmelt triggers floods in a few catchments^{##UREF##18##27##,##UREF##19##28##}. The role of soil moisture in driving river floods is widely recognized in literature^{##UREF##1##2##,##UREF##2##3##,##UREF##8##11##–##UREF##10##13##}, whereas groundwater is not considered in flood related studies. Groundwater plays an important role in maintaining the flow of rivers, but its influence on floods is poorly understood^{##UREF##20##29##}. Groundwater well observations are sparse and may not represent the influence of water storage in deeper saturated zone on floods. Therefore, baseflow is used to understand the role of groundwater storage in controlling floods in Peninsular India.</p>", "<p id=\"Par5\">Annual maximum flows are the largest floods experienced in a year and often represent the most disastrous flood event. Trends in the annual flood magnitudes are estimated to understand the changes in water availability. Impact of reservoirs is also examined in this study to understand the influence of flow regulations in Peninsular India. The natural flow regime of rivers is largely altered due to boom in dam construction across the world during the last century^{##UREF##21##30##}. The regulation of rivers with reservoirs for different purposes such as irrigation, hydropower, water supply and flood control significantly alters the downstream flow by storing and releasing water with certain operation rules. Flow regulations affect the magnitude, frequency and timing of downstream high and low flows^{##UREF##22##31##–##UREF##25##34##}. Therefore, it is important to study the impact of reservoirs on the flow regime using downstream streamflow records. This study presents the analysis of pre- and post-dam construction high flow changes to understand the influence of reservoirs on annual flood characteristics: peak, volume and duration. Understanding the extent to which the reservoir regulations affected the flood characteristics is crucial for designing better reservoir operation rules in Peninsular catchments.</p>", "<p id=\"Par6\">The importance of rainfall, soil moisture and baseflow for generating floods in Peninsular India is investigated first using Kendall’s rank correlation coefficient and Pearson correlation coefficient for different lags at annual timescale. However, high flows with a little lower magnitude than the annual maximum flow can occur in the same year and in the same catchment. In addition, drivers of these floods of different magnitudes can be quite different from annual maximum floods. Event-based approach which extracts the sample using peaks-over-threshold is more robust to assess the importance of flood generating mechanisms^{##UREF##10##13##,##UREF##17##26##}. Therefore, further investigation is performed using Event Coincidence Analysis (ECA), which tests for possible causal influence of flood drivers in triggering the flood events of little lower magnitude than annual floods. Triggering effect is evaluated using the statistics based on trigger coincidences for the condition that extreme rainfall, soil moisture and baseflow are followed by the flood events. ECA results are used to find the dominant driver which has a higher influence on the flood events in Peninsular catchments.</p>" ]

[ "<title>Methods</title>", "<title>Datasets</title>", "<p id=\"Par17\">Daily streamflow time series for 70 catchments in six major river basins of Peninsular India (Fig. ##FIG##0##1##a) are obtained from India Water Resources Information System (<ext-link ext-link-type=\"uri\" xlink:href=\"https://indiawris.gov.in/wris/\">https://indiawris.gov.in/wris/</ext-link>). Gaps in the daily streamflow records are filled using time series methods for synthesizing missing streamflow records^{##UREF##32##41##,##UREF##33##42##}. Daily high resolution rainfall^{##UREF##34##43##} dataset on a grid size of 0.25° is obtained from India Meteorological Department (IMD). European Space Agency Climate Change Initiative (ESA CCI) soil moisture^{##UREF##35##44##–##UREF##37##46##} with daily temporal and 0.25° spatial resolution is used in this study. High resolution Aridity index values are obtained from Global Aridity Index and Potential Evapotranspiration Database—Version 3 (Global-AI_PET_v3)^{##REF##35840601##47##}. Information on location, year of construction, capacity and purpose of reservoirs is collected from Central Water Commission (CWC) report on National Register of Large Dams^{##UREF##38##48##}. Catchments are delineated in Quantum Geographic Information System (QGIS) using Digital Elevation Model (DEM) obtained from Shuttle Radar Topographic Mission (SRTM) at 30 m spatial resolution (<ext-link ext-link-type=\"uri\" xlink:href=\"https://srtm.csi.cgiar.org/srtmdata/\">https://srtm.csi.cgiar.org/srtmdata/</ext-link>). Catchment average rainfall, soil moisture and aridity index are calculated across the selected catchments of Peninsular India. ESA CCI daily soil moisture (COMBINED) data product is available from 1978; therefore a common period of 40 years (1979–2018) is selected based on the availability of data for all the variables.</p>", "<title>Baseflow separation</title>", "<p id=\"Par18\">George and Sekhar^{##UREF##39##49##} finds Ekhardt filter^{##UREF##40##50##} more suitable compared to other digital filters for baseflow separation in Kabini basin, a tributary of Cauvery river in Western Ghats, India. Therefore, Ekhardt filter, a two-parameter recursive filter is used to estimate baseflow in the study area. The filter equation is given bywhere α is the recession constant and is the maximum baseflow index modelled by the algorithm, is the baseflow, and is discharge for time step . Peninsular catchments are underlain by hard rock aquifers; therefore is selected. Recession constant is computed based on the master recession curve (MRC) method described in WMO manual on low-flow estimation and prediction^{##UREF##41##51##}. The beginning of recession is marked below the threshold at least two days after the peak flood discharge. Segment length is computed for each catchment and MRC is obtained by plotting pairs of and . Recession constant α is estimated as the slope of the curve. The procedure is illustrated for a randomly selected Haralahalli catchment of Krishna river basin in Fig. ##SUPPL##0##S5##.</p>", "<title>Trend estimation</title>", "<p id=\"Par19\">Trends in annual maximum streamflow are detected using Mann–Kendall trend test^{##UREF##42##52##} and the slope of linear trends in Peninsular catchments is computed using Sen-Theil slope estimator^{##UREF##43##53##}. In order to facilitate a relative comparison of trends across catchments of different sizes, they are expressed in units of percentage change per decade following previous studies^{##UREF##44##54##–##REF##31462777##56##}, such thatwhere is the trend in %/decade, is the Sen’s slope and is the mean of annual maximum streamflow time series. Decadal trends in the flood drivers are estimated similarly.</p>", "<title>Extracting flood characteristics</title>", "<p id=\"Par20\">Annual flood peak, volume and duration series are extracted using the procedure followed in previous studies^{##UREF##46##57##–##UREF##49##60##}. Annual maxima of streamflow data is the peak flow. Baseflow is used as the criterion to delineate flood hydrograph and derive flood volume and flood duration. Start day of flood runoff is marked as the abrupt rise in discharge (above the baseflow) and flattening of the recession limb (a return to baseflow) is the end day as shown in Fig. ##FIG##2##3##b. Flood duration for the selected year is Flood volume for <italic>i</italic>th year with observed streamflow on <italic>j</italic>th day is computed as followswhere and are the observed daily streamflows on the start and end dates of flood runoff, respectively. In this study, a flood event is defined as the upper part of the hydrograph lying above the fixed threshold as described by Karmakar and Simonovic^{##UREF##48##59##}. Flood duration is and flood volume is estimated for the threshold discharge after deducting the baseflow volume.</p>", "<title>Quantifying the impact of flow regulations</title>", "<p id=\"Par21\">Human activities like construction of reservoirs significantly affect the hydrological system by disturbing the natural flow conditions. Paired-catchment approach is the classical method in catchment hydrology to detect the impact of a disturbance on the flow regime^{##UREF##25##34##,##UREF##50##61##,##UREF##51##62##}. The flow regimes of two nearby catchments with similar physical characteristics are compared in this method by setting one as a benchmark catchment and other as a disturbed catchment. Indian catchments are bigger in size and it is difficult to find adequate number of pairs with the presence of a large number of hydraulic structures in a single catchment. Therefore, the “<italic>pre-post-disturbance</italic>” approach which compares hydrologic extremes before and after a disturbance is used in this study to quantify the impact of human influence. A minimum of 15 years and an optimum of 20 years for each part are required such that the normal, dry and wet years within each period are equally distributed^{##UREF##52##63##,##UREF##53##64##}. Stream networks are delineated for the Peninsular river basins and locations of dams are marked on the network. The streamflow gauging stations which lie downstream of the dams on same flowlines are identified. A comprehensive analysis on changes in flood characteristics (peak, volume and duration) is conducted by dividing the streamflow records into two parts: the undisturbed period and the disturbed period. Length of streamflow records varies between a maximum of 52 years (1967–2018) to a minimum of 40 years (1979–2018) for quantifying the changes in flood characteristics. For a robust assessment of changes only the structures constructed after 1980 are considered so that a good length of records is available before a disturbance. Changes in the flood characteristics are estimated as , where and are the mean characteristics after disturbance and before the disturbance, respectively.</p>", "<title>Event coincidence analysis</title>", "<p id=\"Par22\">Event Coincidence Analysis (ECA) is a recently developed statistical tool exclusively designed for measuring the strength, directionality and time lag of statistical interdependency between two event series^{##REF##22143765##65##,##UREF##54##66##}. Donges et al.^{##UREF##55##67##} used the ECA framework to investigate the role of floods as triggers of epidemic outbreaks with country-level observational data. Manoj et al.^{##UREF##56##68##} employed ECA to identify and quantify the preconditioning of precipitation extremes by soil moisture anomalies over India. ECA is suitable to test for existence, direction and significance of possible relationship between pairs of two event series^{##UREF##56##68##,##UREF##57##69##}\n and . ECA is utilized in this study to test for existence and significance of statistical interrelationship of floods with the flood drivers.</p>", "<p id=\"Par23\">Let be flood events occurring at timings and be the flood drivers (rainfall, soil moisture and baseflow) occurring at times \n and are the number of events of event series and respectively. The event series are assumed to cover a time interval with length , such that and which yields the event rates and Coincidences of events in both the series are counted and the strength of statistical interrelationship is quantified using a measure called “<italic>Trigger Coincidence Rate</italic>” . It measures the fraction of -type events that are followed by at least one -type event. Multiple -type events within the coincidence interval are counted only once.</p>", "<p id=\"Par24\">Trigger coincidence rate^{##UREF##55##67##} is defined aswhere is the coincidence interval and is the time lag parameter. An instantaneous coincidence occurs if events of two event series occur closer in time i.e. if the condition is satisfied. A lagged coincidence occurs when the events shifted by time lag i.e. at time coincide with the -type event and the condition holds. denotes the Heaviside function which conveys information on whether the flood drivers have a triggering effect on flood events or not. The values of vary between 0 (complete absence of triggering effect between and ) and 1 ( events succeed all the events).</p>", "<title>Testing the significance of coincidences</title>", "<p id=\"Par25\">The two event series are assumed to be randomly distributed and mutually independent over the continuous time interval . The occurrences of coincidences are rare and thus and event time series are treated as two independent Poisson processes. This allows derivation of distributions of coincidence rates to test the statistical significance of ECA results. The probability of occurrence of a given number of trigger coincidences between two event series can be approximated by Binomial distribution^{##UREF##54##66##}where is the temporal tolerance and is the time lag between and . Significance test for coincidence measure is based on the null hypothesis that the number of coincidences can be explained by two independent series of randomly distributed events. The -value of empirically observed number of coincidences with respect to the test distribution in Eq. (##FORMU##51##4##) i.e. the probability to obtain a number of coincidences equal to or greater than is given by . Null hypothesis is rejected if the -value is smaller than the defined confidence level α.</p>" ]

[ "<title>Results</title>", "<title>Trend analysis</title>", "<p id=\"Par8\">Annual flood magnitudes have decreased in Peninsular catchments over the period 1979–2018 (Fig. ##FIG##1##2##a). An increase in flood magnitudes is observed only in two catchments (Kurubhata and Kantamal) of Mahanadi river basin and one catchment (Dameracherla) of Godavari river basin. Flood magnitudes are declining drastically in Narmada river basin. Trends in flood magnitudes show a strong association with trends in annual mean baseflow (Fig. ##FIG##1##2##b). Signs of flood magnitude trends are more consistent with the signs of trends in baseflow compared to rainfall and soil moisture. The strength of dependence between trends in flood magnitudes and trends in flood drivers is summarized with Kendall rank correlation coefficient. A high value of Kendall’s is observed for the pairs of trends in annual flood magnitude and trends in mean annual baseflow across all the Peninsular catchments. Trends in annual maximum daily rainfall () and annual mean soil moisture () show a weak correlation with trends in flood magnitudes. These results suggest that floods in Peninsular catchments are strongly correlated with baseflow compared to rainfall and soil moisture.</p>", "<title>Effect of flow regulations on flood characteristics</title>", "<p id=\"Par9\">The impacts of reservoir flow regulations in Peninsular catchments are assessed using “<italic>pre-post-disturbance</italic>” approach in this study. Locations of dams are marked on the streamflow network and the streamflow gauges downstream of these dams on the same flow lines are identified (Fig. ##FIG##2##3##a). Total 31 dams are considered which came after 1980 as per the information from National Register of Large Dams (NRLD) and 25 streamflow gauges are marked which have a good length of flow records available for pre-disturbance period. When there is more than one reservoir upstream of a stream gauge, impact is evaluated for the structure which came first and year of construction decides the length of records for the new dam downstream of the old dam. The pre-dam construction period of new dam begins from the year of construction of the older dam lying upstream and ends at its own year of construction. Annual flood peak, volume and duration are computed for pre- and post-dam construction period as illustrated in Fig. ##FIG##2##3##b. The comparison of mean flood characteristics for the two periods shows that reservoir regulations have strong influence on flood characteristics. Reservoir regulation has increased the flood duration by up to 65% while it has reduced the peak flow and flood volume by ~ 48.5% and ~ 50%, respectively. Floods after the construction of dams last longer but are less severe with reduced peak and volume in Peninsular catchments. These impacts are independent of the purpose of reservoirs. Upper Wardha dam which serves the purpose of flood control along with irrigation and water supply shows reduction in all the flood characteristics (peak − 21.5%, volume − 32.1% and duration − 14%). Flood alleviation effect of reservoirs is observed in different parts of the world^{##UREF##22##31##,##UREF##25##34##,##UREF##27##36##}. Reduction in flood severity indicates a positive effect of dam construction on flood alleviation in Peninsular India.</p>", "<title>Importance of flood drivers</title>", "<p id=\"Par10\">It is well established in literature that antecedent soil moisture conditions play an important role in the hydrological response of a catchment^{##UREF##8##11##,##UREF##9##12##,##UREF##11##16##,##UREF##28##37##}. However, importance of antecedent conditions may extend deeper into the saturated zone as revealed in a recent study by Berghuijs and Slater^{##UREF##20##29##}. Here, we investigated the association of annual floods and flood drivers (baseflow, rainfall and soil moisture) with Pearson correlation coefficient for a range of antecedent periods from 1 to 14 days (Fig. ##FIG##3##4##a). Instantaneous values of baseflow and soil moisture are used, whereas accumulated rainfall from a specific lag to the flooding day are used in the correlation analysis. Baseflow shows a strong correlation with annual maximum flows at all the time lags compared to soil moisture. Baseflow dominates for the first few lags i.e. less than 3–4 days and rainfall dominates at longer antecedent periods for 50 catchments. For remaining 20 catchments (especially from Cauvery river basin), baseflow dominates for more than 5–7 days and rainfall dominates for more longer antecedent periods as shown in Fig. ##SUPPL##0##S2##. A catchment with higher baseflow reflects more wet conditions, which means the chances of rapid runoff are high with the incoming rainfall event. On the other hand, the correlation between accumulated rainfall and flow peaks is relatively high at longer time lags because rainfall not only drives the flood peak but it also contributes to soil moisture and groundwater levels. Accumulated rainfall over a longer period will eventually raise the baseflows and thus more water will be contributed to river flows. Results are shown for a randomly selected catchment Bamini with semi-arid climate in Godavari river basin. Similar correlation pattern is observed across 50 catchments of Peninsular India. At short time scales, flood magnitudes are strongly associated to baseflow than rainfall and soil moisture. This observation suggests that baseflow contributes more water to the Peninsular catchments.</p>", "<p id=\"Par11\">The spatial pattern of correlation between flood magnitudes and baseflow computed 5 days before the flood events is shown in Fig. ##FIG##3##4##b. High positive correlation is observed across all the catchments indicating a strong association of baseflow with floods. This highlights the strong influence of baseflow on floods in Peninsular catchments. Relative contribution of baseflow to peak flows is further investigated using Baseflow Index (BFI) on the flooding day. This will help in quantifying the fraction of peak flow which comes from baseflow. Negative correlation is observed between flood magnitudes and BFI (Fig. ##FIG##3##4##c). This suggests that although baseflow contributes more to river flows (Fig. ##FIG##3##4##b); however, its contribution to the event flow magnitude decreases as surface runoff contributes a higher fraction of flood discharge than baseflow. Additionally, a flood event cannot occur without high rainfall even if the landscape has higher baseflow.</p>", "<p id=\"Par12\">The importance of flood drivers is evaluated using the trigger coincidence rate for the period 1979–2018. The p-values are lower than the confidence level , therefore the null hypothesis of independent random series is rejected for all the catchments. All the trigger coincidence rates are statistically significant. Trigger coincidence rates are high for baseflow compared to rainfall and soil moisture across all the Peninsular catchments (Fig. ##FIG##4##5##). Baseflow 1 day prior to the flood event is significantly influencing the river floods, whereas soil moisture on the previous day has lowest triggering effect on floods (Fig. ##FIG##4##5##a–c). This suggests that high baseflow conditions are coinciding more with the severe 95th percentile flood events. The triggering effect of baseflow is longer-lasting as the trigger coincidences remain high compared to other two flood drivers at a time lag of 5 days (Fig. ##FIG##4##5##d–f). Baseflow on the previous day of flood has a higher trigeering effect than longer time lag. ECA results corroborate the statement that baseflow significantly contributes to floods in Peninsular India.</p>", "<p id=\"Par13\">The role of flood drivers may change with time; therefore, investigation is carried out for two periods to better understand the evolving nature of floods and their association with flood drivers. Data is divided into two equal halves from 1979–1998 to 1999–2018. Triggering effect of flood drivers is assessed for 85th as well as 95th percentile extremes to understand the influence on floods of different magnitudes (Figs. ##SUPPL##0##S3##–##SUPPL##0##S4##). Baseflow has high trigger coincidence rates compared to rainfall and soil moisture irrespective of the flood magnitude and record length. These findings suggest that baseflow plays a critical role in controlling floods in Peninsular India and future floods depend on the pre-existing baseflow conditions during high rainfall events.</p>" ]

[]

[ "<title>Conclusions</title>", "<p id=\"Par14\">The active role of groundwater in storm runoff in streams is discovered decades ago^{##UREF##29##38##,##UREF##30##39##}. Baseflow also exerts significant influence over the entire flood frequency curves^{##UREF##31##40##}. Despite the significant role of groundwater in storm runoff generation, groundwater is rarely considered in flood related studies. Recent studies consider the critical role of soil moisture in modulating floods^{##UREF##1##2##,##UREF##2##3##,##UREF##8##11##–##UREF##10##13##}, whereas groundwater is often overlooked. Present study extends our knowledge on process based controls on floods. Our analysis reveals that pre-existing baseflow conditions play an important role in driving floods. Baseflow is the dominant driver of floods at shorter time lags and rainfall controls flood magnitudes at longer antecedent periods. The effect of baseflow is stronger than soil moisture and lasts for longer antecedent periods in Peninsular catchments.</p>", "<p id=\"Par15\">Presence of reservoir in a catchment significantly influences the natural flow regime by storages and releases. Reservoir regulation has reduced flood severity i.e. flood peak and flood volume but duration of flood events has increased after the construction of dams in Peninsular catchments. This attenuation in flood severity is independent of the purpose of reservoirs. A reduction in flood peak and volume is achieved due to the retention of water in the reservoir and by releasing this excess water over longer durations. Reservoir regulation has positive effect by alleviating the flood severity in Peninsular India.</p>", "<p id=\"Par16\">One potential limitation of present study is that we identified single dominant mechanism of floods in the catchments. However, floods in a catchment can arise through a combination of different mechanisms. Present analysis can be further extended by conditioning on the combination of flood drivers using multivariate statistical tools to accurately estimate their combined effect on river flooding. Incorporating more information on the flood generating mechanisms is the key for improving flood predictions and plan better preventive measures.</p>" ]

[ "<p id=\"Par1\">Extreme rainfall prior to a flood event is often a necessary condition for its occurrence; however, rainfall alone is not always an indicator of flood severity. Antecedent wetness condition of a catchment is another important factor which strongly influences the flood magnitudes. The key role of soil moisture in driving floods is widely recognized; however, antecedent conditions of deeper saturated zone may contribute to river floods. Here, we assess how closely the flood magnitudes are associated to extreme rainfall, soil moisture and baseflow in 70 catchments of Peninsular India for the period 1979–2018. Annual flood magnitudes have declined across most of the catchments. Effect of flow regulations is also assessed to understand the impact of human interventions on flood characteristics. Reservoir regulation has positive effect by reducing the flood peak and volume, whereas the duration of flood events has increased after the construction of dams. Baseflow exhibits similar patterns of trends as floods, whereas trends in rainfall and soil moisture extremes are weakly correlated with trends in flood magnitudes. Baseflow is found to be more strongly influencing the flood magnitudes than soil moisture at various time lags. Further analysis with event coincidence analysis confirms that baseflow has stronger triggering effect on river floods in Peninsular India.</p>", "<title>Subject terms</title>" ]

[ "<title>Study area</title>", "<p id=\"Par7\">Narmada, Tapi, Mahanadi, Godavari, Krishna and Cauvery are six major river basins of Peninsular India. Tapi is the smallest river basin with an area of 65,145 km² and Godavari is the largest which covers an area of 312,812 km². Narmada and Tapi are west flowing rivers which join the Arabian Sea, while other four are east flowing rivers which drain into the Bay of Bengal. Godavari is the longest river of length 1465 km and Cauvery is the shortest river with a length of 560 km in Peninsular India. The locations of 70 selected catchments are shown in Fig. ##FIG##0##1##a. The catchment areas vary in size from 1260 to 307,800 km² (Fig. ##SUPPL##0##S1##). The elevation varies between a minimum of 1 m to a maximum of 937 m (Fig. ##FIG##0##1##b). Spatial variation of mean annual maximum runoff rate averaged over a period of 40 years is shown in Fig. ##FIG##0##1##b. High runoff rates are observed in Narmada, lower and middle Mahanadi, Krishna upper sub-basin, Tungabhadra upper sub-basin and Cauvery upper sub-basin. Aridity Index (AI) defined as the ratio of mean annual precipitation to mean annual potential evapotranspiration is shown in Fig. ##FIG##0##1##c. United Nations Environment Programme (UNEP)^{##UREF##26##35##} provides a climate classification scheme based on the Aridity Index values. Peninsular catchments have semi-arid (AI 0.2–0.5), dry sub-humid (AI 0.5–0.65) and humid (AI &gt; 0.65) climate conditions. Spatial variation of Baseflow Index (BFI), the long-term ratio between baseflow to total streamflow is shown in Fig. ##FIG##0##1##d. The distribution of BFI is relatively even with BFI values between 0.25 and 0.50 for most of the catchments.</p>", "<title>Supplementary Information</title>", "<p>\n</p>" ]

[ "<title>Supplementary Information</title>", "<p>The online version contains supplementary material available at 10.1038/s41598-024-51850-w.</p>", "<title>Acknowledgements</title>", "<p>Funding received from the Ministry of Earth Sciences (MoES), Government of India, through the project “Advanced Research in Hydrology and Knowledge Dissemination”, Project no.: MOES/PAMC/H&amp;C/41/2013-PC-II is greatly acknowledged.</p>", "<title>Author contributions</title>", "<p>First author, S.S. conceptualized the problem, collected and processed the data, performed the entire analysis in R programming language, prepared the first draft of the manuscript and revised it. Second author, P.P.M. played a supervisory role, helped in conceptualization of the problem and preparing the final version of the manuscript.</p>", "<title>Data availability</title>", "<p>India-Water Resources Information System (India-WRIS) daily streamflow data used in this study is available at <ext-link ext-link-type=\"uri\" xlink:href=\"https://indiawris.gov.in/wris/#/RiverMonitoring\">https://indiawris.gov.in/wris/#/RiverMonitoring</ext-link>. India Meteorological Department (IMD) provides daily high resolution gridded rainfall data can be accessed from <ext-link ext-link-type=\"uri\" xlink:href=\"https://www.imdpune.gov.in/cmpg/Griddata/Rainfall_25_NetCDF.html\">https://www.imdpune.gov.in/cmpg/Griddata/Rainfall_25_NetCDF.html</ext-link>. European Space Agency Climate Change Initiative (ESA CCI) soil moisture dataset with daily temporal and 0.25° spatial resolution is available at <ext-link ext-link-type=\"uri\" xlink:href=\"https://esa-soilmoisture-cci.org/data\">https://esa-soilmoisture-cci.org/data</ext-link>. Global Aridity Index is available at 10.6084/m9.figshare.7504448.v5. Shuttle Radar Topographic Mission (SRTM) at 30 m spatial resolution is available at <ext-link ext-link-type=\"uri\" xlink:href=\"https://srtm.csi.cgiar.org/srtmdata/\">https://srtm.csi.cgiar.org/srtmdata/</ext-link>.</p>", "<title>Competing interests</title>", "<p id=\"Par26\">The authors declare no competing interests.</p>" ]

[ "<fig id=\"Fig1\"><label>Figure 1</label><caption><p>(<bold>a</bold>) Locations of streamflow gauges with catchment boundaries in six major river basins of Peninsular India, (<bold>b</bold>) elevation map and mean annual maximum runoff rate (streamflow per unit catchment area), (<bold>c</bold>) Aridity Index (AI) and (<bold>d</bold>) Baseflow Index (BFI). The maps in first row are prepared in QGIS (Version 2.14.0 ‘Essen’ (2016), URL: <ext-link ext-link-type=\"uri\" xlink:href=\"http://qgis.org\">http://qgis.org</ext-link>) and the maps in second row are generated using R (Version 4.2.2 (2022), URL: <ext-link ext-link-type=\"uri\" xlink:href=\"https://www.R-project.org/\">https://www.R-project.org/</ext-link>).</p></caption></fig>", "<fig id=\"Fig2\"><label>Figure 2</label><caption><p>(<bold>a</bold>) Trends in annual flood magnitudes and (<bold>b</bold>) association of trends in flood magnitudes with trends in flood drivers. Flood magnitudes are shrinking across most of the catchments in Peninsular India. Trends in floods are strongly correlated with trends in baseflow (Kendall’s ) and weakly correlated with soil moisture (Kendall’s . The figure is prepared in R (Version 4.2.2 (2022), URL: <ext-link ext-link-type=\"uri\" xlink:href=\"https://www.R-project.org/\">https://www.R-project.org/</ext-link>).</p></caption></fig>", "<fig id=\"Fig3\"><label>Figure 3</label><caption><p>(<bold>a</bold>) Locations of reservoirs and streamflow gauges lying downstream on the same flow lines, (<bold>b</bold>) schematic diagram to derive the flood characteristics and (<bold>c</bold>) effect of flow regulations on flood peak, volume and duration. Floods last longer but became less severe with reduced peak and volume after the construction of dams. The map in first row is prepared in QGIS (Version 2.14.0 ‘Essen’ (2016), URL: <ext-link ext-link-type=\"uri\" xlink:href=\"http://qgis.org\">http://qgis.org</ext-link>), (<bold>b</bold>) is generated using Microsoft Word (Version 2010 <ext-link ext-link-type=\"uri\" xlink:href=\"https://www.office.com/\">https://www.office.com/</ext-link>), and (<bold>c</bold>) is prepared in R (Version 4.2.2 (2022), URL: <ext-link ext-link-type=\"uri\" xlink:href=\"https://www.R-project.org/\">https://www.R-project.org/</ext-link>).</p></caption></fig>", "<fig id=\"Fig4\"><label>Figure 4</label><caption><p>(<bold>a</bold>) Correlation between flood magnitude and flood drives (baseflow, rainfall and soil moisture) for a range of antecedent periods (1–14 days) in a randomly selected Peninsular catchment. Baseflow has a stronger and long lasting association with flood peaks than soil moisture at annual time scale. (<bold>b</bold>) Spatial variation of correlation between flood magnitude and baseflow 5 days before the flood event. (<bold>c</bold>) Spatial variation of correlation between flood magnitude and baseflow index on flooding day. All parts of the figure are generated using R (Version 4.2.2 (2022), URL: <ext-link ext-link-type=\"uri\" xlink:href=\"https://www.R-project.org/\">https://www.R-project.org/</ext-link>).</p></caption></fig>", "<fig id=\"Fig5\"><label>Figure 5</label><caption><p>Triggering effect of baseflow, rainfall and soil moisture on floods defined above 95th percentile threshold. Baseflow and soil moisture are based on their instantaneous values before the flood event; whereas rainfall is the 5 days accumulated rainfall. Tigger coincidence rates are high for baseflow 1 day (first row) and 5 days (second row) prior to the flood event. Baseflow has stronger triggering effect on floods in Peninsular India. The maps are prepared in R (Version 4.2.2 (2022), URL: <ext-link ext-link-type=\"uri\" xlink:href=\"https://www.R-project.org/\">https://www.R-project.org/</ext-link>).</p></caption></fig>" ]

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=0.165$$\\end{document}</tex-math><mml:math id=\"M4\"><mml:mrow><mml:mi>τ</mml:mi><mml:mo>=</mml:mo><mml:mn>0.165</mml:mn></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq5\"><alternatives><tex-math id=\"M5\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\tau =0.0746$$\\end{document}</tex-math><mml:math id=\"M6\"><mml:mrow><mml:mi>τ</mml:mi><mml:mo>=</mml:mo><mml:mn>0.0746</mml:mn></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq1\"><alternatives><tex-math id=\"M7\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} 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id=\"IEq6\"><alternatives><tex-math id=\"M11\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\alpha =0.01$$\\end{document}</tex-math><mml:math id=\"M12\"><mml:mrow><mml:mi>α</mml:mi><mml:mo>=</mml:mo><mml:mn>0.01</mml:mn></mml:mrow></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ1\"><label>1</label><alternatives><tex-math id=\"M13\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$b_{i} = \\frac{{\\left( {1 - BFI_{\\max } } \\right)\\alpha b_{i - 1} + \\left( {1 - \\alpha } \\right)BFI_{\\max } Q_{i} }}{{1 - \\alpha BFI_{\\max } }}$$\\end{document}</tex-math><mml:math id=\"M14\" display=\"block\"><mml:mrow><mml:msub><mml:mi>b</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mfrac><mml:mrow><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:mn>1</mml:mn><mml:mo>-</mml:mo><mml:mi>B</mml:mi><mml:mi>F</mml:mi><mml:msub><mml:mi>I</mml:mi><mml:mo movablelimits=\"true\">max</mml:mo></mml:msub></mml:mrow></mml:mfenced><mml:mi>α</mml:mi><mml:msub><mml:mi>b</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:mn>1</mml:mn><mml:mo>-</mml:mo><mml:mi>α</mml:mi></mml:mrow></mml:mfenced><mml:mi>B</mml:mi><mml:mi>F</mml:mi><mml:msub><mml:mi>I</mml:mi><mml:mo 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"<inline-formula id=\"IEq8\"><alternatives><tex-math id=\"M17\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${b}_{i}$$\\end{document}</tex-math><mml:math id=\"M18\"><mml:msub><mml:mi>b</mml:mi><mml:mi>i</mml:mi></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq9\"><alternatives><tex-math id=\"M19\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${Q}_{i}$$\\end{document}</tex-math><mml:math id=\"M20\"><mml:msub><mml:mi>Q</mml:mi><mml:mi>i</mml:mi></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq10\"><alternatives><tex-math id=\"M21\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$i$$\\end{document}</tex-math><mml:math id=\"M22\"><mml:mi>i</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq11\"><alternatives><tex-math id=\"M23\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${BFI}_{{\\text{max}}}=0.25$$\\end{document}</tex-math><mml:math id=\"M24\"><mml:mrow><mml:msub><mml:mrow><mml:mi mathvariant=\"italic\">BFI</mml:mi></mml:mrow><mml:mtext>max</mml:mtext></mml:msub><mml:mo>=</mml:mo><mml:mn>0.25</mml:mn></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq12\"><alternatives><tex-math id=\"M25\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${Q}_{70}$$\\end{document}</tex-math><mml:math id=\"M26\"><mml:msub><mml:mi>Q</mml:mi><mml:mn>70</mml:mn></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq13\"><alternatives><tex-math id=\"M27\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${Q}_{t-1}$$\\end{document}</tex-math><mml:math id=\"M28\"><mml:msub><mml:mi>Q</mml:mi><mml:mrow><mml:mi>t</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq14\"><alternatives><tex-math id=\"M29\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${Q}_{t}$$\\end{document}</tex-math><mml:math id=\"M30\"><mml:msub><mml:mi>Q</mml:mi><mml:mi>t</mml:mi></mml:msub></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ2\"><label>2</label><alternatives><tex-math id=\"M31\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$T = \\frac{S \\times 10\\;years}{{\\overline{x}}} \\times 100$$\\end{document}</tex-math><mml:math id=\"M32\" display=\"block\"><mml:mrow><mml:mi>T</mml:mi><mml:mo>=</mml:mo><mml:mfrac><mml:mrow><mml:mi>S</mml:mi><mml:mo>×</mml:mo><mml:mn>10</mml:mn><mml:mspace width=\"0.277778em\"/><mml:mi>y</mml:mi><mml:mi>e</mml:mi><mml:mi>a</mml:mi><mml:mi>r</mml:mi><mml:mi>s</mml:mi></mml:mrow><mml:mover><mml:mi>x</mml:mi><mml:mo>¯</mml:mo></mml:mover></mml:mfrac><mml:mo>×</mml:mo><mml:mn>100</mml:mn></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq15\"><alternatives><tex-math id=\"M33\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$T$$\\end{document}</tex-math><mml:math id=\"M34\"><mml:mi>T</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq16\"><alternatives><tex-math id=\"M35\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$S$$\\end{document}</tex-math><mml:math id=\"M36\"><mml:mi>S</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq17\"><alternatives><tex-math id=\"M37\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\overline{x }$$\\end{document}</tex-math><mml:math id=\"M38\"><mml:mover><mml:mi>x</mml:mi><mml:mo>¯</mml:mo></mml:mover></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq18\"><alternatives><tex-math id=\"M39\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$i$$\\end{document}</tex-math><mml:math id=\"M40\"><mml:mi>i</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq19\"><alternatives><tex-math id=\"M41\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${D}_{i}=\\left(E{D}_{i}-S{D}_{i}\\right).$$\\end{document}</tex-math><mml:math id=\"M42\"><mml:mrow><mml:msub><mml:mi>D</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mfenced close=\")\" open=\"(\"><mml:mi>E</mml:mi><mml:msub><mml:mi>D</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>-</mml:mo><mml:mi>S</mml:mi><mml:msub><mml:mi>D</mml:mi><mml:mi>i</mml:mi></mml:msub></mml:mfenced><mml:mo>.</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq20\"><alternatives><tex-math id=\"M43\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${Q}_{ij}$$\\end{document}</tex-math><mml:math id=\"M44\"><mml:msub><mml:mi>Q</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">ij</mml:mi></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ3\"><label>3</label><alternatives><tex-math id=\"M45\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{gathered} V_{i} = \\left( {V_{i}^{Total} - V_{i}^{Base} } \\right) \\hfill \\\\ \\quad \\;\\; = \\mathop \\sum \\limits_{{j = SD_{i} }}^{{ED_{i} }} \\left\\{ {Q_{ij} - \\frac{1}{2}\\left( {Q_{is} + Q_{ie} } \\right)} \\right\\} - \\frac{1}{2}D_{i} \\left( {Q_{is} + Q_{ie} } \\right) \\hfill \\\\ \\end{gathered}$$\\end{document}</tex-math><mml:math id=\"M46\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mrow><mml:msub><mml:mi>V</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:msubsup><mml:mi>V</mml:mi><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"italic\">Total</mml:mi></mml:mrow></mml:msubsup><mml:mo>-</mml:mo><mml:msubsup><mml:mi>V</mml:mi><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"italic\">Base</mml:mi></mml:mrow></mml:msubsup></mml:mrow></mml:mfenced></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mrow/><mml:mspace width=\"1em\"/><mml:mspace width=\"0.277778em\"/><mml:mspace width=\"0.277778em\"/><mml:mo>=</mml:mo><mml:munderover><mml:mo movablelimits=\"false\">∑</mml:mo><mml:mrow><mml:mrow><mml:mi>j</mml:mi><mml:mo>=</mml:mo><mml:mi>S</mml:mi><mml:msub><mml:mi>D</mml:mi><mml:mi>i</mml:mi></mml:msub></mml:mrow></mml:mrow><mml:mrow><mml:mi>E</mml:mi><mml:msub><mml:mi>D</mml:mi><mml:mi>i</mml:mi></mml:msub></mml:mrow></mml:munderover><mml:mfenced close=\"}\" open=\"{\"><mml:mrow><mml:msub><mml:mi>Q</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">ij</mml:mi></mml:mrow></mml:msub><mml:mo>-</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:mn>2</mml:mn></mml:mfrac><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:msub><mml:mi>Q</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">is</mml:mi></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:msub><mml:mi>Q</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">ie</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:mfenced></mml:mrow></mml:mfenced><mml:mo>-</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:mn>2</mml:mn></mml:mfrac><mml:msub><mml:mi>D</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:msub><mml:mi>Q</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">is</mml:mi></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:msub><mml:mi>Q</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">ie</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:mfenced></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow/></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq21\"><alternatives><tex-math id=\"M47\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${Q}_{is}$$\\end{document}</tex-math><mml:math id=\"M48\"><mml:msub><mml:mi>Q</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">is</mml:mi></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq22\"><alternatives><tex-math id=\"M49\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${Q}_{ie}$$\\end{document}</tex-math><mml:math id=\"M50\"><mml:msub><mml:mi>Q</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">ie</mml:mi></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq23\"><alternatives><tex-math id=\"M51\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${Q}_{i\\left(thresh\\right)}=0.5\\times Peak Flow$$\\end{document}</tex-math><mml:math id=\"M52\"><mml:mrow><mml:msub><mml:mi>Q</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mfenced close=\")\" open=\"(\"><mml:mi>t</mml:mi><mml:mi>h</mml:mi><mml:mi>r</mml:mi><mml:mi>e</mml:mi><mml:mi>s</mml:mi><mml:mi>h</mml:mi></mml:mfenced></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mn>0.5</mml:mn><mml:mo>×</mml:mo><mml:mi>P</mml:mi><mml:mi>e</mml:mi><mml:mi>a</mml:mi><mml:mi>k</mml:mi><mml:mi>F</mml:mi><mml:mi>l</mml:mi><mml:mi>o</mml:mi><mml:mi>w</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq24\"><alternatives><tex-math id=\"M53\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${D}_{i\\left(thresh\\right)}$$\\end{document}</tex-math><mml:math id=\"M54\"><mml:msub><mml:mi>D</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mfenced close=\")\" open=\"(\"><mml:mi>t</mml:mi><mml:mi>h</mml:mi><mml:mi>r</mml:mi><mml:mi>e</mml:mi><mml:mi>s</mml:mi><mml:mi>h</mml:mi></mml:mfenced></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq25\"><alternatives><tex-math id=\"M55\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${V}_{i\\left(thresh\\right)}$$\\end{document}</tex-math><mml:math id=\"M56\"><mml:msub><mml:mi>V</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mfenced close=\")\" open=\"(\"><mml:mi>t</mml:mi><mml:mi>h</mml:mi><mml:mi>r</mml:mi><mml:mi>e</mml:mi><mml:mi>s</mml:mi><mml:mi>h</mml:mi></mml:mfenced></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq26\"><alternatives><tex-math id=\"M57\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$({C}_{D}-{C}_{U})/{C}_{U}$$\\end{document}</tex-math><mml:math id=\"M58\"><mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>C</mml:mi><mml:mi>D</mml:mi></mml:msub><mml:mo>-</mml:mo><mml:msub><mml:mi>C</mml:mi><mml:mi>U</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo stretchy=\"false\">/</mml:mo><mml:msub><mml:mi>C</mml:mi><mml:mi>U</mml:mi></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq27\"><alternatives><tex-math id=\"M59\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${C}_{D}$$\\end{document}</tex-math><mml:math id=\"M60\"><mml:msub><mml:mi>C</mml:mi><mml:mi>D</mml:mi></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq28\"><alternatives><tex-math id=\"M61\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${C}_{U}$$\\end{document}</tex-math><mml:math id=\"M62\"><mml:msub><mml:mi>C</mml:mi><mml:mi>U</mml:mi></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq29\"><alternatives><tex-math id=\"M63\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$X$$\\end{document}</tex-math><mml:math id=\"M64\"><mml:mi>X</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq30\"><alternatives><tex-math id=\"M65\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$Y$$\\end{document}</tex-math><mml:math id=\"M66\"><mml:mi>Y</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq31\"><alternatives><tex-math id=\"M67\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$X$$\\end{document}</tex-math><mml:math id=\"M68\"><mml:mi>X</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq32\"><alternatives><tex-math id=\"M69\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\left\\{{t}_{1}^{X},\\dots ,{t}_{{N}_{X}}^{X}\\right\\}$$\\end{document}</tex-math><mml:math id=\"M70\"><mml:mfenced close=\"}\" open=\"{\"><mml:msubsup><mml:mi>t</mml:mi><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mi>X</mml:mi></mml:msubsup><mml:mo>,</mml:mo><mml:mo>⋯</mml:mo><mml:mo>,</mml:mo><mml:msubsup><mml:mi>t</mml:mi><mml:mrow><mml:msub><mml:mi>N</mml:mi><mml:mi>X</mml:mi></mml:msub></mml:mrow><mml:mi>X</mml:mi></mml:msubsup></mml:mfenced></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq33\"><alternatives><tex-math id=\"M71\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$Y$$\\end{document}</tex-math><mml:math id=\"M72\"><mml:mi>Y</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq34\"><alternatives><tex-math id=\"M73\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\left\\{{t}_{1}^{Y},\\dots ,{t}_{{N}_{Y}}^{Y}\\right\\}.$$\\end{document}</tex-math><mml:math id=\"M74\"><mml:mrow><mml:mfenced close=\"}\" open=\"{\"><mml:msubsup><mml:mi>t</mml:mi><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mi>Y</mml:mi></mml:msubsup><mml:mo>,</mml:mo><mml:mo>⋯</mml:mo><mml:mo>,</mml:mo><mml:msubsup><mml:mi>t</mml:mi><mml:mrow><mml:msub><mml:mi>N</mml:mi><mml:mi>Y</mml:mi></mml:msub></mml:mrow><mml:mi>Y</mml:mi></mml:msubsup></mml:mfenced><mml:mo>.</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq35\"><alternatives><tex-math id=\"M75\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${N}_{X}$$\\end{document}</tex-math><mml:math id=\"M76\"><mml:msub><mml:mi>N</mml:mi><mml:mi>X</mml:mi></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq36\"><alternatives><tex-math id=\"M77\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${N}_{Y}$$\\end{document}</tex-math><mml:math id=\"M78\"><mml:msub><mml:mi>N</mml:mi><mml:mi>Y</mml:mi></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq37\"><alternatives><tex-math id=\"M79\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$X$$\\end{document}</tex-math><mml:math id=\"M80\"><mml:mi>X</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq38\"><alternatives><tex-math id=\"M81\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$Y,$$\\end{document}</tex-math><mml:math id=\"M82\"><mml:mrow><mml:mi>Y</mml:mi><mml:mo>,</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq39\"><alternatives><tex-math id=\"M83\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\left({t}_{0},{t}_{e}\\right)$$\\end{document}</tex-math><mml:math id=\"M84\"><mml:mfenced close=\")\" open=\"(\"><mml:msub><mml:mi>t</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>t</mml:mi><mml:mi>e</mml:mi></mml:msub></mml:mfenced></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq40\"><alternatives><tex-math id=\"M85\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$T={t}_{e}-{t}_{0}$$\\end{document}</tex-math><mml:math id=\"M86\"><mml:mrow><mml:mi>T</mml:mi><mml:mo>=</mml:mo><mml:msub><mml:mi>t</mml:mi><mml:mi>e</mml:mi></mml:msub><mml:mo>-</mml:mo><mml:msub><mml:mi>t</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq41\"><alternatives><tex-math id=\"M87\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$t_{0} \\le t_{1}^{X} \\le \\cdots \\le t_{{N_{X} }}^{X} \\le t_{e}$$\\end{document}</tex-math><mml:math id=\"M88\"><mml:mrow><mml:msub><mml:mi>t</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo>≤</mml:mo><mml:msubsup><mml:mi>t</mml:mi><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mi>X</mml:mi></mml:msubsup><mml:mo>≤</mml:mo><mml:mo>⋯</mml:mo><mml:mo>≤</mml:mo><mml:msubsup><mml:mi>t</mml:mi><mml:mrow><mml:msub><mml:mi>N</mml:mi><mml:mi>X</mml:mi></mml:msub></mml:mrow><mml:mi>X</mml:mi></mml:msubsup><mml:mo>≤</mml:mo><mml:msub><mml:mi>t</mml:mi><mml:mi>e</mml:mi></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq42\"><alternatives><tex-math id=\"M89\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$t_{0} \\le t_{1}^{Y} \\le \\cdots \\le t_{{N_{Y} }}^{Y} \\le t_{e}$$\\end{document}</tex-math><mml:math id=\"M90\"><mml:mrow><mml:msub><mml:mi>t</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo>≤</mml:mo><mml:msubsup><mml:mi>t</mml:mi><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mi>Y</mml:mi></mml:msubsup><mml:mo>≤</mml:mo><mml:mo>⋯</mml:mo><mml:mo>≤</mml:mo><mml:msubsup><mml:mi>t</mml:mi><mml:mrow><mml:msub><mml:mi>N</mml:mi><mml:mi>Y</mml:mi></mml:msub></mml:mrow><mml:mi>Y</mml:mi></mml:msubsup><mml:mo>≤</mml:mo><mml:msub><mml:mi>t</mml:mi><mml:mi>e</mml:mi></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq43\"><alternatives><tex-math id=\"M91\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\lambda }_{X}={N}_{X}/T$$\\end{document}</tex-math><mml:math id=\"M92\"><mml:mrow><mml:msub><mml:mi>λ</mml:mi><mml:mi>X</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:msub><mml:mi>N</mml:mi><mml:mi>X</mml:mi></mml:msub><mml:mo stretchy=\"false\">/</mml:mo><mml:mi>T</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq44\"><alternatives><tex-math id=\"M93\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\lambda }_{Y}={N}_{Y}/T.$$\\end{document}</tex-math><mml:math id=\"M94\"><mml:mrow><mml:msub><mml:mi>λ</mml:mi><mml:mi>Y</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:msub><mml:mi>N</mml:mi><mml:mi>Y</mml:mi></mml:msub><mml:mo stretchy=\"false\">/</mml:mo><mml:mi>T</mml:mi><mml:mo>.</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq45\"><alternatives><tex-math id=\"M95\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${(r}_{t})$$\\end{document}</tex-math><mml:math id=\"M96\"><mml:mrow><mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>r</mml:mi></mml:mrow><mml:mi>t</mml:mi></mml:msub><mml:mrow><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq46\"><alternatives><tex-math id=\"M97\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$Y$$\\end{document}</tex-math><mml:math id=\"M98\"><mml:mi>Y</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq47\"><alternatives><tex-math id=\"M99\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$X$$\\end{document}</tex-math><mml:math id=\"M100\"><mml:mi>X</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq48\"><alternatives><tex-math id=\"M101\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$X$$\\end{document}</tex-math><mml:math id=\"M102\"><mml:mi>X</mml:mi></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ4\"><label>4</label><alternatives><tex-math id=\"M103\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$r_{t} \\left( {\\Delta T,\\tau } \\right) = \\frac{1}{{N_{Y} }}\\mathop \\sum \\limits_{j = 1}^{{N_{Y} }} F\\left[ {\\mathop \\sum \\limits_{i = 1}^{{N_{X} }} I_{{\\left[ {0,\\Delta T} \\right]}} \\left( {\\left( {t_{i}^{X} - \\tau } \\right) - t_{j}^{Y} } \\right)} \\right]$$\\end{document}</tex-math><mml:math id=\"M104\" display=\"block\"><mml:mrow><mml:msub><mml:mi>r</mml:mi><mml:mi>t</mml:mi></mml:msub><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:mi mathvariant=\"normal\">Δ</mml:mi><mml:mi>T</mml:mi><mml:mo>,</mml:mo><mml:mi>τ</mml:mi></mml:mrow></mml:mfenced><mml:mo>=</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:msub><mml:mi>N</mml:mi><mml:mi>Y</mml:mi></mml:msub></mml:mfrac><mml:munderover><mml:mo movablelimits=\"false\">∑</mml:mo><mml:mrow><mml:mi>j</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:msub><mml:mi>N</mml:mi><mml:mi>Y</mml:mi></mml:msub></mml:munderover><mml:mi>F</mml:mi><mml:mfenced close=\"]\" open=\"[\"><mml:mrow><mml:munderover><mml:mo movablelimits=\"false\">∑</mml:mo><mml:mrow><mml:mi>i</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:msub><mml:mi>N</mml:mi><mml:mi>X</mml:mi></mml:msub></mml:munderover><mml:msub><mml:mi>I</mml:mi><mml:mfenced close=\"]\" open=\"[\"><mml:mrow><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mi mathvariant=\"normal\">Δ</mml:mi><mml:mi>T</mml:mi></mml:mrow></mml:mfenced></mml:msub><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:msubsup><mml:mi>t</mml:mi><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mi>X</mml:mi></mml:msubsup><mml:mo>-</mml:mo><mml:mi>τ</mml:mi></mml:mrow></mml:mfenced><mml:mo>-</mml:mo><mml:msubsup><mml:mi>t</mml:mi><mml:mrow><mml:mi>j</mml:mi></mml:mrow><mml:mi>Y</mml:mi></mml:msubsup></mml:mrow></mml:mfenced></mml:mrow></mml:mfenced></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq49\"><alternatives><tex-math id=\"M105\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\Delta T$$\\end{document}</tex-math><mml:math id=\"M106\"><mml:mrow><mml:mi mathvariant=\"normal\">Δ</mml:mi><mml:mi>T</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq50\"><alternatives><tex-math id=\"M107\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\tau$$\\end{document}</tex-math><mml:math id=\"M108\"><mml:mi>τ</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq51\"><alternatives><tex-math id=\"M109\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${t}_{i}^{X}-{t}_{j}^{Y}\\le \\Delta T$$\\end{document}</tex-math><mml:math id=\"M110\"><mml:mrow><mml:msubsup><mml:mi>t</mml:mi><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mi>X</mml:mi></mml:msubsup><mml:mo>-</mml:mo><mml:msubsup><mml:mi>t</mml:mi><mml:mrow><mml:mi>j</mml:mi></mml:mrow><mml:mi>Y</mml:mi></mml:msubsup><mml:mo>≤</mml:mo><mml:mi mathvariant=\"normal\">Δ</mml:mi><mml:mi>T</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq52\"><alternatives><tex-math id=\"M111\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$X$$\\end{document}</tex-math><mml:math id=\"M112\"><mml:mi>X</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq53\"><alternatives><tex-math id=\"M113\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\tau$$\\end{document}</tex-math><mml:math id=\"M114\"><mml:mi>τ</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq54\"><alternatives><tex-math id=\"M115\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${(t}_{i}^{X}-\\tau )$$\\end{document}</tex-math><mml:math id=\"M116\"><mml:mrow><mml:msubsup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mi>X</mml:mi></mml:msubsup><mml:mrow><mml:mo>-</mml:mo><mml:mi>τ</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq55\"><alternatives><tex-math id=\"M117\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$Y$$\\end{document}</tex-math><mml:math id=\"M118\"><mml:mi>Y</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq56\"><alternatives><tex-math id=\"M119\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${(t}_{i}^{X}-{\\tau )-t}_{j}^{Y}\\le \\Delta T$$\\end{document}</tex-math><mml:math id=\"M120\"><mml:mrow><mml:msubsup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mi>X</mml:mi></mml:msubsup><mml:mo>-</mml:mo><mml:msubsup><mml:mrow><mml:mi>τ</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>-</mml:mo><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mi>j</mml:mi></mml:mrow><mml:mi>Y</mml:mi></mml:msubsup><mml:mo>≤</mml:mo><mml:mi mathvariant=\"normal\">Δ</mml:mi><mml:mi>T</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq57\"><alternatives><tex-math id=\"M121\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$F\\left(\\cdot \\right)$$\\end{document}</tex-math><mml:math id=\"M122\"><mml:mrow><mml:mi>F</mml:mi><mml:mfenced close=\")\" open=\"(\"><mml:mo>·</mml:mo></mml:mfenced></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq58\"><alternatives><tex-math id=\"M123\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$(Y)$$\\end{document}</tex-math><mml:math id=\"M124\"><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>Y</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq59\"><alternatives><tex-math id=\"M125\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$(X)$$\\end{document}</tex-math><mml:math id=\"M126\"><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq60\"><alternatives><tex-math id=\"M127\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${r}_{t}$$\\end{document}</tex-math><mml:math id=\"M128\"><mml:msub><mml:mi>r</mml:mi><mml:mi>t</mml:mi></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq61\"><alternatives><tex-math id=\"M129\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$X$$\\end{document}</tex-math><mml:math id=\"M130\"><mml:mi>X</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq62\"><alternatives><tex-math id=\"M131\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$Y$$\\end{document}</tex-math><mml:math id=\"M132\"><mml:mi>Y</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq63\"><alternatives><tex-math id=\"M133\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$X$$\\end{document}</tex-math><mml:math id=\"M134\"><mml:mi>X</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq64\"><alternatives><tex-math id=\"M135\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$Y$$\\end{document}</tex-math><mml:math id=\"M136\"><mml:mi>Y</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq65\"><alternatives><tex-math id=\"M137\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$T$$\\end{document}</tex-math><mml:math id=\"M138\"><mml:mi>T</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq66\"><alternatives><tex-math id=\"M139\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$X$$\\end{document}</tex-math><mml:math id=\"M140\"><mml:mi>X</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq67\"><alternatives><tex-math id=\"M141\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$Y$$\\end{document}</tex-math><mml:math id=\"M142\"><mml:mi>Y</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq68\"><alternatives><tex-math id=\"M143\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$(K={N}_{X}{\\cdot r}_{t})$$\\end{document}</tex-math><mml:math id=\"M144\"><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>K</mml:mi><mml:mo>=</mml:mo><mml:msub><mml:mi>N</mml:mi><mml:mi>X</mml:mi></mml:msub><mml:msub><mml:mrow><mml:mo>·</mml:mo><mml:mi>r</mml:mi></mml:mrow><mml:mi>t</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ5\"><label>5</label><alternatives><tex-math id=\"M145\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$P\\left( {K;N_{Y} , 1 - \\left( {1 - p} \\right)^{{N_{X} }} } \\right) = \\left( {\\begin{array}{*{20}c} {N_{Y} } \\\\ K \\\\ \\end{array} } \\right)\\left( {1 - \\left( {1 - \\frac{\\Delta T}{{T - \\tau }}} \\right)^{{N_{X} }} } \\right)^{K} \\left( {\\left( {1 - \\frac{\\Delta T}{{T - \\tau }}} \\right)^{{N_{X} }} } \\right)^{{N_{Y} - K}}$$\\end{document}</tex-math><mml:math id=\"M146\" display=\"block\"><mml:mrow><mml:mi>P</mml:mi><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:mi>K</mml:mi><mml:mo>;</mml:mo><mml:msub><mml:mi>N</mml:mi><mml:mi>Y</mml:mi></mml:msub><mml:mo>,</mml:mo><mml:mn>1</mml:mn><mml:mo>-</mml:mo><mml:msup><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:mn>1</mml:mn><mml:mo>-</mml:mo><mml:mi>p</mml:mi></mml:mrow></mml:mfenced><mml:msub><mml:mi>N</mml:mi><mml:mi>X</mml:mi></mml:msub></mml:msup></mml:mrow></mml:mfenced><mml:mo>=</mml:mo><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:msub><mml:mi>N</mml:mi><mml:mi>Y</mml:mi></mml:msub></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mrow/><mml:mi>K</mml:mi></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced><mml:msup><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:mn>1</mml:mn><mml:mo>-</mml:mo><mml:msup><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:mn>1</mml:mn><mml:mo>-</mml:mo><mml:mfrac><mml:mrow><mml:mi mathvariant=\"normal\">Δ</mml:mi><mml:mi>T</mml:mi></mml:mrow><mml:mrow><mml:mi>T</mml:mi><mml:mo>-</mml:mo><mml:mi>τ</mml:mi></mml:mrow></mml:mfrac></mml:mrow></mml:mfenced><mml:msub><mml:mi>N</mml:mi><mml:mi>X</mml:mi></mml:msub></mml:msup></mml:mrow></mml:mfenced><mml:mi>K</mml:mi></mml:msup><mml:msup><mml:mfenced close=\")\" open=\"(\"><mml:msup><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:mn>1</mml:mn><mml:mo>-</mml:mo><mml:mfrac><mml:mrow><mml:mi mathvariant=\"normal\">Δ</mml:mi><mml:mi>T</mml:mi></mml:mrow><mml:mrow><mml:mi>T</mml:mi><mml:mo>-</mml:mo><mml:mi>τ</mml:mi></mml:mrow></mml:mfrac></mml:mrow></mml:mfenced><mml:msub><mml:mi>N</mml:mi><mml:mi>X</mml:mi></mml:msub></mml:msup></mml:mfenced><mml:mrow><mml:msub><mml:mi>N</mml:mi><mml:mi>Y</mml:mi></mml:msub><mml:mo>-</mml:mo><mml:mi>K</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq69\"><alternatives><tex-math id=\"M147\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\Delta T$$\\end{document}</tex-math><mml:math id=\"M148\"><mml:mrow><mml:mi mathvariant=\"normal\">Δ</mml:mi><mml:mi>T</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq70\"><alternatives><tex-math id=\"M149\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\tau$$\\end{document}</tex-math><mml:math id=\"M150\"><mml:mi>τ</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq71\"><alternatives><tex-math id=\"M151\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$X$$\\end{document}</tex-math><mml:math id=\"M152\"><mml:mi>X</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq72\"><alternatives><tex-math id=\"M153\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$Y$$\\end{document}</tex-math><mml:math id=\"M154\"><mml:mi>Y</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq73\"><alternatives><tex-math id=\"M155\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$p$$\\end{document}</tex-math><mml:math id=\"M156\"><mml:mi>p</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq74\"><alternatives><tex-math id=\"M157\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${K}_{e}$$\\end{document}</tex-math><mml:math id=\"M158\"><mml:msub><mml:mi>K</mml:mi><mml:mi>e</mml:mi></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq75\"><alternatives><tex-math 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[ "<media xlink:href=\"41598_2024_51850_MOESM1_ESM.pdf\"><caption><p>Supplementary Information.</p></caption></media>" ]

{ "acronym": [], "definition": [] }

[ "<title>Introduction</title>", "<p id=\"Par2\">Cancer cells face heightened proteotoxic stress compared to their normal counterparts due to various factors, including the production of mutated proteins, the upregulation of multiprotein complex components induced by aneuploidy [##REF##26144554##1##], and increased protein synthesis driven by oncogenic activation [##UREF##0##2##]. As a result, the survival of cancer cells relies heavily on the intricate machinery responsible for alleviating proteotoxic stress and maintaining proteostasis. This machinery encompasses coordinated processes like protein synthesis, folding, processing, and degradation [##UREF##0##2##, ##UREF##1##3##]. One pivotal player in these proteostasis-associated processes is Valosin-containing protein (VCP/p97), a hexameric AAA+ ATPase [##REF##29153394##4##]. VCP plays a significant role in diverse cellular functions, including endoplasmic reticulum (ER)-associated degradation (ERAD) [##REF##11740563##5##], mitochondrial-associated degradation (MAD) [##UREF##2##6##], and the ubiquitin-proteasome system (UPS) [##REF##28475898##7##]. Notably, VCP is frequently overexpressed in various cancer types and holds promise as both a cancer prognostic biomarker and therapeutic target [##REF##27536557##8##, ##UREF##3##9##]. However, the precise mechanisms by which VCP inhibition selectively eradicates cancer cells while sparing non-cancerous cells have remained elusive.</p>", "<p id=\"Par3\">In this study, we present evidence that VCP inhibition preferentially induces cytotoxicity in breast cancer cells when compared to non-transformed cells, primarily through the induction of paraptosis. Paraptosis is a non-apoptotic cell death mechanism characterized by cytoplasmic vacuolation originating from the ER and/or mitochondria [##REF##11121041##10##]. Since apoptotic pathways are often compromised in drug-resistant cancer cells, leading to therapeutic failures [##REF##25936818##11##], it becomes imperative to explore strategies that promote alternative cell death mechanisms, such as paraptosis, especially in tumors that have progressed despite conventional apoptosis-targeted therapies. A comprehensive understanding of the mechanistic details of cancer cell death is crucial for devising effective therapeutic strategies. Importantly, paraptosis differs from apoptosis in that it does not involve the release of mitochondrial cytochrome c or caspase activation. While various factors, including proteasome inhibition [##REF##24625971##12##–##REF##30529689##14##] and thiol proteostasis impairment [##REF##30323190##15##, ##REF##30796201##16##], and Ca²⁺ imbalance [##REF##33585447##17##], have all been implicated in paraptosis, the detailed molecular basis remains to be fully elucidated.</p>", "<p id=\"Par4\">Our study has illuminated the pivotal role of VCP as a molecular target for inducing paraptosis in cancer cells. Mechanistically, VCP inhibition in breast cancer cells intensifies proteotoxic stress by restoring translation, thereby contributing to the occurrence of paraptosis. This process involves the activating transcription factor 4 (ATF4)/DNA damage-inducible transcript 4 (DDIT4) axis and the mechanistic target of rapamycin complex 2 (mTORC2)/Akt signaling pathways, which play a significant role in translational recovery and subsequent amplification of proteotoxic stress. Furthermore, we emphasize the critical role of eukaryotic translation initiation factor 3 subunit D (eIF3d) as a mediator for translational recovery in cancer cells exposed to proteotoxic stress. In contrast, when VCP is inhibited in non-transformed cells, it triggers translational suppression, ultimately alleviating proteotoxic stress and promoting cell survival. Considering the crucial role of hyperactive Akt, driven by oncogenes, in cancer cell survival and resistance to therapy [##REF##32973135##18##], identifying vulnerabilities within specific subsets of cancer cells can pave the way for tailored therapies targeting oncogene-addicted cancer cells.</p>", "<p id=\"Par5\">In summary, our work suggests that inducing paraptosis through VCP inhibition may open up novel therapeutic avenues for cancer cells characterized by hyperactive Akt.</p>" ]

[ "<title>Materials and methods</title>", "<title>Chemicals and antibodies</title>", "<p id=\"Par25\">Chemicals were purchased from various sources: eeyarestatin-1 (Eer1), LY294002, PD98059, U0126, SP600125, and SB203580 from Calbiochem (EDM Millipore Corp., Billerica, MA, USA); CB-5083 from Biovision (Milpitas, California, USA); NMS-873 from APExBIO (Houston, TX 77014, USA); z-VAD-fmk from R&amp;D Systems (Minneapolis, MN, USA); Necrostatin-1 (Nec-1), 3-methyladenine (3-MA), bafilomycin A1 (Bafilo), chloroquine (CQ), ferrostatin-1 (Ferro), and cycloheximide (CHX) from Sigma-Aldrich (St. Louis, MO, USA); PP242 and Torin1 from Selleckchem (Houston, TX 77014, USA); TRAIL from KOMA BIOTECH (Seoul, South Korea); MitoTracker-Red (MTR), tetramethylrhodamine methyl ester (TMRM), 4′,6-diamidino-2-phenylindole (DAPI), and propidium iodide (PI) from Molecular Probes (Eugene, OR, USA). The following antibodies were employed: VCP (#2648), GFP (#2555), p-eIF2α (#9721), eIF2α (#9722), CHOP (#2895), Nrf1(#8052), p-ERK1/2 (#9101), ERK (#9102), p-Akt (S473) (#9271), Akt (#9272), p-Akt (T308) (#9275), p-p70S6K (#9234), p70S6K (#2708), p-4EBP1 (#9451), 4EBP1 (#9452), Raptor (#2280), Rictor (#2114), and ATF4 (#11815) from Cell Signaling Technology (Danvers, MA, USA); β-actin (sc-47778), cytochrome C (sc-13156), Tom20 (sc-11415), ubiquitin (sc-8017), ATF4 (sc-200), and Mcl-1 (sc-819) from Santa Cruz (Dallas, TX, USA); α-Puromycin (MABE343) from Millipore (Billerica, MA, USA); Calnexin (CNX; PA5-19169) from Invitrogen (Carlsbad, CA, USA); Tim 23 (611222) from BD biotechnology (San Jose, CA, USA); Caspase-3 (ADI-AAP-113) from Enzo Life Sciences (Farmingdale, NY, USA); poly (ADP-ribose) polymerase (PARP; ab32071) and Bap31 (ab37120) from Abcam (Cambridge, UK); Ras (clone RAS10, #05-516) from Millipore; The secondary antibodies used were anti-rabbit IgG HRP (G-21234) and anti-mouse IgG HRP (G-21040) from Molecular Probes, Inc. (Eugene, OR, USA), and anti-rat IgG HRP from Sigma (A9037-1).</p>", "<title>Cell culture</title>", "<p id=\"Par26\">Human breast cancer cell lines, the MCF10A human mammary epithelial cell line, and HEK-293T cells were acquired from the American Type Culture Collection (ATCC, Manassas, VA, USA). All cell lines underwent regular mycoplasma contamination checks, and their authenticity was confirmed through standard morphological examination using a microscope. The cell cultures were as follows: MDA-MB 231 and BT549 cells in RPMI-1640 medium (GIBCO-BRL, Grand Island, NY, USA); T47D and MDA-MB 468 cells in DMEM with high glucose (Hyclone, Logan, UT, USA); MDA-MB 435 S cells in DMEM with low glucose (Hyclone); Hs578T cells in DMEM high-glucose medium supplemented with 10 μg/ml insulin (Sigma-Aldrich, St. Louis, MO, USA); and MCF10A cells in DMEM/F12 medium supplemented with 5% horse serum, insulin, human epidermal growth factor, hydrocortisone, and cholera toxin (Calbiochem).</p>", "<title>Cell viability assay</title>", "<p id=\"Par27\">All experiments were conducted in a low-glucose DMEM medium to exclude the effects of high glucose concentrations. Cells were cultured in 24-well plates (4×10⁴ cells per well), treated as indicated, fixed with methanol/acetone (1:1) at −20 °C for 5 min, washed with PBS, and stained with 1 μg/ml propidium iodide at room temperature for 10 min. Plates were imaged using an IncuCyte device (Essen Bioscience, Ann Arbor, MI, USA) and analyzed with IncuCyte ZOOM 2016B software. The IncuCyte program’s processing definition was set to identify attached (live) cells by their red-stained nuclei. The percentage of live cells was normalized to that of untreated control cells (100%).</p>", "<title>Immunoblot analysis</title>", "<p id=\"Par28\">Immunoblot analysis was performed as described previously [##REF##20036734##33##]. Representative results from at least three independent experiments are displayed, and unprocessed scans of immunoblots are provided as Source Data.</p>", "<title>Immunofluorescence microscopy</title>", "<p id=\"Par29\">Following treatments, cells were fixed with acetone/methanol (1:1) for 5 min at −20 °C or with 4% paraformaldehyde for 10 min at room temperature. Fixed cells were blocked in 5% BSA in PBS for 30 min and incubated overnight at 4 °C with primary antibodies [BAP31 (rabbit, ab37120 from Abcam), Tim23 (mouse, 611222 from BD), CNX (goat, PA5-19169 from Invitrogen), cytochrome <italic>c</italic> (mouse, sc-13156 from Santa Cruz), or Tom20 (mouse, sc-17764 from Santa Cruz)] diluted (1:500) in blocking buffer. Cells were then washed and incubated with diluted (1:1000) anti-mouse or anti-rabbit Alexa Fluor 488 or 594 (Molecular Probes) for 1 h at room temperature. After mounting on slides with ProLong Gold antifade mounting reagent (Molecular Probes), cells were observed with a K1-Fluo confocal laser scanning microscope (Nanoscope Systems, Daejeon, Korea) using an appropriate filter set (excitation bandpass, 488 nm; emission bandpass, 525/50).</p>", "<title>Transmission electron microscopy</title>", "<p id=\"Par30\">Cells were pre-fixed in Karnovsky’s solution (1% paraformaldehyde, 2% glutaraldehyde, 2 mM calcium chloride, 0.1 M cacodylate buffer, pH 7.4) for 2 h, post-fixed in 1% osmium tetroxide and 1.5% potassium ferrocyanide for 1 h, dehydrated with 50–100% alcohol, embedded in Poly/Bed 812 resin (Pelco, Redding, CA, USA), polymerized, and observed under an electron microscope (EM 902 A, Carl Zeiss, Oberkochen, Germany).</p>", "<title>Mouse xenograft studies</title>", "<p id=\"Par31\">Animal experiments adhered to the guidelines and regulations approved by the Institutional Animal Care and Use Committees of Asan Institute for Life Science (approval number 2017-12-091, granted on May 02, 2017). Female BALB/c nude mice (nu/nu, 5 weeks old; Japan SLC, Hamamatsu, Japan) were injected in the right flank with MDA-MB 435 S cells (5 × 10⁶ cells/mouse). Tumors were allowed to grow for 3 weeks until the average tumor volume reached 100–150 mm³. Mice were randomized into three groups (<italic>n</italic> = 5 per group) and received oral administration (O.A.; qd4/3off) of vehicle (PBS containing 0.25% DMSO), 100 mg/kg CB-5083, or 150 mg/kg CB-5083. Researchers were blinded to the group allocations during the experiment and when assessing the outcome. Tumor size was measured twice a week for 2 weeks, and tumor volume was calculated. On the 15^th day, mice were sacrificed, and the tumors were isolated, fixed in 4% paraformaldehyde, and embedded in paraffin. Tissue sections stained with H&amp;E were observed under a K1-Fluo microscope (Nanoscope Systems) and photographed using a complementary metal-oxide-semiconductor (CMOS) camera.</p>", "<title>Construction of plasmids encoding mCherry-VCP WT and mCherry-VCP QQ</title>", "<p id=\"Par32\">mCherry-VCP WT and mCherry-VCP QQ were generated from the plasmids VCP (wt)-AdditionEGFP (#23971) and VCP(DK0)-EGFP (VCP QQ) (#23974) (Addgene, Watertown, MA, USA), respectively, using the pENTRY/pDEST-mCherry system (Invitrogen). The fragments encoding VCP WT and VCP QQ were PCR amplified using the following primers: forward (ATGGCTTCTGGAGCCGATTCA) and reverse (GCCATACAGGTCATCVATCATT). These fragments were used to generate the pENTRY-VCP WT and pENTRY-VCP QQ vectors. Subsequently, mCherry-VCP WT and mCherry-VCP QQ were generated by recombining the pENTRY-VCP WT or pENTRY-VCP QQ vector with a pCS-mCherry vector utilizing the Gateway LR cloning system from Invitrogen.</p>", "<title>Generation and preparation of recombinant adenoviruses expressing VCP WT-EGFP and VCP QQ-EGFP</title>", "<p id=\"Par33\">Replication-incompetent adenovirus expressing VCP WT-EGFP or VCP QQ-EGFP were generated as described previously [##REF##28929492##74##, ##UREF##11##75##]. The DNA fragment encoding the VCP WT-EGFP- or VCP QQ-EGFP was excised from the respective plasmids (VCP WT-EGFP (#23971) and VCP(DKO)-EGFP (#23974), Addgene) using <italic>BamH</italic>1 and <italic>Bgl</italic>II restriction enzymes. These fragments were then ligated with the <italic>BamH</italic>1-digested adenoviral shuttle vector, pCA14. The resulting constructs, pCA14/VCP WT-EGFP and pCA14/VCP QQ-EGFP, were linearized by <italic>Pvu</italic>I digestion. The E1/E3-deleted adenoviral vector, dE1-RGD, was also linearized by <italic>BstB</italic>I digestion. These linearized vectors were co-transformed into <italic>E. coli</italic> BJ5183 competent cells for homologous recombination. The resulting adenoviral plasmids, dE1/VCP WT-EGFP and dE1/VCP QQ-EGFP, were digested with <italic>Pac</italic>I and transfected into 293 A cells. Finally, adenoviruses expressing VCP WT-EGFP or VCP QQ-EGFP were propagated, amplified in 293 A cells, and purified using cesium chloride density gradient centrifugation.</p>", "<title>Small interfering RNA-mediated gene silencing</title>", "<p id=\"Par34\">siRNA Negative Control (siNC) (Stealth RNAi^TM, 12935300) was purchased from Invitrogen (Carlsbad, CA, USA). VCP-targeted siRNAs were acquired from QIAGEN (Hilden Düsseldorf, NRW, Germany). These included siVCP #1 (target sequence AACAGCCATTCTCAAACAGAA), siVCP #2 (target sequence ATCCGTCGAGATCACTTTGAA), and siVCP #3 (target sequence AAGATGGATCTCATTGACCTA). CHOP (<italic>DDIT3</italic>) targeted siRNA (target sequence GAGCUCUGAUUGACCGAAUGGUGAA) was synthesized by Invitrogen. siATF4 (target sequences: CCACUCCAGAUCAUUCCUU, GGAUAUCACUGAAGGAGAU, and GUGAGAAACUGGAUAAGAA, sc-35112) was obtained from Santa Cruz. The siRNA oligonucleotides were annealed and transfected into cells using the RNAiMAX reagent (Invitrogen) following the manufacturer’s instructions. Western blotting was performed to confirm successful siRNA-mediated knockdown.</p>", "<title>Lentivirus-mediated shRNA transduction</title>", "<p id=\"Par35\">To generate the lentiviral vectors encoding short hairpin RNA (shRNA), the pLKO.1 neo plasmid (#13425: Addgene, Cambridge, MA, USA) was digested using <italic>Age</italic>I and <italic>EcoR</italic>I. Two oligonucleotide strands were mixed and incubated at 95 °C for 4 min, and then at 70 °C for 10 min before slowly cooling to room temperature. The annealed oligo pair was ligated into the digested pLKO.1 neo plasmid using T4 ligase at 20 °C for 16 h. The sequences of the oligonucleotides used to knock down each target gene are listed in Supplementary Table ##SUPPL##0##1##. To produce the lentivirus containing each plasmid, HEK-293T cells were transfected with the lentiviral vector in the presence of pMD2.G/psPAX2.0 using linear polyethyleneimine (MW2,500; Polysciences, Warrington, PA, USA). Following transfection, the virus-containing supernatants were filtered, combined with polybrene, and used to infect MDA-MB 435 S cells. qRT-PCR and Western blot analyses were performed to validate the efficiency of transfection. The sequences of the shRNA are provided in Supplementary Table ##SUPPL##0##1##.</p>", "<title>Quantitative Real-Time RT-PCR (qRT-PCR)</title>", "<p id=\"Par36\">Total RNA was extracted using the TRIzol® reagent (Invitrogen). Subsequently, cDNA was synthesized using 1 μg of total RNA with the M-MLV cDNA Synthesis kit (EZ006S; Enzynomics, Daejeon, Korea). Quantitative real-time polymerase chain reaction (qRT-PCR) was conducted using a Bio-Rad Real-Time PCR System (Bio-Rad, Richmond, CA, USA). The results were analyzed using the 2^–ΔΔCt method [##REF##11846609##76##]. Primers for qRT-PCR are listed in Supplementary Table ##SUPPL##0##2##.</p>", "<title>Establishment of MCF10A cell lines stably expressing HRas^G12V and KRas^G12V</title>", "<p id=\"Par37\">To establish cell lines expressing <italic>HRas</italic>^{<italic>G12V</italic>} and <italic>KRas</italic>^{<italic>G12V</italic>}, GP2-293 packaging cells were co-transfected with pVSV-G (#631530: Clontech, Mountain View, CA, USA) along with either pBABE-puro, pBABE puro H-Ras V12, or pBABE puro K-Ras V12 (#9051, #9052, or #1764: Addgene) using a CalPhos™ Mammalian Transfection Kit (#631312, Clontech) following the manufacturer’s instructions. Retroviral supernatants were used to transduce MCF10A cells in the presence of polybrene (5 mg/mL; Millipore, Burlington, MA, USA). Transduced cells were selected with puromycin (Invivogen, San Diego, CA, USA) for 3 weeks. Selected single cells were isolated, and the expression of HRas^G12V and KRas^G12V was confirmed by Western blotting.</p>", "<title>Morphological examination of ER and mitochondria</title>", "<p id=\"Par38\">Cell lines stably expressing fluorescence in the ER lumen (YFP-ER cells), ER membrane (Sec61β-GFP cells), or mitochondria (YFP-Mito cells) [##REF##30796201##16##] were used for morphological studies. YFP-ER cells were stained with 100 nM MitoTracker-Red (MTR) for 10 min to observe both the ER and mitochondria. Confocal microscopy was performed using a K1-Fluo confocal laser scanning microscope (Nanoscope Systems, Daejeon, Korea) with an appropriate filter set (excitation bandpass, 488 nm; emission bandpass, 525/50).</p>", "<title>Analysis of protein synthesis by puromycin labeling</title>", "<p id=\"Par39\">Protein synthesis was monitored using the SUnSET method [##REF##19305406##40##]. Briefly, newly synthesized peptides in cultured cells were labeled by adding 10 μg/ml puromycin for 10 min before cell collection. Whole-cell extracts were prepared for Western blotting using an anti-puromycin antibody (Millipore) and anti-mouse IgG-HRP-linked antibody (Molecular Probes). Fold changes in the protein levels of interest compared to β-actin were calculated following densitometric analysis.</p>", "<title>Statistical analysis</title>", "<p id=\"Par40\">All experiments were repeated at least three times. Data were presented as mean ± standard deviation. Statistical analysis was performed using GraphPad Prism 9 (Graph Pad Software Inc, San Diego, CA, USA). The normality of data was assessed using Kolmogorov–Smirnov tests, and equal variance was assessed using Bartlett’s test. For normally distributed data, statistical differences were determined using analysis of variance (ANOVA), followed by the Bonferroni multiple comparison test. For all tests, <italic>p</italic> &lt; 0.05 was considered significant (ns not significant, *<italic>p</italic> &lt; 0.05, **<italic>p</italic> &lt; 0.01, ***<italic>p</italic> &lt; 0.001, **** <italic>p</italic> &lt; 0.0001).</p>" ]

[ "<title>Results</title>", "<title>VCP is a molecular target of paraptosis</title>", "<p id=\"Par6\">While various natural products and chemicals have been shown to induce paraptosis [##REF##26802901##19##, ##REF##31306970##20##], the molecular mechanisms underlying this process remain unclear. To identify potential molecular targets for inducing paraptosis, we utilized the Connectivity Map (CMap, <ext-link ext-link-type=\"uri\" xlink:href=\"http://cleu.io/cmap\">http://cleu.io/cmap</ext-link>) [##REF##29195078##21##], a database that links pharmacological drugs and genomic data. The CMap dataset includes transcriptome information for 17 paraptosis-inducing chemicals, such as withaferin A [##REF##28033383##22##], pyrrolidine dithiocarbamate (PDTC) [##REF##29329420##23##], 15-deoxy-Δ^12,14-prostaglandin J₂ (15d-PGJ₂) [##REF##19448671##24##], and xanthohumol [##REF##28415750##25##]. We sought genetic perturbations that elicited transcriptional alterations similar to those induced by these paraptosis-inducing chemicals. VCP knockdown emerged as the top-ranked perturbagen, with actions resembling those of the examined paraptosis inducers (Supplementary Fig. ##SUPPL##0##1##).</p>", "<p id=\"Par7\">Given VCP’s crucial role in proteostasis [##REF##29153394##4##–##REF##28475898##7##] and the established connection between proteostatic disruption and paraptosis [##REF##28798402##13##, ##REF##30796201##16##, ##REF##26802901##19##, ##REF##23538442##26##–##UREF##4##28##], our investigation aimed to determine whether VCP knockdown alone could trigger paraptosis. We observed that VCP knockdown, executed using three independent siRNAs, led to cell death accompanied by extensive vacuolation in MDA-MB 435 S cells (Fig. ##FIG##0##1a, b##). Similar results were obtained through the adenovirus-mediated expression of a dominant-negative VCP mutant (VCP QQ; VCP^{E305Q, E578Q}) [##REF##12847084##29##] fused to an enhanced green fluorescent protein (EGFP) (Fig. ##FIG##0##1c, d##) or treatments with various VCP inhibitors, including eeyarestatin-1 (Eer1, an ER membrane-binding domain-containing VCP inhibitor) [##REF##21124757##30##], CB-5083 (an inhibitor of the D2 ATPase domain of VCP) [##REF##26555175##31##], and NMS-873 (an allosteric VCP inhibitor) [##REF##23892893##32##] (Fig. ##FIG##0##1e, f##). Next, we explored whether VCP inhibition induces vacuolation originating from the ER and/or mitochondria, a hallmark of paraptosis. To visualize these organelles, we utilized YFP-ER [##REF##20036734##33##], Sec61β-GFP [##REF##30796201##16##], and YFP-Mito cells [##REF##20036734##33##], which exhibit fluorescence in the ER lumen, ER membrane, and mitochondrial matrix, respectively, along with MitoTracker-Red (MTR) staining. We found that VCP siRNAs (Fig. ##FIG##0##1g##), mCherry-fused VCP QQ mutant (Fig. ##FIG##0##1h##), and three VCP inhibitors (Fig. ##FIG##0##1i##) commonly induced significant dilations of the ER and mitochondria. In particular, Eer1 induced the most dramatic dilation among the tested VCP inhibitors. Electron microscopy further revealed megamitochondria (giant mitochondria) and ER-derived vacuoles in Eer1-treated cells (Fig. ##FIG##0##1j##). Time-lapse imaging in Eer1-treated YFP-ER and YFP-Mito cells confirmed ER or mitochondria swelling and fusion (Supplementary Fig. ##SUPPL##0##2##).</p>", "<p id=\"Par8\">Subsequently, we investigated the involvement of apoptosis in the anticancer effect of VCP inhibition. Unlike the release of cytochrome <italic>c</italic> from mitochondria observed with the apoptosis-inducer tumor necrosis factor-related apoptosis-inducing ligand (TRAIL), Eer1 or CB-5083 treatment, VCP knockdown, or VCP QQ-EGFP expression resulted in cytochrome <italic>c</italic> accumulation within or at the periphery of dilated mitochondria (Fig. ##FIG##1##2a##). Furthermore, caspase-3 and PARP cleavage, which was induced by TRAIL, were not notably observed by Eer1 or CB-5083 treatment (Fig. ##FIG##1##2b##). While z-VAD-fmk (a pan-caspase inhibitor) effectively blocked TRAIL-induced cell death and apoptotic morphologies (Fig. ##FIG##1##2c, d##), it did not affect vacuolation-associated cell death induced by VCP inhibitors (Fig. ##FIG##1##2e, f##). Furthermore, inhibitors of necroptosis (necrostatin-1; Nec1), ferroptosis (ferrostatin-1; Ferro), and an early-phase autophagy inhibitor (3-methyladenine; 3-MA) did not attenuate the cytotoxicity of VCP inhibitors. In contrast, late-stage autophagy inhibitors (bafilomycin A; Bafilo and chloroquine; CQ) enhanced it (Fig. ##FIG##1##2e, f##). In contrast, cycloheximide (CHX), known to block paraptosis [##REF##11121041##10##], effectively inhibited mitochondria- and ER-derived vacuolation and cell death induced by the VCP inhibitors (Fig. ##FIG##1##2g–i##). Together, these results suggest that VCP inhibition predominantly induces paraptosis as a cell death mechanism in cancer cells.</p>", "<title>VCP inhibition triggers paraptosis in various breast cancer cell lines and in vivo xenograft mouse models, sparing non-transformed cells</title>", "<p id=\"Par9\">We further examined the impact of VCP inhibition on other breast cancer cell lines. Treatment with Eer1 or CB-5083 induced cell death accompanied by vacuolation in several breast cancer cells, including BT549, MDA-MB 231, Hs578T, MDA-MB468, and T47D cells (Fig. ##FIG##2##3a, b##). However, both Eer1 and CB-5083 displayed considerably lower cytotoxicity towards MCF10A cells, a non-tumorigenic breast epithelial cell line, without affecting their morphology (Fig. ##FIG##2##3a, b##). Immunocytochemistry of calnexin (CNX), Bap31 (an ER marker protein), and Tim23 (a mitochondrial marker protein) confirmed ER- and mitochondria-derived vacuolation in Eer1- or CB-5083-treated cancer cells (Fig. ##FIG##2##3b, c##).</p>", "<p id=\"Par10\">To assess the in vivo effect of VCP inhibition, nude mice xenografted with MDA-MB 435 S cells were orally treated with saline or CB-5083. CB-5083 dose-dependently reduced tumor volume and weight without causing weight loss in mice (Fig. ##FIG##2##3d–g##). Hematoxylin and eosin (H&amp;E) staining revealed severe vacuolation in the tumor tissues of CB-5083-treated mice (Fig. ##FIG##2##3h##). These results indicate that targeting VCP induces paraptosis both in vitro and in vivo and selectively affects cancer cells over non-transformed cells.</p>", "<title>Oncogene-driven Akt activation sensitizes non-transformed cells to VCP inhibition</title>", "<p id=\"Par11\">We explored whether the preferential sensitivity of cancer cells to VCP inhibition is linked to oncogenic activation. To investigate this, we introduced oncogenes such as KRas^G12V and HRas^G12V into non-transformed cells and examined their response to Eer1 treatment. HRas^G12V-expressing cells (HRas^G12V/MCF10A) showed significantly greater sensitivity to Eer1-induced cytotoxicity than KRas^G12V-expressing cells (KRas^G12V/MCF10A) or Mock/MCF10A cells (Fig. ##FIG##3##4a##). Interestingly, Eer1 treatment induced cell death accompanied by ER and mitochondrial dilations only in HRas^G12V-expressing cells (Fig. ##FIG##3##4a–c##). CB-5083 treatment also induced a similar dilation of the ER and mitochondria in HRas^G12V-expressing cells but not in Mock/MCF10A cells (Fig. ##FIG##3##4c##). Next, we investigated the downstream signaling pathways, including RAF/MEK/ERK and phosphatidylinositol-3-kinase (PI3K)/Akt/mTOR, which are associated with mutant Ras [##REF##32648136##34##]. ERK activation was observed in cells expressing either HRas^G12V or KRas^G12V and further enhanced by Eer1 treatment (Fig. ##FIG##3##4d##). However, Akt activity was markedly increased only in HRas^G12V-expressing cells and further enhanced by Eer1. Inhibition of PI3K/Akt using LY294002 (a PI3K/Akt inhibitor) or MK-2206 (an Akt inhibitor), but not inhibition of MEK (using PD98059 or U0126), blocked Eer1-induced paraptosis in HRas^G12V/MCF10A (Fig. ##FIG##3##4e, f##). These results suggest the critical role of the PI3K/Akt pathway in sensitizing non-transformed cells to VCP inhibition. Next, we further examined the involvement of mTOR complex 1 (mTORC1) and mTOR complex 2 (mTORC2) in VCP inhibition-induced paraptosis. Inhibitors targeting both mTORC1 and mTORC2, such as PP242 and Torin1 (mTORC1/2 inhibitors), but not the mTORC1-specific inhibitor rapamycin, effectively inhibited Eer1-induced paraptosis in HRas^G12V/MCF10A cells (Fig. ##FIG##3##4e, f##). Similar results were obtained in HRas^G12V/MCF10A undergoing CB-5083-induced paraptosis (Fig. ##FIG##3##4f##). This was further supported by the reduction in S6 and eIF4E-binding protein 1 (4E-BP1) phosphorylation, indicative of mTORC1 activity [##REF##15314020##35##], but the increase in Akt phosphorylation at S473, indicative of mTORC2 activity [##REF##15314020##35##], in Eer1-treated HRas^G12V/MCF10A cells (Fig. ##FIG##3##4d##). Collectively, these findings suggest differential roles for mTORC1 or mTORC2 in VCP inhibition-mediated paraptosis.</p>", "<title>Activation of mTORC2/Akt contributes to VCP inhibition-mediated paraptosis</title>", "<p id=\"Par12\">We investigated whether mTOR signaling plays a similar role in cancer cells undergoing VCP inhibition-induced paraptosis. In MDA-MB 435 cells, Eer1 or CB-5083 treatment led to progressive Akt phosphorylation while reducing 4E-BP1 phosphorylation (Fig. ##FIG##4##5a##), indicating mTORC2 activation and mTORC1 inhibition. Knockdown of Rictor (a component of mTORC2) but not that of Raptor (a component of mTORC1) significantly attenuated Eer1- or CB-5083-mediated cytotoxicity and vacuolation (Fig. ##FIG##4##5b, c##). Conversely, overexpression of mTOR and constitutively active Akt (myristoylated Akt (Myr-Akt)) potentiated Eer1- or CB-5083-induced cytotoxicity and vacuolation (Fig. ##FIG##4##5d, e##). Pretreatment of cells with PP242 or LY294002 effectively inhibited Eer1- or CB-5083-induced paraptosis (Fig. ##FIG##4##5f–h##). These results suggest that hyperactive mTORC2/Akt signaling contributes significantly to VCP inhibition-mediated paraptosis.</p>", "<title>The ATF4/DDIT4 axis plays a crucial role in Akt activation and subsequent paraptosis upon VCP inhibition</title>", "<p id=\"Par13\">Proteotoxic stress caused by proteostatic disruption triggers the integrated stress response (ISR) [##REF##28212730##36##], converging on the phosphorylation of eukaryotic initiation factor 2α (eIF2α). This reduces cap-dependent translation while promoting the translation of specific mRNAs, including activating transcription factor 4 (ATF4) [##REF##27629041##37##]. In our study, VCP inhibitors, VCP knockdown, and VCP QQ mutant expression consistently upregulated poly-ubiquitinated proteins and the components of ISR, including phosphorylated eIF2α (p-eIF2α), ATF4, and C/EBP homologous protein (CHOP), in MDA-MB 435 S cells (Fig. ##FIG##5##6a##). Among these, ATF4 was found to be crucially associated with VCP inhibition-mediated paraptosis. ATF4 knockdown effectively inhibited paraptosis induced by Eer1, CB-5083, NMS-873 (Fig. ##FIG##5##6b–e##), VCP knockdown (Supplementary Fig. ##SUPPL##0##3a–c##), or VCP QQ mutant expression (Supplementary Fig. ##SUPPL##0##3d–f##). Knockdown of ATF4 but not CHOP effectively inhibited Eer1-induced paraptosis (Fig. ##FIG##5##6b–e##). These results underscore the role of ATF4 in VCP inhibition-mediated paraptosis. To further explore the role of ATF4 in this process, we performed the transcriptomic analysis in MDA-MB 435 S cells transfected with siNC (non-targeted siRNA) or siATF4 and in the absence or presence of Eer1 (Supplementary Fig. ##SUPPL##0##4a, b##). We identified genes that were responsive to Eer1 (fold change of siNC+Eer1/siNC-Eer1 &gt; 2) and highly dependent on ATF4 (fold change of siATF4+Eer1/siNC+Eer1 &lt; -2) (Supplementary Fig. ##SUPPL##0##4c##). Among these ATF4 downstream targets, we further investigated the role of DNA-damage-inducible transcript 4 (DDIT4), which is known to be associated with mTORC1 inhibition and mTORC2/Akt activation [##REF##23528835##38##, ##REF##34862383##39##]. Our findings revealed that Eer1 upregulated DDIT4, along with ATF4 upregulation and Akt activation (Fig. ##FIG##5##6f##). ATF4 knockdown inhibited Eer1-induced DDIT4 upregulation at the mRNA and protein levels, as well as Akt activation (Fig. ##FIG##5##6g, h##). Additionally, DDIT4 knockdown effectively inhibited Eer1-induced Akt activation (Fig. ##FIG##5##6i##). Furthermore, DDIT4 knockdown significantly blocked Eer1-induced cell death and vacuolation (Fig. ##FIG##5##6j, k##). These results suggest that the ATF4/DDIT4 axis, particularly DDIT4, may mediate Akt activation in VCP inhibition-mediated paraptosis (Fig. ##FIG##5##6l##).</p>", "<title>mTORC2/Akt-mediated translational recovery contributes to cancer-selective cytotoxicity of VCP inhibition</title>", "<p id=\"Par14\">Our investigation into the differential vulnerability of cancer and non-transformed cells to VCP inhibition revealed distinct responses in the ISR between the two cell types. While Eer1 induced robust and sustained poly-ubiquitinated protein accumulation and p-eIF2α phosphorylation in MDA-MB 435 S cells, these responses were delayed and weaker in MCF10A cells, lacking ATF4 upregulation (Fig. ##FIG##6##7a##). The protein synthesis, as assessed by the SUnSET assay [##REF##19305406##40##], was strongly suppressed in Eer1-treated MCF10A cells but showed initial reduction followed by recovery in MDA-MB 435 S cells, concurrent with ATF4 upregulation (Fig. ##FIG##6##7a##). Inhibition of translation with CHX at 4 h post-Eer1 treatment effectively blocked cell death and vacuolation (Fig. ##FIG##6##7b–d##), emphasizing the importance of translational recovery. This translational recovery and ATF4/CHOP upregulation were also observed in HRas^G12V/MCF10A cells but not in Mock/MCF10A cells (Fig. ##FIG##6##7e##), correlating with their sensitivity to VCP inhibition-induced paraptosis. These results suggest that effective translational suppression may prevent the death of non-transformed cells by alleviating VCP inhibition-mediated proteotoxic stress. However, in cancer cells with hyperactive Akt (possibly driven by oncogenic signals such as HRas^G12V), the translational recovery under VCP inhibition-mediated proteotoxic stress may enhance proteotoxicity by increasing the accumulation of misfolded proteins in the ER and mitochondria, leading to paraptosis. Next, we investigated whether the ATF4/DDIT4 axis and mTORC2/Akt signals are linked to translational dysregulation in VCP inhibition-mediated paraptosis. Either ATF4 or DDIT4 knockdown inhibited Eer1-induced translational recovery without affecting eIF2α phosphorylation (Fig. ##FIG##6##7f, g##), suggesting that the ATF4/DDIT4 axis may contribute to VCP inhibition-mediated paraptosis by positively affecting Akt activation and translational recovery. In addition, knockdown of Rictor but not Raptor potently inhibited Eer1-induced Akt activation, translational recovery, and ATF4/CHOP upregulation (Fig. ##FIG##6##7h##). Similar results were obtained by PP242 or LY294002 pretreatment (Fig. ##FIG##6##7i##). These results indicate the importance of the ATF4/DDIT4 axis and mTORC2/Akt signal in VCP inhibition-mediated paraptosis. Interestingly, mTORC2/Akt inhibition suppressed Eer1-induced ATF4 upregulation (Fig. ##FIG##6##7h, i##), and the knockdown of ATF4 or DDIT4 inhibited Eer1-induced Akt activation (Fig. ##FIG##5##6n, o##). Cross-regulation between the ATF4/DDIT4 axis and mTORC2/Akt signaling upon VCP inhibition suggests their cooperative role in translational recovery and proteotoxic stress enhancement.</p>", "<title>eIF3d may critically contribute to translational recovery in VCP inhibition-mediated paraptosis</title>", "<p id=\"Par15\">The mechanism underlying translation recovery in VCP inhibition-mediated paraptosis was further explored. It is known that under stress conditions, mTORC1-dependent cap-dependent mRNA translation is suppressed [##REF##22552098##41##]. However, alternative mechanisms have been proposed to allow protein synthesis to adapt to various stressors. These mechanisms include eukaryotic translation initiation factor 3 subunit D (eIF3d), a subunit of the eIF3 complex with cap-binding activity) [##REF##27462815##42##–##REF##35508137##45##], as well as the m⁶A-pathway, which involves methyltransferase-like 3 (METTL3) [##REF##27117702##46##], ATP Binding Cassette Subfamily F Member 1 (ABCF1) [##REF##29107534##47##], and YTH N6-Methyladenosine RNA Binding Protein F1 (YTHDF1)) [##REF##31996915##48##]. Remarkably, our findings revealed that eIF3d knockdown had a significant impact on Eer1-induced paraptosis (Fig. ##FIG##7##8a–d##), effectively inhibiting translational recovery (Fig. ##FIG##7##8e##). Intriguingly, this effect was not observed with knockdown of eIF4E, METTL3, ABCF1, or YTHDF1 (Fig. ##FIG##7##8a–d##). Furthermore, eIF3d knockdown resulted in the enhancement of eIF2α phosphorylation (Fig. ##FIG##7##8e##). Notably, eIF3d knockdown also suppressed the upregulation of ATF4 at the protein level without downregulating ATF4 mRNA levels (Fig. ##FIG##7##8e, f##). These findings underscore the pivotal role of eIF3d in facilitating translational recovery during VCP inhibition-induced paraptosis.</p>", "<p id=\"Par16\">In summary, the selective cytotoxicity of VCP inhibition towards cancer cells can be attributed to the disruption of proteotoxic stress mitigation pathways involving the ATF4/DDIT4 axis and hyperactive mTORC2/Akt signaling. This process is further modulated by eIF3d-mediated translational recovery, which enhances proteotoxicity selectively in cancer cells undergoing paraptosis upon VCP inhibition (Fig. ##FIG##7##8g##).</p>" ]

[ "<title>Discussion</title>", "<p id=\"Par17\">Identifying cancer-selective targets and understanding their underlying mechanisms are pivotal in developing effective cancer therapies. Among these potential targets, VCP has emerged as both a prognostic biomarker and a prospective therapeutic target in cancer [##REF##27536557##8##, ##UREF##3##9##]. Our study introduces a novel perspective by highlighting VCP’s central role as a molecular target in paraptosis, a distinctive form of programmed cell death. Importantly, our findings demonstrate that inhibiting VCP leads to preferential cell death in breast cancer cells compared to non-transformed cells, primarily through the induction of paraptosis. Genetic and pharmacological intervention of VCP commonly elicits the morphological features of paraptosis, reduction in cell viability, and ISR, demonstrating the crucial role of ATF4 in paraptosis. These results suggest that the gene-level intervention of VCP has the same mechanism of regulating paraptosis as that of VCP inhibitors.</p>", "<p id=\"Par18\">The impairment of VCP-mediated ERAD and MAD processes appears central to this phenomenon. Our experiments revealed that Eer1, a VCP inhibitor, led to increased protein levels of the ERAD substrates (e.g., nuclear respiratory factor 1 (Nrf1) [##REF##21911472##49##] and receptor accessory protein 5 (REEP5)) [##REF##29514927##50##] and the MAD substrates (e.g., myeloid leukemia 1 (Mcl-1) [##REF##27913212##51##] and mitofusin 2 (Mfn2) [##UREF##5##52##]) (see Supplementary Fig. ##SUPPL##0##5##). Inhibition of VCP may result in the progressive accumulation of misfolded proteins within the ER and mitochondria, leading to osmotic pressure changes and subsequent organelle swelling [##REF##16966435##53##]. The fusion of the ER compartments induced by VCP inhibition may disrupt protein synthesis, folding, and transport, further exacerbating proteotoxic stress. Additionally, mitochondrial swelling and fusion at the early phase may act as an adaptive response to maintain mitochondrial membrane integrity [##REF##33585447##17##]. However, excessive megamitochondrial expansion can compromise membrane potential, deplete cellular energy, and ultimately drive paraptotic cell death [##REF##33585447##17##, ##REF##26802901##19##]. Targeting these two organelles during paraptosis represents a unique and promising therapeutic strategy against solid tumors [##REF##26802901##19##].</p>", "<p id=\"Par19\">Proteasomal and VCP inhibitors both induce proteostatic stress [##REF##27536557##8##, ##REF##25385277##54##]. While proteasome inhibitors (PIs) have shown clinical utility in hematological malignancies by inducing apoptosis [##REF##11306489##55##], their effectiveness against solid tumors has been limited [##REF##25385277##54##, ##REF##25303058##56##]. In contrast, various VCP inhibitors have demonstrated potent anti-tumor activities across various hematologic and solid tumor models [##REF##36640759##57##]. The proteasome may not efficiently process ubiquitinated substrates, including those associated with ERAD, MAD, and chromatin-associated degradation, without the assistance of VCP [##UREF##6##58##–##UREF##7##61##]. Therefore, the preferential targeting of solid tumors by VCP inhibitors, compared to PIs, may be attributed to the broader defects in the UPS than those by PIs [##REF##36640759##57##]. Additionally, compared to PIs, VCP inhibitors impact multiple cellular processes, including autophagy [##REF##22450227##62##], endosomal trafficking [##REF##22450227##62##, ##REF##37756124##63##], DNA repair and genome stability [##UREF##8##64##], membrane fusion [##UREF##9##65##], non-proteolytic disassembly of protein phosphatase-1 complex [##REF##26720340##66##, ##REF##37264685##67##], and regulation of PD-L1 expression [##REF##37865914##68##], possibly contributing to their efficacy in solid tumor models [##REF##26555175##31##]. Among the developed VCP inhibitors, ATPase competitive inhibitors, CB-5083 and CB-5339, have reached clinical trials (<ext-link ext-link-type=\"uri\" xlink:href=\"https://clinicaltrials.gov\">https://clinicaltrials.gov</ext-link> trial number NCT02243917 &amp; NCT04372641) by demonstrating effective anti-tumor activity across various tumor models [##REF##26555175##31##, ##REF##28878026##69##, ##REF##35876604##70##]. Understanding the resistance mechanisms is crucial for developing more effective inhibitors or combination therapies. Resistance to VCP inhibitors, primarily attributed to specific mutations in the D2 ring ATPase domain and the linker region connecting the D1 and the D2 domains of VCP, presents a clinical challenge [##REF##36640759##57##].</p>", "<p id=\"Par20\">Recent findings from our laboratory have revealed distinct responses to PI treatment in different cell types [##REF##35114585##27##]. Multiple myeloma (MM) cells were highly susceptible to bortezomib (Bz), inducing apoptosis, while breast cancer cells exhibited resistance. Interestingly, the application of ISRIB, a small molecule known to restore eIF2B-mediated translation during the integrated stress response, protected MM cells from apoptosis while enhancing Bz-mediated cytotoxicity in breast cancer cells by inducing paraptosis. These results suggest that enhancing translation and inducing paraptosis may effectively overcome PI resistance in solid tumor cells.</p>", "<p id=\"Par21\">The present study further underscores that the difference in proteotoxic stress responses between cancer and normal cells could be exploited for therapeutic purposes. Sustained translation attenuation under VCP inhibition can alleviate proteotoxic stress and support the survival of non-transformed cells. However, translation recovery following initial suppression in cancer cells enhances proteotoxic stress, ultimately leading to paraptotic cell death.</p>", "<p id=\"Par22\">The PI3K/Akt/mTOR signaling cascade is hyperactivated in many solid tumors, including breast cancer, contributing to cancer progression and resistance to pro-apoptotic therapies [##REF##24782981##71##, ##REF##33790792##72##]. However, targeting this pathway has shown limited efficacy due to feedback regulation and interference with other signaling pathways [##UREF##10##73##]. Our study reveals that VCP inhibition leads to selective mTORC2 activation and mTORC1 inhibition in cancer cells. In contrast, non-transformed cells exhibit mTORC1 activation without mTORC2 induction upon VCP inhibition. Additionally, our findings demonstrated that mTORC2 activation is essential for the selective action of VCP inhibitors in cancer cells. Inhibition of mTORC2/Akt signals effectively attenuates translational recovery and paraptosis induced by VCP inhibition. Cancer cells with hyperactive mTORC2/Akt signaling are more vulnerable to VCP inhibition, making VCP an attractive target in this context.</p>", "<p id=\"Par23\">The ATF4/DDIT4/mTORC2/Akt signals are known to be required for cell survival under energy-related stresses, such as amino acid deprivation [##REF##34862383##39##]. In our study, the ATF4/DDIT4 axis contributed to Akt activation, translational recovery, and paraptosis upon VCP inhibition. Additionally, eIF3d critically contributed to translational recovery, leading to ATF4 upregulation and enhancing cancer cells’ sensitivity. Therefore, we speculate that in response to proteotoxic stress, such as VCP inhibition, the ATF4/DDIT4/mTORC2/Akt signals and eIF3d may shift the cell fate towards paraptotic cell death.</p>", "<p id=\"Par24\">In conclusion, our study unveils the potential of VCP as a therapeutic target in cancer, emphasizing the selective vulnerability of cancer cells to VCP inhibition-induced paraptosis. This strategy holds promise for overcoming resistance to pro-apoptotic therapies in solid tumors driven by oncogenic PI3K/Akt/mTOR signaling.</p>" ]

[]

[ "<p id=\"Par1\">Valosin-containing protein (VCP)/p97, an AAA+ ATPase critical for maintaining proteostasis, emerges as a promising target for cancer therapy. This study reveals that targeting VCP selectively eliminates breast cancer cells while sparing non-transformed cells by inducing paraptosis, a non-apoptotic cell death mechanism characterized by endoplasmic reticulum and mitochondria dilation. Intriguingly, oncogenic HRas sensitizes non-transformed cells to VCP inhibition-mediated paraptosis. The susceptibility of cancer cells to VCP inhibition is attributed to the non-attenuation and recovery of protein synthesis under proteotoxic stress. Mechanistically, mTORC2/Akt activation and eIF3d-dependent translation contribute to translational rebound and amplification of proteotoxic stress. Furthermore, the ATF4/DDIT4 axis augments VCP inhibition-mediated paraptosis by activating Akt. Given that hyperactive Akt counteracts chemotherapeutic-induced apoptosis, VCP inhibition presents a promising therapeutic avenue to exploit Akt-associated vulnerabilities in cancer cells by triggering paraptosis while safeguarding normal cells.</p>", "<title>Subject terms</title>" ]

[ "<title>Supplementary information</title>", "<p>\n\n\n</p>" ]

[ "<title>Supplementary information</title>", "<p>The online version contains supplementary material available at 10.1038/s41419-024-06434-x.</p>", "<title>Author contributions</title>", "<p>DML, IYK, HJL, MJS, HIL, MYC, JHJ, YHC, SSP, and MY performed experiments and analyse the data. GY, SYJ, EKC, and COY provided technical and material support. DML, EK, and KSC performed study design and writing. All the authors read and approved the manuscript.</p>", "<title>Funding</title>", "<p>This work was supported by the National Research Foundation of Korea (NRF) grants funded by the Korean government (MSIP), Mid-career Research Program (NRF-2023R1A2C2006580) &amp; 2020R1A6A1A03043539)).</p>", "<title>Data availability</title>", "<p>All data and information concerning this study will be provided upon request.</p>", "<title>Competing interests</title>", "<p id=\"Par41\">The authors declare no competing interest.</p>", "<title>Ethics Declaration</title>", "<p id=\"Par42\">The institutional animal care and use committee of the Asan Institute for Life Science approved animal protocols.</p>" ]

[ "<fig id=\"Fig1\"><label>Fig. 1</label><caption><title>VCP impairment induces paraptotic cell death in MDA-MB 435 S cells.</title><p><bold>a</bold>, <bold>b</bold> MDA-MB 435 S cells were transfected with either a negative control siRNA (siNC) or a VCP-targeting siRNA (siVCP) and incubated with fresh medium for 48 h. <bold>c</bold>, <bold>d</bold> MDA-MB 435 S cells infected with adenovirus encoding VCP WT-EGFP or VCP QQ-EGFP for 72 h. <bold>e</bold>, <bold>f</bold> MDA-MB 435 S cells were treated with the indicated concentrations of VCP inhibitors for 24 h <bold>e</bold> or with 10 μM Eer1, 2 μM CB-5083, or 5 μM NMS-873 for 12 h <bold>f</bold>. <bold>g</bold>, <bold>h</bold> YFP-ER, Sec61β-GFP, and YFP-Mito cells were transfected with siNC or siVCP for 48 h <bold>g</bold> or with VCP WT-mCherry or VCP QQ-mCherry for 36 h <bold>h</bold>. <bold>i</bold> YFP-ER cells treated with 10 μM Eer1, 2 μM CB-5083, and 5 μM NMS were stained with MTR. <bold>a</bold> (left), <bold>c</bold> (left), <bold>e</bold> Cell viability was assessed using an IncuCyte system, as described in the Materials and Methods. The percentage of live cells was normalized to that of untreated cells (100%). Cell viability data are presented as the means ± SD of three independent experiments. <italic>n</italic> = 10. The <italic>p</italic>-values in panels <bold>a</bold>, <bold>c</bold>, and <bold>e</bold> were calculated by one-way ANOVA. *<italic>p</italic> &lt; 0.05, **<italic>p</italic> &lt; 0.01, **<italic>p</italic> &lt; 0.001, ****<italic>p</italic> &lt; 0.0001. ns, not significant. <bold>a</bold> (right), <bold>c</bold> (right) Western blotting of VCP using β-actin as a loading control. <bold>b</bold>, <bold>f</bold> Representative phase-contrast microscopic images. <bold>d</bold> Phase-contrast/fluorescence microscopy. <bold>g</bold>–<bold>i</bold> Confocal microscopy. <bold>j</bold> Electron microscopy of cells treated with 10 μM Eer1.</p></caption></fig>", "<fig id=\"Fig2\"><label>Fig. 2</label><caption><title>Apoptosis, necroptosis, ferroptosis, or autophagy may not critically contribute to the anticancer effect of VCP inhibition.</title><p><bold>a</bold> Immunocytochemistry of cytochrome <italic>c</italic> (cyto. <italic>c</italic>) and Tom20 in MDA-MB 435 S cells treated with 200 ng/ml TRAIL, 10 μM Eer1, or 2 μM CB-5083 for the indicated time durations, transfected with siNC or siVCP for 48 h, or infected with adenoviruses encoding VCP WT-EGF or VCP QQ-EGF for 48 h. <bold>b</bold> Western blotting of Caspase-3 and PARP in MDA-MB 435 S cells treated with 10 μM Eer1, 2 μM CB-5083, or 200 ng/μl TRAIL using β-actin as a loading control. The representative blots of two independent experiments are shown. <bold>c</bold>, <bold>d</bold> MDA-MB 435 S cells pretreated with 20 μM z-VAD-fmk were further treated with Eer1 or TRAIL for 24 h. <bold>c</bold> Cellular viability assay. <bold>d</bold> Phase-contrast microscopy. <bold>e, f</bold> MDA-MB 435 S cells pretreated with the indicated death inhibitors were further treated with 10 μM Eer1, 2 μM CB-5083, or 5 μM NMS for 24 h <bold>e</bold> or 12 h <bold>f</bold>. <bold>e</bold> Cell viability assay. <bold>f</bold> Representative phase-contrast microscopic images. <bold>g</bold>, <bold>h</bold> MDA-MB 435 S cells pretreated with CHX were further treated with 10 μM Eer1, 2 μM CB-5083, and 5 μM NMS for 24 h <bold>g</bold> or 12 h <bold>h</bold>. <bold>g</bold> Cell viability assay. <bold>h</bold> Phase-contrast microscopy. <bold>i</bold> Confocal microscopy in YFP-ER cells pretreated with 1 μM CHX, further treated with Eer1, CB-5083, or NMS for 8 h, and stained with MTR. Cell viability data <bold>b</bold>, <bold>e</bold>, <bold>g</bold> represent the means ± SD of three independent experiments. n = 10. The <italic>p</italic>-value was calculated by one-way ANOVA. *<italic>p</italic> &lt; 0.05, **<italic>p</italic> &lt; 0.01, **<italic>p</italic> &lt; 0.001, ****<italic>p</italic> &lt; 0.0001. ns, not significant.</p></caption></fig>", "<fig id=\"Fig3\"><label>Fig. 3</label><caption><title>VCP inhibition is preferentially cytotoxic to cancer cells compared to non-malignant cells and induces paraptosis in vitro and in vivo.</title><p><bold>a</bold>–<bold>c</bold> Cells were treated with the indicated concentrations of Eer1 or CB-5083 for 24 h <bold>a</bold>, 10 μM Eer1 or 2 μM CB-5083 for 12 h <bold>b</bold>, <bold>c</bold>. <bold>a</bold> Cell viability assay. <bold>b</bold> Phase-contrast microscopy. <bold>c</bold> Confocal microscopy of the immunocytochemical staining of Bap31, calnexin (CNX), and Tim23. <bold>d</bold>–<bold>h</bold> Xenograft-bearing mice were treated with the indicated amounts of CB-5083 as described in the Material and Methods. The tumor volume <bold>d</bold> and body weight <bold>e</bold> were measured twice a week for 15 days, and the growth curve was plotted. On the 15^th day, tumors were isolated, photographed <bold>f</bold>, and weighed <bold>g</bold>. <bold>h</bold> H&amp;E staining. Yellow arrows indicate the cellular vacuoles in the tumor tissues of CB-5083-treated mice. Data <bold>a</bold>, <bold>d</bold>, <bold>e</bold>, <bold>g</bold> are presented as the means ± SD of three independent experiments. The <italic>p</italic>-values in panels <bold>a, d</bold>, and <bold>e</bold> were calculated by two-way ANOVA, and the <italic>p</italic>-values in panel <bold>g</bold> were calculated by one-way ANOVA. *<italic>p</italic> &lt; 0.05, **<italic>p</italic> &lt; 0.01, ***<italic>p</italic> &lt; 0.001, ****<italic>p</italic> &lt; 0.0001. ns, not significant.</p></caption></fig>", "<fig id=\"Fig4\"><label>Fig. 4</label><caption><title>PI3K/Akt/mTOR signals may be required for the oncogenic Ras-mediated cell vulnerability to VCP inhibition.</title><p><bold>a</bold>–<bold>d</bold> Parental MCF10A cells, mock vector-transfected, HRas^G12V-, or KRas^G12V-expressing MCF10A cells were treated with Eer1 for 24 h <bold>a</bold> or for the indicated time durations <bold>b</bold>, <bold>d</bold>, or treated with 10 μM Eer1 or 2 μM CB-5083 for 12 h <bold>c</bold>. <bold>e</bold>, <bold>f</bold> HRas^G12V/MCF10A cells pretreated with the indicated inhibitors were further treated with 10 μM Eer1 for 24 h <bold>e</bold>, or treated with 10 μM Eer1 or 2 μM CB-5083 for 12 h <bold>f</bold>. <bold>a</bold> (right), <bold>e</bold> Cell viability assay. <bold>a</bold> (left) Western blotting to confirm the overexpression of HRas^G12V or KRas^G12V using β-actin as a loading control. <bold>b, f</bold> Phase-contrast microscopy. <bold>c</bold> Immunocytochemistry of Bap31, calnexin (CNX), and Tim23. <bold>d</bold> Western blotting. Cell viability data <bold>a</bold>, <bold>e</bold> represent the means ± SD of three independent experiments. <italic>n</italic> = 10. The <italic>p</italic>-values in panels <bold>a</bold> and <bold>e</bold> were calculated by two-way ANOVA and one-way ANOVA, respectively. *<italic>p</italic> &lt; 0.05, **<italic>p</italic> &lt; 0.01, ***<italic>p</italic> &lt; 0.001, ****<italic>p</italic> &lt; 0.0001. ns not significant.</p></caption></fig>", "<fig id=\"Fig5\"><label>Fig. 5</label><caption><title>mTORC2/Akt activation is critical for VCP inhibition-induced paraptosis.</title><p><bold>a</bold> Western blotting of the proteins associated with mTOR signals in 10 μM Eer1- or 2 μM CB-5083-treated MDA-MB 435 S cells for the indicated time durations. Representative blots of two independent experiments are shown. <bold>b</bold>, <bold>c</bold> MDA-MB 435 S cells transduced with the lentivirus encoding shRNA against <italic>Raptor</italic> or <italic>Rictor</italic> gene were treated with 10 μM Eer1 or 2 μM CB-5083 12 h <bold>b</bold> (left) and <bold>c</bold> or for 24 h <bold>b</bold> (right). <bold>b</bold> (left) qRT-PCR using GAPDH as a reference gene. <bold>b</bold> (right) Cell viability assay. <bold>c</bold> Phase-contrast microscopy. <bold>d</bold>, <bold>e</bold> MDA-MB 435 S cells transfected with Mock vector or the plasmid encoding mTOR or Myr-Akt were treated with 10 μM Eer1 or 2 μM CB-5083 for 24 h <bold>d</bold> or 12 h <bold>e</bold>. <bold>d</bold> (left) Western blotting. <bold>d</bold> (middle) Cell viability assay. <bold>d</bold> (right) IC₅₀ of Eer1 or CB-5083 in the respective cells was assessed. <bold>e</bold> Phase-contrast microscopy. <bold>f</bold>, <bold>g</bold> MDA-MB 435 S cells pretreated with the indicated inhibitors were further treated with 10 μM Eer1 or 2 μM CB-5083 for 24 h <bold>f</bold> or 12 h <bold>g</bold>. <bold>f</bold> Cell viability assay. <bold>g</bold> Phase-contrast microscopy. <bold>h</bold> Confocal microscopy in YFP-ER cells pretreated with 1 μM PP242 or 10 μM LY294002, further treated with 10 μM Eer1 or 2 μM CB-5083 for 8 h, and stained with MTR. Cell viability data (<bold>b</bold> (right), <bold>d</bold> (middle), <bold>f</bold> are presented as the means ± SD of three independent experiments. <italic>n</italic> = 10. The <italic>p</italic>-values in panel <bold>b</bold> (right) and <bold>f</bold> were calculated by one-way ANOVA. The <italic>p</italic>-values in panel <bold>d</bold> (middle) were calculated by two-way ANOVA. **<italic>p</italic> &lt; 0.01, ****<italic>p</italic> &lt; 0.0001. ns not significant.</p></caption></fig>", "<fig id=\"Fig6\"><label>Fig. 6</label><caption><title>The ATF4/DDIT4 axis is crucial for VCP inhibition-mediated paraptosis, affecting Akt activation.</title><p><bold>a</bold> Western blotting of the ISR-associated proteins in MDA-MB 435 S cells treated with 10 μM Eer1, 2 μM CB, or 5 μM NMS-873, transfected with siVCP, or infected with VCP QQ-EGFP. <bold>b, c</bold> MDA-MB 435 S cells transfected with siATF4 or siCHOP were treated with 10 μM Eer1 for 24 h <bold>b</bold> or 12 h <bold>c</bold>. <bold>b</bold> Western blotting. The representative blots of two independent experiments are shown. <bold>c</bold> Phase-contrast microscopy. <bold>d</bold> Cell viability assay in MDA-MB 435 S cells transfected with siATF4 or siCHOP were treated with the indicated VCP inhibitors for 24 h. <bold>e</bold> Confocal microscopy in YFP-ER cells transfected with siNC or siATF4, treated with the VCP inhibitors for 16 h, and stained with MTR. <bold>f</bold> Western blotting in MDA-MB 435 S cells treated with 10 μM Eer1. <bold>g</bold>–<bold>k</bold> MDA-MB 435 S cells transfected with siNC or siATF4 <bold>g</bold>, <bold>h</bold>, or those transduced with the lentivirus encoding shNT or shDDIT4 <bold>i</bold>–<bold>k</bold> were further treated with 10 μM Eer1 for 12 h <bold>g, j</bold> (right), <bold>k</bold>, indicated time points <bold>h</bold>, <bold>i</bold>, or 24 h <bold>j</bold> (left). <bold>g</bold>, <bold>j</bold> (right) qRT-PCR of DDIT4 using GAPDH as a reference. <bold>h</bold>, <bold>i</bold> Western blotting. <bold>j</bold> (left) Cell viability assay. <bold>k</bold> Phase-contrast microscopy. <bold>l</bold> Hypothetical upstream signals for VCP inhibition-mediated paraptosis. Cell viability data <bold>d</bold>, <bold>j</bold> are presented as the means ± SD of three independent experiments. <italic>n</italic> = 10. The <italic>p</italic>-values were calculated by two-way ANOVA. ****<italic>p</italic> &lt; 0.0001.</p></caption></fig>", "<fig id=\"Fig7\"><label>Fig. 7</label><caption><title>The ATF4/DDIT4 axis and mTORC2/Akt are critically involved in translational recovery during VCP inhibition-mediated paraptosis.</title><p><bold>a</bold> Western blotting of the ISR-associated proteins and newly synthesized puromycinylated peptides was performed in MCF10A and MDA-MB 435 S cells treated with 1 μM CHX or 10 μM Eer1. Representative blots of two independent experiments are shown. The band intensity of puromycinylated proteins was analyzed using ImageJ. <bold>b</bold>–<bold>d</bold> Eer1-treated MDA-MB 435 S cells were co-treated or post-treated (after Eer1 treatment for 4 h) with 1 μM CHX and further incubated for 24 h. <bold>b</bold> The treatment schedule. <bold>c</bold> Cell viability assay. Data are presented as the means ± SD of three independent experiments. n = 10. The <italic>p</italic>-value was calculated by one-way ANOVA. ****<italic>p</italic> &lt; 0.0001. <bold>d</bold> Phase-contrast microscopy. <bold>e</bold> Mock vector-transfected or HRas^G12V-expressing MCF10A cells were treated with 10 μM Eer1. <bold>f</bold>–<bold>h</bold> MDA-MB 435 S cells transfected with siNC or siATF4 <bold>f</bold> or those transduced with the lentivirus encoding the indicated shRNA <bold>g</bold>, <bold>h</bold> were further treated with Eer1. <bold>i</bold> MDA-MB 435 S cells pretreated with 1 μM PP242 or 10 μM LY294002 were further treated with Eer1. <bold>e</bold>–<bold>i</bold> Western blotting of the indicated proteins was performed with β-actin as a positive control. The representative blots of two independent experiments are shown.</p></caption></fig>", "<fig id=\"Fig8\"><label>Fig. 8</label><caption><title>eIF3d critically contributes to VCP inhibition-mediated translational recovery and subsequent paraptosis.</title><p><bold>a</bold>–<bold>f</bold> MDA-MB 435 S cells transduced with the lentivirus encoding shRNAs against the indicated genes were treated with 10 μM Eer1 for 24 h <bold>a</bold>, 12 h <bold>b</bold>–<bold>d</bold>, indicated time duration <bold>e</bold> or 8 h <bold>f</bold>. <bold>a</bold> Cell viability. Data are presented as the means ± SD of three independent experiments. <italic>n</italic> = 10. The <italic>p</italic>-value was calculated by two-way ANOVA. *<italic>p</italic> &lt; 0.05, ***<italic>p</italic> &lt; 0.001. <bold>b, f</bold> qRT-PCR of the indicated genes using GAPDH as a reference gene. <bold>c</bold> Phase-contrast microscopy. <bold>d</bold> Immunocytochemistry. <bold>e</bold> Western blotting. <bold>g</bold> Hypothetical model for the cause of the preferential cytotoxicity of VCP inhibition to cancer cells with hyperactive Akt. While normal cells with low Akt activity are less sensitive to VCP inhibition due to translational suppression, cancer cells with hyperactive Akt due to oncogenic activation are more vulnerable to VCP inhibition-mediated paraptosis via enhanced proteotoxic stress triggered by ATF4/DDIT4/p-Akt-and eIF3d-mediated translational recovery. Therefore, cellular fates in response to VCP inhibition may depend on their Akt activity.</p></caption></fig>" ]

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[ "<supplementary-material content-type=\"local-data\" id=\"MOESM1\"></supplementary-material>", "<supplementary-material content-type=\"local-data\" id=\"MOESM2\"></supplementary-material>" ]

[ "<fn-group><fn><p>Edited by Professor Boris Zhivotovsky</p></fn><fn><p><bold>Publisher’s note</bold> Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.</p></fn><fn><p>These authors contributed equally: Dong Min Lee, In Young Kim.</p></fn></fn-group>" ]

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{ "acronym": [], "definition": [] }

[ "<title>Introduction</title>", "<p id=\"Par7\">Despite an increased understanding of the physiological processes involved in tumor metastasis, there are limited therapies that have proven clinical efficacy in advanced metastatic cancers such as glioblastoma, ovarian, prostate, pancreatic, and triple-negative breast cancer. The critical role of the TME as both a stimulator and a suppressor of tumor progression and metastasis is now widely recognized^{##UREF##0##1##,##REF##25082194##2##}. A potential TME-targeted therapy has been proposed where metastasis-incompetent tumors generate metastasis-suppressive microenvironments in distant organs by inducing TSP-1 expression in the bone marrow-derived Gr1+ myeloid cells^{##REF##11530335##3##,##REF##23633432##4##}.</p>", "<p id=\"Par8\">A potent inhibitor of tumor metastasis, Prosaposin (Psap), acts via stimulation of p53 and the anti-tumorigenic TSP-1 in bone marrow-derived cells that are recruited to metastatic sites^{##REF##11531279##5##,##REF##19581582##6##}. Within the TME, TSP-1 has been shown to act on two key receptors CD36 and CD47 (4,5). As a mediator of the pro-apoptotic activity of TSP-1, CD36 has been shown to be expressed on greater than 97% of human serous ovarian tumors tested (7). CD36 was also found to be increased in metastatic tumors compared to primary tumors, which are 2-3-fold higher than levels in ovarian and fallopian tube tissue^{##REF##26962158##7##}. CD36 has also been shown to be expressed in multiple human cancer cell lines including those derived from pancreatic, ovarian, breast, and prostate cancer^{##REF##26962158##7##,##REF##21282354##8##}. Another TSP-1 receptor, CD47 is expressed in various types of cancer and has been shown to inhibit the direct killing of cancer cells^{##REF##27283989##9##} by binding to SIRPα on the cell surface of macrophages which represents a “do-not eat-me” signal to prevent phagocytosis by the macrophage^{##REF##19632178##10##}. CD47 also stimulates tumor-initiating cells, sometimes called cancer stem cells, to differentiate into mature cells^{##REF##27283989##9##}. High levels of either CD36 or CD47 are both prognostic indicators of poor outcomes for cancer patients^{##REF##30069479##11##,##REF##32920329##12##}. Taken together, these findings suggest that a drug that stimulates expression of TSP-1 in the TME may have multiple beneficial effects as an anti-cancer agent.</p>", "<p id=\"Par9\">To identify potential anti-cancer agents a proprietary TME screening platform was utilized to evaluate metastatic vs localized tumors and refractory vs responsive tumors. Based on these findings, VT1021 was developed with drug-like properties derived from the active sequence in Psap^{##REF##23633432##4##}. VT1021 exhibited TSP-1-inducing activity and significantly regressed tumors in a PDX model of metastatic ovarian cancer^{##REF##26962158##7##}. The in vivo activity of VT1021 in murine xenograft models with several human solid tumor indications is presented in this report. This first-in-human phase 1 study was designed to determine RP2D, investigate the safety, pharmacokinetics (PK), and efficacy as well as confirm the mechanism of action of the novel, first-in-class, dual inhibitor of CD36 and CD47, VT1021, in patients with advanced solid tumors. Here, we select RP2D for VT1021, demonstrate that it is safe and well tolerated at all dosing levels, achieving a disease control rate (DCR) of 42.9% with clear validation of the proposed mechanism of action via the stimulation of TSP-1 in the TME.</p>" ]

[ "<title>Methods</title>", "<title>Clinical study design</title>", "<p id=\"Par10\">NCT03364400 was a phase 1, first-in-human, multicenter, open-label, dose escalation, and expansion study of VT1021 designed and sponsored by Vigeo Therapeutics, Inc. Data for the dose escalation phase are reported here. The data cut-off date was December 8, 2021. For the dose escalation portion of the study, the first patient was enrolled on November 28, 2017, and the last patient was enrolled on January 27, 2020.</p>", "<p id=\"Par11\">The primary objective of the escalation phase was to determine the RP2D for VT1021. The secondary objectives were to characterize the adverse event (AE) profile, determine the PK, and describe preliminary evidence of efficacy, if feasible, by using objective response rate (ORR), disease control rate (DCR), and progression-free survival (PFS) based on Response Evaluation Criteria in Solid Tumors (RECIST) v1.1. Exploratory objectives included pharmacodynamic (PD) assessment of expression levels of CD36, CD47, TSP-1 and selected immune cells by immunohistochemistry (IHC) on pairs of pre- and on-study biopsies.</p>", "<p id=\"Par12\">Eligible patients had advanced solid tumors that were refractory to, or intolerant of, existing therapies known to provide clinical benefit for their condition. Patients were aged ≥18 years and had Eastern Cooperative Oncology Group (ECOG) performance status of ≤2. Patients had evaluable or measurable disease by RECIST v1.1. Patients had to have adequate marrow reserve, liver and renal function. Key exclusion criteria included diagnosis of another malignancy within the past 2 years, history of a major surgical procedure or a significant traumatic injury within 14 days prior to commencing study drug, treatment with investigational therapy(ies) within 5 half-lives of the investigational therapy prior to the first scheduled day of dosing with VT1021, evidence of symptomatic brain metastases and use of other concurrent chemotherapy, immunotherapy, radiotherapy, or investigational anti-cancer therapy. Full eligibility criteria are available in the Protocol (Supplementary Information).</p>", "<p id=\"Par13\">Dose escalation was a variation to the traditional 3 + 3 study design. The dose escalation consisted of the administration of VT1021 intravenously twice weekly at doses of 0.5, 1.0, 2.0, 3.3, 5.1, 6.6, 8.8, 11.8 or 15.6 mg/kg (Fig. ##FIG##0##1##). The starting dose was 1 mg/kg, established based on pre-clinical toxicity studies. Safety was evaluated using CTCAE version 5.0. Each dose level would enroll at least one patient. If no dose-limiting toxicity (DLT) was observed, the next patient would be enrolled at the next higher dose level. If one DLT was observed, a minimum of 3 patients must be treated at the same dose level. Dose escalation was to continue until at least 2 patients in a cohort of 6 experienced DLT. Patients received VT1021 by intravenous infusion twice weekly IV on a 28-day cycle. The extent of disease was evaluated by imaging studies at the end of Cycle 2 and after every 2 cycles thereafter. Treatment would continue until disease progression, unacceptable toxicity or another withdrawal criteria was met. Intra-patient dose escalation was permitted upon meeting pre-specified criteria. RP2D was defined as the dose level where ≤33% patients experienced DLT. DLTs (defined in the study protocol, Supplementary Information) were assessed during the first 28 days of treatment.</p>", "<p id=\"Par14\">To decrease the risk of infusion reactions during the first week of dosing a premedication regimen was implemented. Prior to receiving each infusion of VT1021 patients were required to receive premedication with either dexamethasone by mouth 6–12 h pre-infusion or methylprednisolone 0.5 to 2 h prior to start of infusion and antihistamine (H1 antagonist), acetaminophen, and H2 blockers at the discretion of the investigator. In lieu of the premedication regimen clinical investigators were allowed to administer premedication regimes as per institutional guidelines. The premedication corticosteroid dose was to be decreased, tapered, or eliminated at the Investigator’s discretion after the first week of dosing.</p>", "<p id=\"Par15\">Patient assessment and follow-up procedures can be found in the schedule of assessments in the study protocol Appendix ##SUPPL##1##1## (Supplementary Information). Per protocol, regular safety assessments were performed in a population of patients who have received at least one dose of VT1021, including but not limited to physical examinations, ECOG/Karnofsky PS, electrocardiograms, and laboratory parameters. Clinical response was evaluated by using RECIST v1.1 and iRECIST in a population of patients who have at least completed one cycle of VT1021 treatment, per protocol. Blood samples were collected for PK analysis on Days 1, 4, 8, 11, 15, 18, 22, 25 and 50 at pre-dose, and at 0, 2-, 4-, 6- and 24-h post-infusion (Fig. ##FIG##1##2##). Plasma concentrations were determined with a validated assay using liquid chromatography- mass spectrometry.</p>", "<p id=\"Par16\">For patients who have signed a consent form, a pre-study biopsy or archival tumor specimen obtained within 6 months prior to study initiation was collected. In addition, on-study biopsies were collected at the end of Cycle 1 Week 4 or at any time during Cycle 2. Biopsies could be obtained after cycle 2 at the discretion of the Investigator. Paired pre-study and on-study biopsies were analyzed for expression of CD36, CD47, TSP-1, and immune cell populations by both IHC (Figs. ##FIG##2##3##–##FIG##4##5##) and MIBI (Multiplexed Ion Beam Imaging). MIBI is performed by staining formalin-fixed paraffin-embedded (FFPE) tissue with a panel of metal-labeled antibodies and then imaging the tissue using time-of-flight secondary ion mass spectrometry (ToF-SIMS)^{##REF##31633026##13##}. The masses of detected species are then assigned to target biomolecules given the unique metal isotope label of each antibody, creating multiplexed images. All antibodies in the panel have been MIBI validated on human FFPE tissue.</p>", "<p id=\"Par17\">All relevant ethical regulations were followed during the study. The methods were performed in accordance with relevant guidelines and regulations and approved by the Food and Drug Administration (FDA). Written informed consent was obtained from all patients who participated in the study. The Institutional Review Boards (IRBs) in all participating institutions have approved the study protocol. The institutions participated in the study are Northwestern University Medical School, Chicago, IL, Horizon Oncology Center, Lafayette, IN, South Texas Accelerated Research Therapeutics, San Antonio, TX, and Beth Israel Deaconess Hospital, Boston, MA.</p>", "<p id=\"Par18\">We used the CONSORT checklist when writing our report^{##UREF##1##14##}.</p>", "<title>Inclusion criteria</title>", "<p id=\"Par19\">To qualify for enrollment, all the following criteria must be met: (1) Patient must provide written informed consent. (2) Patient is ≥18 years of age. (3) For the Dose Escalation Phase: Patients with advanced solid tumors that are refractory to, or intolerant of, existing therapies known to provide clinical benefit for their condition. (4) Patient has evaluable or measurable disease by RECIST v1.1. (5) Patient has a performance status (PS) of 0–1 on the Eastern Cooperative Oncology Group (ECOG) scale. (6) Patient is at least 21 days removed from therapeutic radiation or chemotherapy prior to the first scheduled day of dosing with VT1021 and has recovered to Grade ≤ 1 (National Cancer Institute [NCI] Common Terminology Criteria for Adverse Events [CTCAE] v5.0) from all clinically significant toxicities related to prior therapies. (a) For patients receiving nitrosoureas or mitomycin C, the window is 6 weeks. (b) For patients receiving monoclonal antibody therapy, the window is at least one half-life or 4 weeks (whichever is shorter). (7) Patient has adequate organ function defined as: (a) Absolute neutrophil count (ANC) ≥ 1.5 × 10⁹/L (1500/µL) and absolute lymphocyte count (ALC) ≥ 7 × 10⁹/L (700/µL). (b) Platelet ≥100 × 10⁹/L. (c) Hemoglobin ≥9 g/dL. (d) Activated partial thromboplastin time/ prothrombin time/international normalized ratio (aPTT/PT/INR) ≤ 1.5 × upper limit of normal (ULN) unless the patient is on anticoagulants in which case therapeutically acceptable values (as determined by the investigator) meet eligibility requirements. (e) Aspartate aminotransferase (AST) or alanine aminotransferase (ALT) ≤ 2.5 × ULN. In the case of known (i.e., radiological or biopsy documented) liver metastasis, serum transaminase levels must be ≤5 × ULN. (f) Total serum bilirubin ≤1.5 × ULN (except for patients with known Gilbert’s Syndrome ≤3 × ULN is permitted). (g) Renal: Serum creatinine &lt;2.0 × ULN and creatinine clearance ≥50 L/min/1.73 m². (h) Serum albumin &gt;3 gm/dL. (8) Patient agrees to use acceptable methods of contraception during the study and for at least 90 days after the last dose of VT1021 if sexually active and able to bear or beget children.</p>", "<title>Exclusion criteria</title>", "<p id=\"Par20\">The presence of any of the following will exclude the patient from the study: (1) Diagnosis of another malignancy within the past 2 years (excluding a history of carcinoma in situ of the cervix, superficial non-melanoma skin cancer, or superficial bladder cancer that has been adequately treated, or stage 1 prostate cancer that does not require treatment or requires only treatment with luteinizing hormone-releasing hormone agonists or antagonists if initiated at least 90 days prior to the first dose of VT1021). (2) History of a major surgical procedure or a significant traumatic injury within 14 days prior to commencing study drug, or the anticipation of the need for a major surgical procedure during the course of the study. (3) Treatment with investigational therapy(ies) within 5 half-lives of the investigational therapy prior to the first scheduled day of dosing with VT1021, or 4 weeks if the half-life of the investigational agent is not known, whichever is shorter. (4) Concurrent serious (as determined by the Principal Investigator [PI]) medical conditions, including, but not limited to, New York Heart Association (NYHA) class III or IV congestive heart failure, history of congenital prolonged QT syndrome, uncontrolled infection, active hepatitis B, hepatitis C or human immunodeficiency virus (HIV), or other significant co-morbid conditions that, in the opinion of the Investigator, would impair study participation or cooperation. (5) Pregnant or planning to become pregnant or breast feed while on study. (6) Evidence of symptomatic brain metastases. Patients with treated (surgically excised or irradiated) and stable brain metastases are eligible, assuming the patient has adequately recovered from treatment, the treatment was at least 28 days prior to initiation of study drug, and baseline brain computed tomography (CT) with contrast or magnetic resonance imaging (MRI) within 14 days of initiation of study drug, is negative for new or worsening brain metastases. (7) Other concurrent chemotherapy, immunotherapy, radiotherapy, or investigational anti-cancer therapy. (8) Requirement for palliative radiotherapy to lesions that are defined as target lesions by RECIST/RANO criteria at the time-of study entry. (9) Known hypersensitivity to any of the components of VT1021 (sodium phosphate dibasic anhydrous, sodium phosphate monobasic monohydrate, mannitol, polysorbate 80) or a severe reaction to PS20- or PS80-containing drugs or investigational agents (e.g. amiodarone, Vitamin K, etoposide, docetaxel, cancer vaccine, protein biotherapeutics [like monoclonal antibodies], erythropoietin-stimulating agents, fosaprepitant). (10) Chronic, systemically administered glucocorticoids in doses equivalent to &gt;5 mg prednisone daily. Topical, inhalational, ophthalmic, intraarticular, and intranasal glucocorticoids are permitted. Isolated or intermittent use of systemically administered glucocorticoids to treat complications of malignancy, use as a premedication, or as a onetime prep for an imaging procedure is permitted. If patient was on &gt;5 mg prednisone/day equivalent, last dose must have been at least 7 days prior to the first planned dose of study drug. (11) Patients with active hepatitis B (e.g., hepatitis B surface antigen [HBsAg] reactive) are excluded, however, patients with past hepatitis B virus (HBV) infection or resolved HBV infection (defined as the presence of hepatitis B core antibody [HBcAb] and absence of HBsAg) may be enrolled provided that prior testing/known status for HBV deoxyribonucleic acid (DNA) is negative. Patients with active hepatitis C (e.g., hepatitis C virus [HCV] ribonucleic acid [RNA] [qualitative] are detected) are excluded, however, patients with cured hepatitis C (negative HCV RNA prior test/known status) may be enrolled.</p>", "<title>Statistical analysis</title>", "<p id=\"Par21\">The disease control rate (DCR) used for clinical outcomes was calculated as the percentage of patients with advanced cancer whose therapeutic intervention has led to a complete response, partial response, or stable disease. The 90% Confidence Interval (CI) was calculated using the exact (Clopper-Pearson) interval.</p>", "<p id=\"Par22\">Statistical analysis was performed with Graphpad Prism 9.3.1, <italic>p</italic> values were calculated by unpaired two-sample <italic>t</italic>-test, graphs of a point with error bars are used to indicate the average values and standard error of the mean (SEM).</p>", "<title>Reporting summary</title>", "<p id=\"Par23\">Further information on research design is available in the ##SUPPL##5##Nature Portfolio Reporting Summary## linked to this article.</p>" ]

[ "<title>Results</title>", "<p id=\"Par24\">The RP2D for VT1021 was determined to be 11.8 mg/kg, twice weekly dosing. The consideration was based on the combination of assessment of safety, tolerability, as well as PK exposure across all dose levels. The individual assessments of these parameters are described in the following sections.</p>", "<title>Clinical patient population, treatment, and disposition</title>", "<p id=\"Par25\">Thirty-eight patients, who received at least 1 dose of VT1021, were enrolled (Fig. ##FIG##0##1##). Patient demographics and disease characteristics by dose cohort are shown in Table ##TAB##0##1##. The median age was 65 years (range 40–84) and 100% of patients had an ECOG performance status of ≤2. A variety of tumor types enrolled most commonly being ovarian cancer (8 patients, 21%) and pancreatic cancer (7 patients, 19%); other tumor types had three or fewer patients (Table ##TAB##0##1##).</p>", "<p id=\"Par26\">The patient population was heavily pre-treated; all patients received prior antineoplastic therapy with 78.9% of patients receiving ≥3 prior treatment regimens for advanced/metastatic disease, with 42% receiving prior radiotherapy.</p>", "<p id=\"Par27\">Intra-patient dose escalation was permitted and a total of 4 patients were dose-escalated: one patient with colorectal cancer was dose-escalated from 0.5 to 1.0 mg/kg at cycle 8, one patient with uterine cancer was dose-escalated from 5.1 to 6.6 mg/kg at cycle 7, and two patients, one with thymoma and the other with pseudomyxoma peritonei, were dose-escalated from 8.8 to 11.8 mg/kg at cycles 7 and 8, respectively. One patient with ovarian cancer dose de-escalated from 6.6 to 5.1 mg/kg at cycle 2 due to concern of worsening baseline peripheral neuropathy from grade 1 to 2, which was later attributed to previous platinum chemotherapy.</p>", "<p id=\"Par28\">To achieve the minimum of 3 evaluable patients dose levels 1.0 mg/kg, 5.1 mg/kg and 15.6 mg/kg were over-enrolled. Some of the patients withdrew voluntarily, others withdrew to go to hospice care, at least 1 patient withdrew due to an infusion reaction during the first dose.</p>", "<p id=\"Par29\">The first patient dosed at the protocol-defined starting dose of 1 mg/kg experienced a grade 3 infusion reaction on the third dose. To ensure safety, 3 patients were treated at the lower dose level of 0.5 mg/kg and the protocol was amended to require premedication of steroids and/or antihistamines (at the PI’s discretion) prior to the first infusion. No infusion reaction was reported in 3 patients treated at 0.5 mg/kg and dose escalation resumed. Every patient received premedication per protocol and the majority of the patients were tapered off the premedication after 1–2 weeks of VT1021 treatment.</p>", "<p id=\"Par30\">Of the patients that were discontinued from the study, reasons for discontinuation included disease progression (65.8%), patient or physician decision (13.2%), AEs (10.5%) and death (7.9%). None of the deaths were attributed to VT1021 treatment. Four patients died on treatment, and the cause of death were hepatic failure due to disease progression (at 5.1 mg/kg dose level), hepatic failure due to disease progression (at 6.6 mg/kg dose level), tumor hemorrhage (at 15.6 mg/kg dose level) and septic shock/unrelated to protocol (at 15.6 mg/kg dose level). Three patients died after treatment discontinuation within 30 days following the last dose of VT1021, and the cause of death were disease progression (at 3.3 mg/kg dose level), multi-organ failure (at 11.8 mg/kg dose level), and disease progression (at 15.6 mg/kg dose level).</p>", "<title>Safety and tolerability</title>", "<p id=\"Par31\">Overall, VT1021 had a clean safety profile. The incidence of ≥grade 3 AEs suspected to be related to the study drug was very low (7.9%). Thirty-seven patients (97.4%) experienced at least one treatment-emergent adverse event (TEAE) and 17 patients (44.7%) experienced grade ≥3 TEAEs shown in Table ##TAB##1##2## (TEAEs in ≥5% of patients). A TEAE is defined as any event that occurs on or after the first dose of study drug administration or any pre-existing event which worsened in severity after dosing. There were 5 patients (13.2%) with fatal TEAEs, none of which were classified as drug-related. AEs suspected to be related to the study treatment (RTEAEs) were experienced by 18 patients (47.4%) shown in Table ##TAB##2##3## (RTEAEs in ≥5% of patients); the most frequent RTEAEs (≥10% of patients) were fatigue (6 patients, 15.8%), nausea (4 patients, 10.5%) and infusion-related reaction (4 patients, 10.5%). Grade 3 RTEAEs were reported in 3 patients (7.9%) where there was a single occurrence each of infusion-related reaction, anemia, and increased aspartate aminotransferase (AST), blood bilirubin and creatinine. Study drug was held for grade 3 elevation in AST and blood bilirubin and the patient was discontinued from treatment for clinical progression of disease. Study drug was discontinued for the patient with grade 3 infusion reaction and not re-started for the patient with anemia and increased blood creatinine who subsequently experienced an SAE of sepsis.</p>", "<title>DLTs, PK and RP2D</title>", "<p id=\"Par32\">Throughout the course of the dose escalation trial no patient experienced a DLT and thus MTD was not achieved. Because no MTD was reached, the recommended phase 2 dose (RP2D) was determined based on the pharmacokinetic (PK) profile. Table ##TAB##3##4## shows the PK parameters for VT1021 by dose cohort and Fig. ##FIG##1##2## shows the median concentration-time profiles by dose. VT1021 plasma exposures increased dose proportionally from 0.5 to 8.8 mg/kg based on mean C_max, AUC, and CL values and the exposures from 8.8 mg/kg to 15.6 mg/kg were similar. VT1021 did not appear to accumulate in plasma with repeated dosing, which is consistent with the dosing frequency and short terminal half-life values observed (average 1.2 to 1.3 h across all doses and sampling days).</p>", "<p id=\"Par33\">Based on this data, the RP2D of 11.8 mg/kg twice weekly was selected based on the observation that PK exposure levels were similar from 8.8 to 15.6 mg/kg with no increased dose-related AEs or toxicities were observed.</p>", "<title>Efficacy</title>", "<p id=\"Par34\">While efficacy was not a primary readout for the dose escalation trial, VT1021 did demonstrate single-agent activity in multiple patients. Out of 38 patients who received at least one dose of VT1021 in the escalation phase, 28 patients were considered evaluable based on the criteria of completing at least one cycle of treatment with tumor imaging during cycle 2. One patient with metastatic thymoma (Stage 4) achieved confirmed partial response (PR) and remained on treatment for 504 days. Eleven patients had stable disease (SD) in 9 different solid tumor indications, resulting in a disease control rate (DCR) of 42.9% (Table ##TAB##4##5##).</p>", "<p id=\"Par35\">To better understand the biological activity of VT1021, pre-study biopsies from 25 evaluable patients were analyzed by immunohistochemistry (IHC). Specifically, the expression levels of CD36 and CD47, the two major cell surface receptors for TSP-1, were assayed. The intensity of each marker was analyzed by Image J/Fiji. Biopsies were scored as being either low, medium or high (representative images are shown in Fig. ##FIG##2##3##). The scores were measured by both the percentage of cells with positive staining of the biomarkers, and by the level of intensity of staining signal. Moreover, patients were further classified as being dual high for both CD36 and CD47, not dual high, or unknown. The percent change in target lesion from baseline (based on the length of the long axis), correlation to dual high CD36 and CD47 status and duration of exposure to VT1021 for the evaluable patients are shown in Fig. ##FIG##3##4##. Nine of 25 patients with available biopsies were scored as dual high for CD36 and CD47 (36%) (Fig. ##FIG##3##4a##). Overall, for all evaluable patients the median duration on treatment was 53 days. Out of the 9 patients with dual high CD36 and CD47 expression, 8 achieved SD (89%) with a mean treatment duration of 148 days (Fig. ##FIG##3##4b##).</p>", "<title>Biomarker analyses</title>", "<p id=\"Par36\">Since the mechanism of action (MOA) of VT1021 is mediated by the induction of TSP-1 in MDSCs^{##REF##23633432##4##,##REF##26962158##7##}, we sought to analyze the expression of TSP-1 in pre- and on-study biopsies. The rationale was that induction of TSP-1 expression would functionally reprogram the TME in patient tumors. Although not required by the study protocol, paired pre- and on-study tumor tissue samples were voluntarily obtained from 7 patients and were analyzed by IHC for expression of CD36, CD47 and TSP-1. The seven patients who provided paired biopsy samples were the following: pancreatic cancer at 5.1 mg/kg, prostate cancer at 5.1 mg/kg, uterine carcinosarcoma at 5.1 mg/kg, kidney cancer at 8.8 mg/kg, appendiceal carcinoma at 11.8 mg/kg, and two ovarian cancer at 15.6 mg/kg. VT1021 induced expression of TSP-1 in the TME in all on-study biopsies analyzed, with one representative image shown in Fig. ##FIG##4##5##. Significantly, analysis of CD36 and CD47 expression revealed no change in pre- vs on-study biopsies (Fig. ##FIG##4##5##).</p>", "<p id=\"Par37\">Additionally, we assessed the composition of the TME to determine whether VT1021 was able to reprogram the recruited immune and inflammatory cells. Tumor tissue samples from 4 patients were analyzed for quantitative and qualitative changes in MDSCs, T cells and macrophages after VT1021 treatment. The four patients whose paired biopsy samples were used for quantitative biomarker analysis were the following: uterine carcinosarcoma at 5.1 mg/kg, kidney cancer at 8.8 mg/kg, and two ovarian cancer at 15.6 mg/kg. Analysis of the 4 pairs of biopsies revealed that TSP-1 expression was induced in MDSCs in on-study biopsies compared to pre-study (Fig. ##FIG##4##5##). Moreover, 3 out of 4 on-study biopsies displayed increased CD8+ CTLs and iNOS+ M1 macrophages with concomitant decreases in FoxP3+ regulatory T cells and CD163 + M2 macrophages. Figure ##FIG##4##5## depicts a representative example of changes observed in the TME in a patient with metastatic renal cell carcinoma (RCC) who achieved SD in the 8.8 mg/kg cohort and was on treatment for 105 days. Similar TSP-1 induction and macrophages repolarization results were also found in the MIBI study (Supplementary Information).</p>" ]

[ "<title>Discussion</title>", "<p id=\"Par38\">We report here the first-in-human experience of VT1021 in patients with advanced solid tumors who were refractory to multiple lines and various classes of systemic therapies. The study rigorously assessed the safety, PK/PD, and clinical activities of this first-in-class agent, which targets the cell surface molecules CD36 and CD47 simultaneously, via induction of TSP-1^{##UREF##2##15##}. Treatment with VT1021 in this population was safe and well tolerated. The major drug-related toxicity was infusion reaction noted at the protocol-defined starting dose which resolved following pre-medications with steroids and/or antihistamines. The incidence of ≥grade 3 AEs suspected to be related to the study drug was very low (7.9%). There were several patients that died on the study however none were attributed to the study drug, determined by the clinical principal investigators. The patient population in the escalation phase was all heavily pre-treated with more than four previous lines of therapy and multiple metastases. The PK parameters were characterized for each dose cohort and were observed to increase proportionally from 0.5 mg/kg to 8.8 mg/kg while the exposures from 8.8 mg/kg to 15.6 mg/kg were similar. Because the MTD was not reached, the RP2D of 11.8 mg/kg was selected based on PK exposure. The dosing schedule of 11.8 mg/kg twice weekly was further evaluated in tumor type-specific expansion cohorts, namely GBM, ovarian and pancreatic cancer which will be reported when survival data is fully mature. Out of 28 evaluable patients reported in this study, one patient achieved PR and 11 patients had SD with a DCR of 42.9%. Of the SD patients who provided biopsies, 72.7% had dual high expression of both CD36 and CD47. Biomarker analyses in tumor biopsies confirmed the mechanism of action of VT1021 to induce expression of TSP-1 in MDSCs and reprogram the TME from immunologically cold to hot. Taken together these findings support the clinical advancement of VT1021 into phase 1b/ II single agent and/or combinatorial studies.</p>", "<p id=\"Par39\">The clinical activity of VT1021 in the dose escalation portion of this phase 1 study indicates the potential efficacy in select solid tumor indications which typically harbor an immunologically cold TME and for which treatment with drugs such as checkpoint inhibitors have shown very little benefit^{##REF##34439292##16##}. One patient with metastatic thymoma, the only patient with this indication in this study, achieved PR after the second cycle of treatment, and was on study for 504 days, while 11 patients with other solid tumors including pseudomyxoma peritonei (<italic>n</italic> = 1), leiomyosarcoma (<italic>n</italic> = 1), appendiceal (<italic>n</italic> = 1), uterine (<italic>n</italic> = 1), pancreatic (<italic>n</italic> = 1), uterine carcinosarcoma (<italic>n</italic> = 1), kidney (<italic>n</italic> = 1), colorectal (<italic>n</italic> = 2), and ovarian (<italic>n</italic> = 2) had SD. Additionally, exploratory analysis suggests that dual high expression of CD36 and CD47 may predict response. Among 9 patients with dual high expression of CD36 and CD47, 8 patients achieved SD.</p>", "<p id=\"Par40\">Exploratory pharmacodynamic studies on paired tumor biopsies (pre-study and on-study) confirmed the mechanism of action of VT1021. The induction of TSP-1 was observed in all the tested biopsy samples. Although the number of available biopsy pairs was low, modulation of the TME from immunologically cold to hot, another hallmark of VT1021 activity, was also observed by augmented levels of active tumor-killing immune cells and lower levels of immunosuppressive cells in a majority of on-study biopsies. Although VT1021 thus far has been shown to be safe and well tolerated in patients with advanced solid tumors, there are several limitations to this study which will be addressed in future clinical trials such as increasing the number of patients in the treatment group, requiring pre-treatment biopsies from all patients prior to enrollment, and optimizing the dose of single-agent VT1021.</p>", "<p id=\"Par41\">VT1021 is the first clinical-stage molecule that functions by stimulating the expression of TSP-1 in the TME. The stimulation of TSP-1 simultaneously targets both CD36 and CD47 harnessing the full anti-tumor activity of TSP-1. Other drugs have attempted to exploit the anti-tumor activity of TSP-1 by utilizing small regions of the protein^{##REF##18006769##17##,##REF##18981463##18##} developed to target CD36 and CD47 individually^{##UREF##2##15##}. VT1021, however, induces the production of endogenous, localized, full-length TSP-1 in MDSCs, potentially improving TSP-1-dependent efficacy. Expression of full-length TSP-1 causes tumor reduction by CD36-dependent induction of apoptosis in tumor cells and endothelial cells. TSP-1 also blocks the CD47-SIRPα “do-not-eat-me” macrophage checkpoint to enable phagocytosis of tumor cells^{##REF##31410189##19##,##REF##28442746##20##}.</p>", "<p id=\"Par42\">The unique ability of VT1021 to target both CD36 and CD47 concurrently underscores the novel, first-in-class status of this molecule. The expansion phase of this study in selected solid tumor cohorts has recently been completed and results from this, as well as exploration of potential predictive and pharmacodynamic biomarkers, will be reported separately once survival data is more mature. VT1021 is currently in a global registration-ready clinical study (AGILE) for both newly diagnosed and recurrent GBM patients. Additional studies have been planned for other solid tumor indications, as single agent, and as part of the combination regimens with standard of care chemotherapies and immune checkpoint inhibitors.</p>" ]

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[ "<title>Background</title>", "<p id=\"Par1\">VT1021 is a cyclic peptide that induces the expression of thrombospondin-1 (TSP-1) in myeloid-derived suppressor cells (MDSCs) recruited to the tumor microenvironment (TME). TSP-1 reprograms the TME via binding to CD36 and CD47 to induce tumor and endothelial cell apoptosis as well as immune modulation in the TME.</p>", "<title>Methods</title>", "<p id=\"Par2\">Study VT1021-01 (ClinicalTrials.gov ID NCT03364400) used a modified 3 + 3 design. The primary objective was to determine the recommended Phase 2 dose (RP2D) in patients with advanced solid tumors. Safety, tolerability, and pharmacokinetics (PK) were assessed. Patients were dosed twice weekly intravenously in 9 cohorts (0.5–15.6 mg/kg). Safety was evaluated using CTCAE version 5.0 and the anti-tumor activity was evaluated by RECIST version 1.1.</p>", "<title>Results</title>", "<p id=\"Par3\">The RP2D of VT1021 is established at 11.8 mg/kg. VT1021 is well tolerated with no dose-limiting toxicities reported (0/38). The most frequent drug-related adverse events are fatigue (15.8%), nausea (10.5%), and infusion-related reactions (10.5%). Exposure increases proportionally from 0.5 to 8.8 mg/kg. The disease control rate (DCR) is 42.9% with 12 of 28 patients deriving clinical benefit including a partial response (PR) in one thymoma patient (504 days).</p>", "<title>Conclusions</title>", "<p id=\"Par4\">VT1021 is safe and well-tolerated across all doses tested. RP2D has been selected for future clinical studies. PR and SD with tumor shrinkage are observed in multiple patients underscoring the single-agent potential of VT1021. Expansion studies in GBM, pancreatic cancer and other solid tumors at the RP2D have been completed and results will be communicated in a separate report.</p>", "<title>Plain language summary</title>", "<p id=\"Par5\">It may be possible to treat cancers with therapies that modify the tumor microenvironment. This is the environment in the body in which tumors survive and grow and is composed of different types of cells. One such potential therapy is VT1021. Here, we conduct the first clinical trial to test this therapy in patients. We identify the optimal dose of the treatment to take into further studies, finding that VT1021 is safe and well tolerated by patients. We see some signs that the treatment is working in some patients and see evidence of modification of the tumor microenvironment. These findings help to inform further clinical trials of VT1021 to determine whether it is safe and effective in larger cohorts of patients.</p>", "<p id=\"Par6\">Mahalingam et al. report findings from a first-in-human dose escalation study of the tumor microenvironment modulator VT1021 in patients with advanced solid tumors. VT1021 is found to be safe and well tolerated and the recommended phase II dose is established based on pharmacokinetic/dynamic properties and preliminary clinical activities.</p>", "<title>Subject terms</title>" ]

[ "<title>Supplementary information</title>", "<p>\n\n\n\n\n\n\n</p>" ]

[ "<title>Supplementary information</title>", "<p>The online version contains supplementary material available at 10.1038/s43856-024-00433-x.</p>", "<title>Author contributions</title>", "<p>Patient enrollment and study monitoring: D.M., W.H., A.P., A.B. Development of methodology for biomarker studies: S.W., M.Y.V., J.J.C., J.W. Acquisition of data: S.W., M.Y.V., H.P., J.C., M.C. Analysis and interpretation of data: S.W., M.Y.V., J.J.C., R.S.W., J.C., J.M., M.C., J.W. Study supervision: M.C., J.W.</p>", "<title>Peer review</title>", "<title>Peer review information</title>", "<p id=\"Par43\"><italic>Communications Medicine</italic> thanks Jan Rekowski, Renuka Iyer, and the other, anonymous, reviewer(s) for their contribution to the peer review of this work. A peer review file is available.</p>", "<title>Data availability</title>", "<p>All data supporting the findings of this study are available within the paper and its Supplementary Information. Individual participant data that underlie the results reported in this paper after deidentified will be shared upon request. Study protocol has been provided in Supplementary Information. Source data for the figures are available as Supplementary Data ##SUPPL##2##1## and Supplementary Data ##SUPPL##3##2##. Additional data is available from the corresponding author upon reasonable request. Data requests submitted by researchers who provide a methodologically sound proposal will be accepted beginning immediately following publication and ending 5 years following publication.</p>", "<title>Competing interests</title>", "<p id=\"Par44\">Vigeo Therapeutics designed and sponsored the clinical study in this article. R.S.W. is a co-founder of, and consultant for, Vigeo Therapeutics, which has licensed technology from Boston Children’s Hospital. M.Y.V., J.J.C., S.W., H.P., J.C., J.M., M.C., and J.W. are employees of Vigeo Therapeutics. D.M., W.H., A.P., and A.B. declare no competing interests.</p>" ]

[ "<fig id=\"Fig1\"><label>Fig. 1</label><caption><p>Study schema and participants of the phase 1 clinical trial of VT1021.</p></caption></fig>", "<fig id=\"Fig2\"><label>Fig. 2</label><caption><title>Concentration-time profiles for VT1021, by dose cohort.</title><p><bold>a</bold> Cycle 1 day 1. <bold>b</bold> Cycle 1 day 4. <bold>c</bold> Cycle 2 day 50.</p></caption></fig>", "<fig id=\"Fig3\"><label>Fig. 3</label><caption><title>Representative images of CD36 and CD47 expression intensity by Immunohistochemistry.</title><p><bold>a</bold> Low CD36 expression from a patient with pancreatic cancer, <bold>b</bold> Medium CD36 expression from a patient with ovarian cancer. <bold>c</bold> Medium CD47 expression from a patient with non-small cell lung cancer. <bold>d</bold> High expression of CD36 and <bold>e</bold> high expression of CD47 from a patient with uterine cancer. No low CD47 expression was observed in any of the biopsies. Bar denotes 250 µm.</p></caption></fig>", "<fig id=\"Fig4\"><label>Fig. 4</label><caption><title>Percentage change from baseline in target lesions and duration of treatment with VT1021.</title><p><bold>a</bold> Percentage change from baseline was determined for target tumor lesions obtained at best response calculated as the percentage change from baseline. The best overall response is shown for each patient according to RECIST v1.1. Expression intensities of CD36 and CD47 are indicated as dual high (green), not dual high (gray), or unknown (blue). <bold>b</bold> Duration of exposure to VT1021 was determined as the period from cycle 1 day date to the end of treatment date. Dose cohorts are indicated by color. *Patients dose-escalated from 8.8 to 11.8 mg/kg. PR partial response, SD stable disease, PD progressive disease.</p></caption></fig>", "<fig id=\"Fig5\"><label>Fig. 5</label><caption><title>VT1021 Reprograms the tumor microenvironment by inducing Thrombospodin-1 (TSP-1) expression.</title><p>Immunohistochemistry was performed on pre- and on-study tumor biopsies (both from the liver) of a patient with metastatic Renal Cell Carcinoma, dosed at 8.8 mg/kg VT1021. Upper Panel: CD36, CD47, TSP-1, TSP-1/CD11b, and myeloid-derived suppressor cells (CD11bCD14). Lower Panel: Cytotoxic T cells (CD3 + /CD8 + ), Tregs (CD3 + /FoxP3 + ), M1 macrophages (CD68 + /iNOS + ) and M2 macrophages (CD68 + /CD163 + ). Bar denotes 100 µm. iNOS inducible nitric oxide synthase.</p></caption></fig>" ]

[ "<table-wrap id=\"Tab1\"><label>Table 1</label><caption><p>Baseline patient demographics and characteristics, by treatment group.</p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th/><th>0.5 mg/kg <italic>n</italic> = 3</th><th>1.0 mg/kg <italic>n</italic> = 6</th><th>2.0 mg/kg <italic>n</italic> = 4</th><th>3.3 mg/kg <italic>n</italic> = 3</th><th>5.1 mg/kg <italic>n</italic> = 6</th><th>6.6 mg/kg <italic>n</italic> = 4</th><th>8.8 mg/kg <italic>n</italic> = 3</th><th>11.8 mg/kg <italic>n</italic> = 3</th><th>15.6 mg/kg <italic>n</italic> = 6</th><th>All patients <italic>n</italic> = 38</th></tr></thead><tbody><tr><td>Median age, years (range)</td><td>69 (63–82)</td><td>70.5 (46–73)</td><td>46.5 (40–67)</td><td>78 (67–83)</td><td>65 (57–84)</td><td>64.5 (60–83)</td><td>64 (64–66)</td><td>58 (54–61)</td><td>65 (54–76)</td><td>65 (40–84)</td></tr><tr><td>Sex (male), <italic>n</italic> (%)</td><td>2 (66.7)</td><td>4 (66.7)</td><td>1 (25.0)</td><td>1 (33.3)</td><td>3 (50.0)</td><td>0</td><td>1 (33.3)</td><td>2 (66.7)</td><td>2 (33.3)</td><td>16 (42.1)</td></tr><tr><td colspan=\"11\">Race, <italic>n</italic> (%)</td></tr><tr><td>Caucasian</td><td>3 (100)</td><td>5 (83.3)</td><td>2 (50.0)</td><td>3 (100)</td><td>4 (66.7)</td><td>4 (100)</td><td>3 (100)</td><td>3 (100)</td><td>5 (83.3)</td><td>32 (84.2)</td></tr><tr><td>Black</td><td>0</td><td>1 (16.7)</td><td>2 (50.0)</td><td>0</td><td>1 (16.7)</td><td>0</td><td>0</td><td>0</td><td>1 (16.7)</td><td>5 (13.2)</td></tr><tr><td>Asian</td><td>0</td><td>0</td><td>0</td><td>0</td><td>1 (16.7)</td><td>0</td><td>0</td><td>0</td><td>0</td><td>1 (2.6)</td></tr><tr><td colspan=\"11\">ECOG PS, <italic>n</italic> (%)</td></tr><tr><td>0</td><td>0</td><td>0</td><td>2 (50.0)</td><td>1 (33.3)</td><td>3 (50.0)</td><td>2 (50.0)</td><td>0</td><td>1 (33.3)</td><td>0</td><td>9 (23.7)</td></tr><tr><td>1</td><td>3 (100)</td><td>6 (100)</td><td>1 (25.0)</td><td>2 (66.7)</td><td>3 (50.0)</td><td>2 (50.0)</td><td>3 (100)</td><td>2 (66.7)</td><td>6 (100)</td><td>28 (73.7)</td></tr><tr><td>2</td><td>0</td><td>0</td><td>1 (25.0)</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>1 (2.6)</td></tr><tr><td>Primary tumor type, <italic>n</italic> (%)</td><td/><td/><td/><td/><td/><td/><td/><td/><td/><td>(<italic>n</italic> = 38)</td></tr><tr><td>Adenoid cystic carcinoma</td><td>0</td><td>0</td><td>1 (25.0)</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>1 (2.6)</td></tr><tr><td>Anal cancer</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>1 (16.7)</td><td>1 (2.6)</td></tr><tr><td>Appendiceal cancer</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>1 (33.3)</td><td>0</td><td>1 (2.6)</td></tr><tr><td>Uterine carcinosarcoma</td><td>0</td><td>0</td><td>0</td><td>0</td><td>1 (16.7)</td><td>0</td><td>0</td><td>0</td><td>0</td><td>1 (2.6)</td></tr><tr><td>Cholangiocarcinoma</td><td>0</td><td>0</td><td>1 (25.0)</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>1 (2.6)</td></tr><tr><td>Colorectal cancer</td><td>1 (33.3)</td><td>0</td><td>0</td><td>0</td><td>1 (16.7)</td><td>0</td><td>0</td><td>0</td><td>1 (16.7)</td><td>3 (7.9)</td></tr><tr><td>Esophageal cancer</td><td>0</td><td>0</td><td>0</td><td>0</td><td>1 (16.7)</td><td>0</td><td>0</td><td>1 (33.3)</td><td>0</td><td>2 (5.3)</td></tr><tr><td>Head and neck cancer</td><td>0</td><td>1 (16.7)</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>1 (2.6)</td></tr><tr><td>Leiomyosarcoma</td><td>0</td><td>1 (16.7)</td><td>1 (25.0)</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>2 (5.3)</td></tr><tr><td>Non-small cell lung cancer</td><td>1 (33.3)</td><td>1 (16.7)</td><td>0</td><td>1 (33.3)</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>3 (7.9)</td></tr><tr><td>Ovarian cancer</td><td>1 (33.3)</td><td>0</td><td>0</td><td>0</td><td>1 (16.7)</td><td>3 (75.0)</td><td>0</td><td>0</td><td>3 (50.0)</td><td>8 (21.1)</td></tr><tr><td>Pancreatic cancer</td><td>0</td><td>2 (33.3)</td><td>0</td><td>1 (33.3)</td><td>1 (16.7)</td><td>1 (25.0)</td><td>0</td><td>1 (33.3)</td><td>1 (16.7)</td><td>7 (18.4)</td></tr><tr><td>Prostate cancer</td><td>0</td><td>0</td><td>0</td><td>1 (33.3)</td><td>1 (16.7)</td><td>0</td><td>0</td><td>0</td><td>0</td><td>2 (5.3)</td></tr><tr><td>Pseudomyxoma peritonei</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>1 (33.3)</td><td>0</td><td>0</td><td>1 (2.6)</td></tr><tr><td>Renal cell carcinoma</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>1 (33.3)</td><td>0</td><td>0</td><td>1 (2.6)</td></tr><tr><td>Small cell lung cancer</td><td>0</td><td>1 (16.7)</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>1 (2.6)</td></tr><tr><td>Testicular cancer</td><td>0</td><td>0</td><td>1 (25.0)</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>1 (2.6)</td></tr><tr><td>Thymoma</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>1 (33.3)</td><td>0</td><td>0</td><td>1 (2.6)</td></tr><tr><td colspan=\"11\">Prior treatment regimens, <italic>n</italic> (%)</td></tr><tr><td>1</td><td>1 (33.3)</td><td>1 (16.7)</td><td>0</td><td>0</td><td>0</td><td>0</td><td>2 (66.7)</td><td>0</td><td>0</td><td>4 (10.5)</td></tr><tr><td>2</td><td>0</td><td>0</td><td>0</td><td>1 (33.3)</td><td>1 (16.7)</td><td>0</td><td>0</td><td>1 (33.3)</td><td>1 (16.7)</td><td>4 (10.5)</td></tr><tr><td>≥3</td><td>2 (66.7)</td><td>5 (83.3)</td><td>4 (100.0)</td><td>2 (66.7)</td><td>5 (83.3)</td><td>4 (100.0)</td><td>1 (33.3)</td><td>2 (66.7)</td><td>5 (83.3)</td><td>30 (78.9)</td></tr><tr><td colspan=\"11\">Prior Radiotherapy, <italic>n</italic> (%)</td></tr><tr><td>Yes</td><td>1 (33.3)</td><td>5 (83.3)</td><td>2 (50.0)</td><td>1 (33.3)</td><td>0</td><td>1 (25.0)</td><td>1 (33.3)</td><td>2 (66.7)</td><td>3 (50.0)</td><td>16 (42.1)</td></tr><tr><td>No</td><td>2 (66.7)</td><td>1 (16.7)</td><td>2 (50.0)</td><td>2 (66.7)</td><td>6 (100)</td><td>3 (75.0)</td><td>2 (66.7)</td><td>1 (33.3)</td><td>3 (50.0)</td><td>22 (57.9)</td></tr></tbody></table></table-wrap>", "<table-wrap id=\"Tab2\"><label>Table 2</label><caption><p>Treatment-emergent adverse events.</p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th>Preferred term, <italic>n</italic> (%)</th><th colspan=\"2\">0.5 mg/kg <italic>n</italic> = 3</th><th colspan=\"2\">1.0 mg/kg <italic>n</italic> = 6</th><th colspan=\"2\">2.0 mg/kg <italic>n</italic> = 4</th><th colspan=\"2\">3.3 mg/kg <italic>n</italic> = 3</th><th colspan=\"2\">5.1 mg/kg <italic>n</italic> = 6</th><th colspan=\"2\">6.6 mg/kg <italic>n</italic> = 4</th><th colspan=\"2\">8.8 mg/kg <italic>n</italic> = 3</th><th colspan=\"2\">11.8 mg/kg <italic>n</italic> = 3</th><th colspan=\"2\">15.6 mg/kg <italic>n</italic> = 6</th><th colspan=\"2\">All patients <italic>n</italic> = 38</th></tr><tr><th/><th>All</th><th>Gr ≥ 3</th><th>All</th><th>Gr ≥ 3</th><th>All</th><th>Gr ≥ 3</th><th>All</th><th>Gr ≥ 3</th><th>All</th><th>Gr ≥ 3</th><th>All</th><th>Gr ≥ 3</th><th>All</th><th>Gr ≥ 3</th><th>All</th><th>Gr ≥ 3</th><th>All</th><th>Gr ≥ 3</th><th>All</th><th>Gr ≥ 3</th></tr></thead><tbody><tr><td>Total</td><td>3 (100)</td><td>0</td><td>6 (100)</td><td>2 (33.3)</td><td>3 (75.0)</td><td>2 (50.0)</td><td>3 (100)</td><td>1 (33.3)</td><td>6 (100)</td><td>5 (83.3)</td><td>4 (100)</td><td>1 (25.0)</td><td>3 (100)</td><td>1 (33.3)</td><td>3 (100)</td><td>1 (33.3)</td><td>6 (100)</td><td>4 (66.7)</td><td>37 (97.4)</td><td>17 (44.7)</td></tr><tr><td>Fatigue</td><td>0</td><td>0</td><td>1 (16.7)</td><td>0</td><td>1 (25.0)</td><td>0</td><td>1 (33.3)</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>2 (66.7)</td><td>1 (33.3)</td><td>3 (50.0)</td><td>0</td><td>8 (21.1)</td><td>1 (2.6)</td></tr><tr><td>Abdominal pain</td><td>0</td><td>0</td><td>2 (33.3)</td><td>1 (16.7)</td><td>1 (25.0)</td><td>0</td><td>0</td><td>0</td><td>1 (16.7)</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>2 (66.7)</td><td>1 (33.3)</td><td>1 (16.7)</td><td>0</td><td>7 (18.4)</td><td>2 (5.3)</td></tr><tr><td>Constipation</td><td>0</td><td>0</td><td>1 (16.7)</td><td>1 (16.7)</td><td>0</td><td>0</td><td>0</td><td>0</td><td>1 (16.7)</td><td>0</td><td>1 (25.0)</td><td>0</td><td>1 (33.3)</td><td>0</td><td>2 (66.7)</td><td>0</td><td>1 (16.7)</td><td>0</td><td>7 (18.4)</td><td>1 (2.6)</td></tr><tr><td>Nausea</td><td>1 (33.3)</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>2 (66.7)</td><td>0</td><td>1 (33.3)</td><td>0</td><td>2 (33.3)</td><td>0</td><td>6 (15.8)</td><td>0</td></tr><tr><td>Anemia</td><td>0</td><td>0</td><td>1 (16.7)</td><td>1 (16.7)</td><td>1 (25.0)</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>1 (33.3)</td><td>0</td><td>1 (33.3)</td><td>1 (33.3)</td><td>1 (16.7)</td><td>1 (16.7)</td><td>5 (13.2)</td><td>3 (7.9)</td></tr><tr><td>Urinary tract infection</td><td>0</td><td>0</td><td>1 (16.7)</td><td>0</td><td>0</td><td>0</td><td>1 (33.3)</td><td>0</td><td>0</td><td>0</td><td>2 (50.0)</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>1 (16.7)</td><td>0</td><td>5 (13.2)</td><td>0</td></tr><tr><td>Arthralgia</td><td>0</td><td>0</td><td>0</td><td>0</td><td>1 (25.0)</td><td>0</td><td>0</td><td>0</td><td>1 (16.7)</td><td>0</td><td>2 (50.0)</td><td>0</td><td>1 (33.3)</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>5 (13.2)</td><td>0</td></tr><tr><td>Dyspnoea</td><td>1 (33.3)</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>1 (16.7)</td><td>0</td><td>0</td><td>0</td><td>2 (66.7)</td><td>0</td><td>1 (33.3)</td><td>1 (33.3)</td><td>0</td><td>0</td><td>5 (13.2)</td><td>1 (2.6)</td></tr><tr><td>Infusion-related reaction</td><td>0</td><td>0</td><td>2 (33.3)</td><td>1 (16.7)</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>1 (25.0)</td><td>0</td><td>1 (33.3)</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>4 (10.5)</td><td>1 (2.6)</td></tr><tr><td>Blood bilirubin increased</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>1 (16.7)</td><td>0</td><td>1 (25.0)</td><td>1 (25.0)</td><td>0</td><td>0</td><td>1 (33.3)</td><td>1 (33.3)</td><td>1 (16.7)</td><td>0</td><td>4 (10.5)</td><td>2 (5.3)</td></tr><tr><td>Decreased appetite</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>1 (16.7)</td><td>0</td><td>1 (25.0)</td><td>0</td><td>1 (33.3)</td><td>0</td><td>0</td><td>0</td><td>1 (16.7)</td><td>0</td><td>4 (10.5)</td><td>0</td></tr><tr><td>Hyperuricaemia</td><td>1 (33.3)</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>1 (16.7)</td><td>0</td><td>1 (25.0)</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>1 (16.7)</td><td>0</td><td>4 (10.5)</td><td>0</td></tr><tr><td>Hypokalaemia</td><td>0</td><td>0</td><td>1 (16.7)</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>1 (16.7)</td><td>0</td><td>1 (25.0)</td><td>0</td><td>1 (33.3)</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>4 (10.5)</td><td>0</td></tr><tr><td>Hypomagnesaemia</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>2 (33.3)</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>2 (33.3)</td><td>0</td><td>4 (10.5)</td><td>0</td></tr><tr><td>Headache</td><td>1 (33.3)</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>1 (33.3)</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>1 (33.3)</td><td>0</td><td>0</td><td>0</td><td>1 (16.7)</td><td>0</td><td>4 (10.5)</td><td>0</td></tr><tr><td>Abdominal distension</td><td>0</td><td>0</td><td>0</td><td>0</td><td>1 (25.0)</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>1 (33.3)</td><td>1 (33.3)</td><td>0</td><td>0</td><td>1 (16.7)</td><td>1 (16.7)</td><td>3 (7.9)</td><td>2 (5.3)</td></tr><tr><td>Vomiting</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>1 (33.3)</td><td>0</td><td>0</td><td>0</td><td>2 (33.3)</td><td>0</td><td>3 (7.9)</td><td>0</td></tr><tr><td>Oedema peripheral</td><td>0</td><td>0</td><td>1 (16.7)</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>1 (33.3)</td><td>0</td><td>1 (33.3)</td><td>0</td><td>0</td><td>0</td><td>3 (7.9)</td><td>0</td></tr><tr><td>Oral candidiasis</td><td>0</td><td>0</td><td>1 (16.7)</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>1 (25.0)</td><td>0</td><td>1 (33.3)</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>3 (7.9)</td><td>0</td></tr><tr><td>Aspartate aminotransferase increased</td><td>0</td><td>0</td><td>1 (16.7)</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>1 (16.7)</td><td>0</td><td>1 (25.0)</td><td>1 (25.0)</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>3 (7.9)</td><td>1 (2.6)</td></tr><tr><td>Back pain</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>1 (16.7)</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>2 (33.3)</td><td>0</td><td>3 (7.9)</td><td>0</td></tr><tr><td>Diarrhea</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>1 (16.7)</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>1 (33.3)</td><td>0</td><td>0</td><td>0</td><td>2 (5.3)</td><td>0</td></tr><tr><td>Chills</td><td>0</td><td>0</td><td>0</td><td>0</td><td>1 (25.0)</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>1 (16.7)</td><td>0</td><td>2 (5.3)</td><td>0</td></tr><tr><td>Pyrexia</td><td>0</td><td>0</td><td>0</td><td>0</td><td>1 (25.0)</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>1 (16.7)</td><td>0</td><td>2 (5.3)</td><td>0</td></tr><tr><td>Hepatic failure</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>1 (16.7)</td><td>1 (16.7)</td><td>1 (25.0)</td><td>1 (25.0)</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>2 (5.3)</td><td>2 (5.3)</td></tr><tr><td>Sinusitis</td><td>0</td><td>0</td><td>0</td><td>0</td><td>1 (25.0)</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>1 (16.7)</td><td>0</td><td>2 (5.3)</td><td>0</td></tr><tr><td>Alanine aminotransferase increased</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>1 (16.7)</td><td>0</td><td>1 (25.0)</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>2 (5.3)</td><td>0</td></tr><tr><td>Blood alkaline phosphatase increased</td><td>0</td><td>0</td><td>1 (16.7)</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>1 (16.7)</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>2 (5.3)</td><td>0</td></tr><tr><td>Blood creatine increased</td><td>0</td><td>0</td><td>1 (16.7)</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>1 (16.7)</td><td>1 (16.7)</td><td>2 (5.3)</td><td>1 (2.6)</td></tr><tr><td>White blood cell count decreased</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>1 (25.0)</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>1 (16.7)</td><td>0</td><td>2 (5.3)</td><td>0</td></tr><tr><td>Dizziness</td><td>0</td><td>0</td><td>1 (16.7)</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>1 (16.7)</td><td>0</td><td>2 (5.3)</td><td>0</td></tr><tr><td>Peripheral sensory neuropathy</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>1 (25.0)</td><td>0</td><td>0</td><td>0</td><td>1 (33.3)</td><td>0</td><td>0</td><td>0</td><td>2 (5.3)</td><td>0</td></tr><tr><td>Anxiety</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>1 (16.7)</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>1 (33.3)</td><td>1 (33.3)</td><td>0</td><td>0</td><td>2 (5.3)</td><td>1 (2.6)</td></tr><tr><td>Vaginal hemorrhage</td><td>0</td><td>0</td><td>0</td><td>0</td><td>1 (25.0)</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>1 (16.7)</td><td>0</td><td>2 (5.3)</td><td>0</td></tr><tr><td>Wheezing</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>1 (16.7)</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>1 (33.3)</td><td>0</td><td>0</td><td>0</td><td>2 (5.3)</td><td>0</td></tr><tr><td>Rash maculo-papular</td><td>1 (33.3)</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>1 (25.0)</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>2 (5.3)</td><td>0</td></tr><tr><td>Deep vein thrombosis</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>1 (33.3)</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>1 (33.3)</td><td>0</td><td>0</td><td>0</td><td>2 (5.3)</td><td>0</td></tr></tbody></table></table-wrap>", "<table-wrap id=\"Tab3\"><label>Table 3</label><caption><p>Related Treatment-emergent adverse events.</p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th>Preferred term, <italic>n</italic> (%)</th><th colspan=\"2\">0.5 mg/kg <italic>n</italic> = 3</th><th colspan=\"2\">1.0 mg/kg <italic>n</italic> = 6</th><th colspan=\"2\">2.0 mg/kg <italic>n</italic> = 4</th><th colspan=\"2\">3.3 mg/kg <italic>n</italic> = 3</th><th colspan=\"2\">5.1 mg/kg <italic>n</italic> = 6</th><th colspan=\"2\">6.6 mg/kg <italic>n</italic> = 4</th><th colspan=\"2\">8.8 mg/kg <italic>n</italic> = 3</th><th colspan=\"2\">11.8 mg/kg <italic>n</italic> = 3</th><th colspan=\"2\">15.6 mg/kg <italic>n</italic> = 6</th><th colspan=\"2\">All Patients <italic>n</italic> = 38</th></tr><tr><th/><th>All</th><th>Gr ≥ 3</th><th>All</th><th>Gr ≥ 3</th><th>All</th><th>Gr ≥ 3</th><th>All</th><th>Gr ≥ 3</th><th>All</th><th>Gr ≥ 3</th><th>All</th><th>Gr ≥ 3</th><th>All</th><th>Gr ≥ 3</th><th>All</th><th>Gr ≥ 3</th><th>All</th><th>Gr ≥ 3</th><th>All</th><th>Gr ≥ 3</th></tr></thead><tbody><tr><td>Total</td><td>1 (33.3)</td><td>0</td><td>3 (50.0)</td><td>1 (16.7)</td><td>1 (25.0)</td><td>0</td><td>1 (33.3)</td><td>0</td><td>2 (33.3)</td><td>0</td><td>3 (75.0)</td><td>1 (25.0)</td><td>1 (33.3)</td><td>0</td><td>3 (100)</td><td>0</td><td>3 (50.0)</td><td>1 (16.7)</td><td>18 (47.4)</td><td>3 (7.9)</td></tr><tr><td>Fatigue</td><td>0</td><td>0</td><td>0</td><td>0</td><td>1 (25.0)</td><td>0</td><td>1 (33.3)</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>1 (33.3)</td><td>0</td><td>3 (50.0)</td><td>0</td><td>6 (15.8)</td><td>0</td></tr><tr><td>Nausea</td><td>1 (33.3)</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>1 (33.3)</td><td>0</td><td>2 (33.3)</td><td>0</td><td>4 (10.5)</td><td>0</td></tr><tr><td>Infusion-related reaction</td><td>0</td><td>0</td><td>2 (33.3)</td><td>1 (16.7)</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>1 (25.0)</td><td>0</td><td>1 (33.3)</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>4 (10.5)</td><td>1 (2.6)</td></tr><tr><td>Hypomagnesaemia</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>2 (33.3)</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>1 (16.7)</td><td>0</td><td>3 (7.9)</td><td>0</td></tr><tr><td>Aspartate aminotransferase increased</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>1 (16.7)</td><td>0</td><td>1 (25.0)</td><td>1 (25.0)</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>2 (5.3)</td><td>1 (2.6)</td></tr><tr><td>Blood bilirubin increased</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>1 (16.7)</td><td>0</td><td>1 (25.0)</td><td>1 (25.0)</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>2 (5.3)</td><td>1 (2.6)</td></tr><tr><td>Hyperuricaemia</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>1 (16.7)</td><td>0</td><td>1 (25.0)</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>2 (5.3)</td><td>0</td></tr><tr><td>Dizziness</td><td>0</td><td>0</td><td>1 (16.7)</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>1 (16.7)</td><td>0</td><td>2 (5.3)</td><td>0</td></tr></tbody></table></table-wrap>", "<table-wrap id=\"Tab4\"><label>Table 4</label><caption><p>PK parameters by dose cohort.</p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th/><th>0.5 mg/kg <italic>n</italic> = 3</th><th>1.0 mg/kg <italic>n</italic> = 6</th><th>2.0 mg/kg <italic>n</italic> = 4</th><th>3.3 mg/kg <italic>n</italic> = 3</th><th>5.1 mg/kg <italic>n</italic> = 6</th><th>6.6 mg/kg <italic>n</italic> = 4</th><th>8.8 mg/kg <italic>n</italic> = 3</th><th>11.8 mg/kg <italic>n</italic> = 3</th><th>15.6 mg/kg <italic>n</italic> = 6</th></tr></thead><tbody><tr><td colspan=\"10\">Cycle 1 Day 1</td></tr><tr><td><italic>n</italic></td><td>3</td><td>5</td><td>4</td><td>3</td><td>6</td><td>4</td><td>3</td><td>3</td><td>4</td></tr><tr><td>C_max (ng/mL)</td><td>825 (199)</td><td>2790 (940)</td><td>5220 (1500)</td><td>6960 (5750)</td><td>16293 (3911)</td><td>15800 (4557)</td><td>29100 (7440)</td><td>26033 (13059)</td><td>43500 (12700)</td></tr><tr><td><italic>n</italic></td><td>3</td><td>5</td><td>4</td><td>3</td><td>6</td><td>4</td><td>3</td><td>3</td><td>4</td></tr><tr><td>t_max (hr)</td><td>0.5 (0)</td><td>0.5 (0)</td><td>0.5 (0)</td><td>0.5 (0)</td><td>0.5 (0)</td><td>0.5 (0)</td><td>0.5 (0)</td><td>0.5 (0)</td><td>0.5 (0)</td></tr><tr><td><italic>n</italic></td><td>3</td><td>5</td><td>4</td><td>3</td><td>6</td><td>4</td><td>3</td><td>3</td><td>4</td></tr><tr><td>AUC_0-6 (hr*ng/mL)</td><td>1230 (251)</td><td>4480 (1470)</td><td>7300 (1930)</td><td>10900 (8380)</td><td>26267 (9872)</td><td>30625 (16504)</td><td>51400 (19800)</td><td>45733 (28033)</td><td>69500 (20100)</td></tr><tr><td><italic>n</italic></td><td>3</td><td>5</td><td>4</td><td>3</td><td>6</td><td>3</td><td>3</td><td>3</td><td>4</td></tr><tr><td>t_1/2 (hr)</td><td>1.14 (0.233)</td><td>1.61 (0.729)</td><td>1.27 (0.254)</td><td>1.69 (1.05)</td><td>1.28 (0.264)</td><td>1.14 (0.061)</td><td>1.52 (0.427)</td><td>1.21 (0.273)</td><td>1.28 (0.312)</td></tr><tr><td colspan=\"10\">Cycle 1 Day 4</td></tr><tr><td><italic>n</italic></td><td>3</td><td>5</td><td>4</td><td>3</td><td>6</td><td>4</td><td>3</td><td>3</td><td>4</td></tr><tr><td>C_max (ng/mL)</td><td>1320 (331)</td><td>3610 (1190)</td><td>6590 (2320)</td><td>8170 (6940)</td><td>20267 (7507)</td><td>19175 (2690)</td><td>24400 (8880)</td><td>25467 (12987)</td><td>48200 (6760)</td></tr><tr><td><italic>n</italic></td><td>3</td><td>5</td><td>4</td><td>3</td><td>6</td><td>4</td><td>3</td><td>3</td><td>4</td></tr><tr><td>t_max (hr)</td><td>0.5 (0)</td><td/><td/><td/><td/><td/><td/><td/><td/></tr><tr><td><italic>n</italic></td><td>3</td><td>5</td><td>4</td><td>3</td><td>6</td><td>4</td><td>3</td><td>3</td><td>4</td></tr><tr><td>AUC_0-last (hr*ng/mL)</td><td>1810 (465)</td><td>5860 (1870)</td><td>9160 (3130)</td><td>13000 (11200)</td><td>32450 (17605)</td><td>35850 (14061)</td><td>40800 (15200)</td><td>44200 (26008)</td><td>73100 (10800)</td></tr><tr><td><italic>n</italic></td><td>3</td><td>5</td><td>4</td><td>2</td><td>6</td><td>3</td><td>3</td><td>3</td><td>4</td></tr><tr><td>t_1/2 (hr)</td><td>1.18 (0.115)</td><td>1.94 (0.984)</td><td>1.22 (0.178)</td><td>2.14 (1.35)</td><td>1.35 (0.470)</td><td>1.08 (0.051)</td><td>1.26 (0.229)</td><td>1.21 (0.171)</td><td>1.22 (0.229)</td></tr><tr><td colspan=\"10\">Cycle 2 Day 50</td></tr><tr><td><italic>n</italic></td><td>3</td><td>2</td><td>1</td><td>3</td><td>2</td><td>2</td><td>3</td><td>2</td><td>2</td></tr><tr><td>C_max (ng/mL)</td><td>966 (225)</td><td>3820 (453)</td><td>6710</td><td>11400 (1970)</td><td>15700 (849)</td><td>17900 (9260)</td><td>34200 (1120)</td><td>23650 (16617)</td><td>39500 (22600)</td></tr><tr><td><italic>n</italic></td><td>3</td><td>2</td><td>1</td><td>3</td><td>2</td><td>2</td><td>3</td><td>2</td><td>2</td></tr><tr><td>t_max (hr)</td><td>0.5 (0)</td><td>0.5 (0)</td><td>0.5</td><td>0.5 (0)</td><td>0.5 (0)</td><td>0.5 (0)</td><td>0.5 (0)</td><td>0.5 (0)</td><td>0.5 (0)</td></tr><tr><td><italic>n</italic></td><td>3</td><td>2</td><td>1</td><td>3</td><td>2</td><td>2</td><td>3</td><td>2</td><td>2</td></tr><tr><td>AUC_0-6 (hr*ng/mL)</td><td>1360 (265)</td><td>5360 (530)</td><td>8790</td><td>19500 (7100)</td><td>24300 (566)</td><td>29700 (15300)</td><td>53200 (6470)</td><td>38100 (31537)</td><td>55500 (31100)</td></tr><tr><td><italic>n</italic></td><td>3</td><td>2</td><td>1</td><td>3</td><td>2</td><td>2</td><td>3</td><td>2</td><td>2</td></tr><tr><td>t_1/2 (hr)</td><td>1.21 (0.293)</td><td>1.06 (0.051)</td><td>1.02</td><td>1.99 (1.14)</td><td>1.13 (0.206)</td><td>1.18 (0.153)</td><td>1.21 (0.228)</td><td>1.05 (0.255)</td><td>1.06 (0.131)</td></tr></tbody></table></table-wrap>", "<table-wrap id=\"Tab5\"><label>Table 5</label><caption><p>Best overall response (assessed by investigators according to RECIST v1.1).</p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th/><th>All patients <italic>N</italic> = 28</th></tr></thead><tbody><tr><td>Best overall response, <italic>n</italic> (%)</td><td>1 (3.6)</td></tr><tr><td>Complete response</td><td>0</td></tr><tr><td>Partial response (confirmed)</td><td>1 (3.6)</td></tr><tr><td>Stable disease</td><td>11 (39.3)</td></tr><tr><td>Progressive disease</td><td>16 (57.1)</td></tr><tr><td>Disease control rate, % (90% CI)</td><td>42.9 (27–60)</td></tr></tbody></table></table-wrap>" ]

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[ "<supplementary-material content-type=\"local-data\" id=\"MOESM1\"></supplementary-material>", "<supplementary-material content-type=\"local-data\" id=\"MOESM2\"></supplementary-material>", "<supplementary-material content-type=\"local-data\" id=\"MOESM3\"></supplementary-material>", "<supplementary-material content-type=\"local-data\" id=\"MOESM4\"></supplementary-material>", "<supplementary-material content-type=\"local-data\" id=\"MOESM5\"></supplementary-material>", "<supplementary-material content-type=\"local-data\" id=\"MOESM6\"></supplementary-material>" ]

[ "<table-wrap-foot><p><italic>ECOG PS</italic> Eastern Cooperative Oncology Group performance status.</p></table-wrap-foot>", "<table-wrap-foot><p>Treatment-Emergent Adverse Events is defined as any event that occurs on or after the first dose of study drug administration or any pre-existing event which worsened in severity after dosing. Safety Population - Dose Escalation = Patients with at least one dose. Dose groups represent Patients’ initial dose. Patients with multiple unique events are counted once per each unique preferred term and System Organ Class. Coding used MedDRA version 23.0 Any event not graded will have a default value of grade 3.</p></table-wrap-foot>", "<table-wrap-foot><p>TEAE is defined as any event that occurs on or after the first dose of study drug administration or any pre-existing event which worsened in severity after dosing. Safety Population - Dose Escalation = Patients with at least one dose. Dose groups represent Patients’ initial dose. Patients with multiple unique events are counted once per each unique preferred term and System Organ Class. Coding used MedDRA version 23.0. Any event not graded will have a default value of grade 3.</p></table-wrap-foot>", "<table-wrap-foot><p>Values shown are mean and (standard deviation).</p><p><italic>AUC</italic>_{<italic>0-6</italic>} area under the curve (AUC) from start of infusion to 6 h post end of-infusion, <italic>C</italic>_{<italic>max</italic>} maximum concentration, <italic>n</italic> number of concentration vs time profiles included in the summary statistics, <italic>t</italic>_{<italic>1/2</italic>} terminal phase half-life, <italic>t</italic>_{<italic>max</italic>} time-of maximum concentration.</p></table-wrap-foot>", "<table-wrap-foot><p>The 90% Confidence Interval (CI) was calculated using the exact (Clopper-Pearson) interval. Best overall response is calculated by including “complete responses” and “partial responses”. Disease control rate is calculated by including “complete responses”, “partial responses” and “stable diseases”.</p><p><italic>RECIST</italic> Response Evaluation Criteria in Solid Tumors.</p></table-wrap-foot>", "<fn-group><fn><p><bold>Publisher’s note</bold> Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.</p></fn></fn-group>" ]

[ "<graphic xlink:href=\"43856_2024_433_Fig1_HTML\" id=\"d32e355\"/>", "<graphic xlink:href=\"43856_2024_433_Fig2_HTML\" id=\"d32e382\"/>", "<graphic xlink:href=\"43856_2024_433_Fig3_HTML\" id=\"d32e417\"/>", "<graphic xlink:href=\"43856_2024_433_Fig4_HTML\" id=\"d32e431\"/>", "<graphic xlink:href=\"43856_2024_433_Fig5_HTML\" id=\"d32e439\"/>" ]

[ "<media xlink:href=\"43856_2024_433_MOESM1_ESM.docx\"><caption><p>Description of Additional Supplementary Files</p></caption></media>", "<media xlink:href=\"43856_2024_433_MOESM2_ESM.pdf\"><caption><p>Supplementary Information</p></caption></media>", "<media xlink:href=\"43856_2024_433_MOESM3_ESM.xlsx\"><caption><p>Supplementary Data 1</p></caption></media>", "<media xlink:href=\"43856_2024_433_MOESM4_ESM.xlsx\"><caption><p>Supplementary Data 2</p></caption></media>", "<media xlink:href=\"43856_2024_433_MOESM5_ESM.pdf\"><caption><p>Peer Review File</p></caption></media>", "<media xlink:href=\"43856_2024_433_MOESM6_ESM.pdf\"><caption><p>Reporting Summary</p></caption></media>" ]

{ "acronym": [], "definition": [] }

[ "<title>Introduction</title>", "<p id=\"Par2\">In humans, the superfamily of Cytochrome P450 (CYP) enzymes comprises 18 families of heme-containing proteins that belong to the group of oxidoreductases. Enzymes of only three CYP families are involved in hepatic drug metabolism. The name CYP450 derives from its characteristic light absorbance attribute that is caused by the inherent heme group^{##REF##11178272##1##}. Due to their almost unique properties, reaction mechanism and long evolutionary history, these enzymes have been studied intensely in various aspects since their discovery^{##REF##26002730##2##,##REF##34206277##3##}. These studies had great practical benefits and allow for example the use of most drugs in medicine^{##UREF##0##4##}.</p>", "<p id=\"Par3\">Eukaryotic CYPs can be found in different cell organelles, mainly in the endoplasmic reticulum^{##REF##29653695##5##,##REF##32951814##6##}. Since CYPs are the only human enzymes capable of catalyzing hydroxylation of non-activated carbon atoms, they have a very broad and overlapping substrate specificity and additionally they form a variety of isoforms^{##REF##11911841##7##}. In hepatocytes, CYPs mainly process xenobiotics, thereby conducting the first step of their body excretion^{##REF##20232918##8##}. Many drugs, which are ultimately also xenobiotics, are also metabolized by CYPs. The importance of CYPs for drug development and validation is therefore far-reaching and has already been described in detail in various reviews^{##REF##28124606##9##–##REF##23444277##11##}. Due to individual CYP polymorphism, CYPs are also an important factor in the implementation of individualized medicine, particularly in the dosage of pharmaceuticals^{##REF##23467454##12##}. The majority of hepatically metabolized drugs involves the CYP enzymes CYP1A2, CYP2B6, CYP2C9, CYP2C19, CYP2D6 and CYP3A4/5 accounting for more than 79% of drug oxidation^{##REF##18695978##13##,##REF##22680629##14##}. Some drugs are designed as prodrugs to be bioactivated by CYPs in order to form an active component^{##UREF##1##15##}. On the other hand, CYPs significantly determine the half-life of certain individual drugs. In both cases, the drug concentration in an organism is highly dependent on CYPs. Additionally, drugs themselves can act as activators and inhibitors for CYPs. Therefore, interference of the drugs with the enzymatic pathway of CYPs must be considered. Consequently this interference may result in malfunction of the metabolism mechanisms and therefore leading to severe side-effects^{##REF##22680629##14##}.</p>", "<p id=\"Par4\">Since CYPs have an extensive influence on drug metabolism, a reliant screening mechanism with specific CYPs for drug discovery would facilitate the investigation of the metabolic fate of novel drugs as well as of CYP-inhibitors and modulators. CYP inhibition as well as induction can lead to failures of several drugs and a consequent withdrawal from the market. This issue has also been addressed in a comprehensive publication^{##REF##22680629##14##}. To circumvent such issues, screenings can be performed by using human liver cell microsomes from primary hepatocytes, representing the physiological CYP spectrum or more CYP specific by using microsomes derived from genetically modified cells that only express one CYP, so called mono CYP microsomes. The usage of primary hepatocyte microsomes for CYP analysis as well as drug metabolization studies bears the disadvantage to address all variants of endogenous CYPs simultaneously, thereby impeding mono CYP analysis. The recombinant expression of CYPs has already been extensively investigated^{##UREF##2##16##}. However, mammalian CYPs in particular are difficult to produce. Restrictions are caused by the complex co-factor requirements, like the heme group and the availability of the redox partner, cytochrome P450 oxidoreductase (CPR)^{##REF##23086197##17##}.</p>", "<p id=\"Par5\">Bacterial systems such as <italic>Escherichia coli</italic> are probably the most popular platform for recombinant protein expression, due to their straightforward handling and high growth rate. However, the synthesis of complex eukaryotic proteins like CYPs is unfavorable since they can only be expressed in a modified soluble form^{##UREF##2##16##,##REF##22001938##18##}. In contrast to prokaryotes, yeast as well as higher eukaryotes like insect and mammal cells, possess organelles like the endoplasmic reticulum and the Golgi apparatus, enabling proper anchoring of membrane bound proteins. Usually the redox partner NADPH-Cytochrome-P450-oxidoreductase (CPR) is co-expressed to ensure CYP activity^{##REF##17826178##19##–##REF##11368792##21##}. Additionally the co-expression of the chaperon led to an increased yield of active protein for some CYPs by supporting the folding mechanism^{##REF##23886957##22##}. The co-expression of auxiliary factors also plays a role likewise in prokaryotic and in eukaryotic recombinant expression systems.</p>", "<p id=\"Par6\">For industrial purposes a common expression host for recombinant eukaryotic CYPs is <italic>Saccharomyces cerevisiae</italic>, for example for the large scale production of the antimalarial artemisinin through the coexpression of a CPR, CYP71AV1 from <italic>Artemisia annual</italic> and other enzymes^{##REF##23575629##23##}. Also, mammalian CYPs have been used to design a biosynthetic pathway, including 4 CYPs, in yeast for the generation of hydrocortisone from simple carbon sources^{##REF##12514739##24##}. When it comes to mammalian expression systems several liver cell-lines have been shown to be capable of CYP overexpression. However, these cell lines come with the disadvantage of background CYP activity. This problem can be circumvented by functional overexpression of CYPs together with CPR in CHO or HEK cells as shown in recent studies^{##REF##37681872##25##,##REF##32843676##26##}. However, harnessing the advantages of eukaryotic systems for the cell-free synthesis of CYPs remains an unexplored field. Cell-free protein synthesis (CFPS) has the potential for flexible and adjustable analysis of individual CYPs. Taking advantage of endogenous membrane structures eukaryotic cell-free systems have been successfully used for the synthesis of other membrane localized proteins^{##REF##27684475##27##}. In contrast to cell-based expression systems, the CYP translation process is directly accessible. This open system can be directly manipulated and allows the straightforward supplementation of additional components to the reaction like heme and heme precursors (δ-aminolevulinic acid, glucose, glycine, as well as different iron species) as well as heme-producing enzymes to receive active CYPs^{##REF##23172243##28##}. Additionally, it is possible to modify the cells that are used for lysate production similar to cell-based systems. Hereby, CPR can be integrated into the endogenous microsomes in advance to create the suitable reaction environment for CYPs. The stable modification of eukaryotic CHO cells with CPR has been realized earlier^{##REF##9434754##20##}. The desired CYP enzyme can be synthesized in the translationally active modified CHO-CPR lysate in a straightforward manner. Subsequently, various screening assays can be performed without any purification or further processing steps in 96 and 384- well plates. In this context advantages and limits of cell-free synthesis for drug development have been addressed recently^{##REF##32198631##29##}.</p>" ]

[ "<title>Methods</title>", "<title>Template generation</title>", "<p id=\"Par8\">Templates for the synthesis of CYPs in cell-free systems were generated by Biocat GmbH. The protein encoding sequence and further regulatory factors for CAP-independent protein synthesis by using a Cricket paralysis virus-IRES^{##REF##24376523##30##} (Gene number 714916-1/2/3, 724709-12) was integrated in a pUC57-1.8k-vector backbone.</p>", "<title>Cell fermentation, lysis and lysate procession</title>", "<p id=\"Par9\">Suspension adapted Chinese Hamster Ovary cells (CHO-K1) were routinely cultivated in ProCHO5 medium (Lonza Group AG, Basel, Switzerland) supplemented with 6 mM <sc>l</sc>-alanyl-<sc>l</sc>-glutamine (Merck, Darmstadt, Germany). CHO suspension cells were cultured in non-baffled flasks (Corning, New York, USA) at 37 °C and 5 vol-% CO₂ at 100 rpm on an orbital shaker. CHO cells were grown in suspension cultures in shaking flask to a maximal volume of 500 mL or in a 5 L bioreactor. CHO cells were harvested at a density of approximately 4 × 10⁶ cells/mL. During incubation in the fermenter, viability, oxygen concentration, pH and cell density were monitored. Cell washing, lysis and lysate processing were performed as described earlier^{##REF##27684475##27##,##REF##24018795##31##,##REF##35939898##32##}. In short, cells were centrifuges at 200×<italic>g</italic> for 10 min, and the pellet washed with 40 mM HEPES–KOH (pH 7.5), 100 mM NaOAc and 4 mM DTT. The pellet was then resuspended in the same buffer at a density of approximately 5 × 10⁸ cells/mL. Cell-disruption was performed by syringing the suspension through a 20-gauge needle. After a final centrifugation step at 10,000×<italic>g</italic> for 10 min the supernatant was applied to a size-exclusion chromatography column (Sephadex G-25, GE Healthcare, Freiburg, Germany) and elution fractions with high RNA content were pooled. Residing mRNA was digested by addition of 10 U/mL micrococcal nuclease S7 (Roche, Mannheim, Germany) and 1 mM CaCl₂. After incubation for 2 min 6.7 mM EDTA (f.c.) were added. Finally, the lysate was immediately shock-frozen and stored at − 80 °C.</p>", "<p id=\"Par10\">Lysates were prepared from CHO-K1 cells. Additional to the CHO-K1 wild type cell line, lentiviral modified CHO cells that either express human CPR (CHO-CPR) or human CPR together with CYP3A4 (CHO-CPR/CYP3A4) were used. Blasticidin (Biovision GmbH, Ilmenau, Germany) (3 µg/mL, resistance of the CPR expression vector) or Blasticidin and Zeocin (Abcam, Cambridge, UK) (300 µg/mL, resistance of the CYP3A4 expression vector) were added to the culture medium, to maintain the expression of human CPR or CPR/CYP3A4 in the corresponding CHO cell lines. The lysis process for the generation of translationally active lysates was the same as for wild type CHO-K1 cells.</p>", "<title>Cell-free protein synthesis</title>", "<p id=\"Par11\">Synthesis of proteins in translationally active lysates derived from cultured Chinese hamster ovary (CHO) cells and its modified variants CHO-CPR and CHO-CPR/CYP3A4 cells, was performed in batch based systems as previously described^{##REF##27684475##27##}. Accordingly designed, plasmids suitable for cell-free protein synthesis (CFPS), coding for the CYP of interest, were applied as template. T7-RNA-Polymerase, amino acids, an energy regeneration system and other supplements were added to the translationally active lysates with the additional supplementation of 5 µM heme (porcine) (Alfa Aesar Haverhill, Massachusetts, USA) to the reaction and a reaction temperature of 24 °C was set unless noted otherwise. For the isolation of microsomes the translation mixture (TM) was centrifuged at 16,000×<italic>g</italic> for 10 min at 4 °C. The pellet was resuspended in the same volume of PBS to receive the microsomal fraction (MF). The microsomal fraction comprises the endogenous microsomes derived from the endoplasmic reticulum including the de novo synthesized membrane bound proteins.</p>", "<title>Protein yield determination</title>", "<p id=\"Par12\">To validate successful cell-free protein synthesis, radioactive labeling of de novo synthesized proteins with ¹⁴C-leucine was performed enabling qualitative characterization by autoradiography and quantitative analysis through scintillation counting as described earlier^{##REF##22982167##33##}. Disintegrations per minute (dpm) were measured by liquid scintillation counting performed using the Hidex 600 SL (Hidex). Protein yields were calculated based on the dpm, the molecular weight of the synthesized protein, the specific radioactivity A_spec (Eq. ##FORMU##0##1##) and the total number of leucines in the target protein (Eq. ##FORMU##1##2##).</p>", "<title>Acetone precipitation, SDS-PAGE and Autoradiography</title>", "<p id=\"Par13\">Sodium dodecyl sulfate polyacrylamide gel electrophoresis (SDS-PAGE) and autoradiography were used to analyze homogeneity and molecular weight of in vitro translated proteins. 45 μL water were added to 5 µL of the sample and precipitated with 150 μL ice cold acetone at 4 °C for at least 15 min. Precipitated proteins were pelleted at 16,000 × g for 10 min at 4 °C. Protein pellets were dried for 1 h at 45 °C and re-suspended in 20 μL LDS sample buffer. The samples were loaded onto 10% SDS-PAGE gels. SDS-PAGE was performed at 150 V for 1 h. The gels were stained for 1 h using SimplyBlue—SafeStain, and destained in water over night. The gels were dried (Unigeldryer) for 70 min at 70 °C. The dried gels were put on a phosphor screen for at least three days. Radiolabeled proteins were visualized on the Amersham Typhoon laser scanner (GE Healthcare).</p>", "<title>Western blot</title>", "<p id=\"Par14\">Western blotting and subsequent antibody detection were used for the identification of endogenous and de novo synthesized CYP3A4 and CPR in the translation mixture of the cell-free synthesis reaction. SDS-PAGE was performed like described above. Proteins were blotted on a PVDF membrane with an iBlot device (Thermo Fisher Scientific, Waltham, Massachusetts, USA). The membrane was washed three times with TBS and subsequently blocked with 2% Bovine Serum Albumin (BSA) (Carl Roth GmbH + Co. KG, Karlsruhe, Germany) over night at 4 °C. After three washing steps with TBS/T, the membrane was incubated with the primary antibody at a concentration of 0.4 μg/mL in 2% BSA for three hours at room temperature. The blot was washed three times with TBS/T and incubated with a secondary Horse Radish Peroxidase (HRP) linked antibody at a final concentration of 0.5 μg/mL in 2% BSA at room temperature for one hour. Three final washing steps in TBS/T were performed. Chemiluminescent signals were detected after incubation with ECL detection reagent. The primary antibody used for the detection of CPR was “CYPOR (F-10): sc-25270” (Santa Cruz Biotechnology, Dallas, Texas, USA), the primary antibody used for the detection of CYP3A4 was “CYP3A4 (HL3): sc-53850” (Santa Cruz Biotechnology, Dallas, Texas, USA).</p>", "<title>Fluorescence microscopy</title>", "<p id=\"Par15\">Confocal laser scanning microscopy was used to analyze protein translocation. In preparation, the microsomal fraction was separated from the rest of the translation mixture as described above. 5 µL of the MF were diluted in 15 µL PBS and transferred on chambered Coverslips (ibidi GmbH, Gräfelfing, Germany), The samples were analyzed by confocal laser scanning microscopy using a LSM 510 Meta (Zeiss). Therefore, samples were excited with an argon laser at 488 nm, and the emission signals were recorded with a bandpass filter in the wavelength range from 505 to 550 nm. Photobleaching was performed using an argon laser at 488 nm with 100% laser intensity. After photobleaching pictures were taken each minute for 14 min.</p>", "<title>CPR activity assay</title>", "<p id=\"Par16\">CPR activity was determined by the NADPH dependent conversion of the water-soluble tetrazolium salt WST-8 using the “Cytochrome P450 Reductase Activity Assay Kit” (Abcam, Cambridge, UK). The assay was performed according to the manufacturers protocol. The activity was determined directly in the translationally active lysate of wild type (wt) CHO cells and CHO-CPR cells. Additionally, the microsomal fraction was isolated as described previously. The activity was quantified using a calibration curve that was generated with supplements supplied by the kit.</p>", "<title>CYP activity assays</title>", "<p id=\"Par17\">For CYP activity measurement, “P450-Glo™ Assays” (Promega, Madison, Wisconsin, USA) were used. CYP1A2 activity was detected by Methoxy-Luciferin (Luciferin-ME) turnover (V8772), CYP2B6 was detected by Dimethoxybutyl-Luciferin (Luciferin-2B6) turnover (V8321) and CYP3A4 was detected by Luciferin isopropyl acetal (Luciferin-IPA) turnover (V9001). The reaction was performed according to the Promega “P450-Glo™ Assays” protocol except the CYP reaction time was prolonged to 1 h unless otherwise noted. The reaction temperature was set at 37 °C. The NADPH Regeneration System (V9510) was used for the supply of NADPH during the assay. Three control approaches were performed, one with buffer control, one designed as no template control and one control, using human liver cell microsomes (Gibco™ Human Microsomes, 50 Donors) (Thermo Fisher Scientific, Waltham, Massachusetts, USA) as positive control. Human microsomes were tested at a final concentration of 0.4 mg/mL. If not otherwise noted 5 µL of the microsomal fraction of the cell-free reaction were applied as samples to the activity assay. For response condition adjustments, CYP activities were usually expressed as percentage of the highest CYP activity during the assay. For CYP activity quantification, a standard curve was prepared using beetle luciferin (Promega, Madison, Wisconsin, USA) according to the protocol.</p>", "<title>Indirect substrate screening</title>", "<p id=\"Par18\">Luciferase-based assays were also used as a preliminary screening procedure for the turnover of various potential CYP substrates. The substrates testosterone (Sigma-Aldrich Chemie GmbH, Taufkirchen, Germany), midazolam (Sigma-Aldrich Chemie GmbH, Taufkirchen, Germany), efavirenz (Fisher Scientific GmbH, Schwerte, Germany), and phenacetin (Fisher Scientific GmbH, Schwerte, Germany) were solved at a concentration of 3 mM in 100% methanol. Cholesterol as a steroid that is not known to be a CYP substrate of the selected CYPs, was used as control substrate and was prepared in the same way as the substrates. Cell-free CYP synthesis and isolation of the microsomal fraction was performed as described above. The luciferase based CYP activity assay was performed using 5 µL of the microsomal fraction of the cell-free synthesis of CYP1A2, CYP2B6 and CYP3A4. A final concentration of 200 µM of the analyzed substrate was added to the CYP reaction in parallel to the specific luciferase substrate. A vehicle control with methanol was performed to exclude an influence of the solvent. Changes in the turnover of the luciferase product indicate an interaction of the test substrate with the tested CYP. Changes in luminescence signal were expressed in percentage with reference to the result from the batch without addition of a substrate.</p>", "<title>Statistical analysis</title>", "<p id=\"Par19\">Excel Data Analysis tools were used for statistical analysis, especially to test for statistical significance between two independent samples. After F-test for variance, a variance corresponding t-test was performed for records that were normally distributed.</p>" ]

[ "<title>Results</title>", "<p id=\"Par20\">CYP3A4 is known to be involved in the metabolism of most approved medications. Consequently, it was selected as a model protein for initiating cell-free synthesis of cytochrome P450s in eukaryotic lysates.</p>", "<title>Generation of a modified CHO-CPR Lysate</title>", "<p id=\"Par21\">Modified CHO-K1 cell-lines were cultivated similar to wild-type CHO-K1 cells described earlier^{##REF##27684475##27##}. Using a lentivirus vector system, CHO-CPR and CHO-CPR/CYP3A4 cell clones were generated. A doubling rate of about 48 h of CHO-CPR and CHO-CPR/CYP3A4 compared to the wild type cell line with a doubling rate of about 24 h slowed down the process, but did not prevent the achievement of sufficiently high cell densities. The different cell-lines were harvested in the exponential growth phase. Typical growth conditions in the fermenter are shown exemplarily for CHO-CPR cells (Appendix). To retain translocationally active microsomes in the lysate, cells were mildly disrupted using a 20-gauge syringe. After buffer exchange and supplementation of the raw lysate, translational active lysates of the modified cell lines were prepared similar to the process of wild type cell lysate generation. With total target protein yields of around 40 µg/mL at standard conditions, the protein translation in the modified lysates is in the same range as observed for the typical CHO based cell free protein synthesis. After validation of translational activity, the lysates from CHO-CPR cells and CHO-CPR/CYP3A4 cells were additionally analyzed for their CPR and CYP activity.</p>", "<title>Validation of CPR activity in the generated CHO-CPR-lysates</title>", "<p id=\"Par22\">CPR acts as co-enzyme for the analyzed CYPs, therefore it is mandatory for their activity. By engineering CHO-K1 cells a more than threefold increased CPR activity could be detected (Fig. ##FIG##0##1##A). The processed lysates from CHO-CPR cells were centrifuged to separate the endogenous microsomes from the soluble components of the lysate at 16,000×<italic>g</italic> for 10 min. Activity in the CPR-Assay was drastically improved compared to CHO wild type cells. The activity can be detected in particular in the microsomal fraction with about 90% of the total activity in the translation mixture (Fig. ##FIG##0##1##B). While this increase in activity can be linked to the overexpression of CPR, the residual low activity measured in the supernatant fraction could as well stem from soluble cytoplasmic reductases such as novel reductase^{##REF##10625700##34##}. Moreover, the signal in the supernatant might result from non-pelleted smaller vesicles in the CHO system, that need a higher centrifugation speed^{##REF##28916746##35##}.</p>", "<p id=\"Par23\">To further characterize the CPR activity of the lysates on CYPs, cell-free synthesized CYP3A4 was produced in the translationally active lysates. CYP3A4 served as model protein for the cell-free synthesis of CYPs.</p>", "<title>Cell-free synthesis of CYP3A4 in CHO-CPR lysates</title>", "<p id=\"Par24\">CYP3A4 was produced in a batch-based cell-free synthesis. The synthesis was performed using three different lysates: a wild type CHO lysate, a lysate from the aforementioned CPR-expressing CHO cell line and a lysate from CHO cell line expressing CPR as well as CYP3A4. In each lysate a negative control cell-free reaction without the addition of any DNA template (no-template-control = NTC) was performed. The presence of CPR and CYP3A4 in the translation mixture from each batch reaction was visualized via antibody detection on a western-blot (Fig. ##FIG##1##2##A,B). In the anti-CPR western blot, well-defined bands are detectable at approximately 90 kDa. However, these are less prominent in the wild-type CHO lysate compared to the modified lysates (Fig. ##FIG##1##2##A). A well-defined band at ~ 55 kDa and a second 50 kDa side-band in the anti CYP3A4 western blot can be detected in any sample where CYP3A4 has been synthesized in a cell-free manner (Fig. ##FIG##1##2##B). In addition, a much weaker band at ~ 55 kDa can be detected in the NTC of CHO-CPR-CYP3A4 lysates. Besides the western-blot autoradiography was used to visualize to cell-free synthesized CYP3A4 labeled with ¹⁴C-leucine (Fig. ##FIG##1##2##C). Similar to the anti-CYP3A4 western-blot, a well-defined band at the level of about 55 kDa with a 50 kDa side-band was detected in the samples containing the DNA template. However, no bands in any NTC were observed.</p>", "<p id=\"Par25\">A CYP3A4 specific luminescent assay (Luminescent Assays and Screening Systems for Measuring CYP Activity (Promega, Madison, USA)) was performed for enzyme activity determination. CHO-WT lysate already shows production of active CYP, while producing a low background signal. This activity could be increased by co-synthesizing CPR in the cell-free reaction (Fig. ##SUPPL##0##A2##). However, by far the most CYP3A4 activity was measured in CHO-CPR Lysate after cell-free CYP3A4 synthesis, that notably exceeds the activity of CYP3A4 in the CHO-WT-lysate (Fig. ##FIG##1##2##D). Alternatively, the cell-free synthesis of CYP3A4 in an insect lysate was explored. This led to comparable activities, while exhibiting a higher background signal (Fig. ##SUPPL##0##A2##).</p>", "<title>Adaptations of reaction conditions</title>", "<p id=\"Par26\">Heme is a cofactor of CYPs and is therefore indispensable for its function. Adaption of the amount of supplemented heme to the cell free reaction is therefore mandatory. Heme was supplemented in different concentrations to different batches of the cell-free reaction. The CYP activity in the MF was determined by the Luciferase based CYP3A4 activity assay. A heme concentration of 5 µM resulted in the highest CYP3A4 activity, which was more than twofold higher compared to the control without supplementation (Fig. ##FIG##2##3##).</p>", "<p id=\"Par27\">The supplementation of higher concentrations of heme results in equally 60% reduced activity compared to the 5 µM heme supplemented sample. The concentration of 5 µM heme was used in all subsequent batches. For an increased CYP activity, the cell-free reaction temperature was adapted to 24 °C instead of 30° usual for CHO-CFPS (Fig. ##SUPPL##0##A3##).</p>", "<title>Localization, yield and activity of cell-free produced CYP3A4</title>", "<p id=\"Par28\">CYPs are membrane associated proteins, but in contrast to most trans-membrane proteins they are only N-terminally anchored to the membrane and have a partially lipophilic surface that is oriented towards the membrane. The translocation process therefore differs from other membrane proteins that have already been produced successfully by CHO-based cell-free protein synthesis (CFPS). Consequently, localization and the influence of signal sequences are an important issue for the cell-free synthesis of CYPs. The localization of the cell-free produced CYPs was analyzed using confocal laser scanning microscopy. For this purpose, templates for CYP3A4-eYFP fusion proteins were generated. Additionally, a template containing a melittin signal sequence upstream of the transmembrane segment (Mel-CYP3A4-eYFP) was generated. Both templates were used for cell-free protein synthesis in the modified CHO-lysates. Fluorescence microscopy reveals a distinct difference of CYPs produced with the Mel containing template compared to the template without the Mel signal sequence. The CYPs harboring the signal sequence are preferentially localized at the endogenous microsomes (Fig. ##FIG##3##4##A). According to the yield determination, the addition of a melittin signal sequence led to an increased rate of translocation of 40% (Fig. ##FIG##3##4##C). However, despite the higher protein yields the volumetric activity only increased slightly (Fig. ##FIG##3##4##B).</p>", "<p id=\"Par29\">The co-localization was further analyzed by comparing the microsomal fraction of the CYP-sample with the microsomal fraction of an NTC to which the supernatant fraction of the sample was added and incubated for about an hour. To determine if functional posttranslational translocation into the microsomes in fact was present, this sample was analyzed using fluorescence microscopy (Fig. ##SUPPL##0##A4##) and the activity assay. Usually, a co-translational translocation would be expected; however, fluorescence microscopy reveals a similar image as in the microsomal fraction. The overall intensity of the fluorescence signal seems to be lower than in the MF. The CYPs of the supernatant fraction are not active despite being co-localized with the microsomes of the NTC batch (Fig. ##FIG##4##5##B) and despite displaying a higher target protein yield than the CYPs in the microsomal fraction (Fig. ##FIG##4##5##A).</p>", "<p id=\"Par30\">Since yields of active protein could not be improved the data implies, that a notable amount Mel-CYP3A4 is produced inactive, further experiments were performed using CYP without the melittin signal peptide.</p>", "<title>Synthesis of different CYPs and turnover of pharmaceutically relevant CYP substrates</title>", "<p id=\"Par31\">The application of cell-free protein synthesis enables the time-saving synthesis and analysis of different proteins via template exchange. Besides CYP3A4, CYP1A2 and CYP2B6 were synthesized in the modified CHO cell-free system using the same adapted reaction conditions. Yield determination was executed by scintillation counting of ¹⁴C-labeled protein. Additional activity assays were performed using the corresponding luciferase based assay (Fig. ##FIG##5##6##A). All CYPs were active in the microsomal fraction with almost zero background. Cell-free produced CYP2B6 had the highest activity (15 µU/mL) followed by CYP3A4 (4 µU/mL) and CYP1A2 (2 µU/mL).</p>", "<p id=\"Par32\">An indirect activity assay using the luciferase-based assay will identify potential CYP substrates and inhibitors in a screening procedure. For this purpose, various known pharmaceutically relevant CYP substrates (testosterone, midazolam, efavirenz and phenacetin) were used as a proof of principle. CYP1A2, CYP2B6 and CYP3A4 were cell-free synthesized in modified CHO-CPR lysates. Microsomes containing the CYPs were isolated and applied to the corresponding luciferase-based activity assay. CYP substrates to be analyzed were added to the mono-CYP microsomes in the activity assay. All substrates were added at a final concentration of 200 µM each. A sample without additional substrate (vehicle) and an additional sample with cholesterol as non-interacting control substance with the respective CYPs, were prepared as a reference. Due to competitive turnover, adding of CYP substrates should lead to a decrease of luciferase signal due to competitive substrate turnover in batches with interacting substrates (Fig. ##FIG##5##6##B).</p>", "<p id=\"Par33\">In the competitive assay CYP1A2, CYP2B6 and CYP3A4 were analyzed by their activity changes after adding different CYP substrates. CYP1A2 assay luciferase activity was reduced by all tested substrates while midazolam and efavirenz had the highest impact. CYP2B6 assay luciferase activity reduction was only observed after addition of efavirenz. The supplementation of testosterone led to a 100% increase of monooxygenase activity for CYP2B6. CYP3A4 assay luciferase activity was drastically reduced by testosterone, midazolam and efavirenz.</p>" ]

[ "<title>Discussion</title>", "<p id=\"Par34\">Recombinant expression of membrane proteins has been challenging for many years^{##REF##29702227##36##,##UREF##3##37##}. More than half of all pharmacologically relevant proteins are membrane-bound^{##REF##29075003##38##}. Therefore, an outstanding interest in the development of efficient procedures to produce a wide variety of functional membrane proteins exists. Recent progress in CFPS lead to the successful synthesis of various toxic and membrane bound proteins accessible for research and development^{##REF##32198631##29##,##REF##24931371##39##–##REF##24370776##42##}. However, there are only a few studies on CYPs, one of the pharmaceutically most relevant groups of membrane proteins. Recombinant expression of these heme-containing, membrane bound oxidoreductases was attempted frequently in several research studies with some success^{##UREF##2##16##,##REF##9434754##20##}, but partially limited due to the lack of cofactors and a suitable membrane environment, especially for prokaryotic expression systems^{##REF##23172243##28##}. However, several commercially available products indicate that there is currently unpublished progress and certainly a demanding interest in CYP production, for example by companies such as Hypha discovery, Xenotech, Merck and Thermo Fisher. Until now, research on cell-free protein synthesis based CYP production is only poorly covered. Cell-free protein synthesis based on vesicle containing eukaryotic cell-extracts allows for the precise development of convenient CYP substrate screening systems. In this context the availability of ER originating and CPR harboring endogenous microsomes, which can be programmed with individual CYPs by cell-free synthesis, is of outstanding advantage.</p>", "<p id=\"Par35\">The electron transfer of CPR is mandatory for the activity of CYPs, therefore, a closer look at CPR localization and activity in translationally active lysates is of fundamental importance^{##REF##23086197##17##,##REF##23737303##43##}. Despite CPR activity was detected in the wild type-CHO cells and their lysates per se, an increased CPR activity was detected in CHO lysates derived from the CHO-CPR cell line overexpressing human CPR. The use of CHO cells specifically designed for CYP synthesis and in particular to produce CPR-enriched CHO-CPR lysates, led to a threefold boost of CPR activity due to its overexpression. The microsomes in the CHO lysates originate from the ER of the cells in which CPR and most CYPs are naturally located^{##REF##25288196##44##}. Therefore, a natural-like translocation that led to correct localization and folding of CPR in the microsomes can be assumed. Consequently, the generated CHO-CPR lysates are optimally suited for the production of a variety of CYPs. In our study CYP3A4 was used as model CYP for the characterization of the generated CHO-CPR lysates, since it is the most frequently analyzed CYP and responsible for the majority of phase-I xenobiotic and especially drug metabolization in the human liver^{##REF##23444277##11##}. Cell-free synthesis of CYP3A4 in the modified lysates led to a fourfold increase of total CYP3A4 activity per volume cell-free reaction compared to synthesis in conventional lysates.</p>", "<p id=\"Par36\">The exploitation of fast and convenient high-throughput screening systems for biomolecules is one of the most remarkable advantages of open cell-free systems^{##UREF##4##45##}. This platform technology enables, for example, the synthesis of different CYPs as well as different CYP variants without time-consuming cloning and fermentation steps^{##REF##31681738##41##}. Cell-free protein synthesis based on CHO lysates in this context is a promising technology for various applications, including in vitro drug screening platforms, CYP-specific metabolite phenotyping and synthesis, pharmacologically relevant toxicological studies through to diagnostic applications. The use of CHO cell-lines in protein production is widely established in manifold processes that require a highly evolved eukaryotic expression system. CHO cell-based systems enable the synthesis of complex membrane embedded proteins. For the first time this is shown here in a hybrid model qualifying cell-based and cell-free protein synthesis methods side by side. The lack of CYP background activity in CHO cells^{##REF##37681872##25##} is an additional advantage of this particular cell line for defined CYP applications. Luciferase-based assays are well suited to quantify ratios of CYP activities in different approaches^{##UREF##5##46##} and quantifications of substrate turnover can additionally be determined by using mass spectrometry^{##REF##18474896##47##}. Western blot analysis shows that the amount of cell-free synthesized CYP is significantly higher than in the parallel cell-based approach using CYP-overexpressing CHO-CPR/CYP3A4 cells. To clearly determine if the double band that is observed in the cell-free samples stems from an alternative translation start, a premature termination or has other causes would ultimately need mass spectrometric analysis of the target protein. The difference in the overall synthesis level seems to be even more pronounced than the difference in activity. This may be due to incomplete membrane integration, misfolding or aggregation of a certain amount of cell-free synthesized CYP. Consequently, there is high potential for adaptation of the reaction parameters resulting in the optimal CYP synthesis conditions with further increase CYP specific monooxygenase activity. A prerequisite for an even more efficient synthesis of membrane proteins is a better understanding of the mechanism of translocation in a eukaryotic cell-free protein synthesis system. Translocon interactions and the entire translation process during co-translational translocation, which is essential for the correct localization and the best possible activity of CYPs, are of particular importance^{##REF##8616892##48##,##REF##28564553##49##}. Additionally the lipid composition has a significant influence on CYP activity, especially in the context of the enzyme`s hydrophobic substrates^{##REF##30861250##50##}.</p>", "<p id=\"Par37\">Besides CPR, heme is the most important co-factor of CYPs, that is mandatory for CYP function^{##REF##20860521##51##}. Sufficient availability of heme during cell-free synthesis reaction is of key importance. However, high heme concentration can lead to a decrease in protein activity due to its hydrophobicity and reactivity^{##REF##12130498##52##,##REF##10522552##53##}, which also has a negative effect on the total amount of active CYPs. A certain basic concentration of heme might already be present in the cell-free system, since basal CYP activities can be measured even without the further addition of heme^{##REF##10522552##53##}. Interestingly above 5 µM a plateau below the optimum is reached. A similar observation was made for the synthesis of unspecific peroxygenases in an insect-cell-free system^{##REF##36394036##54##}. Since in both cases the heme supplementation had no influence on the translation efficiency in the analyzed concentration range, there seems to be a more intricate underlying mechanism potentially affecting protein folding. By using confocal microscopy, the co-localization of fluorescently labeled CYPs and microsomes can be detected. An addition of the melittin signal sequence to the template increased the effect of apparent translocation that could be observed during microscopy. The target protein yield in the microsomal fraction determined by radioactive labeling confirms an increased CYP concentration in the microsomal fraction using the melittin signal sequence. This is in accordance to results observed for several other cell-free synthesized secretory and membrane proteins^{##REF##25821419##55##,##REF##35573250##56##}. However, the addition of the melittin signal sequence led only to a minor increase of total volume activity of CYP3A4 but lead to an accumulation of inactive CYP. Translocation efficiency is therefore probably not the limiting factor for more efficient cell-free CYP production. Future studies may identify the remaining restrictions thereby increasing the amount of holo CYPs.</p>", "<p id=\"Par38\">One of the main goals of cell-free CYP synthesis is the development of a screening system^{##REF##31681738##41##}, allowing the parallel analysis of different CYPs. As a proof of principle the human CYP1A2, and CYP2B6 were synthesized showing the straightforward expandability of the cell-free system to CYPs from other gene families. With 10% (CYP1A2), 5% (CYP2B6) and 20% (CYP3A4) participation in CYP metabolism, these CYPs are among the most important representatives of their respective gene families in research and industry^{##REF##29653695##5##}. Transcriptome data suggests, that no homologs of these three human CYPs are expressed in CHO cells^{##REF##34363982##57##}. Accordingly, as for CYP3A4, no significant activity of the other CYPs was measured in lysates of parental CHO or CHO-CPR before the CYP synthesis. The absence of a background CYP activity also demonstrates for CYP1A2 and CYP2B6 how well the CHO cell-free system is suited for specific CYP synthesis and thereby for the generation of mono-CYP microsomes.</p>", "<p id=\"Par39\">The turnover of pharmaceutically relevant CYP substrates by cell-free produced CYPs could be detected indirectly, by analyzing the competitive turnover in the luciferase substrate-based CYP assays. Interactions of a tested CYP with defined substances results in a change in the luciferase assay activity, which was observed here for all three CYPs for several known substrates/substances.</p>", "<p id=\"Par40\">The sterol hormone testosterone is probably the best studied substrate especially concerning CYP3A4^{##UREF##6##58##,##REF##11714865##59##}. Interactions of CYP3A4 with midazolam and efavirenz^{##REF##15496645##60##,##REF##30517006##61##} were also confirmed during the assay. Similar to CYP3A4, several substrates influenced the activity of CYP1A2. This is also in accordance with previous studies^{##REF##19702529##62##} and confirms the successful cell-free synthesis of this CYP isoform. CYP2B6 activity was influenced by efavirenz, a well-known CYP2B6 substrate^{##REF##22471906##63##}. In contrast to other substrates, the CYP substrate testosterone had an activity increasing effect on CYP2B6. This atypical kinetic characteristic of substrate activation by testosterone has already been observed earlier for CYP2B6, due to autoactivation of the enzyme^{##REF##9732386##64##,##REF##11243870##65##}. Upon initial inspection, an anomaly in the assay appears to be present. However, as the values exhibited reproducibility, it was inferred that this discrepancy is attributed to an autoactivation of the enzyme upon substrate binding, resulting in increased Luc substrate turnover.</p>" ]

[ "<title>Conclusion</title>", "<p id=\"Par41\">The high demand of active CYPs requires a straightforward method for the synthesis of members of this enzyme superfamily. Cell-free protein synthesis enables the synthesis of specific active CYPs using a timesaving procedure. By creating a vesicle containing protein production platform from modified CPR overexpressing CHO cells, the generation of mono-CYP microsomes for a huge variety of future applications becomes feasible. However, this synthesis methodology represents a technological innovation in the field of the production of membrane-attached enzymes. Consequently, there is still a huge potential to be addressed, especially regarding the optimization of the translocation process. So far, it was already possible to use cell-free synthesized CYPs for analytical set-ups. Extensive screening procedures regarding mutations, isoforms and genetic variants, but also detailed substrate and inducer/inhibitor screenings are now facilitated by using CFPS. These promising initial results can be a starting point for various fundamental and applied research projects.</p>" ]

[ "<p id=\"Par1\">Cytochromes P450 (CYPs) are a group of monooxygenases that can be found in almost all kinds of organisms. For CYPs to receive electrons from co-substrate NADPH, the activity of NADPH-Cytochrome-P450-oxidoreductase (CPR) is required as well. In humans, CYPs are an integral part of liver-based phase-1 biotransformation, which is essential for the metabolization of multiple xenobiotics and drugs. Consequently, CYPs are important players during drug development and therefore these enzymes are implemented in diverse screening applications. For these applications it is usually advantageous to use mono CYP microsomes containing only the CYP of interest. The generation of mono-CYP containing mammalian cells and vesicles is difficult since endogenous CYPs are present in many cell types that contain the necessary co-factors. By obtaining translationally active lysates from a modified CHO-CPR cell line, it is now possible to generate mono CYPs in a cell-free protein synthesis process in a straightforward manner. As a proof of principle, the synthesis of active human CYPs from three different CYP450 gene families (CYP1A2, CYP2B6 and CYP3A4), which are of outstanding interest in industry and academia was demonstrated. Luciferase based activity assays confirm the activity of the produced CYPs and enable the individual adaptation of the synthesis process for efficient cell-free enzyme production. Furthermore, they allow for substrate and inhibitor screenings not only for wild-type CYPs but also for mutants and further CYP isoforms and variants. As an example, the turnover of selected CYP substrates by cell-free synthesized CYPs was demonstrated via an indirect luciferase assay-based screening setup.</p>", "<title>Subject terms</title>", "<p>Open Access funding enabled and organized by Projekt DEAL.</p>" ]

[ "<title>Aim of the work</title>", "<p id=\"Par7\">Cytochrome P450 enzymes are one of the best-studied classes of enzymes, but recombinant production is challenging due to their membrane localization and enzymatic coupling. The use of vesicle-based cell-free protein synthesis, which enables the fast and efficient production of various membrane proteins, can provide an alternative way of producing defined active human CYPs. The aim of this study is to outline a protein synthesis platform that enables the synthesis of all kinds of CYPs within only a few hours. For this purpose, modified CHO cell lysates containing the necessary CYP co-factors were generated and characterized. In these lysates, CYPs from different gene families were synthesized. CYP1A2, CYP2B6 and CYP3A4 are prominent representatives of the three most important human CYP families and are therefore utilized as a proof of concept in this study. Finally, cell-free synthesized CYPs are used to demonstrate the straightforward applicability of the system for screening procedures.</p>", "<title>Supplementary Information</title>", "<p>\n</p>" ]

[ "<title>Supplementary Information</title>", "<p>The online version contains supplementary material available at 10.1038/s41598-024-51781-6.</p>", "<title>Acknowledgements</title>", "<p>For the lysate preparation, the authors would like to thank D. Wenzel (Fraunhofer IZI-BB, Potsdam-Golm, Germany).</p>", "<title>Author contributions</title>", "<p>J.F.K. Conceptualization, Data curation, Formal analysis, Investigation, Methodology, Supervision, Validation, Visualization, Writing-original draft; C.S. Resources, Methodology, Writing-review &amp; editing; A.Z. Supervision, Writing-review &amp; editing; D.A.W. Resources, Supervision; R.M.W. Data curation, Formal analysis, Investigation, Writing-review &amp; editing; J.H.K. Writing-review &amp; editing, Resources; S.K. Conceptualization, Data curation, Funding acquisition, Methodology, Project administration, Resources, Supervision, Writing-review &amp; editing.</p>", "<title>Funding</title>", "<p>Open Access funding enabled and organized by Projekt DEAL. This research was funded by the Ministry of Science, Research and Culture (MWFK, Brandenburg, Germany), project PZ-Syn (project number F241-03-FhG/005/001).</p>", "<title>Data availability</title>", "<p>All data generated or analyzed during this study are included in this published article (and its Supplementary Information files).</p>", "<title>Competing interests</title>", "<p id=\"Par42\">The authors declare no competing interests.</p>" ]

[ "<fig id=\"Fig1\"><label>Figure 1</label><caption><p>Validation of the CPR activity in the generated lysates by Cytochrome P450 reductase activity assay kit (colorimetric) (ab204704). The assay was performed according to the manufactures protocol. Diphenyleneiodonium chloride was used as inhibitor control. (<bold>A</bold>) CHO-CPR cells were lysed and processed according to protocols for the generation of translationally active lysates. Two lysates were generated: CHO-CPR and CHO-K1 wt cell lysates. (<bold>B</bold>) CHO-CPR lysate was fractionated by centrifugation at 16,000×<italic>g</italic> for 10 min. CPR activity of the supernatant, the microsomal (Pellet) fraction and the processed lysate (lysate before centrifugation) were compared. Standard deviations were calculated from triplicate analysis (n = 3).</p></caption></fig>", "<fig id=\"Fig2\"><label>Figure 2</label><caption><p>Characterization of translationally active CHO lysates derived from genetically modified CHO cells after cell-free synthesis of CYP3A4. CYP3A4 (57 kDa) and CPR (82 kDa) were identified in the translation mixture through SDS-PAGE (10%) and subsequent western blotting with anti CPR antibodies (<bold>A</bold>) and anti CYP3A4 antibodies (<bold>B</bold>) followed by a secondary HRP linked antibody. Cell-free synthesis of CYP3A4 was compared to no-template-controls (NTC) in each sample. An autoradiograph (<bold>C</bold>) allows the detection of ¹⁴C-Leucine labeled cell-free synthesized proteins in the cell-free reaction. The activity of CYP3A4 per µg synthesized protein in the different lysates was determined by an IPA-Luciferin assay after CYP3A4 cell-free synthesis. Synthesized protein was quantified through ¹⁴C-labeling. Background activity from the NTC was subtracted. Measurements were performed as triplicates (n = 3). Blots and autoradiographs visible in each individual sub-image were created simultaneously and treated equally.</p></caption></fig>", "<fig id=\"Fig3\"><label>Figure 3</label><caption><p>Influence of heme concentration on CYP3A4 activity during cell-free protein synthesis. The synthesis reaction was performed in a batch mode for 3 h. Activity was determined using an IPA luciferase activity assay with a sample size of 2 µL. Standard deviations were calculated from triplicate analysis (n = 3).</p></caption></fig>", "<fig id=\"Fig4\"><label>Figure 4</label><caption><p>(<bold>A</bold>) Confocal microscopy images of cell-free synthesized CYP3A4-eYFP and Mel-CYP3A4-eYFP. The proteins were synthesized in batch mode for 3 h. The microsomal fraction of the cell-free reaction was analyzed. The fluorescent image, a brightfield image and an overlay of both images are shown. NTC = no template control; translation reaction without DNA template. (<bold>B</bold>) Determination of CYP3A4-eYFP and Mel-CYP3A4-eYFP yield and enzyme activity after cell-free synthesis in the translation mixture (TM), the microsomal fraction (MF) and the supernatant fraction (SN). Enzyme activity was determined by an IPA-luciferase assay (Promega). (<bold>C</bold>) Additionally, the yield of cell-free produced proteins was determined via radioactive labeling followed by TCA precipitation and scintillation counting. Standard deviations were calculated from triplicate analysis (n = 3).</p></caption></fig>", "<fig id=\"Fig5\"><label>Figure 5</label><caption><p>Analysis of posttranslational translocation of CYP3A4. Proteins were synthesized in the batch mode for 3 h. The microsomal fraction was analyzed and compared to the microsomal fraction of an NTC to which the supernatant fraction of the CYP batch was added. (<bold>A</bold>) Yield determination of cell-free produced proteins via radioactive labeling followed by TCA precipitation and scintillation counting. (<bold>B</bold>) Relative activity of CYP3A4 in the supernatant fraction (SN), the microsomal fraction (MF) and in the microsomal fraction of an NTC to which the supernatant fraction of a 3 h CYP batch synthesis was added and incubated for about an hour. 3 µL per well of the samples were applied in the assay Standard deviations for B and C were calculated from triplicate analysis (n = 3).</p></caption></fig>", "<fig id=\"Fig6\"><label>Figure 6</label><caption><p>(<bold>A</bold>) Activity determination of different CYPs in the microsomal fraction (MF) after 16,000×<italic>g</italic> centrifugation. (<bold>B</bold>) Screening of different pharmaceutically relevant CYP substrates in relation to a sample without additional substrate. Different CYP substrates were added during the CYP activity assay at a concentration of 200 µM. Cholesterol as a steroid that is not known to be a CYP substrate of the selected CYPs, was used as control substrate. A reduction of Luc signal compared to the vehicle control (without substrate) implies the competitive turnover of the CYP substrate and an inhibition of the CYP activity during the assay. Standard deviations were calculated from triplicate analysis (n = 3).</p></caption></fig>" ]

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[ "<disp-formula id=\"Equ1\"><label>1</label><alternatives><tex-math id=\"M1\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${A}_{spec}= \\frac{{c}_{14C-Leu} \\cdot {A}_{14C-Leu Stock}}{{c}_{total Leu}}$$\\end{document}</tex-math><mml:math id=\"M2\" display=\"block\"><mml:mrow><mml:msub><mml:mi>A</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">spec</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mfrac><mml:mrow><mml:msub><mml:mi>c</mml:mi><mml:mrow><mml:mn>14</mml:mn><mml:mi>C</mml:mi><mml:mo>-</mml:mo><mml:mi>L</mml:mi><mml:mi>e</mml:mi><mml:mi>u</mml:mi></mml:mrow></mml:msub><mml:mo>·</mml:mo><mml:msub><mml:mi>A</mml:mi><mml:mrow><mml:mn>14</mml:mn><mml:mi>C</mml:mi><mml:mo>-</mml:mo><mml:mi>L</mml:mi><mml:mi>e</mml:mi><mml:mi>u</mml:mi><mml:mi>S</mml:mi><mml:mi>t</mml:mi><mml:mi>o</mml:mi><mml:mi>c</mml:mi><mml:mi>k</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:msub><mml:mi>c</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">totalLeu</mml:mi></mml:mrow></mml:msub></mml:mfrac></mml:mrow></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ2\"><label>2</label><alternatives><tex-math id=\"M3\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$concentration\\left[\\frac{\\upmu g}{mL}\\right]=\\frac{measured counts \\left[\\frac{dpm}{mL}\\right]\\cdot molecular weight \\left[\\frac{\\upmu g}{pmol}\\right]}{{A}_{spec}\\left[\\frac{dpm}{pmol}\\right]\\cdot {\\#}_{leucines in the protein}}$$\\end{document}</tex-math><mml:math id=\"M4\" display=\"block\"><mml:mrow><mml:mi>c</mml:mi><mml:mi>o</mml:mi><mml:mi>n</mml:mi><mml:mi>c</mml:mi><mml:mi>e</mml:mi><mml:mi>n</mml:mi><mml:mi>t</mml:mi><mml:mi>r</mml:mi><mml:mi>a</mml:mi><mml:mi>t</mml:mi><mml:mi>i</mml:mi><mml:mi>o</mml:mi><mml:mi>n</mml:mi><mml:mfenced close=\"]\" open=\"[\"><mml:mfrac><mml:mrow><mml:mi mathvariant=\"normal\">μ</mml:mi><mml:mi>g</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"italic\">mL</mml:mi></mml:mrow></mml:mfrac></mml:mfenced><mml:mo>=</mml:mo><mml:mfrac><mml:mrow><mml:mi>m</mml:mi><mml:mi>e</mml:mi><mml:mi>a</mml:mi><mml:mi>s</mml:mi><mml:mi>u</mml:mi><mml:mi>r</mml:mi><mml:mi>e</mml:mi><mml:mi>d</mml:mi><mml:mi>c</mml:mi><mml:mi>o</mml:mi><mml:mi>u</mml:mi><mml:mi>n</mml:mi><mml:mi>t</mml:mi><mml:mi>s</mml:mi><mml:mfenced close=\"]\" open=\"[\"><mml:mfrac><mml:mrow><mml:mi mathvariant=\"italic\">dpm</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"italic\">mL</mml:mi></mml:mrow></mml:mfrac></mml:mfenced><mml:mo>·</mml:mo><mml:mi>m</mml:mi><mml:mi>o</mml:mi><mml:mi>l</mml:mi><mml:mi>e</mml:mi><mml:mi>c</mml:mi><mml:mi>u</mml:mi><mml:mi>l</mml:mi><mml:mi>a</mml:mi><mml:mi>r</mml:mi><mml:mi>w</mml:mi><mml:mi>e</mml:mi><mml:mi>i</mml:mi><mml:mi>g</mml:mi><mml:mi>h</mml:mi><mml:mi>t</mml:mi><mml:mfenced close=\"]\" open=\"[\"><mml:mfrac><mml:mrow><mml:mi mathvariant=\"normal\">μ</mml:mi><mml:mi>g</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"italic\">pmol</mml:mi></mml:mrow></mml:mfrac></mml:mfenced></mml:mrow><mml:mrow><mml:msub><mml:mi>A</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">spec</mml:mi></mml:mrow></mml:msub><mml:mfenced close=\"]\" open=\"[\"><mml:mfrac><mml:mrow><mml:mi mathvariant=\"italic\">dpm</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"italic\">pmol</mml:mi></mml:mrow></mml:mfrac></mml:mfenced><mml:mo>·</mml:mo><mml:msub><mml:mo>#</mml:mo><mml:mrow><mml:mi mathvariant=\"italic\">leucinesintheprotein</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:mfrac></mml:mrow></mml:math></alternatives></disp-formula>" ]

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[ "<supplementary-material content-type=\"local-data\" id=\"MOESM1\"></supplementary-material>" ]

[ "<fn-group><fn><p><bold>Publisher's note</bold></p><p>Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.</p></fn></fn-group>" ]

[ "<graphic xlink:href=\"41598_2024_51781_Article_Equ1.gif\" position=\"anchor\"/>", "<graphic xlink:href=\"41598_2024_51781_Article_Equ2.gif\" position=\"anchor\"/>", "<graphic xlink:href=\"41598_2024_51781_Fig1_HTML\" id=\"MO1\"/>", "<graphic xlink:href=\"41598_2024_51781_Fig2_HTML\" id=\"MO2\"/>", "<graphic xlink:href=\"41598_2024_51781_Fig3_HTML\" id=\"MO3\"/>", "<graphic xlink:href=\"41598_2024_51781_Fig4_HTML\" id=\"MO4\"/>", "<graphic xlink:href=\"41598_2024_51781_Fig5_HTML\" id=\"MO5\"/>", "<graphic xlink:href=\"41598_2024_51781_Fig6_HTML\" id=\"MO6\"/>" ]

[ "<media xlink:href=\"41598_2024_51781_MOESM1_ESM.pdf\"><caption><p>Supplementary Figures.</p></caption></media>" ]

{ "acronym": [], "definition": [] }

[ "<title>Introduction</title>", "<p id=\"Par2\">Dye-contaminated wastewater affects to be toxic to aquatic organisms because it has an aromatic structure difficult to degrade, and the colored particles may block the transmission of light into the water body. As a result, aquatic plants and algae are unable to photosynthesize. Furthermore, the lack of oxygen in water sources affects life in water and destroys the scenery which is offensive to the onlookers^{##REF##35026583##1##}. Many industries of dye, pigment, paint, paper, printing, cosmetics, and textile widely use dyes in their product manufacturing, especially direct dyes are popularly used for long-lasting cellulose and lignin dyeing^{##UREF##0##2##}. Direct red 28 (DR28) dye is also popularly used for dyeing cotton in many industries, so wastewater with contaminated DR28 dyes is recommended to be treated before discharging for environmental safety.</p>", "<p id=\"Par3\">The treatment methods of dyes are coagulation-flocculation, chemical oxidation, electrochemistry, ion exchange, ozonation, photochemistry, adsorption, and biological process^{##UREF##1##3##}. However, adsorption is a favored method for adsorbing dyes because it is the effective method, easy operation, suitable cost, and offering several adsorbents^{##UREF##2##4##}. In addition, the main criteria of good adsorbents are required as environmentally friendly adsorbent, easy access, cheap cost, and cost-effective use, so the agricultural waste is one option that corresponds to these requirements above. Many agricultural wastes have been used for removing several dyes shown in Table ##TAB##0##1##. Many studies reported in Table ##TAB##0##1## have applied the sugarcane bagasse to eliminate dyes of reactive blue 19, methyl red, and basic red 2, reactive blue 4^{##UREF##3##5##–##REF##36211078##8##}, so they can affirm the sugarcane bagasse's ability to adsorb several dyes. However, the development of sugarcane bagasse to deal with the specific pollutant targets with the high concentration strength of industrial wastewater also needs more investigation.</p>", "<p id=\"Par4\">Many methods of acid treatment, alkaline treatment, and metal oxide modifications are used to increase the abilities of sugarcane bagasse materials for dye removals also illustrated in Table ##TAB##0##1##. In previous studies, sugarcane bagasse beads modified with titanium dioxide (TiO₂), magnesium oxide (MgO), aluminum oxide (Al₂O₃), and zinc oxide (ZnO) have been used for removing RB4 dye^{##REF##36211078##8##,##UREF##6##9##}; however, no one used them to remove DR28 dye. As a result, their comparison results need to confirm the abilities of sugarcane bagasse beads modified with those metal oxides for removing several anionic dyes. Therefore, this current study attempts to investigate the abilities of sugarcane bagasse beads with or without metal oxide modifications for removing DR28 dye to understand how the addition of metal oxide with different types affects DR28 dye, and which one offers the highest DR28 dye removal.</p>", "<p id=\"Par5\">In this study, sugarcane bagasse beads (SBB), sugarcane bagasse beads modified with titanium dioxide (SBBT), sugarcane bagasse beads modified with magnesium oxide (SBBM), sugarcane bagasse beads modified with aluminum oxide (SBBA), and sugarcane bagasse beads modified with zinc oxide (SBBZ) were synthesized for investigating their characterizations and DR28 dye removal efficiencies. Brunauer–Emmett–Teller (BET), Field emission scanning electron microscopy and focus ion beam (FESEM-FIB), Energy dispersive X-ray spectrometer (EDX), and Fourier transform infrared spectroscopy (FT-IR) were used for identifying their specific surface area, pore volumes, pore sizes, surface structures, chemical elements, and chemical functional groups. In addition, their points of zero charge (pH_pzc) were also investigated to recognize their surface charges. The affecting factors of dosage, contact time, temperature, pH, and concentration were examined by batch tests, and their adsorption isotherms and kinetics were also determined by nonlinear models of Langmuir, Freundlich, Temkin, Dubinin–Radushkevich, pseudo-first-order kinetic, pseudo-second-order kinetic, Elovich, and intra-particle diffusion for describing their adsorption patterns and mechanisms. The thermodynamic study was also investigated to understand the temperature effect on their DR28 dye removals.</p>" ]

[ "<title>Material and method</title>", "<title>Raw material and preparation</title>", "<p id=\"Par6\">Sugarcane bagasse was taken from the local market in Khon Kaen province, Thailand. Before use, it was washed with tap water to remove contaminations, and then it was dried in a hot air oven (Binder, FED 53, Germany) at 80 °C for 24 h. Then, it was ground, sieved in size of 125 µm, and kept in a desiccator called sugarcane bagasse powder (SBP)^{##REF##36211078##8##}.</p>", "<title>Chemicals</title>", "<p id=\"Par7\">All chemicals used in this study were analytical grades (AR) without purification. They were titanium dioxide (TiO₂) (Loba, India), magnesium oxide (MgO) (RCI Labscan, Thailand), aluminum oxide (Al₂O₃) (Kemaus, New Zealand), zinc oxide (ZnO) (QRëC, New Zealand), sodium alginate (NaC₆H₇O₆) (Merck, Germany), calcium chloride dihydrate (CaCl₂·2H₂O) (RCI Labscan, Thailand), direct red 28 (DR28) dye (C₃₂H₂₂N₆Na₂O₆S₂) (Sigma-Aldrich, Germany), 0.1 M HCl (RCI Labscan, Thailand), and 0.1 M NaOH (RCI Labscan, Thailand). The pH adjustments used 0.5% nitric acid (HNO₃) (Merck, Germany) and 0.5% NaOH (RCI Labscan, Thailand).</p>", "<title>Dye solution preparation</title>", "<p id=\"Par8\">The dye solutions are prepared from the stock solution of direct red 28 (DR28) dye of 100 mg/L concentration.</p>", "<title>Material synthesis</title>", "<p id=\"Par9\">The material synthesis methods are mentioned from the study of Ngamsurach et al.^{##REF##36211078##8##}, Praipipat et al.^{##UREF##6##9##}, and Praipipat et al.^{##UREF##13##18##}, and the flow diagrams are illustrated in Fig. ##FIG##0##1##. The details are described below:</p>", "<title>The synthesis of sugarcane bagasse beads (SBB)</title>", "<p id=\"Par10\">Firstly, 10 g of SBP were added to a 1000 mL beaker containing 400 mL of 2% NaC₆H₇O₆, then they were heated by a hot plate (Ingenieurbüro CAT, M. Zipperer GmbH, M 6, Germany) at 60 °C with a stable stirring speed of 200 rpm until homogeneous mixed. Next, they were contained into a syringe with a needle (1.2 mm × 25 mm), and they were dropwise into 250 mL of 0.1 M CaCl₂·2H₂O and soaked for 24 h for a bead setting. Then, they were filtrated, rinsed with DI water, and air-dried at room temperature for 12 h. Finally, they were kept in a desiccator before use called sugarcane bagasse beads (SBB).</p>", "<title>The synthesis of sugarcane bagasse beads modified with titanium dioxide (SBBT) or magnesium oxide (SBBM) or aluminum oxide (SBBA) or zinc oxide (SBBZ)</title>", "<p id=\"Par11\">Firstly, 10 g of SBP were added to a 250 mL Erlenmeyer flask containing 160 mL of 5% (w/v) TiO₂ or MgO or Al₂O₃ or ZnO solution prepared by the deionized water, and they were homogeneously mixed by an orbital shaker (GFL, 3020, Germany) of 200 rpm for 3 h. Next, they were filtered, air-dried at room temperature for 12 h, and kept in a desiccator called sugarcane bagasse powder mixed with TiO₂ or MgO or Al₂O₃ or ZnO (SBPT or SBPM or SBPA, or SBPZ). Then, SBPT or SBPM or SBPA, or SBPZ were added to a 1000 mL beaker containing 400 mL of 2% NaC₆H₇O₆, then they were heated by a hot plate at 60 °C with a stable stirring speed of 200 rpm until homogeneous mixed. Next, they were contained into a syringe with a needle (1.2 mm × 25 mm), and they were dropwise into 250 mL of 0.1 M CaCl₂·2H₂O and soaked for 24 h for a bead setting. Then, they were filtrated, rinsed with DI water, and air-dried at room temperature for 12 h. Finally, they were kept in a desiccator before use called sugarcane bagasse modified with titanium dioxide beads (SBBT), sugarcane bagasse modified with magnesium oxide beads (SBBM), sugarcane bagasse modified with aluminum oxide beads (SBBA), and sugarcane bagasse modified with zinc oxide beads (SBBZ).</p>", "<title>Material characterizations</title>", "<p id=\"Par12\">The material characterizations on the specific surface area, pore volumes, pore sizes, surface structures, chemical elements, and chemical functional groups of SBB, SBBT, SBBM, SBBA, and SBBZ were investigated by Brunauer–Emmett–Teller (BET), Field emission scanning electron microscopy and focus ion beam (FESEM-FIB) with Energy dispersive X-ray spectrometer (EDX) (FEI, Helios NanoLab G3 CX, USA), and Fourier transform infrared spectroscopy (FT-IR) (Bruker, TENSOR27, Hong Kong).</p>", "<title><italic>The point of zero charge (pH</italic>_{<italic>pzc</italic>}<italic>)</italic></title>", "<p id=\"Par13\">The method of the points of zero charge of SBB, SBBT, SBBM, SBBA, and SBBZ for DR28 dye adsorptions is mentioned from the studies of Praipipat et al.^{##UREF##13##18##,##REF##37495622##19##} which was the pH drift method by preparing 0.1 M NaCl solutions with pH values from 2 to 12 by using 0.1 M HCl and 0.1 M NaOH. Then, 2 g/L of SBB or SBBT or SBBM or SBBA, or SBBZ were added to 50 mL of 0.1 M NaCl solution contained in 250 mL Erlenmeyer flask, and it was shaken at 150 rpm for 24 h at room temperature by an orbital shaker. Finally, the final pH of the sample was measured by a pH meter (Mettler Toledo, SevenGo with InLab 413/IP67, Switzerland) and calculated ∆pH (pH_final–pH_initial) to determine the point of zero charge (pH_pzc).</p>", "<title>Batch experiments</title>", "<p id=\"Par14\">The affecting factors of dose (5–30 g/L), contact time (3–18 h), temperature (20–50 °C), pH (3–11), and concentration (30–90 mg/L) with the control condition of initial DR28 dye concentration of 50 mg/L, a sample volume of 100 mL, and a shaking speed of 150 rpm by using an incubator shaker (New Brunswick, Innova 42, USA)^{##REF##36211078##8##,##UREF##6##9##,##REF##36406531##20##} on DR28 dye removal efficiencies of SBB, SBBT, SBBM, SBBA, and SBBZ were investigated through a series of batch experiments which referred from the previous study of Praipipat et al.^{##UREF##13##18##} Their optimum conditions were chosen from the lowest dose or contact time or temperature or pH or concentration with obtaining the highest DR28 dye removal efficiencies^{##UREF##6##9##}. UV–VIS Spectrophotometer (UH5300, Hitachi, Japan) with a wavelength of 497 nm was used for analyzing dye concentrations, and the triplicate experiments were investigated to verify the results and report the average value. Dye removal efficiency in the percentage and dye adsorption capacity is calculated following Eqs. (##FORMU##0##1##)–(##FORMU##1##2##):where <italic>C</italic>_e is the dye concentration at equilibrium (mg/L), <italic>C</italic>₀ is the initial dye concentration (mg/L)<italic>, q</italic>_e is the capacity of dye adsorption on adsorbent material at equilibrium (mg/g)<italic>, V</italic> is the sample volume (L), and <italic>m</italic> is the amount of adsorbent material (g).</p>", "<title>Adsorption isotherms</title>", "<p id=\"Par15\">The adsorption patterns of SBB, SBBT, SBBM, SBBA, and SBBZ were determined by using nonlinear Langmuir, Freundlich, Temkin, and Dubinin–Radushkevich models. Langmuir model is monolayer adsorption, and Freundlich model represents multilayer adsorption^{##UREF##14##21##,##UREF##15##22##}. Temkin model refers to the heat of adsorption with decreasing from the increase of coverage adsorbent, and Dubinin–Radushkevich model is used to determine the adsorption mechanism between physisorption and chemisorption^{##UREF##16##23##,##UREF##17##24##}. Their adsorption isotherms were calculated by Eqs. (##FORMU##2##3##)–(##FORMU##5##6##)^{##REF##36406531##20##–##UREF##17##24##}:</p>", "<p id=\"Par16\">Langmuir isotherm:</p>", "<p id=\"Par17\">Freundlich isotherm:</p>", "<p id=\"Par18\">Temkin isotherm:</p>", "<p id=\"Par19\">Dubinin–Radushkevich isotherm:where <italic>q</italic>_e is the capacity of dye adsorption on adsorbent material at equilibrium (mg/g), <italic>q</italic>_m is the maximum capacity of dye adsorption on adsorbent material (mg/g),<italic> C</italic>_e is the equilibrium of dye concentration (mg/L), <italic>K</italic>_L is Langmuir adsorption constant (L/mg), <italic>K</italic>_F is Freundlich constant of adsorption capacity (mg/g)(L/mg)^1/n, and <italic>n</italic> is the constant depicting of the adsorption intensity. <italic>R</italic> is the universal gas constant (8.314 J/mol K), <italic>T</italic> is the absolute temperature (K), <italic>b</italic>_T is the constant related to the heat of adsorption (J/mol), <italic>A</italic>_T is the equilibrium binding constant corresponding to maximum binding energy (L/mg), <italic>K</italic>_DR is the activity coefficient related to mean adsorption energy (mol²/J²), and <italic>ε</italic> is the Polanyi potential (J/mol). Their graphs are plotted by <italic>q</italic>_e versus <italic>C</italic>_e.</p>", "<p id=\"Par20\">For adsorption isotherm experiments, 25 g/L and 18 h of SBB, or 15 g/L and 18 h of SBBT or 20 g/L and 6 h of SBBM, or 15 g/L and 12 h of SBBA, or 25 g/L and 12 h of SBBZ have added to 250 mL Erlenmeyer flasks with variable DR28 dye concentrations from 30 to 90 mg/L. The control condition of SBB or SBBT or SBBM or SBBA or SBBZ was a sample volume of 100 mL, a shaking speed of 150 rpm, pH 3, and a temperature of 35 °C.</p>", "<title>Adsorption kinetics</title>", "<p id=\"Par21\">The adsorption rate and mechanism of SBB, SBBT, SBBM, SBBA, and SBBZ were determined by using nonlinear pseudo-first-order kinetic, pseudo-second-order kinetic, Elovich, and intra-particle diffusion models. The pseudo-first-order and pseudo-second-order kinetic models are the physisorption and chemisorption processes^{##UREF##18##25##,##UREF##19##26##}. Elovich model is the chemical adsorption process with a heterogeneous surface, and the intra-particle diffusion model refers to the rate limiting in the adsorption process^{##UREF##20##27##,##UREF##21##28##}. Their adsorption kinetics were calculated by Eqs. (##FORMU##6##7##)–(10)^{##UREF##18##25##–##UREF##21##28##}:</p>", "<p id=\"Par22\">Pseudo-first-order kinetic model:</p>", "<p id=\"Par23\">Pseudo-second-order kinetic model:</p>", "<p id=\"Par24\">Elovich model:</p>", "<p id=\"Par25\">Intra-particle diffusion model:where <italic>q</italic>_e is the capacity of dye adsorption on adsorbent material at equilibrium (mg/g)<italic>, </italic><italic>q</italic>_t is the capacity of dye adsorption on adsorbent material at the time (<italic>t</italic>) (mg/g), <italic>k</italic>₁ is a pseudo-first-order rate constant (min⁻¹), and <italic>k</italic>₂ is a pseudo-second-order rate constant (g/mg min). <italic>α</italic> is the initial adsorption rate (mg/g min) and <italic>β</italic> is the extent of surface coverage (g/mg). <italic>k</italic>_i is the intra-particle diffusion rate constant (mg/g min^0.5) and <italic>C</italic>_i is the constant that gives an idea about the thickness of the boundary layer (mg/g)^{##REF##37495622##19##,##REF##36702856##29##}. Their graphs are plotted by <italic>q</italic>_t versus <italic>t</italic>.</p>", "<p id=\"Par26\">For the kinetic experiments, 25 g/L of SBB or 15 g/L of SBBT or 20 g/L of SBBM or 15 g/L of SBBA, or 25 g/L of SBBZ were added to a 1000 mL beaker. The control condition of SBB or SBBT or SBBM or SBBA, or SBBZ was a sample volume of 1000 mL, DR28 dye concentrations of 50 mg/L, a shaking speed of 150 rpm, pH 3, and a contact time of 24 h^{##UREF##13##18##}.</p>", "<title>Thermodynamic study</title>", "<p id=\"Par27\">The temperature effect on DR28 dye adsorption capacities of SBB, SBBT, SBBM, SBBA, and SBBZ were investigated through thermodynamic studies in a range of 293.15–323.15 K, and their results were explained by three thermodynamic parameters of Gibb free energy (∆<italic>G</italic>°), standard enthalpy change (∆<italic>H</italic>°), and standard entropy change (∆<italic>S</italic>°). Equations (##FORMU##10##11##)–(##FORMU##11##13##) were used to calculate their parameters^{##UREF##13##18##}.where <italic>R</italic> is the universal gas constant (8.314 J/mol K), <italic>T</italic> is the absolute temperature (K), and <italic>K</italic>_c is the equilibrium constant (L/mg). The values of ∆<italic>H</italic>° and ∆<italic>S</italic>° were calculated from the slope and intercept of the linear graph between ln <italic>K</italic>_c (<italic>K</italic>_c = <italic>q</italic>_e/<italic>C</italic>_e) and 1/<italic>T</italic>, and ∆<italic>G</italic>° is calculated from Eq. (##FORMU##12##13##).</p>", "<p id=\"Par28\">For the thermodynamic experiments, 25 g/L and 18 h of SBB, or 15 g/L and 18 h of SBBT or 20 g/L and 6 h of SBBM, or 15 g/L and 12 h of SBBA, or 25 g/L and 12 h of SBBZ were applied with temperatures of 293.15–323.15 K with the control condition of DR28 dye concentration of 50 mg/L, a sample volume of 100 mL, pH 3, and a shaking speed of 150 rpm^{##REF##36406531##20##}.</p>" ]

[ "<title>Result and discussion</title>", "<title>BET</title>", "<p id=\"Par29\">The specific surface area, pore volumes, and pore sizes of SBB, SBBT, SBBM, SBBA, and SBBZ are illustrated in Table ##TAB##1##2##. Their specific surface area and pore volume could be arranged from high to low of SBBM &gt; SBBT &gt; SBBA &gt; SBBZ &gt; SBB, and SBBM demonstrated the highest surface area and pore volume among other materials. Since magnesium oxide (MgO), titanium dioxide (TiO₂), aluminum oxide (Al₂O₃), and zinc oxide (ZnO) have a high specific surface area by themselves, the specific surface area of prepared materials by those metal oxides have higher specific surface area than raw material. Moreover, the previous studies reported the specific surface area of MgO, TiO₂, Al₂O₃, and ZnO were 60, 50, 40, and 30 m²/g, and they could be arranged in order from high to low of MgO &gt; TiO₂ &gt; Al₂O₃ &gt; ZnO^{##UREF##22##30##,##UREF##23##31##}. As a result, it could support why SBBM had a higher surface area than other materials. Therefore, metal oxides of TiO₂, MgO, Al₂O₃, and ZnO increased the specific area and pore volumes of materials from the formations of those metal oxides with sugarcane bagasse supported more active sites for capturing DR28 dye adsorptions similar reported by previous studies used the same metal oxides^{##UREF##6##9##,##UREF##13##18##,##REF##36406531##20##}. Moreover, other metal oxides of zinc oxide, iron(III) oxide-hydroxide, and goethite have also been used in previous studies supported this study that the raw materials with adding metal oxides increased the surface area and pore volume^{##UREF##13##18##,##REF##36229648##32##–##UREF##26##36##}. Since their pore sizes were more than 2 nm, they were classified as mesoporous materials by the International Union of Pure and Applied Chemistry (IUPAC) classification^{##UREF##27##37##}.</p>", "<title>FESEM-FIB and EDX</title>", "<p id=\"Par30\">For FESEM-FIB analysis, the surface morphologies at 1,500X magnification with 100 µm of SBB, SBBT, SBBM, SBBA, and SBBZ are demonstrated in Fig. ##FIG##1##2##a–e. The surfaces of SBB, SBBM, SBBT, and SBBA were scaly sheet surfaces and structures with an irregular shape similar to other studies reported^{##REF##36211078##8##,##UREF##6##9##}, whereas SBBZ had a coarse surface similar found in a previous study^{##REF##36211078##8##}.</p>", "<p id=\"Par31\">For EDX analysis, the chemical elements of SBB, SBBT, SBBM, SBBA, and SBBZ are illustrated in Table ##TAB##2##3##, and their EDX mapping distributions are also demonstrated in Fig. ##FIG##1##2##f–j. Five main chemical elements of oxygen (O), carbon (C), calcium (Ca), chloride (Cl), and sodium (Na) were observed in all materials, whereas titanium (Ti), magnesium (Mg), aluminum (Al), and zinc (Zn) only detected in SBBT, SBBM, SBBA, and SBBZ, respectively because of addition of those metal oxides. In addition, the observations of Na, Ca, and Ca in all materials might be from the chemicals of sodium alginate and calcium chloride used in bead formations.</p>", "<title>FT-IR</title>", "<p id=\"Par32\">The chemical functional groups of SBB, SBBT, SBBM, SBBA, and SBBZ are illustrated in Fig. ##FIG##2##3##a–e which they observed five main chemical functional groups of O–H, C–H, C=O, C=C, and C–O–C similar found in previous studies^{##REF##36211078##8##,##UREF##6##9##,##REF##36702856##29##}. For O–H, it was the stretching water molecule, hydroxide groups of alcohol, phenol, and carboxylic acids^{##UREF##6##9##}, and they were found in a range of 3310–3700 cm⁻¹. For C–H, it referred to the bending of alkane (CH₂), alkene (CH₃), and aliphatic and aromatic groups of cellulose^{##REF##16843656##38##} observed in a range of 2896–2960 cm⁻¹. In addition, C–H also represented the stretching of CH₃ in a range of 1330–1430 cm⁻¹, and C–H was the bending of lignin and aromatic ring^{##UREF##28##39##} in a range of 720–750 cm⁻¹. For C=O, it was the stretching of the carbonyl group, aldehyde, and ketone^{##UREF##28##39##} illustrated in a range of 1720–1740 cm⁻¹. For C=C, it was the stretching of the aromatic ring in the lignin structure and the stretching of hemicellulose and cellulose^{##REF##36702856##29##} which were found in ranges of 1500–1610 cm⁻¹ and 810–900 cm⁻¹, respectively. For C–O–C, it referred to the stretching of hemicellulose, cellulose, and sodium alginate^{##REF##36211078##8##} in a range of 1020–1090 cm⁻¹. Moreover, the functional groups of Ti–O–Ti, Mg–O, Al–O, and Zn–O were observed in SBBT, SBBM, SBBA, and SBBZ from the addition of titanium dioxide, magnesium oxide, aluminum oxide, and zinc oxide^{##UREF##13##18##} which were found at 663.49, 655.77, 654.35, and 678.92 cm⁻¹, respectively.</p>", "<title><bold><italic>The point of zero charge (pH</italic></bold>_{<bold><italic>pzc</italic></bold>}<bold><italic>)</italic></bold></title>", "<p id=\"Par33\">The surface charges of SBB, SBBT, SBBM, SBBA, and SBBZ were determined by the point of zero charge (pH_pzc) to expect which pH is preferred for DR28 dye adsorption of each material. Figure ##FIG##3##4## is illustrated the pH_pzc of SBB, SBBT, SBBM, SBBA, and SBBZ which were 6.57, 7.31, 10.11, 7.25, and 7.77, and SBBM illustrated the highest pH_pzc among other materials similar found in a previous study^{##UREF##13##18##}. Since the anionic dye should be adsorbed at a pH of solution (pH_solution) less than pH_pzc because of the positively charged material surface, it can catch up DR28 dye molecule. On the other hand, DR28 dye adsorption is not favored at a pH_solution higher than pH_pzc because of the negatively charged material surface and the repulsion of the DR28 dye molecule. Therefore, DR28 dye adsorptions of each material should take place at a pH of solution less than its pH_pzc (pH_solution &lt; pH_pzc)^{##UREF##13##18##,##REF##36631520##40##}.</p>", "<title>Batch experiments</title>", "<title>The effect of dosage</title>", "<p id=\"Par34\">The effect of dosage from 5 to 30 g/L was designed to investigate how many grams of each material are needed for adsorbing DR28 dye at a concentration of 50 mg/L, a sample volume of 100 mL, a contact time of 12 h, a pH 7, a temperature of 30 °C, and a shaking speed of 150 rpm^{##UREF##6##9##} to obtain the highest DR28 dye removal efficiency, and the results are shown in Fig. ##FIG##4##5##a. DR28 dye removal efficiencies of SBB, SBBT, SBBM, SBBA, and SBBZ were increased with increasing material dosage from 5 to 30 g/L because of increasing of active sites for adsorbing DR28 dye similarly reported by other studies^{##UREF##30##41##,##UREF##31##42##}. Furthermore, the highest DR28 dye removal efficiencies were found at 25 g/L (81.90%), 15 g/L (85.23%), 20 g/L (92.67%), 15 g/L (87.30%), and 25 g/L (83.73%) for SBB, SBBT, SBBM, SBBA, and SBBZ, respectively. Therefore, they were used as the optimum dosages for the effect of contact time.</p>", "<title>The effect of contact time</title>", "<p id=\"Par35\">The effect of contact time from 3 to 18 h was used to determine how much contact time of each material is enough for adsorbing DR28 dye at a concentration of 50 mg/L, a sample volume of 100 mL, a pH 7, a temperature of 30 °C, a shaking speed of 150 rpm^{##UREF##6##9##}, and the optimum contact dosage to achieve the highest DR28 dye removal efficiency, and the results are shown in Fig. ##FIG##4##5##b. DR28 dye removal efficiencies of SBB, SBBT, SBBM, SBBA, and SBBZ were increased with increasing contact time from 3 to 18 h until their saturated adsorptions with discovering constant contact time were the optimum contact time^{##UREF##13##18##}. The highest DR28 dye removal efficiencies were found at 18 h (79.41%), 18 h (84.59%), 6 h (93.16%), 12 h (86.71%), and 12 h (82.94%) for SBB, SBBT, SBBM, SBBA, and SBBZ, respectively. Therefore, they were used as the optimum contact time for the effect of temperature.</p>", "<title>The effect of temperature</title>", "<p id=\"Par36\">The effect of temperature from 20 to 50 °C was examined how many temperatures of each material are good for adsorbing DR28 dye at a concentration of 50 mg/L, a sample volume of 100 mL, a pH 7, a shaking speed of 150 rpm^{##UREF##6##9##}, and the optimum dosage and contact time to get the highest DR28 dye removal efficiency, and the results are shown in Fig. ##FIG##4##5##c. DR28 dye removal efficiencies of SBB, SBBT, SBBM, SBBA, and SBBZ were increased with the increases of temperature from 20 to 35 °C, and then they a little decreased. The highest DR28 dye removal efficiencies were found at 35 °C in all materials with 80.43%, 85.02%, 94.33%, 87.33%, and 83.75% for SBB, SBBT, SBBM, SBBA, and SBBZ, respectively. Therefore, a temperature of 35 °C was the optimum temperature for the effect of pH.</p>", "<title>The effect of pH</title>", "<p id=\"Par37\">The effect of pH from 3 to 11 was used to examine the influence of pH on DR28 dye removal efficiencies of SBB, SBBT, SBBM, SBBA, and SBBZ to find the optimum pH for adsorb DR28 dye at a concentration of 50 mg/L, a sample volume of 100 mL, a shaking speed of 150 rpm^{##UREF##6##9##}, and the optimum dosage, contact time, and temperature to get the highest DR28 dye removal efficiency, and the results are shown in Fig. ##FIG##4##5##d. For pK_a and pH of solution (pH_solution), if the pH_solution is higher than pK_a (pH_solution &gt; pK_a), the dye molecule is in an anionic form. On the opposite, if the pH_solution is less than pK_a (pH_solution &lt; pK_a), the dye molecule is in a cationic form. Since the pK_a of DR28 dye is 4.1^{##UREF##32##43##}, the DR28 dye molecule should adsorb at pH_solution &gt; pK_a. From the results of the point of zero charges (pH_pzc), their DR28 dye adsorptions should occur at pH_solution &lt; pH_pzc. As a result, the high DR28 dye adsorption of each material should be observed at pK_a &lt; pH_solution &lt; pH_pzc. In Fig. ##FIG##4##5##d, their DR28 dye adsorptions were highly adsorbed at pH 3–5, and the highest DR28 dye removal efficiency was found at pH 3 with 79.56%, 84.35%, 93.83%, 86.87%, and 82.58% for SBB, SBBT, SBBM, SBBA, and SBBZ, respectively which might support by the pK_a of carboxyl group (–COOH) in materials which is 3–5^{##UREF##33##44##}. In addition, these results also agreed with the prior studies that found the highest anionic dye removal efficiencies at pH 3^{##REF##36211078##8##,##UREF##6##9##,##UREF##13##18##,##REF##36631520##40##}. Therefore, pH 3 was the optimum pH for the effect of concentration.</p>", "<title>The effect of concentration</title>", "<p id=\"Par38\">The effect of concentration from 30 to 90 mg/L observed how many concentrations of each material could adsorb DR28 dye at a sample volume of 100 mL a shaking speed of 150 rpm^{##UREF##6##9##}, and the optimum dosage, contact time, temperature, and pH to get the highest DR28 dye removal efficiency, and the results are shown in Fig. ##FIG##4##5##e. DR28 dye removal efficiencies of SBB, SBBT, SBBM, SBBA, and SBBZ were decreased with increasing concentration because the decrease of active sites for adsorbing DR28 dye similar to other studies^{##UREF##13##18##}. Their DR28 dye removal efficiencies from 30 to 90 mg/L were 67.27–84.39%, 75.73–89.39%, 83.84–96.02%, 78.09–90.94%, and 70.47–87.50% for SBB, SBBT, SBBM, SBBA, and SBBZ, respectively, and their DR28 dye removal efficiencies at 50 mg/L were 81.51%, 85.44%, 94.27%, 88.31%, and 83.51% for SBB, SBBT, SBBM, SBBA, and SBBZ, respectively.</p>", "<p id=\"Par39\">Finally, the optimum conditions in dosage, contact time, temperature, pH, and concentration of SBB, SBBT, SBBM, SBBA, and SBBZ were 25 g/L, 18 h, 35 °C, pH 3, 50 mg/L, 15 g/L, 18 h, 35 °C, pH 3, 50 mg/L, 20 g/L, 6 h, 35 °C, pH 3, 50 mg/L, 15 g/L, 12 h, 35 °C, pH 3, 50 mg/L, and 25 g/L, 12 h, 35 °C, pH 3, 50 mg/L, respectively. DR28 dye removal efficiencies could be arranged in order from high to low of SBBM &gt; SBBA &gt; SBBT &gt; SBBZ &gt; SBB, and SBBM had the highest DR28 dye removal efficiency with spending less material dosage and contact time than other materials similarly found by previous study with sugarcane bagasse fly ash beads modified with the same types of metal oxide with this study for DR28 dye adsorptions in aqueous solution^{##UREF##13##18##}. Moreover, these results also corresponded to the results of BET analysis that SBBM had a higher surface area with smaller pore size than other materials, so it could adsorb DR 28 dye more than others. Therefore, the addition of metal oxides of magnesium oxide (MgO), titanium dioxide (TiO₂), aluminum oxide (Al₂O₃), and zinc oxide (ZnO) increased material efficiencies for adsorbing DR28 dye, and SBBM was a high-potential material to further use for industrial wastewater treatment.</p>", "<p id=\"Par40\">For the comparison with other anionic dye removals, the previous studies have used sugarcane bagasse or sugarcane bagasse fly ash beads with or without metal modifications of iron (III) oxide-hydroxide, ZnO, TiO₂, MgO, and Al₂O₃ for removing reactive blue 4 (RB4) and DR28 dyes^{##REF##36211078##8##,##UREF##6##9##,##UREF##13##18##}, and the results demonstrated sugarcane bagasse and sugarcane bagasse fly ash beads mixed MgO had the highest RB4 and DR28 dye removals than other materials. These results corresponded to this study that SBBM illustrated the highest DR28 dye removal, so it could confirm that sugarcane bagasse beads with or without metal modifications especially MgO could remove various anionic dyes of RB4 and DR28.</p>", "<title>Adsorption isotherms</title>", "<p id=\"Par41\">The adsorption patterns of SBB, SBBT, SBBM, SBBA, and SBBZ are described by various adsorption isotherms of Langmuir, Freundlich, Temkin, and Dubinin–Radushkevich models. Their graphs are plotted by <italic>q</italic>_e versus<italic> C</italic>_e. The results are shown in Fig. ##FIG##5##6##a–e, and Table ##TAB##3##4## displayed their equilibrium isotherm parameters.</p>", "<p id=\"Par42\">The <italic>R</italic>² value is normally used for determining which adsorption isotherm better explains the adsorption pattern, and the higher <italic>R</italic>² is chosen. As a result, SBB corresponded to Langmuir model relating to the physical adsorption with a high <italic>R</italic>² of 0.997, whereas SBBT, SBBM, SBBA, and SBBZ corresponded to Freundlich model relating to the chemisorption with heterogeneous adsorption with high <italic>R</italic>² values of 0.998, 0.992, 0.997, and 0.994, respectively similar found in a previous study^{##UREF##13##18##}.</p>", "<p id=\"Par43\">Finally, the comparison of the maximum dye adsorption capacity (<italic>q</italic>_m) of this study with other agriculture wastes for DR28 dye removals is demonstrated in Table ##TAB##4##5##. The <italic>q</italic>_{<italic>m</italic>} values of SBB, SBBT, SBBM, SBBA, and SBBZ were higher than cabbage (2.31 mg/g) and rice husk (1.28–2.04 mg/g)^{##UREF##11##15##,##UREF##34##45##}, and the <italic>q</italic>_{<italic>m</italic>} value of SBBM had higher than prior studies in Table ##TAB##4##5## expect the studies of Rehman et al.^{##UREF##35##46##}, Ibrahim and Sani^{##UREF##36##47##}, and Masoudian et al.^{##REF##31712690##48##}.</p>", "<title>Adsorption kinetics</title>", "<p id=\"Par44\">The adsorption rates and mechanisms of SBB, SBBT, SBBM, SBBA, and SBBZ are determined by several adsorption kinetics of pseudo-first-order kinetic, pseudo-second-order kinetic, Elovich, and intra-particle diffusion models. Their graphs are plotted by <italic>q</italic>_t versus <italic>t</italic>. The results are shown in Fig. ##FIG##6##7##a–e, and Table ##TAB##5##6## reported their equilibrium kinetic parameters.</p>", "<p id=\"Par45\">Similar to adsorption isotherm, the <italic>R</italic>² value is normally used for determining which adsorption kinetic better describes the adsorption rate and mechanism, and the higher <italic>R</italic>² is preferred. Since the <italic>R</italic>² values of SBB, SBBT, SBBM, SBBA, and SBBZ in a pseudo-second-order kinetic model demonstrated the highest values of 0.997, 0.997, 0.999, 0.999, and 0.994, respectively, their adsorption rates and mechanisms were well described by chemisorption with the heterogeneous process agreed with a previous study reported^{##UREF##13##18##}. In addition, the kinetic parameter of <italic>q</italic>_e is used for comparing their DR28 dye adsorption capacities. The <italic>q</italic>_e of SBBM was higher than other materials, so it could adsorb DR28 dye more than other materials agreed with the batch experiment results. Furthermore, the equilibrium DR28 dye adsorption capacities of SBB, SBBT, SBBM, SBBA, and SBBZ demonstrated in Fig. ##FIG##6##7##f which reached the equilibrium within 60 min indicated their fast kinetic reaction rates.</p>", "<title>Thermodynamic study</title>", "<p id=\"Par46\">The results of thermodynamic studies in a range of 293.15–323.15 K of SBB, SBBT, SBBM, SBBA, and SBBZ on DR28 dye removals are demonstrated in Table ##TAB##6##7## and Fig. ##FIG##7##8##a–e. Their ∆<italic>G</italic>° had negative values in all temperatures which meant they were a favorable adsorption process of a spontaneous nature. For ∆<italic>H</italic>°, all materials had positive values which meant their DR28 dye adsorption processes were endothermic^{##UREF##13##18##}, and their ∆<italic>S</italic>° had positive values which meant the randomness during the adsorption process was increased^{##UREF##29##51##}. Therefore, the increasing temperature was favorable for DR28 dye adsorptions onto all materials.</p>" ]

[ "<title>Result and discussion</title>", "<title>BET</title>", "<p id=\"Par29\">The specific surface area, pore volumes, and pore sizes of SBB, SBBT, SBBM, SBBA, and SBBZ are illustrated in Table ##TAB##1##2##. Their specific surface area and pore volume could be arranged from high to low of SBBM &gt; SBBT &gt; SBBA &gt; SBBZ &gt; SBB, and SBBM demonstrated the highest surface area and pore volume among other materials. Since magnesium oxide (MgO), titanium dioxide (TiO₂), aluminum oxide (Al₂O₃), and zinc oxide (ZnO) have a high specific surface area by themselves, the specific surface area of prepared materials by those metal oxides have higher specific surface area than raw material. Moreover, the previous studies reported the specific surface area of MgO, TiO₂, Al₂O₃, and ZnO were 60, 50, 40, and 30 m²/g, and they could be arranged in order from high to low of MgO &gt; TiO₂ &gt; Al₂O₃ &gt; ZnO^{##UREF##22##30##,##UREF##23##31##}. As a result, it could support why SBBM had a higher surface area than other materials. Therefore, metal oxides of TiO₂, MgO, Al₂O₃, and ZnO increased the specific area and pore volumes of materials from the formations of those metal oxides with sugarcane bagasse supported more active sites for capturing DR28 dye adsorptions similar reported by previous studies used the same metal oxides^{##UREF##6##9##,##UREF##13##18##,##REF##36406531##20##}. Moreover, other metal oxides of zinc oxide, iron(III) oxide-hydroxide, and goethite have also been used in previous studies supported this study that the raw materials with adding metal oxides increased the surface area and pore volume^{##UREF##13##18##,##REF##36229648##32##–##UREF##26##36##}. Since their pore sizes were more than 2 nm, they were classified as mesoporous materials by the International Union of Pure and Applied Chemistry (IUPAC) classification^{##UREF##27##37##}.</p>", "<title>FESEM-FIB and EDX</title>", "<p id=\"Par30\">For FESEM-FIB analysis, the surface morphologies at 1,500X magnification with 100 µm of SBB, SBBT, SBBM, SBBA, and SBBZ are demonstrated in Fig. ##FIG##1##2##a–e. The surfaces of SBB, SBBM, SBBT, and SBBA were scaly sheet surfaces and structures with an irregular shape similar to other studies reported^{##REF##36211078##8##,##UREF##6##9##}, whereas SBBZ had a coarse surface similar found in a previous study^{##REF##36211078##8##}.</p>", "<p id=\"Par31\">For EDX analysis, the chemical elements of SBB, SBBT, SBBM, SBBA, and SBBZ are illustrated in Table ##TAB##2##3##, and their EDX mapping distributions are also demonstrated in Fig. ##FIG##1##2##f–j. Five main chemical elements of oxygen (O), carbon (C), calcium (Ca), chloride (Cl), and sodium (Na) were observed in all materials, whereas titanium (Ti), magnesium (Mg), aluminum (Al), and zinc (Zn) only detected in SBBT, SBBM, SBBA, and SBBZ, respectively because of addition of those metal oxides. In addition, the observations of Na, Ca, and Ca in all materials might be from the chemicals of sodium alginate and calcium chloride used in bead formations.</p>", "<title>FT-IR</title>", "<p id=\"Par32\">The chemical functional groups of SBB, SBBT, SBBM, SBBA, and SBBZ are illustrated in Fig. ##FIG##2##3##a–e which they observed five main chemical functional groups of O–H, C–H, C=O, C=C, and C–O–C similar found in previous studies^{##REF##36211078##8##,##UREF##6##9##,##REF##36702856##29##}. For O–H, it was the stretching water molecule, hydroxide groups of alcohol, phenol, and carboxylic acids^{##UREF##6##9##}, and they were found in a range of 3310–3700 cm⁻¹. For C–H, it referred to the bending of alkane (CH₂), alkene (CH₃), and aliphatic and aromatic groups of cellulose^{##REF##16843656##38##} observed in a range of 2896–2960 cm⁻¹. In addition, C–H also represented the stretching of CH₃ in a range of 1330–1430 cm⁻¹, and C–H was the bending of lignin and aromatic ring^{##UREF##28##39##} in a range of 720–750 cm⁻¹. For C=O, it was the stretching of the carbonyl group, aldehyde, and ketone^{##UREF##28##39##} illustrated in a range of 1720–1740 cm⁻¹. For C=C, it was the stretching of the aromatic ring in the lignin structure and the stretching of hemicellulose and cellulose^{##REF##36702856##29##} which were found in ranges of 1500–1610 cm⁻¹ and 810–900 cm⁻¹, respectively. For C–O–C, it referred to the stretching of hemicellulose, cellulose, and sodium alginate^{##REF##36211078##8##} in a range of 1020–1090 cm⁻¹. Moreover, the functional groups of Ti–O–Ti, Mg–O, Al–O, and Zn–O were observed in SBBT, SBBM, SBBA, and SBBZ from the addition of titanium dioxide, magnesium oxide, aluminum oxide, and zinc oxide^{##UREF##13##18##} which were found at 663.49, 655.77, 654.35, and 678.92 cm⁻¹, respectively.</p>", "<title><bold><italic>The point of zero charge (pH</italic></bold>_{<bold><italic>pzc</italic></bold>}<bold><italic>)</italic></bold></title>", "<p id=\"Par33\">The surface charges of SBB, SBBT, SBBM, SBBA, and SBBZ were determined by the point of zero charge (pH_pzc) to expect which pH is preferred for DR28 dye adsorption of each material. Figure ##FIG##3##4## is illustrated the pH_pzc of SBB, SBBT, SBBM, SBBA, and SBBZ which were 6.57, 7.31, 10.11, 7.25, and 7.77, and SBBM illustrated the highest pH_pzc among other materials similar found in a previous study^{##UREF##13##18##}. Since the anionic dye should be adsorbed at a pH of solution (pH_solution) less than pH_pzc because of the positively charged material surface, it can catch up DR28 dye molecule. On the other hand, DR28 dye adsorption is not favored at a pH_solution higher than pH_pzc because of the negatively charged material surface and the repulsion of the DR28 dye molecule. Therefore, DR28 dye adsorptions of each material should take place at a pH of solution less than its pH_pzc (pH_solution &lt; pH_pzc)^{##UREF##13##18##,##REF##36631520##40##}.</p>", "<title>Batch experiments</title>", "<title>The effect of dosage</title>", "<p id=\"Par34\">The effect of dosage from 5 to 30 g/L was designed to investigate how many grams of each material are needed for adsorbing DR28 dye at a concentration of 50 mg/L, a sample volume of 100 mL, a contact time of 12 h, a pH 7, a temperature of 30 °C, and a shaking speed of 150 rpm^{##UREF##6##9##} to obtain the highest DR28 dye removal efficiency, and the results are shown in Fig. ##FIG##4##5##a. DR28 dye removal efficiencies of SBB, SBBT, SBBM, SBBA, and SBBZ were increased with increasing material dosage from 5 to 30 g/L because of increasing of active sites for adsorbing DR28 dye similarly reported by other studies^{##UREF##30##41##,##UREF##31##42##}. Furthermore, the highest DR28 dye removal efficiencies were found at 25 g/L (81.90%), 15 g/L (85.23%), 20 g/L (92.67%), 15 g/L (87.30%), and 25 g/L (83.73%) for SBB, SBBT, SBBM, SBBA, and SBBZ, respectively. Therefore, they were used as the optimum dosages for the effect of contact time.</p>", "<title>The effect of contact time</title>", "<p id=\"Par35\">The effect of contact time from 3 to 18 h was used to determine how much contact time of each material is enough for adsorbing DR28 dye at a concentration of 50 mg/L, a sample volume of 100 mL, a pH 7, a temperature of 30 °C, a shaking speed of 150 rpm^{##UREF##6##9##}, and the optimum contact dosage to achieve the highest DR28 dye removal efficiency, and the results are shown in Fig. ##FIG##4##5##b. DR28 dye removal efficiencies of SBB, SBBT, SBBM, SBBA, and SBBZ were increased with increasing contact time from 3 to 18 h until their saturated adsorptions with discovering constant contact time were the optimum contact time^{##UREF##13##18##}. The highest DR28 dye removal efficiencies were found at 18 h (79.41%), 18 h (84.59%), 6 h (93.16%), 12 h (86.71%), and 12 h (82.94%) for SBB, SBBT, SBBM, SBBA, and SBBZ, respectively. Therefore, they were used as the optimum contact time for the effect of temperature.</p>", "<title>The effect of temperature</title>", "<p id=\"Par36\">The effect of temperature from 20 to 50 °C was examined how many temperatures of each material are good for adsorbing DR28 dye at a concentration of 50 mg/L, a sample volume of 100 mL, a pH 7, a shaking speed of 150 rpm^{##UREF##6##9##}, and the optimum dosage and contact time to get the highest DR28 dye removal efficiency, and the results are shown in Fig. ##FIG##4##5##c. DR28 dye removal efficiencies of SBB, SBBT, SBBM, SBBA, and SBBZ were increased with the increases of temperature from 20 to 35 °C, and then they a little decreased. The highest DR28 dye removal efficiencies were found at 35 °C in all materials with 80.43%, 85.02%, 94.33%, 87.33%, and 83.75% for SBB, SBBT, SBBM, SBBA, and SBBZ, respectively. Therefore, a temperature of 35 °C was the optimum temperature for the effect of pH.</p>", "<title>The effect of pH</title>", "<p id=\"Par37\">The effect of pH from 3 to 11 was used to examine the influence of pH on DR28 dye removal efficiencies of SBB, SBBT, SBBM, SBBA, and SBBZ to find the optimum pH for adsorb DR28 dye at a concentration of 50 mg/L, a sample volume of 100 mL, a shaking speed of 150 rpm^{##UREF##6##9##}, and the optimum dosage, contact time, and temperature to get the highest DR28 dye removal efficiency, and the results are shown in Fig. ##FIG##4##5##d. For pK_a and pH of solution (pH_solution), if the pH_solution is higher than pK_a (pH_solution &gt; pK_a), the dye molecule is in an anionic form. On the opposite, if the pH_solution is less than pK_a (pH_solution &lt; pK_a), the dye molecule is in a cationic form. Since the pK_a of DR28 dye is 4.1^{##UREF##32##43##}, the DR28 dye molecule should adsorb at pH_solution &gt; pK_a. From the results of the point of zero charges (pH_pzc), their DR28 dye adsorptions should occur at pH_solution &lt; pH_pzc. As a result, the high DR28 dye adsorption of each material should be observed at pK_a &lt; pH_solution &lt; pH_pzc. In Fig. ##FIG##4##5##d, their DR28 dye adsorptions were highly adsorbed at pH 3–5, and the highest DR28 dye removal efficiency was found at pH 3 with 79.56%, 84.35%, 93.83%, 86.87%, and 82.58% for SBB, SBBT, SBBM, SBBA, and SBBZ, respectively which might support by the pK_a of carboxyl group (–COOH) in materials which is 3–5^{##UREF##33##44##}. In addition, these results also agreed with the prior studies that found the highest anionic dye removal efficiencies at pH 3^{##REF##36211078##8##,##UREF##6##9##,##UREF##13##18##,##REF##36631520##40##}. Therefore, pH 3 was the optimum pH for the effect of concentration.</p>", "<title>The effect of concentration</title>", "<p id=\"Par38\">The effect of concentration from 30 to 90 mg/L observed how many concentrations of each material could adsorb DR28 dye at a sample volume of 100 mL a shaking speed of 150 rpm^{##UREF##6##9##}, and the optimum dosage, contact time, temperature, and pH to get the highest DR28 dye removal efficiency, and the results are shown in Fig. ##FIG##4##5##e. DR28 dye removal efficiencies of SBB, SBBT, SBBM, SBBA, and SBBZ were decreased with increasing concentration because the decrease of active sites for adsorbing DR28 dye similar to other studies^{##UREF##13##18##}. Their DR28 dye removal efficiencies from 30 to 90 mg/L were 67.27–84.39%, 75.73–89.39%, 83.84–96.02%, 78.09–90.94%, and 70.47–87.50% for SBB, SBBT, SBBM, SBBA, and SBBZ, respectively, and their DR28 dye removal efficiencies at 50 mg/L were 81.51%, 85.44%, 94.27%, 88.31%, and 83.51% for SBB, SBBT, SBBM, SBBA, and SBBZ, respectively.</p>", "<p id=\"Par39\">Finally, the optimum conditions in dosage, contact time, temperature, pH, and concentration of SBB, SBBT, SBBM, SBBA, and SBBZ were 25 g/L, 18 h, 35 °C, pH 3, 50 mg/L, 15 g/L, 18 h, 35 °C, pH 3, 50 mg/L, 20 g/L, 6 h, 35 °C, pH 3, 50 mg/L, 15 g/L, 12 h, 35 °C, pH 3, 50 mg/L, and 25 g/L, 12 h, 35 °C, pH 3, 50 mg/L, respectively. DR28 dye removal efficiencies could be arranged in order from high to low of SBBM &gt; SBBA &gt; SBBT &gt; SBBZ &gt; SBB, and SBBM had the highest DR28 dye removal efficiency with spending less material dosage and contact time than other materials similarly found by previous study with sugarcane bagasse fly ash beads modified with the same types of metal oxide with this study for DR28 dye adsorptions in aqueous solution^{##UREF##13##18##}. Moreover, these results also corresponded to the results of BET analysis that SBBM had a higher surface area with smaller pore size than other materials, so it could adsorb DR 28 dye more than others. Therefore, the addition of metal oxides of magnesium oxide (MgO), titanium dioxide (TiO₂), aluminum oxide (Al₂O₃), and zinc oxide (ZnO) increased material efficiencies for adsorbing DR28 dye, and SBBM was a high-potential material to further use for industrial wastewater treatment.</p>", "<p id=\"Par40\">For the comparison with other anionic dye removals, the previous studies have used sugarcane bagasse or sugarcane bagasse fly ash beads with or without metal modifications of iron (III) oxide-hydroxide, ZnO, TiO₂, MgO, and Al₂O₃ for removing reactive blue 4 (RB4) and DR28 dyes^{##REF##36211078##8##,##UREF##6##9##,##UREF##13##18##}, and the results demonstrated sugarcane bagasse and sugarcane bagasse fly ash beads mixed MgO had the highest RB4 and DR28 dye removals than other materials. These results corresponded to this study that SBBM illustrated the highest DR28 dye removal, so it could confirm that sugarcane bagasse beads with or without metal modifications especially MgO could remove various anionic dyes of RB4 and DR28.</p>", "<title>Adsorption isotherms</title>", "<p id=\"Par41\">The adsorption patterns of SBB, SBBT, SBBM, SBBA, and SBBZ are described by various adsorption isotherms of Langmuir, Freundlich, Temkin, and Dubinin–Radushkevich models. Their graphs are plotted by <italic>q</italic>_e versus<italic> C</italic>_e. The results are shown in Fig. ##FIG##5##6##a–e, and Table ##TAB##3##4## displayed their equilibrium isotherm parameters.</p>", "<p id=\"Par42\">The <italic>R</italic>² value is normally used for determining which adsorption isotherm better explains the adsorption pattern, and the higher <italic>R</italic>² is chosen. As a result, SBB corresponded to Langmuir model relating to the physical adsorption with a high <italic>R</italic>² of 0.997, whereas SBBT, SBBM, SBBA, and SBBZ corresponded to Freundlich model relating to the chemisorption with heterogeneous adsorption with high <italic>R</italic>² values of 0.998, 0.992, 0.997, and 0.994, respectively similar found in a previous study^{##UREF##13##18##}.</p>", "<p id=\"Par43\">Finally, the comparison of the maximum dye adsorption capacity (<italic>q</italic>_m) of this study with other agriculture wastes for DR28 dye removals is demonstrated in Table ##TAB##4##5##. The <italic>q</italic>_{<italic>m</italic>} values of SBB, SBBT, SBBM, SBBA, and SBBZ were higher than cabbage (2.31 mg/g) and rice husk (1.28–2.04 mg/g)^{##UREF##11##15##,##UREF##34##45##}, and the <italic>q</italic>_{<italic>m</italic>} value of SBBM had higher than prior studies in Table ##TAB##4##5## expect the studies of Rehman et al.^{##UREF##35##46##}, Ibrahim and Sani^{##UREF##36##47##}, and Masoudian et al.^{##REF##31712690##48##}.</p>", "<title>Adsorption kinetics</title>", "<p id=\"Par44\">The adsorption rates and mechanisms of SBB, SBBT, SBBM, SBBA, and SBBZ are determined by several adsorption kinetics of pseudo-first-order kinetic, pseudo-second-order kinetic, Elovich, and intra-particle diffusion models. Their graphs are plotted by <italic>q</italic>_t versus <italic>t</italic>. The results are shown in Fig. ##FIG##6##7##a–e, and Table ##TAB##5##6## reported their equilibrium kinetic parameters.</p>", "<p id=\"Par45\">Similar to adsorption isotherm, the <italic>R</italic>² value is normally used for determining which adsorption kinetic better describes the adsorption rate and mechanism, and the higher <italic>R</italic>² is preferred. Since the <italic>R</italic>² values of SBB, SBBT, SBBM, SBBA, and SBBZ in a pseudo-second-order kinetic model demonstrated the highest values of 0.997, 0.997, 0.999, 0.999, and 0.994, respectively, their adsorption rates and mechanisms were well described by chemisorption with the heterogeneous process agreed with a previous study reported^{##UREF##13##18##}. In addition, the kinetic parameter of <italic>q</italic>_e is used for comparing their DR28 dye adsorption capacities. The <italic>q</italic>_e of SBBM was higher than other materials, so it could adsorb DR28 dye more than other materials agreed with the batch experiment results. Furthermore, the equilibrium DR28 dye adsorption capacities of SBB, SBBT, SBBM, SBBA, and SBBZ demonstrated in Fig. ##FIG##6##7##f which reached the equilibrium within 60 min indicated their fast kinetic reaction rates.</p>", "<title>Thermodynamic study</title>", "<p id=\"Par46\">The results of thermodynamic studies in a range of 293.15–323.15 K of SBB, SBBT, SBBM, SBBA, and SBBZ on DR28 dye removals are demonstrated in Table ##TAB##6##7## and Fig. ##FIG##7##8##a–e. Their ∆<italic>G</italic>° had negative values in all temperatures which meant they were a favorable adsorption process of a spontaneous nature. For ∆<italic>H</italic>°, all materials had positive values which meant their DR28 dye adsorption processes were endothermic^{##UREF##13##18##}, and their ∆<italic>S</italic>° had positive values which meant the randomness during the adsorption process was increased^{##UREF##29##51##}. Therefore, the increasing temperature was favorable for DR28 dye adsorptions onto all materials.</p>" ]

[ "<title>Conclusion</title>", "<p id=\"Par48\">Five adsorbent materials of sugarcane bagasse beads (SBB), sugarcane bagasse modified with titanium dioxide beads (SBBT), sugarcane bagasse modified with magnesium oxide beads (SBBM), sugarcane bagasse modified with aluminum oxide beads (SBBA), and sugarcane bagasse modified with zinc oxide beads (SBBZ) were synthesized from sugarcane bagasse and various metal oxides for investigating their DR28 dye removal efficiencies. SBBM had the highest specific surface area and pore volume, whereas its pore size was the smallest among other materials. The surfaces of SBB, SBBM, SBBT, and SBBA were scaly sheet surfaces and structures with an irregular shape, whereas SBBZ was a coarse surface. Five main chemical elements of oxygen (O), carbon (C), calcium (Ca), chloride (Cl), and sodium (Na) were observed in all materials, whereas titanium (Ti), magnesium (Mg), aluminum (Al), and zinc (Zn) only detected in SBBT, SBBM, SBBA, and SBBZ, respectively. Five main chemical functional groups of O–H, C–H, C=O, C=C, and C–O–C were found in all materials, and Ti–O–Ti, Mg–O, Al–O, and Zn–O were observed in SBBT, SBBM, SBBA, and SBBZ. The points of zero charge (pH_pzc) of SBB, SBBT, SBBM, SBBA, and SBBZ were 6.57, 7.31, 10.11, 7.25, and 7.77, respectively. All materials could adsorb DR28 dye at a concentration of 50 mg/L by more than 81%, and SBBM illustrated the highest DR28 dye removal efficiency of 94.27%. For adsorption isotherm, Langmuir model was a suitable model for SBB corresponding to physical adsorption, whereas Freundlich model was an appropriate model to explain the adsorption pattern of SBBT, SBBM, SBBA, and SBBZ relating to physicochemical adsorption. For adsorption kinetic, a pseudo-second-order kinetic model was the best-fit model for all materials well explained by the chemisorption mechanism. Since the ∆<italic>G</italic>° of all materials had negative values, they were a favorable adsorption process of a spontaneous nature. While their ∆<italic>H</italic>° had positive values which meant they were an endothermic process. For ∆<italic>S</italic>°, they had positive values which meant the randomness during the adsorption process was increased. Therefore, all materials were potential materials for adsorbing DR28 dye, especially SBBM.</p>", "<p id=\"Par49\">For future works, the real wastewater might be applied to confirm their abilities for DR28 dye adsorptions. In addition, other anionic dyes might be investigated for possible adsorption by SBB, SBBT, SBBM, SBBA, and SBBZ. Moreover, the continuous flow study should study for the possible application in the industrial wastewater system. Furthermore, the leaching of metal oxides from SBBT, SBBM, SBBA, and SBBZ after the adsorption process might be suggested to investigate and confirm their no contaminations in treated wastewater.</p>" ]

[ "<p id=\"Par1\">The direct red 28 (DR28) dye contamination in wastewater blocks the transmission of light into the water body resulting in the inability to photosynthesize by aquatic life. In addition, it is difficult to break down and persist in the environment, and it is also harmful to aquatic life and water quality because of its aromatic structure. Thus, wastewater contaminated with dyes is required to treat before releasing into the water body. Sugarcane bagasse beads (SBB), sugarcane bagasse modified with titanium dioxide beads (SBBT), sugarcane bagasse modified with magnesium oxide beads (SBBM), sugarcane bagasse modified with aluminum oxide beads (SBBA), and sugarcane bagasse modified with zinc oxide beads (SBBZ) for DR28 dye removal in aqueous solution, and they were characterized with several techniques of BET, FESEM-FIB, EDX, FT-IR, and the point of zero charges (pH_pzc). Their DR28 dye removal efficiencies were examined through batch tests, adsorption isotherms, and kinetics. SBBM had the highest specific surface area and pore volume, whereas its pore size was the smallest among other materials. The surfaces of SBB, SBBM, SBBT, and SBBA were scaly sheet surfaces with an irregular shape, whereas SBBZ was a coarse surface. Oxygen, carbon, calcium, chloride, sodium, O–H, C–H, C=O, C=C, and C–O–C were found in all materials. The pH_pzc of SBB, SBBT, SBBM, SBBA, and SBBZ were 6.57, 7.31, 10.11, 7.25, and 7.77. All materials could adsorb DR28 dye at 50 mg/L by more than 81%, and SBBM had the highest DR28 dye removal efficiency of 94.27%. Langmuir model was an appropriate model for SBB, whereas Freundlich model was a suitable model for other materials. A pseudo-second-order kinetic model well described their adsorption mechanisms. Their adsorptions of the DR28 dye were endothermic and spontaneous. Therefore, they were potential materials for adsorbing DR28 dye, especially SBBM.</p>", "<title>Subject terms</title>" ]

[ "<title>The possible mechanisms for DR28 dye adsorptions</title>", "<p id=\"Par47\">The possible mechanisms for DR28 dye adsorptions of SBB, SBBT, SBBM, SBBA, and SBBZ are demonstrated in Fig. ##FIG##8##9## which modified the idea from the study of Ngamsurach et al.^{##REF##36211078##8##} and Praipipat et al.^{##UREF##6##9##,##UREF##13##18##}. Their main chemical functional groups of O–H, C–H, C=O, C=C, and C–O–C<bold>,</bold> and the complex molecules of Ti–O–Ti, Mg–O, Al–O–Al, and Zn–O connected with their hydroxyl group (O–H) played a main role for DR28 dye adsorptions. The possible mechanisms of electrostatic attraction, hydrogen bonding interaction, and n–π bonding interaction are used for explaining DR28 dye adsorptions by SBB, SBBT, SBBM, SBBA, and SBBZ demonstrated in Fig. ##FIG##8##9##.</p>" ]

[ "<title>Author contributions</title>", "<p>P.P.: Supervision, Project administration, Conceptualization, Funding acquisition, Investigation, Methodology, Validation, Visualization, Writing—Original Draft, Writing-Review and Editing. P.N.: Visualization, Writing—Original Draft. N.L.: Investigation. C.K.: Investigation. P.B.: Investigation. W.N.: Investigation.</p>", "<title>Funding</title>", "<p>The authors are grateful for the financial support received from The Office of the Higher Education Commission and The Thailand Research Fund grant (MRG6080114), Coordinating Center for Thai Government Science and Technology Scholarship Students (CSTS) and National Science and Technology Development Agency (NSTDA) Fund grant (SCHNR2016-122), and Research and Technology Transfer Affairs of Khon Kaen University.</p>", "<title>Data availability</title>", "<p>The datasets used and/or analyzed during the current study are available from the corresponding author upon reasonable request.</p>", "<title>Competing interests</title>", "<p id=\"Par50\">The authors declare no competing interests.</p>" ]

[ "<fig id=\"Fig1\"><label>Figure 1</label><caption><p>The synthesis of SBB, SBBT, SBBM, SBBA, and SBBZ.</p></caption></fig>", "<fig id=\"Fig2\"><label>Figure 2</label><caption><p>The surface morphologies and the chemical distributions by EDX mapping of (<bold>a</bold>, <bold>f</bold>) SBB, (<bold>b</bold>, <bold>g</bold>) SBBT, (<bold>c</bold>, <bold>h</bold>) SBBM, (<bold>d</bold>, <bold>i</bold>) SBBA, and (<bold>e</bold>, <bold>j</bold>) SBBZ.</p></caption></fig>", "<fig id=\"Fig3\"><label>Figure 3</label><caption><p>FT-IR spectra of (<bold>a</bold>) SBB, (<bold>b</bold>) SBBT, (<bold>c</bold>) SBBM, (<bold>d</bold>) SBBA, and (<bold>e</bold>) SBBZ.</p></caption></fig>", "<fig id=\"Fig4\"><label>Figure 4</label><caption><p>The points of zero charge of SBB, SBBT, SBBM, SBBA, and SBBZ.</p></caption></fig>", "<fig id=\"Fig5\"><label>Figure 5</label><caption><p>The batch experiments of SBB, SBBT, SBBM, SBBA, and SBBZ in (<bold>a</bold>) dose, (<bold>b</bold>) contact time, (<bold>c</bold>) temperature, (<bold>d</bold>) pH, and (<bold>e</bold>) concentration for DR28 dye adsorptions.</p></caption></fig>", "<fig id=\"Fig6\"><label>Figure 6</label><caption><p>The adsorption isotherms of (<bold>a</bold>) SBB, (<bold>b</bold>) SBBT, (<bold>c</bold>) SBBM, (<bold>d</bold>) SBBA, and (<bold>e</bold>) SBBZ for DR28 dye adsorptions.</p></caption></fig>", "<fig id=\"Fig7\"><label>Figure 7</label><caption><p>The adsorption kinetics of (<bold>a</bold>) SBB, (<bold>b</bold>) SBBT, (<bold>c</bold>) SBBM, (<bold>d</bold>) SBBA, (<bold>e</bold>) SBBZ for DR28 dye adsorptions, and (<bold>f</bold>) the equilibrium DR28 dye adsorption capacities of SBB, SBBT, SBBM, SBBA, and SBBZ.</p></caption></fig>", "<fig id=\"Fig8\"><label>Figure 8</label><caption><p>The thermodynamic plots for DR28 dye adsorptions by (<bold>a</bold>) SBB, (<bold>b</bold>) SBBT, (<bold>c</bold>) SBBM, (<bold>d</bold>) SBBA, and (<bold>e</bold>) SBBZ.</p></caption></fig>", "<fig id=\"Fig9\"><label>Figure 9</label><caption><p>The possible mechanisms for DR28 dye adsorptions by SBB, SBBT, SBBM, SBBA, and SBBZ.</p></caption></fig>" ]

[ "<table-wrap id=\"Tab1\"><label>Table 1</label><caption><p>The agricultural wastes with or without modifications for removing various dyes<bold>.</bold></p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th align=\"left\">Materials</th><th align=\"left\">Dyes</th><th align=\"left\">Dose (g)</th><th align=\"left\">Time (min)</th><th align=\"left\">Temp (°C )</th><th align=\"left\">pH</th><th align=\"left\">Conc. (mg/L)</th><th align=\"left\">Volume (mL)</th><th align=\"left\"><italic>q</italic>_m (mg/g)</th><th align=\"left\">Refs</th></tr></thead><tbody><tr><td align=\"left\" colspan=\"10\">Non-modification</td></tr><tr><td align=\"left\"> White dragon fruit peel</td><td align=\"left\">Direct red 28</td><td align=\"left\">0.02</td><td align=\"left\">180</td><td align=\"left\">25</td><td align=\"left\">5.02</td><td align=\"left\">0–1,000</td><td align=\"left\">10</td><td char=\".\" align=\"char\">59.30</td><td align=\"left\">^{##UREF##7##10##}</td></tr><tr><td align=\"left\"> Potato peel</td><td align=\"left\">Direct red 80</td><td align=\"left\">2</td><td align=\"left\">60</td><td align=\"left\">30</td><td align=\"left\">2</td><td align=\"left\">20–200</td><td align=\"left\">100</td><td char=\".\" align=\"char\">27.78</td><td align=\"left\">^{##REF##33767044##11##}</td></tr><tr><td align=\"left\"> Potato peel</td><td align=\"left\">Methylene blue</td><td align=\"left\">2</td><td align=\"left\">10</td><td align=\"left\">30</td><td align=\"left\">7</td><td align=\"left\">20–200</td><td align=\"left\">100</td><td char=\".\" align=\"char\">97.08</td><td align=\"left\">^{##REF##33767044##11##}</td></tr><tr><td align=\"left\"> Groundnut shell (activated carbon)</td><td align=\"left\">Direct red 1</td><td align=\"left\">0.1</td><td align=\"left\">20</td><td align=\"left\">28</td><td align=\"left\">2</td><td align=\"left\">10–50</td><td align=\"left\">100</td><td char=\".\" align=\"char\">5.11</td><td align=\"left\">^{##UREF##8##12##}</td></tr><tr><td align=\"left\"> Almond shell</td><td align=\"left\">Crystal violet</td><td align=\"left\">0.2</td><td align=\"left\">90</td><td align=\"left\">20</td><td align=\"left\">6</td><td align=\"left\">20–200</td><td align=\"left\">40</td><td char=\".\" align=\"char\">12.20</td><td align=\"left\">^{##UREF##9##13##}</td></tr><tr><td align=\"left\"> Corn silk</td><td align=\"left\">Reactive blue 19</td><td align=\"left\">0.25</td><td align=\"left\">1440</td><td align=\"left\">25</td><td align=\"left\">2</td><td align=\"left\">10–500</td><td align=\"left\">50</td><td char=\".\" align=\"char\">60.60</td><td align=\"left\">^{##UREF##10##14##}</td></tr><tr><td align=\"left\"> Corn silk</td><td align=\"left\">Reactive red 218</td><td align=\"left\">0.25</td><td align=\"left\">1440</td><td align=\"left\">25</td><td align=\"left\">2</td><td align=\"left\">10–500</td><td align=\"left\">50</td><td char=\".\" align=\"char\">51.60</td><td align=\"left\">^{##UREF##10##14##}</td></tr><tr><td align=\"left\"> Rice husk</td><td align=\"left\">Direct red 28</td><td align=\"left\">0.5</td><td align=\"left\">10</td><td align=\"left\">30</td><td align=\"left\">4</td><td align=\"left\">20–100</td><td align=\"left\">15</td><td char=\".\" align=\"char\">1.58</td><td align=\"left\">^{##UREF##11##15##}</td></tr><tr><td align=\"left\"> Sugarcane bagasse (biochar)</td><td align=\"left\">Reactive blue 19</td><td align=\"left\">0.2</td><td align=\"left\">360</td><td align=\"left\">25</td><td align=\"left\">2</td><td align=\"left\">50–1000</td><td align=\"left\">50</td><td char=\".\" align=\"char\">58.10</td><td align=\"left\">^{##UREF##3##5##}</td></tr><tr><td align=\"left\"> Sugarcane bagasse</td><td align=\"left\">Methyl red</td><td align=\"left\">0.4</td><td align=\"left\">180</td><td align=\"left\">26</td><td align=\"left\">6</td><td align=\"left\">50–200</td><td align=\"left\">100</td><td char=\".\" align=\"char\">5.66</td><td align=\"left\">^{##UREF##4##6##}</td></tr><tr><td align=\"left\"> Sugarcane bagasse</td><td align=\"left\">Basic red 2</td><td align=\"left\">1</td><td align=\"left\">1020</td><td align=\"left\">25</td><td align=\"left\">10</td><td align=\"left\">5–40</td><td align=\"left\">250</td><td char=\".\" align=\"char\">58.85</td><td align=\"left\">^{##UREF##5##7##}</td></tr><tr><td align=\"left\"> Bagasse (beads)</td><td align=\"left\">Reactive blue 4</td><td align=\"left\">2</td><td align=\"left\">720</td><td align=\"left\">70</td><td align=\"left\">3</td><td align=\"left\">30–90</td><td align=\"left\">100</td><td char=\".\" align=\"char\">3.17</td><td align=\"left\">^{##REF##36211078##8##}</td></tr><tr><td align=\"left\" colspan=\"10\">Modification</td></tr><tr><td align=\"left\"> Sugarcane bagasse treated by phosphoric acid (H₃PO₄)</td><td align=\"left\">Methyl red</td><td align=\"left\">0.4</td><td align=\"left\">180</td><td align=\"left\">26</td><td align=\"left\">6</td><td align=\"left\">50–250</td><td align=\"left\">100</td><td char=\".\" align=\"char\">10.98</td><td align=\"left\">^{##UREF##4##6##}</td></tr><tr><td align=\"left\"> Sugarcane bagasse treated by sulfuric acid (H₂SO₄)</td><td align=\"left\">Basic red 2</td><td align=\"left\">1</td><td align=\"left\">1020</td><td align=\"left\">25</td><td align=\"left\">10</td><td align=\"left\">5–40</td><td align=\"left\">250</td><td char=\".\" align=\"char\">54.82</td><td align=\"left\">^{##UREF##5##7##}</td></tr><tr><td align=\"left\"> Sugarcane bagasse treated by NaOH</td><td align=\"left\">Basic red 2</td><td align=\"left\">1</td><td align=\"left\">1020</td><td align=\"left\">25</td><td align=\"left\">10</td><td align=\"left\">5–40</td><td align=\"left\">250</td><td char=\".\" align=\"char\">62.88</td><td align=\"left\">^{##UREF##5##7##}</td></tr><tr><td align=\"left\"> Sugarcane bagasse MgO/N-doped active carbon</td><td align=\"left\">Methyl orange</td><td align=\"left\">0.05</td><td align=\"left\">100</td><td align=\"left\">30</td><td align=\"left\">–</td><td align=\"left\">130–170</td><td align=\"left\">150</td><td char=\".\" align=\"char\">384.61</td><td align=\"left\">^{##REF##31761628##16##}</td></tr><tr><td align=\"left\"> Sugarcane bagasse modified with iron oxide (Fe₃O₄)</td><td align=\"left\">Methylene Blue</td><td align=\"left\">16.5</td><td align=\"left\">360</td><td align=\"left\">25</td><td align=\"left\">8.4</td><td align=\"left\">1–10</td><td align=\"left\">25</td><td char=\".\" align=\"char\">12.28</td><td align=\"left\">^{##UREF##12##17##}</td></tr><tr><td align=\"left\"> Bagasse beads with mixed iron (III) oxide-hydroxide</td><td align=\"left\">Reactive blue 4</td><td align=\"left\">3</td><td align=\"left\">540</td><td align=\"left\">70</td><td align=\"left\">3</td><td align=\"left\">30–90</td><td align=\"left\">100</td><td char=\".\" align=\"char\">3.77</td><td align=\"left\">^{##REF##36211078##8##}</td></tr><tr><td align=\"left\"> Bagasse beads with mixed zinc oxide</td><td align=\"left\">Reactive blue 4</td><td align=\"left\">3</td><td align=\"left\">720</td><td align=\"left\">60</td><td align=\"left\">3</td><td align=\"left\">30–90</td><td align=\"left\">100</td><td char=\".\" align=\"char\">3.18</td><td align=\"left\">^{##REF##36211078##8##}</td></tr><tr><td align=\"left\"> Bagasse beads mixed titanium dioxide</td><td align=\"left\">Reactive blue 4</td><td align=\"left\">1.5</td><td align=\"left\">900</td><td align=\"left\">30</td><td align=\"left\">3</td><td align=\"left\">30–90</td><td align=\"left\">100</td><td char=\".\" align=\"char\">6.04</td><td align=\"left\">^{##UREF##6##9##}</td></tr><tr><td align=\"left\"> Bagasse beads mixed magnesium oxide</td><td align=\"left\">Reactive blue 4</td><td align=\"left\">2.5</td><td align=\"left\">900</td><td align=\"left\">30</td><td align=\"left\">3</td><td align=\"left\">30–90</td><td align=\"left\">100</td><td char=\".\" align=\"char\">5.55</td><td align=\"left\">^{##UREF##6##9##}</td></tr><tr><td align=\"left\"> Bagasse beads mixed aluminum oxide</td><td align=\"left\">Reactive blue 4</td><td align=\"left\">3</td><td align=\"left\">900</td><td align=\"left\">30</td><td align=\"left\">3</td><td align=\"left\">30–90</td><td align=\"left\">100</td><td char=\".\" align=\"char\">3.41</td><td align=\"left\">^{##UREF##6##9##}</td></tr></tbody></table></table-wrap>", "<table-wrap id=\"Tab2\"><label>Table 2</label><caption><p>The specific surface area, pore volumes, and pore sizes of SBB, SBBT, SBBM, SBBA, and SBBZ.</p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th align=\"left\">Materials</th><th align=\"left\">Specific surface area (m²/g)*<break/> ± SD</th><th align=\"left\">Pore volume (cm³/g)**<break/> ± SD</th><th align=\"left\">Pore size (nm)**<break/> ± SD</th></tr></thead><tbody><tr><td align=\"left\">SBB</td><td char=\".\" align=\"char\">14.734 ± 0.011</td><td char=\".\" align=\"char\">0.039 ± 0.002</td><td char=\".\" align=\"char\">6.495 ± 0.014</td></tr><tr><td align=\"left\">SBBT</td><td char=\".\" align=\"char\">51.376 ± 0.014</td><td char=\".\" align=\"char\">0.063 ± 0.002</td><td char=\".\" align=\"char\">3.685 ± 0.015</td></tr><tr><td align=\"left\">SBBM</td><td char=\".\" align=\"char\">57.241 ± 0.012</td><td char=\".\" align=\"char\">0.071 ± 0.001</td><td char=\".\" align=\"char\">3.364 ± 0.017</td></tr><tr><td align=\"left\">SBBA</td><td char=\".\" align=\"char\">43.485 ± 0.010</td><td char=\".\" align=\"char\">0.051 ± 0.003</td><td char=\".\" align=\"char\">4.052 ± 0.013</td></tr><tr><td align=\"left\">SBBZ</td><td char=\".\" align=\"char\">39.197 ± 0.016</td><td char=\".\" align=\"char\">0.045 ± 0.002</td><td char=\".\" align=\"char\">4.319 ± 0.012</td></tr></tbody></table></table-wrap>", "<table-wrap id=\"Tab3\"><label>Table 3</label><caption><p>The chemical elements of SBB, SBBT, SBBM, SBBA, and SBBZ.</p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th align=\"left\" rowspan=\"2\">Materials</th><th align=\"left\" colspan=\"9\">Chemical elements (%wt) ± SD</th></tr><tr><th align=\"left\">O</th><th align=\"left\">C</th><th align=\"left\">Ca</th><th align=\"left\">Cl</th><th align=\"left\">Na</th><th align=\"left\">Ti</th><th align=\"left\">Mg</th><th align=\"left\">Al</th><th align=\"left\">Zn</th></tr></thead><tbody><tr><td align=\"left\">SBB</td><td char=\".\" align=\"char\">45.6 ± 0.2</td><td char=\".\" align=\"char\">41.3 ± 0.1</td><td char=\".\" align=\"char\">8.1 ± 0.1</td><td char=\".\" align=\"char\">4.4 ± 0.1</td><td char=\".\" align=\"char\">0.6 ± 0.1</td><td char=\".\" align=\"char\">–</td><td char=\".\" align=\"char\">–</td><td char=\".\" align=\"char\">–</td><td char=\".\" align=\"char\">–</td></tr><tr><td align=\"left\">SBBT</td><td char=\".\" align=\"char\">42.7 ± 0.3</td><td char=\".\" align=\"char\">28.8 ± 0.2</td><td char=\".\" align=\"char\">6.5 ± 0.2</td><td char=\".\" align=\"char\">2.4 ± 0.2</td><td char=\".\" align=\"char\">0.9 ± 0.1</td><td char=\".\" align=\"char\">18.7 ± 0.2</td><td char=\".\" align=\"char\">–</td><td char=\".\" align=\"char\">–</td><td char=\".\" align=\"char\">–</td></tr><tr><td align=\"left\">SBBM</td><td char=\".\" align=\"char\">41.6 ± 0.1</td><td char=\".\" align=\"char\">33.3 ± 0.1</td><td char=\".\" align=\"char\">7.4 ± 0.3</td><td char=\".\" align=\"char\">3.5 ± 0.1</td><td char=\".\" align=\"char\">0.7 ± 0.1</td><td char=\".\" align=\"char\">–</td><td char=\".\" align=\"char\">13.5 ± 0.2</td><td char=\".\" align=\"char\">–</td><td char=\".\" align=\"char\">–</td></tr><tr><td align=\"left\">SBBA</td><td char=\".\" align=\"char\">35.4 ± 0.2</td><td char=\".\" align=\"char\">35.2 ± 0.2</td><td char=\".\" align=\"char\">6.7 ± 0.4</td><td char=\".\" align=\"char\">2.6 ± 0.1</td><td char=\".\" align=\"char\">0.8 ± 0.1</td><td char=\".\" align=\"char\">–</td><td char=\".\" align=\"char\">–</td><td char=\".\" align=\"char\">19.3 ± 0.2</td><td char=\".\" align=\"char\">–</td></tr><tr><td align=\"left\">SBBZ</td><td char=\".\" align=\"char\">32.2 ± 0.3</td><td char=\".\" align=\"char\">30.5 ± 0.2</td><td char=\".\" align=\"char\">7.9 ± 0.3</td><td char=\".\" align=\"char\">3.2 ± 0.2</td><td char=\".\" align=\"char\">0.9 ± 0.2</td><td char=\".\" align=\"char\">–</td><td char=\".\" align=\"char\">–</td><td char=\".\" align=\"char\">–</td><td char=\".\" align=\"char\">25.3 ± 0.3</td></tr></tbody></table></table-wrap>", "<table-wrap id=\"Tab4\"><label>Table 4</label><caption><p>The equilibrium isotherm parameters of SBB, SBBT, SBBM, SBBA, and SBBZ for DR28 dye adsorptions.</p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th align=\"left\">Isotherm models</th><th align=\"left\">Parameters</th><th align=\"left\">SBB</th><th align=\"left\">SBBT</th><th align=\"left\">SBBM</th><th align=\"left\">SBBA</th><th align=\"left\">SBBZ</th></tr></thead><tbody><tr><td align=\"left\" rowspan=\"5\">Langmuir</td><td align=\"left\"><italic>q</italic>_m (mg/g)</td><td char=\".\" align=\"char\">3.243</td><td char=\".\" align=\"char\">6.249</td><td char=\".\" align=\"char\">6.578</td><td char=\".\" align=\"char\">6.409</td><td char=\".\" align=\"char\">3.293</td></tr><tr><td align=\"left\"><italic>K</italic>_L (L/mg)</td><td char=\".\" align=\"char\">0.101</td><td char=\".\" align=\"char\">0.119</td><td char=\".\" align=\"char\">0.095</td><td char=\".\" align=\"char\">0.128</td><td char=\".\" align=\"char\">0.131</td></tr><tr><td align=\"left\"><italic>R</italic>²</td><td char=\".\" align=\"char\">0.997</td><td char=\".\" align=\"char\">0.989</td><td char=\".\" align=\"char\">0.983</td><td char=\".\" align=\"char\">0.988</td><td char=\".\" align=\"char\">0.983</td></tr><tr><td align=\"left\"><italic>R</italic>²_adj</td><td char=\".\" align=\"char\">0.996</td><td char=\".\" align=\"char\">0.987</td><td char=\".\" align=\"char\">0.980</td><td char=\".\" align=\"char\">0.986</td><td char=\".\" align=\"char\">0.979</td></tr><tr><td align=\"left\">RMSE</td><td char=\".\" align=\"char\">0.038</td><td char=\".\" align=\"char\">0.118</td><td char=\".\" align=\"char\">0.101</td><td char=\".\" align=\"char\">0.142</td><td char=\".\" align=\"char\">0.142</td></tr><tr><td align=\"left\" rowspan=\"5\">Freundlich</td><td align=\"left\">1<italic>/n</italic></td><td char=\".\" align=\"char\">0.428</td><td char=\".\" align=\"char\">0.464</td><td char=\".\" align=\"char\">0.594</td><td char=\".\" align=\"char\">0.471</td><td char=\".\" align=\"char\">0.427</td></tr><tr><td align=\"left\"><italic>K</italic>_F (mg/g)(L/mg)^1/n</td><td char=\".\" align=\"char\">0.588</td><td char=\".\" align=\"char\">1.100</td><td char=\".\" align=\"char\">0.804</td><td char=\".\" align=\"char\">1.168</td><td char=\".\" align=\"char\">0.643</td></tr><tr><td align=\"left\"><italic>R</italic>²</td><td char=\".\" align=\"char\">0.971</td><td char=\".\" align=\"char\">0.998</td><td char=\".\" align=\"char\">0.992</td><td char=\".\" align=\"char\">0.997</td><td char=\".\" align=\"char\">0.994</td></tr><tr><td align=\"left\"><italic>R</italic>²_adj</td><td char=\".\" align=\"char\">0.965</td><td char=\".\" align=\"char\">0.997</td><td char=\".\" align=\"char\">0.991</td><td char=\".\" align=\"char\">0.996</td><td char=\".\" align=\"char\">0.993</td></tr><tr><td align=\"left\">RMSE</td><td char=\".\" align=\"char\">0.096</td><td char=\".\" align=\"char\">0.058</td><td char=\".\" align=\"char\">0.086</td><td char=\".\" align=\"char\">0.071</td><td char=\".\" align=\"char\">0.045</td></tr><tr><td align=\"left\" rowspan=\"5\">Temkin</td><td align=\"left\"><italic>b</italic>_T (J/mol)</td><td char=\".\" align=\"char\">3234.628</td><td char=\".\" align=\"char\">1826.036</td><td char=\".\" align=\"char\">1761.978</td><td char=\".\" align=\"char\">1789.554</td><td char=\".\" align=\"char\">3635.924</td></tr><tr><td align=\"left\"><italic>A</italic>_T (L/g)</td><td char=\".\" align=\"char\">0.823</td><td char=\".\" align=\"char\">1.139</td><td char=\".\" align=\"char\">0.919</td><td char=\".\" align=\"char\">1.246</td><td char=\".\" align=\"char\">1.239</td></tr><tr><td align=\"left\"><italic>R</italic>²</td><td char=\".\" align=\"char\">0.991</td><td char=\".\" align=\"char\">0.990</td><td char=\".\" align=\"char\">0.988</td><td char=\".\" align=\"char\">0.988</td><td char=\".\" align=\"char\">0.972</td></tr><tr><td align=\"left\"><italic>R</italic>²_adj</td><td char=\".\" align=\"char\">0.990</td><td char=\".\" align=\"char\">0.988</td><td char=\".\" align=\"char\">0.986</td><td char=\".\" align=\"char\">0.985</td><td char=\".\" align=\"char\">0.966</td></tr><tr><td align=\"left\">RMSE</td><td char=\".\" align=\"char\">0.069</td><td char=\".\" align=\"char\">0.133</td><td char=\".\" align=\"char\">0.144</td><td char=\".\" align=\"char\">0.165</td><td char=\".\" align=\"char\">0.152</td></tr><tr><td align=\"left\" rowspan=\"6\">Dubinin–Radushkevich</td><td align=\"left\"><italic>q</italic>_{<italic>m</italic>} (mg/g)</td><td char=\".\" align=\"char\">2.216</td><td char=\".\" align=\"char\">3.871</td><td char=\".\" align=\"char\">3.371</td><td char=\".\" align=\"char\">4.097</td><td char=\".\" align=\"char\">2.138</td></tr><tr><td align=\"left\"><italic>K</italic>_DR (mol²/J²)</td><td char=\".\" align=\"char\">3.793</td><td char=\".\" align=\"char\">1.881</td><td char=\".\" align=\"char\">1.632</td><td char=\".\" align=\"char\">1.452</td><td char=\".\" align=\"char\">1.613</td></tr><tr><td align=\"left\"><italic>E</italic> (kJ/mol)</td><td char=\".\" align=\"char\">0.363</td><td char=\".\" align=\"char\">0.516</td><td char=\".\" align=\"char\">0.554</td><td char=\".\" align=\"char\">0.587</td><td char=\".\" align=\"char\">0.557</td></tr><tr><td align=\"left\"><italic>R</italic>²</td><td char=\".\" align=\"char\">0.921</td><td char=\".\" align=\"char\">0.841</td><td char=\".\" align=\"char\">0.906</td><td char=\".\" align=\"char\">0.860</td><td char=\".\" align=\"char\">0.746</td></tr><tr><td align=\"left\"><italic>R</italic>²_adj</td><td char=\".\" align=\"char\">0.905</td><td char=\".\" align=\"char\">0.809</td><td char=\".\" align=\"char\">0.887</td><td char=\".\" align=\"char\">0.832</td><td char=\".\" align=\"char\">0.695</td></tr><tr><td align=\"left\">RMSE</td><td char=\".\" align=\"char\">0.168</td><td char=\".\" align=\"char\">0.480</td><td char=\".\" align=\"char\">0.352</td><td char=\".\" align=\"char\">0.490</td><td char=\".\" align=\"char\">0.323</td></tr></tbody></table></table-wrap>", "<table-wrap id=\"Tab5\"><label>Table 5</label><caption><p>The comparison of the maximum dye adsorption capacity (<italic>q</italic>_m) with various agriculture wastes for DR28 dye removals.</p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th align=\"left\">Materials</th><th align=\"left\"><italic>q</italic>_m (mg/g)</th><th align=\"left\">References</th></tr></thead><tbody><tr><td align=\"left\">Pine bark</td><td char=\".\" align=\"char\">3.92</td><td align=\"left\">^{##UREF##37##49##}</td></tr><tr><td align=\"left\">Sugarcane bagasse treated with propionic acid</td><td char=\".\" align=\"char\">3.54</td><td align=\"left\">^{##UREF##38##50##}</td></tr><tr><td align=\"left\">Cabbage</td><td char=\".\" align=\"char\">2.31</td><td align=\"left\">^{##UREF##34##45##}</td></tr><tr><td align=\"left\">Potato peel</td><td char=\".\" align=\"char\">6.90</td><td align=\"left\">^{##UREF##35##46##}</td></tr><tr><td align=\"left\">Pea peels</td><td char=\".\" align=\"char\">16.40</td><td align=\"left\">^{##UREF##35##46##}</td></tr><tr><td align=\"left\">Watermelon rind</td><td char=\".\" align=\"char\">24.75</td><td align=\"left\">^{##UREF##36##47##}</td></tr><tr><td align=\"left\">Watermelon rind modified by titanium oxide</td><td char=\".\" align=\"char\">15.30</td><td align=\"left\">^{##REF##31712690##48##}</td></tr><tr><td align=\"left\">Rice husk</td><td char=\".\" align=\"char\">1.58</td><td align=\"left\">^{##UREF##11##15##}</td></tr><tr><td align=\"left\">Rice husk (biochar)</td><td char=\".\" align=\"char\">1.28</td><td align=\"left\">^{##UREF##11##15##}</td></tr><tr><td align=\"left\"><p>Rice husk (biochar) modified with</p><p>potassium hydroxide (KOH)</p></td><td char=\".\" align=\"char\">2.04</td><td align=\"left\">^{##UREF##11##15##}</td></tr><tr><td align=\"left\">SBB</td><td char=\".\" align=\"char\">3.24</td><td align=\"left\">This study</td></tr><tr><td align=\"left\">SBBT</td><td char=\".\" align=\"char\">6.25</td><td align=\"left\">This study</td></tr><tr><td align=\"left\">SBBM</td><td char=\".\" align=\"char\">6.58</td><td align=\"left\">This study</td></tr><tr><td align=\"left\">SBBA</td><td char=\".\" align=\"char\">6.41</td><td align=\"left\">This study</td></tr><tr><td align=\"left\">SBBZ</td><td char=\".\" align=\"char\">3.29</td><td align=\"left\">This study</td></tr></tbody></table></table-wrap>", "<table-wrap id=\"Tab6\"><label>Table 6</label><caption><p>The adsorption kinetic parameters of SBB, SBBT, SBBM, SBBA, and SBBZ for DR28 dye adsorptions.</p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th align=\"left\">Kinetic models</th><th align=\"left\">Parameters</th><th align=\"left\">SBB</th><th align=\"left\">SBBT</th><th align=\"left\">SBBM</th><th align=\"left\">SBBA</th><th align=\"left\">SBBZ</th></tr></thead><tbody><tr><td align=\"left\" rowspan=\"5\">Pseudo-first-order</td><td align=\"left\"><italic>q</italic>_e (mg/g)</td><td char=\".\" align=\"char\">0.438</td><td char=\".\" align=\"char\">1.091</td><td char=\".\" align=\"char\">1.819</td><td char=\".\" align=\"char\">1.252</td><td char=\".\" align=\"char\">0.447</td></tr><tr><td align=\"left\"><italic>k</italic>₁ (min⁻¹)</td><td char=\".\" align=\"char\">0.049</td><td char=\".\" align=\"char\">0.002</td><td char=\".\" align=\"char\">0.003</td><td char=\".\" align=\"char\">0.003</td><td char=\".\" align=\"char\">0.007</td></tr><tr><td align=\"left\"><italic>R</italic>²</td><td char=\".\" align=\"char\">0.917</td><td char=\".\" align=\"char\">0.702</td><td char=\".\" align=\"char\">0.731</td><td char=\".\" align=\"char\">0.721</td><td char=\".\" align=\"char\">0.941</td></tr><tr><td align=\"left\"><italic>R</italic>²_adj</td><td char=\".\" align=\"char\">0.915</td><td char=\".\" align=\"char\">0.696</td><td char=\".\" align=\"char\">0.725</td><td char=\".\" align=\"char\">0.715</td><td char=\".\" align=\"char\">0.940</td></tr><tr><td align=\"left\">RMSE</td><td char=\".\" align=\"char\">1.244</td><td char=\".\" align=\"char\">0.511</td><td char=\".\" align=\"char\">0.577</td><td char=\".\" align=\"char\">0.274</td><td char=\".\" align=\"char\">0.095</td></tr><tr><td align=\"left\" rowspan=\"5\">Pseudo-second-order</td><td align=\"left\"><italic>q</italic>_e (mg/g)</td><td char=\".\" align=\"char\">1.922</td><td char=\".\" align=\"char\">3.178</td><td char=\".\" align=\"char\">3.526</td><td char=\".\" align=\"char\">3.254</td><td char=\".\" align=\"char\">2.151</td></tr><tr><td align=\"left\"><italic>k</italic>₂ (g/mg·min)</td><td char=\".\" align=\"char\">0.012</td><td char=\".\" align=\"char\">0.019</td><td char=\".\" align=\"char\">0.018</td><td char=\".\" align=\"char\">0.083</td><td char=\".\" align=\"char\">0.118</td></tr><tr><td align=\"left\"><italic>R</italic>²</td><td char=\".\" align=\"char\">0.997</td><td char=\".\" align=\"char\">0.997</td><td char=\".\" align=\"char\">0.999</td><td char=\".\" align=\"char\">0.999</td><td char=\".\" align=\"char\">0.994</td></tr><tr><td align=\"left\"><italic>R</italic>²_adj</td><td char=\".\" align=\"char\">0.996</td><td char=\".\" align=\"char\">0.9960</td><td char=\".\" align=\"char\">0.999</td><td char=\".\" align=\"char\">0.999</td><td char=\".\" align=\"char\">0.993</td></tr><tr><td align=\"left\">RMSE</td><td char=\".\" align=\"char\">0.094</td><td char=\".\" align=\"char\">0.060</td><td char=\".\" align=\"char\">0.018</td><td char=\".\" align=\"char\">0.017</td><td char=\".\" align=\"char\">0.049</td></tr><tr><td align=\"left\" rowspan=\"5\">Elovich</td><td align=\"left\"><italic>α</italic> (mg/g·min)</td><td char=\".\" align=\"char\">1.325</td><td char=\".\" align=\"char\">1.045</td><td char=\".\" align=\"char\">11.910</td><td char=\".\" align=\"char\">9.548</td><td char=\".\" align=\"char\">2.094</td></tr><tr><td align=\"left\"><italic>β</italic> (g/mg)</td><td char=\".\" align=\"char\">3.864</td><td char=\".\" align=\"char\">2.225</td><td char=\".\" align=\"char\">4.346</td><td char=\".\" align=\"char\">3.240</td><td char=\".\" align=\"char\">3.961</td></tr><tr><td align=\"left\"><italic>R</italic>²</td><td char=\".\" align=\"char\">0.714</td><td char=\".\" align=\"char\">0.922</td><td char=\".\" align=\"char\">0.769</td><td char=\".\" align=\"char\">0.814</td><td char=\".\" align=\"char\">0.837</td></tr><tr><td align=\"left\"><italic>R</italic>²_adj</td><td char=\".\" align=\"char\">0.708</td><td char=\".\" align=\"char\">0.921</td><td char=\".\" align=\"char\">0.764</td><td char=\".\" align=\"char\">0.810</td><td char=\".\" align=\"char\">0.833</td></tr><tr><td align=\"left\">RMSE</td><td char=\".\" align=\"char\">22.506</td><td char=\".\" align=\"char\">0.225</td><td char=\".\" align=\"char\">0.254</td><td char=\".\" align=\"char\">0.224</td><td char=\".\" align=\"char\">0.180</td></tr><tr><td align=\"left\" rowspan=\"5\">Intra-particle diffusion</td><td align=\"left\"><italic>k</italic>_i (mg/g·min^0.5)</td><td char=\".\" align=\"char\">0.030</td><td char=\".\" align=\"char\">0.058</td><td char=\".\" align=\"char\">0.029</td><td char=\".\" align=\"char\">0.033</td><td char=\".\" align=\"char\">0.013</td></tr><tr><td align=\"left\"><italic>C</italic>_i (mg/g)</td><td char=\".\" align=\"char\">0.972</td><td char=\".\" align=\"char\">1.627</td><td char=\".\" align=\"char\">1.734</td><td char=\".\" align=\"char\">2.329</td><td char=\".\" align=\"char\">1.881</td></tr><tr><td align=\"left\"><italic>R</italic>²</td><td char=\".\" align=\"char\">0.770</td><td char=\".\" align=\"char\">0.636</td><td char=\".\" align=\"char\">0.794</td><td char=\".\" align=\"char\">0.762</td><td char=\".\" align=\"char\">0.727</td></tr><tr><td align=\"left\"><italic>R</italic>²_adj</td><td char=\".\" align=\"char\">0.765</td><td char=\".\" align=\"char\">0.628</td><td char=\".\" align=\"char\">0.790</td><td char=\".\" align=\"char\">0.757</td><td char=\".\" align=\"char\">0.721</td></tr><tr><td align=\"left\">RMSE</td><td char=\".\" align=\"char\">0.295</td><td char=\".\" align=\"char\">0.387</td><td char=\".\" align=\"char\">0.448</td><td char=\".\" align=\"char\">0.412</td><td char=\".\" align=\"char\">0.291</td></tr></tbody></table></table-wrap>", "<table-wrap id=\"Tab7\"><label>Table 7</label><caption><p>Thermodynamic parameters of SBB, SBBT, SBBM, SBBA, and SBBZ<bold>.</bold></p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th align=\"left\"/><th align=\"left\">SBB</th><th align=\"left\">SBBT</th><th align=\"left\">SBBM</th><th align=\"left\">SBBA</th><th align=\"left\">SBBZ</th></tr></thead><tbody><tr><td align=\"left\" colspan=\"6\">Δ<italic>G</italic>° (J/mol)</td></tr><tr><td align=\"left\">293.15 K</td><td align=\"left\">− 1312.45</td><td align=\"left\">− 1760.65</td><td align=\"left\">− 2391.08</td><td align=\"left\">− 1962.24</td><td align=\"left\">− 1531.55</td></tr><tr><td align=\"left\">298.15 K</td><td align=\"left\">− 1349.72</td><td align=\"left\">− 1807.15</td><td align=\"left\">− 2451.84</td><td align=\"left\">− 2015.61</td><td align=\"left\">− 1575.55</td></tr><tr><td align=\"left\">303.15 K</td><td align=\"left\">− 1387.98</td><td align=\"left\">− 1857.25</td><td align=\"left\">− 2513.91</td><td align=\"left\">− 2067.68</td><td align=\"left\">− 1618.91</td></tr><tr><td align=\"left\">308.15 K</td><td align=\"left\">− 1427.26</td><td align=\"left\">− 1903.46</td><td align=\"left\">− 2579.93</td><td align=\"left\">− 2140.78</td><td align=\"left\">− 1663.39</td></tr><tr><td align=\"left\">313.15 K</td><td align=\"left\">− 1468.15</td><td align=\"left\">− 1953.38</td><td align=\"left\">− 2642.27</td><td align=\"left\">− 2175.51</td><td align=\"left\">− 1709.05</td></tr><tr><td align=\"left\">318.15 K</td><td align=\"left\">− 1509.13</td><td align=\"left\">− 2004.55</td><td align=\"left\">− 2708.70</td><td align=\"left\">− 2231.39</td><td align=\"left\">− 1755.93</td></tr><tr><td align=\"left\">323.15 K</td><td align=\"left\">− 1551.83</td><td align=\"left\">− 2057.05</td><td align=\"left\">− 2776.74</td><td align=\"left\">− 2288.66</td><td align=\"left\">− 1804.11</td></tr><tr><td align=\"left\">Δ<italic>H</italic>° (J/mol)</td><td align=\"left\">1026.28</td><td align=\"left\">1128.54</td><td align=\"left\">1375.55</td><td align=\"left\">1216.67</td><td align=\"left\">1123.89</td></tr><tr><td align=\"left\">Δ<italic>S</italic>° (J/mol K)</td><td align=\"left\">7.97</td><td align=\"left\">9.85</td><td align=\"left\">12.84</td><td align=\"left\">10.84</td><td align=\"left\">9.05</td></tr></tbody></table></table-wrap>" ]

[ "<disp-formula id=\"Equ1\"><label>1</label><alternatives><tex-math id=\"M1\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\text{Dye removal efficiency }}\\left( \\% \\right) \\, = \\, \\left( {\\left( {C_{0} - C_{{\\text{e}}} } \\right)/C_{0} } \\right) \\, \\times \\, 100$$\\end{document}</tex-math><mml:math id=\"M2\" display=\"block\"><mml:mrow><mml:mrow><mml:mtext>Dye removal efficiency</mml:mtext><mml:mspace width=\"0.333333em\"/></mml:mrow><mml:mfenced close=\")\" open=\"(\"><mml:mo>%</mml:mo></mml:mfenced><mml:mspace width=\"0.166667em\"/><mml:mo>=</mml:mo><mml:mspace width=\"0.166667em\"/><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:msub><mml:mi>C</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo>-</mml:mo><mml:msub><mml:mi>C</mml:mi><mml:mtext>e</mml:mtext></mml:msub></mml:mrow></mml:mfenced><mml:mo stretchy=\"false\">/</mml:mo><mml:msub><mml:mi>C</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:mrow></mml:mfenced><mml:mspace width=\"0.166667em\"/><mml:mo>×</mml:mo><mml:mspace width=\"0.166667em\"/><mml:mn>100</mml:mn></mml:mrow></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ2\"><label>2</label><alternatives><tex-math id=\"M3\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\text{Dye adsorption capacity }}\\left( {q_{e} } \\right) \\, = (C_{0} - C_{{\\text{e}}} )V/m$$\\end{document}</tex-math><mml:math id=\"M4\" display=\"block\"><mml:mrow><mml:mrow><mml:mtext>Dye adsorption capacity</mml:mtext><mml:mspace width=\"0.333333em\"/></mml:mrow><mml:mfenced close=\")\" open=\"(\"><mml:msub><mml:mi>q</mml:mi><mml:mi>e</mml:mi></mml:msub></mml:mfenced><mml:mspace width=\"0.166667em\"/><mml:mo>=</mml:mo><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>C</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo>-</mml:mo><mml:msub><mml:mi>C</mml:mi><mml:mtext>e</mml:mtext></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mi>V</mml:mi><mml:mo stretchy=\"false\">/</mml:mo><mml:mi>m</mml:mi></mml:mrow></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ3\"><label>3</label><alternatives><tex-math id=\"M5\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$q_{e} = \\, q_{{\\text{m}}} K_{L} C_{{\\text{e}}} /1 + K_{{\\text{L}}} C_{{\\text{e}}}$$\\end{document}</tex-math><mml:math id=\"M6\" display=\"block\"><mml:mrow><mml:msub><mml:mi>q</mml:mi><mml:mi>e</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mspace width=\"0.166667em\"/><mml:msub><mml:mi>q</mml:mi><mml:mtext>m</mml:mtext></mml:msub><mml:msub><mml:mi>K</mml:mi><mml:mi>L</mml:mi></mml:msub><mml:msub><mml:mi>C</mml:mi><mml:mtext>e</mml:mtext></mml:msub><mml:mo stretchy=\"false\">/</mml:mo><mml:mn>1</mml:mn><mml:mo>+</mml:mo><mml:msub><mml:mi>K</mml:mi><mml:mtext>L</mml:mtext></mml:msub><mml:msub><mml:mi>C</mml:mi><mml:mtext>e</mml:mtext></mml:msub></mml:mrow></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ4\"><label>4</label><alternatives><tex-math id=\"M7\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$q_{{\\text{e}}} = \\, K_{{\\text{F}}} C_{{\\text{e}}}^{1/n}$$\\end{document}</tex-math><mml:math id=\"M8\" display=\"block\"><mml:mrow><mml:msub><mml:mi>q</mml:mi><mml:mtext>e</mml:mtext></mml:msub><mml:mo>=</mml:mo><mml:mspace width=\"0.166667em\"/><mml:msub><mml:mi>K</mml:mi><mml:mtext>F</mml:mtext></mml:msub><mml:msubsup><mml:mi>C</mml:mi><mml:mrow><mml:mtext>e</mml:mtext></mml:mrow><mml:mrow><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">/</mml:mo><mml:mi>n</mml:mi></mml:mrow></mml:msubsup></mml:mrow></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ5\"><label>5</label><alternatives><tex-math id=\"M9\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$q_{{\\text{e}}} = RT/b_{{\\text{T}}} {\\text{ln}}A_{{\\text{T}}} C_{{\\text{e}}}$$\\end{document}</tex-math><mml:math id=\"M10\" display=\"block\"><mml:mrow><mml:msub><mml:mi>q</mml:mi><mml:mtext>e</mml:mtext></mml:msub><mml:mo>=</mml:mo><mml:mi>R</mml:mi><mml:mi>T</mml:mi><mml:mo stretchy=\"false\">/</mml:mo><mml:msub><mml:mi>b</mml:mi><mml:mtext>T</mml:mtext></mml:msub><mml:mtext>ln</mml:mtext><mml:msub><mml:mi>A</mml:mi><mml:mtext>T</mml:mtext></mml:msub><mml:msub><mml:mi>C</mml:mi><mml:mtext>e</mml:mtext></mml:msub></mml:mrow></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ6\"><label>6</label><alternatives><tex-math id=\"M11\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$q_{e} = q_{{\\text{m}}} {\\text{exp}}( - K_{{{\\text{DR}}}} \\varepsilon^{2} )$$\\end{document}</tex-math><mml:math id=\"M12\" display=\"block\"><mml:mrow><mml:msub><mml:mi>q</mml:mi><mml:mi>e</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:msub><mml:mi>q</mml:mi><mml:mtext>m</mml:mtext></mml:msub><mml:mtext>exp</mml:mtext><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mo>-</mml:mo><mml:msub><mml:mi>K</mml:mi><mml:mtext>DR</mml:mtext></mml:msub><mml:msup><mml:mi>ε</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ7\"><label>7</label><alternatives><tex-math id=\"M13\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$q_{{\\text{t}}} = q_{{\\text{e}}} (1 - e^{{ - k^{{k_{1} t}} }} )$$\\end{document}</tex-math><mml:math id=\"M14\" display=\"block\"><mml:mrow><mml:msub><mml:mi>q</mml:mi><mml:mtext>t</mml:mtext></mml:msub><mml:mo>=</mml:mo><mml:msub><mml:mi>q</mml:mi><mml:mtext>e</mml:mtext></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>1</mml:mn><mml:mo>-</mml:mo><mml:msup><mml:mi>e</mml:mi><mml:mrow><mml:mo>-</mml:mo><mml:msup><mml:mi>k</mml:mi><mml:mrow><mml:msub><mml:mi>k</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mi>t</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:msup><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ8\"><label>8</label><alternatives><tex-math id=\"M15\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$q_{{\\text{t}}} = \\, k_{2} q_{{\\text{e}}}^{2} t/\\left( {1 + \\, q_{{\\text{e}}} k_{2} t} \\right)$$\\end{document}</tex-math><mml:math id=\"M16\" display=\"block\"><mml:mrow><mml:msub><mml:mi>q</mml:mi><mml:mtext>t</mml:mtext></mml:msub><mml:mo>=</mml:mo><mml:mspace width=\"0.166667em\"/><mml:msub><mml:mi>k</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:msubsup><mml:mi>q</mml:mi><mml:mrow><mml:mtext>e</mml:mtext></mml:mrow><mml:mn>2</mml:mn></mml:msubsup><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">/</mml:mo><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:mn>1</mml:mn><mml:mo>+</mml:mo><mml:mspace width=\"0.166667em\"/><mml:msub><mml:mi>q</mml:mi><mml:mtext>e</mml:mtext></mml:msub><mml:msub><mml:mi>k</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mi>t</mml:mi></mml:mrow></mml:mfenced></mml:mrow></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ9\"><label>9</label><alternatives><tex-math id=\"M17\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$q_{t} = \\beta \\;{\\text{ln}}\\;t + \\beta \\;{\\text{ln}}\\;\\alpha$$\\end{document}</tex-math><mml:math id=\"M18\" display=\"block\"><mml:mrow><mml:msub><mml:mi>q</mml:mi><mml:mi>t</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mi>β</mml:mi><mml:mspace width=\"0.277778em\"/><mml:mtext>ln</mml:mtext><mml:mspace width=\"0.277778em\"/><mml:mi>t</mml:mi><mml:mo>+</mml:mo><mml:mi>β</mml:mi><mml:mspace width=\"0.277778em\"/><mml:mtext>ln</mml:mtext><mml:mspace width=\"0.277778em\"/><mml:mi>α</mml:mi></mml:mrow></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ10\"><label>10</label><alternatives><tex-math id=\"M19\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$q_{{\\text{t}}} = k_{{\\text{i}}} t^{0.5} + C_{{\\text{i}}}$$\\end{document}</tex-math><mml:math id=\"M20\" display=\"block\"><mml:mrow><mml:msub><mml:mi>q</mml:mi><mml:mtext>t</mml:mtext></mml:msub><mml:mo>=</mml:mo><mml:msub><mml:mi>k</mml:mi><mml:mtext>i</mml:mtext></mml:msub><mml:msup><mml:mi>t</mml:mi><mml:mrow><mml:mn>0.5</mml:mn></mml:mrow></mml:msup><mml:mo>+</mml:mo><mml:msub><mml:mi>C</mml:mi><mml:mtext>i</mml:mtext></mml:msub></mml:mrow></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ11\"><label>11</label><alternatives><tex-math id=\"M21\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\Delta G^\\circ \\, = \\, - RT\\;{\\text{ln}}\\;K_{{\\text{c}}}$$\\end{document}</tex-math><mml:math id=\"M22\" display=\"block\"><mml:mrow><mml:mi mathvariant=\"normal\">Δ</mml:mi><mml:msup><mml:mi>G</mml:mi><mml:mo>∘</mml:mo></mml:msup><mml:mspace width=\"0.166667em\"/><mml:mo>=</mml:mo><mml:mspace width=\"0.166667em\"/><mml:mo>-</mml:mo><mml:mi>R</mml:mi><mml:mi>T</mml:mi><mml:mspace width=\"0.277778em\"/><mml:mtext>ln</mml:mtext><mml:mspace width=\"0.277778em\"/><mml:msub><mml:mi>K</mml:mi><mml:mtext>c</mml:mtext></mml:msub></mml:mrow></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ12\"><label>12</label><alternatives><tex-math id=\"M23\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\text{ln}}\\;K_{{\\text{c}}} = \\, - \\Delta H^\\circ /RT + \\, \\Delta S^\\circ /R$$\\end{document}</tex-math><mml:math id=\"M24\" display=\"block\"><mml:mrow><mml:mtext>ln</mml:mtext><mml:mspace width=\"0.277778em\"/><mml:msub><mml:mi>K</mml:mi><mml:mtext>c</mml:mtext></mml:msub><mml:mo>=</mml:mo><mml:mspace width=\"0.166667em\"/><mml:mo>-</mml:mo><mml:mi mathvariant=\"normal\">Δ</mml:mi><mml:msup><mml:mi>H</mml:mi><mml:mo>∘</mml:mo></mml:msup><mml:mo stretchy=\"false\">/</mml:mo><mml:mi>R</mml:mi><mml:mi>T</mml:mi><mml:mo>+</mml:mo><mml:mspace width=\"0.166667em\"/><mml:mi mathvariant=\"normal\">Δ</mml:mi><mml:msup><mml:mi>S</mml:mi><mml:mo>∘</mml:mo></mml:msup><mml:mo stretchy=\"false\">/</mml:mo><mml:mi>R</mml:mi></mml:mrow></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ13\"><label>13</label><alternatives><tex-math id=\"M25\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\Delta G^\\circ \\, = \\, \\Delta H^\\circ \\, - T\\Delta S^\\circ$$\\end{document}</tex-math><mml:math id=\"M26\" display=\"block\"><mml:mrow><mml:mi mathvariant=\"normal\">Δ</mml:mi><mml:msup><mml:mi>G</mml:mi><mml:mo>∘</mml:mo></mml:msup><mml:mspace width=\"0.166667em\"/><mml:mo>=</mml:mo><mml:mspace width=\"0.166667em\"/><mml:mi mathvariant=\"normal\">Δ</mml:mi><mml:msup><mml:mi>H</mml:mi><mml:mo>∘</mml:mo></mml:msup><mml:mspace width=\"0.166667em\"/><mml:mo>-</mml:mo><mml:mi>T</mml:mi><mml:mi mathvariant=\"normal\">Δ</mml:mi><mml:msup><mml:mi>S</mml:mi><mml:mo>∘</mml:mo></mml:msup></mml:mrow></mml:math></alternatives></disp-formula>" ]

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[ "<table-wrap-foot><p>*BET specific surface area.</p><p>**Barrett–Joyner–Halenda (BJH) method.</p></table-wrap-foot>", "<fn-group><fn><p><bold>Publisher's note</bold></p><p>Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.</p></fn></fn-group>" ]

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[ "<title>Introduction</title>", "<p id=\"Par2\">Nearly half of China’s coal resource reserves and output are attributed to thick coal seams^{##UREF##0##1##–##UREF##2##3##}. The fully mechanized longwall top coal caving (LTCC) mining technology is one of the main technologies used for mining thick seams^{##UREF##3##4##–##UREF##6##7##}. Improper timing of the caving-opening closure in the caving process of the LTCC mining face can result in excessive or insufficient caving, leading to resource wastage or compromising coal quality^{##UREF##7##8##,##UREF##8##9##}. Reasonable time for closing the caving-opening depends on the mixing degree of coal gangue^{##UREF##9##10##–##UREF##11##12##}. Therefore, accurate, real-time and effective technology for identifying coal gangue mixing degree is an effective measure to realize automation and intelligent mining in coal mines, as well as promoting the intelligent industrialization of coal industry^{##UREF##12##13##–##REF##38203112##15##}. It can effectively reduce labor cost, enhance the safety of coal mining operations, reduce equipment maintenance expenses, and significantly improve the coal mining extraction rate, ultimately leading to higher productivity and efficiency^{##UREF##6##7##,##UREF##14##16##,##UREF##15##17##}.</p>", "<p id=\"Par3\">In recent years, many experts and scholars have conducted extensive scientific research in the field of coal and gangue detection^{##UREF##16##18##}. Qingjun Song et al. proposed various methods to identify the vibration sound signal emitted by hydraulic support tail beam in the process of coal and gangue collapsed, aiming to realize detection of coal and gangue^{##UREF##14##16##,##UREF##17##19##}. Bingxiang Huang et al. proposed a coal gangue detection method utilizing near-infrared spectroscopy^{##UREF##18##20##}. Jiachen Wang et al. employed the top coal tracker to monitor the movement of top coal and achieve automated coal drawing in combination with coal gangue image recognition. Chuangyou Liu et al. proposed an coal gangue identification method using active microwave irradiation infrared detection. Zengcai Wang et al. proposed γ Detection of coal gangue mixing rate in top coal caving mining by radiographic method^{##UREF##11##12##,##UREF##15##17##,##UREF##19##21##,##UREF##20##22##}. Liansheng Li et al. put forward a method for identifying coal and gangue based on density difference^{##UREF##21##23##}, and Feng Xing proposed a non-contact microwave detection technology to detect the mixing degree of coal and gangue^{##UREF##22##24##}. Jingjing Deng et al. employed terahertz technology to generate images of coal gangue mixture for the purpose of coal gangue detection^{##UREF##23##25##–##UREF##26##28##}. Yuming Huo optimized the parameters of intelligent coal drawing process by establishing a predictive model of the periodic coal drawing time^{##UREF##27##29##,##UREF##28##30##}.</p>", "<p id=\"Par4\">The research above on the recognition of coal and gangue in LTCC mining, primarily adopt the principles of image grayscale, sound spectrum, vibration spectrum, natural γ method, and various composite monitoring methods. However, it is evident from the research objects and results that this study is still in the preliminary stage, primarily due to the complex structural characteristics of extremely thick coal seams in China, which often contain multiple layers of dirt bands^{##UREF##29##31##,##UREF##30##32##}. The presence of dirt bands results in a mixed flow containing top coal, dirt bands and roof rocks flowing out of the top-coal caving-opening during the top coal caving process of the working face^{##UREF##31##33##}.</p>", "<p id=\"Par5\">That is, the coal gangue detection method in LTCC mining should not only address the detection of coal gangue mixing ratio in coal seams with a simple structure, but also accommodate complex structure coal seams^{##UREF##30##32##,##UREF##32##34##}. In order to achieve the objective of real-time and accurate recognition of the mixing degree between coal and gangue in LTCC mining, this paper proposes an accurate recognition method based on natural γ ray, which utilizes the radiation difference characteristics of coal and rock natural γ ray. A low radiation level radioactive measurement method is employed to determine the instantaneous mixing ratio of coal and gangue mixture during the top coal caving process, thereby laying the foundation for realizing the intellectualization of LTCC mining.</p>" ]

[ "<title>Methods</title>", "<title>Distribution characteristics of natural radionuclides in thick coal seams</title>", "<title>Natural radionuclide</title>", "<p id=\"Par6\">Natural radionuclides are formed during interstellar processes, such as the Big Bang, and continue to exist. They were transported to Earth during its formation. Currently, there are three natural radioactive series (uranium, actinium and thorium) and some non-series radionuclides in nature, however, only the former three can significantly impact radiometric measurement, as illustrated in Table ##TAB##0##1##.</p>", "<title>Deposition characteristics of natural radionuclides in coal beds</title>", "<p id=\"Par7\">Natural radionuclides are present in various types of rocks, including coal-bearing strata^{##UREF##33##35##–##REF##36615527##37##}. In LTCC mining, the natural ray coal gangue identification technology heavily relies on the immediate roof of the coal seam. As such, the roof deposition characteristics of thick and extra-thick coal seams were studied, and the characteristics of their natural radiation intensity were analyzed.</p>", "<p id=\"Par8\">Factors influencing the content and distribution of radionuclides in sedimentary rocks comprise the sediment source, composition and structure of the rocks, sedimentary conditions and sedimentary environment, radionuclide content of parent rock, duration of radionuclide presence, sediment grain size, and distance from the original location. Thus, the abundance of radionuclides in rocks is influenced by factors related to their formation mode, location and temporal aspects. For natural rocks, the distribution of radionuclides is as follows:<list list-type=\"order\"><list-item><p id=\"Par9\">Rocks or minerals of a similar nature exhibit comparable levels of radionuclide abundances.</p></list-item><list-item><p id=\"Par10\">There are significant variations in the abundance of radionuclides among different rocks or minerals.</p></list-item></list></p>", "<p id=\"Par11\">These aforementioned laws possess statistical characteristics and exist objectively, forming the basis for the coal gangue identification technology by natural gamma-ray.</p>", "<p id=\"Par12\">To sum up, the coal mine roof exhibits varying radiation characteristics due to the diverse composition of sedimentary rocks, resulting in significant variations in the levels of uranium, thorium and potassium.</p>", "<p id=\"Par13\">The radionuclides content in the roof primarily correlates with the grain size of the sediment, the amount of organic substances in the sedimentary environment, the sedimentary environment and conditions, the sedimentary time and other factors. Consequently, the following general rules apply:<list list-type=\"order\"><list-item><p id=\"Par14\">The content of radionuclides in rocks of the same type is similar. The content of radionuclides in different rocks varies greatly.</p></list-item><list-item><p id=\"Par15\">In the coal bearing rock series, the lowest radioactive intensity is coal, while the radioactivity of conglomerate, coarse sandstone, medium sandstone, fine sandstone, siltstone, sandy mudstone, shale and mudstone gradually increases. The smaller the particle size of diagenetic material, the greater the mud content and the stronger the radiation.</p></list-item><list-item><p id=\"Par16\">Inland roof rock exhibit lesser radiation compared to offshore roof rock. The presence of asphaltene mudstone, phosphorite and organic matter in the offshore sedimentary rocks contributes to the effective absorption of radionuclides during sedimentation, resulting in a generally higher radiation level compared to inland type rocks.</p></list-item><list-item><p id=\"Par17\">The shorter coal forming time generally means stronger roof radiation. Thick coal seams are mostly lignite with low metamorphic degree, and their roof formation time is less than bituminous coal and anthracite, so their roof radioactivity is relatively large. Therefore, for similar roof rocks, the shorter formation time means the greater radiation intensity, which is beneficial to the application of natural ray coal gangue detection technology.</p></list-item><list-item><p id=\"Par18\">Sedimentary rocks containing diagenetic minerals such as potash exhibit high radiation content. In order to identify the source of natural radiation accurately, it is necessary to analyze the composition of diagenetic minerals during measuring the radiation of coal mine roof.</p></list-item></list></p>", "<p id=\"Par19\">As depicted in Fig. ##FIG##0##1## that the radioactivity content of coal is the lowest. If the shielding effect of loose coal is considered, the radioactivity of coal can be ignored. The radioactivity of potassium salt is the highest. The radioactivity of common roof rocks in coal mines such as sandstone is more than 5 times that of coal. It is with a large different, therefore, it is feasible to obtain the content of gangue in the mixture by measuring the radiation intensity in the mixture of coal and gangue.</p>", "<title>Characteristics of roof rock property of LTCC face in extra thick coal seam in China</title>", "<p id=\"Par20\">According to the statistics of China National Knowledge Infrastructure Net documents, the immediate roof lithology of 94 LTCC faces in China is shown in Fig. ##FIG##1##2##.</p>", "<p id=\"Par21\">As depicted in Fig. ##FIG##1##2##, among the 90 working faces surveyed, mudstone is the most prevalent roof rock type, representing 58.5% of the total statistical data, followed by sandstone at 30.9%, while conglomerate and limestone hav substantially lower occurrences. Among the sandstones, fine grained siltstone represented the primary type of sanedstone, accounting for 51.7% of the sandstone roof.</p>", "<p id=\"Par22\">According to Fig. ##FIG##0##1##, about 86% thick coal seams in China have significant differences in immediate roof radiation intensity from coal seams, so natural γ X-ray technology has wide applicability in coal gangue identification.</p>", "<title>Radiation characteristics of coal and rock strata in typical thick coal seam mining areas in China</title>", "<p id=\"Par23\">It can be seen from the above analysis that the radionuclide content of sedimentary rocks is related to the content of clay minerals, formation time, environment of sedimentary area and other factors. Therefore, representative typical mining areas such as Dongsheng, Datong, Yanzhou, Shuozhou and Longkou are selected to analyze and study the radiation characteristics of coal and roof rock in thick and extra thick coal seams.</p>", "<p id=\"Par24\">As depicted in Fig. ##FIG##2##3##, ① there are radioactive elements in coal and rock, and the radiation intensity of coal samples is generally small, even less than their own shielding capacity. Therefore, in this paper, the radiation content of coal is considered as 0. The radiation intensity of roof rock is much higher than that of coal sample, and the difference can be several times or even dozens of times. Therefore, the mixing degree of coal and gangue can be identified through the difference of radiation characteristics between coal and rock; ② The difference of radiation between different rocks is huge, so it can be realized to distinguish and identify the dirt band and roof rock in the complex structure coal seam through the difference of radiation characteristics between different rocks; ③ The radiation intensity of rock samples from the same sedimentary rock stratum in the same coal field is similar, so for the same working face or even the same coal seam, there is no need to frequently adjust the parameters in the process of using the ray coal gangue identification technology.</p>", "<title>Basic principle of natural γ-ray coal and gangue recognition</title>", "<p id=\"Par25\">The principle of natural γ-rays coal and gangue recognition is summarized as follows:</p>", "<p id=\"Par26\">Based on the radiation differentiation characteristics of natural γ-rays in coal and rock, the method of low radiation level radioactivity measurement is adopted to identify the instantaneous mixing rate of coal and gangue flow in coal releasing process. Combined with the time-series characteristics of caving flow of top coal in fully mechanized caving mining and the energy spectrum characteristics of different strata, the automatic identification of coal and gangue in fully mechanized caving of thick coal seam with complex structure containing multiple gangue is realized.<list list-type=\"order\"><list-item><p id=\"Par27\">In the process of top coal caving, the gangue outflow from the caving-opening has a changing law from nothing to existence, from less to more.</p></list-item><list-item><p id=\"Par28\">In the process of top coal caving, the natural radiation intensity in the coal and gangue mixture changes from weak to strong, and the content of gangue in the mixed flow can be determined.</p></list-item><list-item><p id=\"Par29\">By detecting the instantaneous radiation intensity of the coal and gangue mixed flow to determine the rate of gangue and thus determine the time to close the coal drain.</p></list-item><list-item><p id=\"Par30\">For coal beds with complex structure containing one to multiple layers of gangue, the influence of gangue inclusion on the accuracy of coal and gangue recognition can be excluded according to the different energy spectrum characteristics of gangue inclusion and the caving time sequence characteristics, which is, the coal and waste collection at different levels will be caved in the sequence of time according to the different positions from the caving-opening in the process of caving.</p></list-item></list></p>", "<p id=\"Par31\">The schematic diagram of natural γ-ray coal and gangue recognition is shown in Fig. ##FIG##3##4##.</p>", "<title>Development of coal and gangue identification system</title>", "<p id=\"Par32\">The developed mine intrinsic safe coal and gangue natural ray real-time detection system includes two parts: detector and data acquisition display terminal. The detector is composed of NaI crystal(Ф100 × 100), photomultipler, shell, data processing terminal and explosion-proof shell. The data processing terminals include coal and gangue identification system APP and intrinsic safe Android mobile phones. The data acquisition display terminal and the detector will be connected wirelessly for data transmission. During the measurement at the LTCC face under the shaft, the single chip microcomputer will be used for data analysis, processing and display. The system and related indicators are shown in Table ##TAB##1##2##.</p>", "<p id=\"Par33\">Due to the difference between the background radiation of the underground environment and that of the laboratory environment, parameter debugging is required before the dynamic monitoring of the coal drawing mouth to adapt to the underground radiation environment.</p>", "<p id=\"Par34\">The detector is placed in the material chamber of 1303 working face. The chamber floor is a coal seam covered with sand and gravel. Adjust the detector detection face upward and downward once, and the bottom shall be padded 20 cm above the ground, as shown in Fig. ##FIG##4##5##.</p>", "<p id=\"Par35\">(1) Influence of supply voltage.</p>", "<p id=\"Par36\">In order to test the influence of different power supply voltages of the circuit on the detection efficiency of the detector, adjust the counting conditions when the power supply voltage of the detector is 11 V and 12 V, as shown in Fig. ##FIG##5##6##.</p>", "<p id=\"Par37\">As depicted in Fig. ##FIG##5##6## that under the situation that other conditions remain unchanged and only the power supply voltage is changed, the count value of the detector is relatively stable, which can verify that the circuit can work normally under the power supply condition of more than 11 v.</p>", "<p id=\"Par38\">(2) Threshold debugging.</p>", "<p id=\"Par39\">The threshold adjustment range is 0–1 v, and the adjustment amplitude is 0.02 v. The test results are shown in Fig. ##FIG##6##7##.</p>", "<p id=\"Par40\">As depicted in Fig. ##FIG##6##7##, (1) with the increase of threshold value, the counting value of the detector shows a downward trend; (2) under the threshold value of 0–0.08 v, the counting of the detector is too large, and after exceeding 0.08 v, the counting of the detector is in the normal range; (3) the detection surface of the detector is directional, that is, the counting of the detector is not only related to the position of the detector, but also related to the direction of the detection surface.</p>", "<p id=\"Par41\">(3) Background comparison between ground and underground environments.</p>", "<p id=\"Par42\">In order to compare the radiation difference under different environmental conditions above and below the surface, the background counts of the detector are counted at different thresholds, as shown in Fig. ##FIG##7##8##.</p>", "<p id=\"Par43\">As depicted in Fig. ##FIG##8##9## that after exceeding 0.08 v, the underground environment detector counts in the normal range, and the technology slowly decreases with the increase of threshold; In the surface environment, the count is in the normal range only when the threshold value is 0.5 v. This shows that the radiation field of the ground environment is far more complex than that of the underground environment. As the radiation content of rock is not affected by the change of detection location, the comparison between the presence of gangue and the absence of gangue will be more obvious during underground detection than that on the ground.</p>", "<p id=\"Par44\">In order to verify the detection effect of gangue radiation in the underground environment, the static test is carried out in the material roadway chamber of the underground working face. Put the detector in the chamber, with the detection face upward and the bottom pad 20 cm above the ground.</p>", "<p id=\"Par45\">During the test, first count the background under different threshold values (0.08–1 V), then place gangue (about 10 kg), and count again under different threshold values (0.08–1 V). Figure ##FIG##8##9## is the specific data.</p>", "<p id=\"Par46\">As depicted in Fig. ##FIG##8##9## that the radiation intensity of the same pile of gangue decreases with the increase of the threshold value, and the background value also descends. Because the background radiation of the underground environment comes from the radioactive elements in the roof and floor rocks and the radioactive elements in the air, of which the radioactive elements in the roof and floor rocks account for the majority. Therefore, the energy spectrum of the background is close to that of the gangue, and the adjustment of the threshold value will affect the counts of both.</p>", "<p id=\"Par47\">In order to determine the obvious threshold area for detection, the net count and background value of gangue placed under the same threshold are divided to obtain the counting increase of gangue placed under different threshold conditions, as shown in Fig. ##FIG##9##10##.</p>", "<p id=\"Par48\">As depicted in Fig. ##FIG##9##10## that before the threshold value 0.2 V, the counting amplitude fluctuates greatly. Between 0.2 and 0.9 V, the counting amplitude increases steadily and to a certain extent. At the threshold value of 1 V, the amplitude decreases significantly. Therefore, under the underground environment, the threshold value can be set between 0.8 and 0.9 V, for effective detection efficiency.</p>", "<p id=\"Par49\">Based on the formed identification method of coal-gangue-rock in LTCC of extra thick coal seams, and taking the LTCC working faces of Lilou Coal Mine (thick coal seam with simple structure), Xiaoyu Coal Mine (thick coal seam with single layer dirt band) and Tashan Coal Mine (thick coal seam with complex structure) as specific conditions, the on-site installation scheme of detectors is designed to test the sensitivity, signal stability and adaptability to the environment of detectors; The response characteristics of the detector to the dirt band and roof rock are analyzed to provide a basis for determining the identification parameters.</p>", "<p id=\"Par50\">KZT12 intrinsically safe coal gangue identification detector for mining has three installation positions: under the support tail beam with the detection direction facing the coal scupper, above the rear scraper conveyor with the detection direction facing the coal flow, and under the support tail beam with the detection direction facing the coal scupper, as shown in Fig. ##FIG##10##11##A–C.</p>" ]

[ "<title>Result</title>", "<title>Application of coal gangue identification system in simple structure thick coal seam fully mechanized caving face</title>", "<p id=\"Par51\">Working face 1303 (coal seam 3) of Lilou Coal Mine is located in the middle and lower part of Shanxi Formation. Most of the coal seams (coal seam 3) are relatively stable with simple structure. The average thickness of the coal seam is 6.98 m, the mining height is 3.6 m, the drawing height is 3.38 m, and the average dip angle of the coal seam is 13°. The immediate roof of the coal seam is sandy mudstone with a thickness of 0.96 m, and the basic top is fine sandstone with a thickness of 15.12 m.</p>", "<p id=\"Par52\">KZT12 intrinsically safe coal gangue identification detector is installed under the tail beam of the support, with the detection direction facing the coal chute, as shown in Fig. ##FIG##9##10##A.</p>", "<p id=\"Par53\">The radiation data and filtering data (Kalman filtering method) of coal gangue monitored during coal drawing are shown in Fig. ##FIG##11##12##.</p>", "<p id=\"Par54\">As depicted in Fig. ##FIG##11##12## that during top coal drawing, the detector has a relatively sensitive response to the occurrence and content of gangue. The detected radiation data has obvious periodicity, which can be divided into two stages: stage of pure coal and mixing stage of top coal and gangue. In the pure coal stage, there is pure coal near the caving-opening. Since there are almost no radioactive nuclides in the coal, the radiation curve of the radiation data detected by the detector fluctuates near the background value, but the fluctuation range is small. As the top coal being exhausted, the immediate roof rock gradually mixes into the caving-opening. At this stage, the radiation intensity increases significantly, indicating that a large amount of gangue is mixed,, the caving-opening operation is terminated in combination with the basis of “close the caving-opening when see the gangue”, The radiation curve gradually decreases and returns to the background level with the closing of the caving-opening.</p>", "<title>Application test of coal gangue identification system in LTCC face with single layer thick coal seam of dirt band</title>", "<p id=\"Par55\">The coal seam of Working Face 8202 in Xiaoyu Coal Mine has a stable occurrence with little change. The thickness of the coal seam is 9.2–10.2 m, with an average thickness of 9.7 m. The top coal contains a layer of dirt band. The coal cutting height of the shearer is 3.2 m, the coal drawing height is 6.5 m, and the mining drawing ratio is 1:2.03.</p>", "<p id=\"Par56\">KZT12 intrinsically safe coal gangue identification detector is installed above the rear scraper conveyor, with the detection direction facing the coal flow, as shown in Fig. ##FIG##9##10##B.</p>", "<p id=\"Par57\">The radiation data and filtering data (Kalman filtering method) of coal gangue monitored during coal drawing are shown in Fig. ##FIG##12##13##. For the convenience of analysis, the occurrence of dirt bands in the top coal is compared with the data detected at the caving-opening.</p>", "<p id=\"Par58\">As depicted in Fig. ##FIG##12##13## that during top coal drawing, the detector has a relatively sensitive response to the occurrence and content of dirt band/roof rock. The process of top coal drawing lasts for 200 s, and there is a radiation wave peak. The radiation data detected has obvious periodicity, which can be divided into three stages: pure coal stage, gangue mixing stage and immediate roof rock mixing stage. In the pure coal stage, there is pure coal near the caving-opening. Since there is almost no radionuclide in the coal, the radiation data detected by the detector is at the same level with the background, that is, the radiation intensity fluctuates within 42–78 cps within 0–70 s after the caving-opening is opened, and there is no obvious upward or downward trend.</p>", "<p id=\"Par59\">With the process of top coal drawing, the gangue containing in the top coal will reach the coal drawing opening. At this stage, the data detected by the detector will show an upward trend. With the full discharge of the gangue, the data detected by the detector will gradually decline, that is, in 70–95 s, the radiation intensity will first rise from 55 to 98cps, and then gradually decline; In the following 95–160 s, the coal drawing is a pure coal stage, and the radiation intensity is relatively stable; As the top coal is exhausted, after 160 s, the immediate roof rock starts to enter the coal drawing opening. At this stage, combined with the coal drawing basis of “see the gangue then close the caving outlet”, the coal drawing operation is terminated. The radiation curve gradually decreases and returns to the background level as the coal drawing opening is closed.</p>", "<title>Application test of gangue identification system in thick seam with complex structure</title>", "<p id=\"Par60\">The average thickness of 8205 LTCC coal seam in Tashan Coal Mine is 15.09 m, the mining height is 3.8 m, the caving height is 11.29 m, and the mining caving ratio is 1:2.97. The cycle progress is 0.8 m, and the coal drawing step is 0.8 m. The coal seam contains 4–8 layers of dirt bands, with an average of 6 layers. The thickness of a single layer varies from 0.23 to 0.65 m. The lithology of the dirt bands is magmatic rock, sandy mudstone, mudstone, carbonaceous mudstone, and kaolinite. Most of the upper part of the coal seam is metamorphosed and silicified due to lamprophyre intrusion.</p>", "<p id=\"Par61\">KZT12 intrinsically safe coal gangue identification detector is installed under the tail beam of the support, with the detection direction facing the coal chute, as shown in Fig. ##FIG##9##10##C.</p>", "<p id=\"Par62\">The radiation data and filtering data (Kalman filtering method) of coal gangue monitored during coal drawing are shown in Fig. ##FIG##13##14##. For the convenience of analysis, the occurrence of dirt bands in the top coal is compared with the data detected at the coal drawing hole.</p>", "<p id=\"Par63\">As depicted in Fig. ##FIG##13##14## that during coal drawing, the detector has a relatively sensitive response to the occurrence and content of dirt band/roof gangue. The process of coal drawing at the caving-opening lasts for 220 s, and there are four radiation peaks. Due to the existence of dirt bands, the radiation data detected has obvious periodicity, which can be divided into three stages: pure coal stage, dirt band mixing stage and immediate roof mixing stage. The dirt band mixing stage can be subdivided according to the dirt band position. At the same time, the radiation performance of different lithology is different. The numbers of ① in the figure are the numbers of the dirt bands in the top coal, which are sorted from bottom to top. In the pure coal stage, there is pure coal near the coal cave outlet. Since there is almost no radionuclide in the coal, the radiation data detected by the detector at this stage is equal to the background, that is, the radiation intensity fluctuates within the range of 40–87 cps within 0–35 s after the coal cave outlet is opened, and there is no obvious upward or downward trend.</p>", "<p id=\"Par64\">As the top coal caving process enters the mixed stage of coal and gangue, and the dirt band in the 1st layer of top coal reaches the coal caving outlet first. At this time, the data detected by the detector shows an upward trend. With the exhaustion of the 1st layer of dirt band, the data detected by the detector gradually decreases. That is, the radiation intensity first increases from 55 to 138 CPS and then gradually decreases within 35–58 s. Then the dirt band in layer ② arrives at the coal caving outlet, and the data detected by the detector shows an upward trend again. Because the distance between the dirt band layer ② and the dirt band layer ③ is relatively close, the time periods for these two layers to enter the top-coal caving-opening. Therefore, there is no obvious feature in the entry sequence on the radiation intensity curve, that is, for the 63–107 s period, the radiation intensity curve first slowly rises from 71 to 143 cps during the 63–95 s period,, and then decreases from 143 to 79 cps during the 95–107 s period, It can be found that the main reason for the long duration of the rising section during the mixing of the whole dirt band layer ② and the dirt band layer ③ is that the mixing amount of the dirt band layer ① is gradually decreasing, and the mixing amount of the dirt band layer ③ is gradually increasing as the mixing amount of the dirt band layer ② is gradually decreasing; The main reason for the short duration in the descending section is that the dirt band layers ①, ② and ③ are decreasing, and even the dirt band layers ① is no longer discharged through the coal caving outlet; At about 110s, the dirt band ④ is mixed in. During 110–137 s, during 110–115, the radiation intensity fluctuates within the range of 85cps to 107cps without obvious upward or downward trend, It is analyzed that at this time, the mixing amount of the dirt band layer ② and ③ decreases gradually, while the mixing amount of the dirt band layer ④ increases gradually, resulting in no obvious upward or downward trend of the radiation intensity; During 115–122 s, the radiation intensity curve rises from 66 to 134 cps. The analysis is that at this time, the mixing amount of the dirt band ④ increases to the maximum, while the mixing amount of the dirt band ② and ③ is very small; Then, during 122–138 s, the radiation intensity curve decreased from 136 to 71 cps, which was analyzed as a result of the gradual decrease in the mixing amount of the dirt band ④.</p>", "<p id=\"Par65\">From 140 s to the end of 220 s coal drawing, the radiation intensity curve shows a slow upward trend. According to the comprehensive histogram of the working face and the histogram obtained by drilling before the experiment, The coal in the range from the dirt band layer ④ in the top coal to the top plate is mostly metamorphosed and silicified due to the intrusion of lamprophyre. This part of coal seam is defined as the dirt band layer ⑤, and this coal drawing stage is defined as the mixing stage of lamprophyre/ immediate roof, Therefore, with the mixing of this part of coal seams, the radiation intensity curve has an obvious upward trend at this time, and the recovery value of this part of coal is already small, so coal drawing will stop.</p>", "<p id=\"Par66\">In the mixing stage of the dirt bands, due to the different location, thickness and lithology of the dirt bands in each layer, they show different radiation characteristics during the releasing process: ① The closer the position of the dirt band is to the caving-opening, the earlier the time of the caving-opening, which follows the time sequence of the top caving-opening; ② When the thickness of the dirt band is relatively large, the radiation monitoring curve corresponding to the mixing of the dirt band in the process of releasing is obviously rising and then slowly falling after maintaining the intensity for a certain time, That is, the dirt band is thick and will last for a period of time after being mixed into the coal drawing flow. When the dirt band is thin, the radiation monitoring curve corresponding to it shows a fluctuation phenomenon of one or more sections rising and then falling immediately from the mixing of the dirt band to the end during the releasing process, and the wave crest height of each section of the fluctuation has a certain difference, That is, the gangue layer is thin, which can not be continuously mixed into the coal drawing flow, and there is intermittent phenomenon, that is, the influence of different thickness of gangue on the radiation monitoring data is different, that is, the peak width of the monitoring curve is different; ③ The radiation intensity of the gangue with different lithology is different, so the influence on the radiation monitoring data is different during the mixing process. That is, the peak height of the monitoring curve is different. ④Two layers of dirt bands near each other will be mixed and overlapped.</p>", "<p id=\"Par67\">In conclusion, the KZT12 intrinsically safe coal gangue identification detector has a high sensitivity to the amount of gangue discharged at the coal caving outlet, and can determine the gangue mixing ratio at the top coal caving process in real time.</p>" ]

[ "<title>Discussion</title>", "<p id=\"Par68\">LTCC mining technology is one of the main technologies for safe and efficient mining of extra-thick coal seams in China. At present, LTCC mining technology has been applied in most of the extra-thick coal seams in China, and has made breakthroughs and development in theory and key engineering application. However, the top-coal caving process in Full-mechanized top-coal caving still relies on manual control according to the principle of \" close caveing-opening when see gangue\", which is an artificial process. It is difficult to avoid the situation that resources are wasted and the quality of coal is affected due to over-or less-drowning in the process of top-coal caving. Moreover, the number of top-coal caving supports in the fully-mechanized top-coal caving face is large, the working environment of the top-coal caving procedure is poor, and the labor intensity and working efficiency of the manual control of the top-coal caving-opening are high.</p>", "<p id=\"Par69\">In recent years, the intellectualization of coal mines has been developed rapidly. Taking intelligent fully mechanized mining as the technical core, it has improved the intelligent level of coal mines and provided technical support for the high-quality development of the coal industry. The development of intelligent fully mechanized mining technology has promoted the research on intelligent technology of LTCC. Because coal gangue identification is the key technology to realize automatic top coal caving and LTCC intelligent mining, domestic experts, scholars and scientific research institutions have carried out a lot of research work for this purpose and made gratifying progress. However, the instability of coal seam thickness and the existence of dust, dust falling water mist, brightness, space noise and other complex environments in the coal drawing space have brought great difficulties to the accuracy and reliability of coal gangue identification, which is also the reason why continue research have been carried out. For more than 10 years, the has successively carried out the application of near-infrared ray, dual energy γ Ray and nature γ based on the analysis and comparison of its reliability and feasibility, the coal gangue recognition based on low level radiation γ ray is proposed. The method of ray coal gangue recognition has been studied theoretically and experimentally, and tested and analyzed on the spot in Lilou Coal Mine, Tashan Coal Mine and Xiaoyu Coal Mine. The preliminary application shows that the proposed method and equipment for identifying coal gangue based on natural gamma ray can fully meet the requirements of real-time monitoring of coal gangue mixing degree at the top coal drawing process. Of course, the prerequisite for the application of coal gangue recognition method based on low level radiation γ ray is that the immediate roof rock needs to contain certain radioactive elements. According to the research in section ‘Radiation characteristics of coal and rock strata in typical thick coal seam mining areas in China’ of this paper, it has been proved that radioactive elements exist in the immediate roof of most thick coal seams in China.</p>", "<p id=\"Par70\">Based on the complex conditions of the top coal caving opening in the LTCC mining face, in order to fully apply this technology to the top coal caving face, the author believes that the later research should focus on the following two points: (1) Installation position of coal gangue identification detector. Because of the different positions of detectors, first of all, the detection range of the detectors to the coal drawing mouth is different, and because of the influence of the floor and the gangue in the goaf, the noise source of the detectors also needs to be analyzed. Second, the monitoring range of the top coal is different due to the different positions of the detectors, which relates to the number of detectors installed and the way they cooperate with the electro-hydraulic control system. (2) The shape and size of the detector. In view of the complexity of the working environment of the coal chute and the space limitation, in order to achieve a good detection effect, in addition to the installation position, the size of the detector should also meet the detection requirements as far as possible. At the same time, in order to meet the safety requirements, in combination with the special structure of the hydraulic support, it is necessary to customize the detector with a special shape suitable for the coal chute space, which brings research challenges to the design and fabrication of detectors and the acquisition of signals. The team has carried out research on the above two points and believes that in the near future, better results will be displayed.</p>" ]

[ "<title>Conclusions</title>", "<p id=\"Par71\">\n<list list-type=\"order\"><list-item><p id=\"Par72\">Among the common gangue and roof rocks, the radioactive intensity of carbonaceous mudstone is the highest, followed by sandy mudstone and kaolin mudstone, while the radioactive intensity of lamprophyre is relatively minimum. When the gangue and the immediate roof lithology are the same, the radioactive intensity is similar. Coal has the lowest radioactivity. Through the analysis of the radiation intensity of the coal seam and roof in the typical thick seam LTCC mining face, it is feasible to use the natural γ ray method to identify the coal and gangue.</p></list-item><list-item><p id=\"Par73\">Based on the radiation characteristics of natural γ rays in the process of releasing coal and gangue in extra thick coal seams and the change law of radiation intensity value when the immediate roof is mixed γ value of radiation intensity is taken as the identification parameter, and the identification method of coal and gangue in LTCC mining of extra thick coal seam is proposed.</p></list-item><list-item><p id=\"Par74\">The KZT12 intrinsically safe coal gangue identification detector developed for mining has the ability to monitor the different mixing degrees of coal and gangue mixture in real time, and has good applicability to the identification of coal and gangue in the process of fully mechanized caving mining of extremely thick coal seams with complex structures. Because of nature γ ray has a strong penetrability, the detector is less affected by water mist, dust, light, etc. during mining, and has strong environmental adaptability, and can realize volume monitoring.</p></list-item><list-item><p id=\"Par75\">The automatic recognition method and system of coal and gangue in extra thick coal seams have been formed. The field test and analysis of coal gangue recognition have been carried out in the fully mechanized caving mining working faces of Lilou Coal Mine, Xiaoyu Coal Mine and Tashan Coal Mine, and remarkable technical effects have been achieved, laying a foundation for further research and application of intelligent fully mechanized caving mining in extra thick coal seams under different conditions.</p></list-item></list></p>" ]

[ "<p id=\"Par1\">To address the technical limitations of automatic coal and gangue detection technology in fully mechanized top coal caving mining operations, the low radiation level radioactivity measurement method is utilized to assess the degree of coal-gangue mixture in top coal caving process. This approach is based on the distinguishing radiation characteristics of natural γ-rays between coal and gangue. This study analyzed the distribution characteristics of natural γ-rays in coal and rock layers of thick coal seams and the applicability of this method, introduced the basic principle of coal-gangue detection technology based on natural γ-ray, developed the test system about automatic coal-gangue detection, studied the radiation characteristics of coal and gangue, proposed determination model of the coal-gangue mixed degree, combined with the time sequence characteristics of the top coal’s releasing flow and the energy spectrum characteristics of different layers of rock, realized the precise coal-gangue detection technology in complex structure thick coal seam with multiple gangue. Field tests were conducted in Lilou, Xiaoyu and Tashan Coal Mine. The test results were well corroborated with the research results and achieved the expected results, which laid the foundation for the field application of intelligent coal mining.</p>", "<title>Subject terms</title>" ]

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[ "<title>Acknowledgements</title>", "<p>We would like to thank all the authors of this article for their work and the relevant leaders and technicians of Tashan, Xiaoyu and Lilou Coal Mines for their support and help.</p>", "<title>Author contributions</title>", "<p>C.L. put forward the research ideas and methods, N.Z. carried out the theoretical analysis and experiment. All authors wrote and reviewed the manuscript. All authors listed have made a substantial, direct, and intellectual contribution to the work and approved it for publication.</p>", "<title>Funding</title>", "<p>Financial support for this work, provided by the Natural Science Foundation of China (Grant No. 52174137), China Postdoctoral Science Foundation (Grant Nos. 2020T130697, 2019M661994), and State Key Laboratory of Mining Response and Disaster Prevention and Control in Deep Coal Mines Open Foundation (Grant No. SKLMRDPC20KF13), are gratefully acknowledged.</p>", "<title>Data availability</title>", "<p>The original contributions presented in the study are included in this article, further inquiries can be directed to the corresponding author.</p>", "<title>Competing interests</title>", "<p id=\"Par76\">The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.</p>" ]

[ "<fig id=\"Fig1\"><label>Figure 1</label><caption><p>Comparison chart of sedimentary radioactive content.</p></caption></fig>", "<fig id=\"Fig2\"><label>Figure 2</label><caption><p>Immediate roof lithologic distribution of full-mechanized caving mining face.</p></caption></fig>", "<fig id=\"Fig3\"><label>Figure 3</label><caption><p>Radiation characteristics of typical thick coal measures.</p></caption></fig>", "<fig id=\"Fig4\"><label>Figure 4</label><caption><p>Schematic diagram of natural γ-ray coal and gangue identification.</p></caption></fig>", "<fig id=\"Fig5\"><label>Figure 5</label><caption><p>Detector parameters adjustment chamber placement.</p></caption></fig>", "<fig id=\"Fig6\"><label>Figure 6</label><caption><p>Comparison of detector counting under different supply voltages.</p></caption></fig>", "<fig id=\"Fig7\"><label>Figure 7</label><caption><p>Detector background CPS values under different thresholds.</p></caption></fig>", "<fig id=\"Fig8\"><label>Figure 8</label><caption><p>Radiation difference under different environmental conditions above and below surface.</p></caption></fig>", "<fig id=\"Fig9\"><label>Figure 9</label><caption><p>Radiation intensity test of gangue in coal mine environment.</p></caption></fig>", "<fig id=\"Fig10\"><label>Figure 10</label><caption><p>Ratio of net count to background value.</p></caption></fig>", "<fig id=\"Fig11\"><label>Figure 11</label><caption><p>(<bold>A</bold>) Detector is installed under the support tail beam with the detection direction facing the coal chute. (<bold>B</bold>) Detector is installed above the rear scraper conveyor with the detection direction facing the coal flow. (<bold>C</bold>) Detector is installed under the support tail beam with the detection direction facing the coal chute.</p></caption></fig>", "<fig id=\"Fig12\"><label>Figure 12</label><caption><p>Radiation curve of top coal drawing process.</p></caption></fig>", "<fig id=\"Fig13\"><label>Figure 13</label><caption><p>Radiation curve of top coal drawing process and its comparison with bar chart.</p></caption></fig>", "<fig id=\"Fig14\"><label>Figure 14</label><caption><p>Radiation curve of top-coal drawing process and its comparison with bar chart.</p></caption></fig>" ]

[ "<table-wrap id=\"Tab1\"><label>Table 1</label><caption><p>The characteristics of three kinds of natural radioactive substances.</p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th align=\"left\">Nuclide</th><th align=\"left\">Number of nuclide decays/g s</th><th align=\"left\">Number of photons produced by each decay of nuclide</th><th align=\"left\">Average photon energy/MeV</th></tr></thead><tbody><tr><td align=\"left\">U</td><td align=\"left\">1.23 × 10⁴</td><td char=\".\" align=\"char\">2.24</td><td char=\".\" align=\"char\">0.80</td></tr><tr><td align=\"left\">Th</td><td align=\"left\">4.02 × 10³</td><td char=\".\" align=\"char\">2.51</td><td char=\".\" align=\"char\">0.93</td></tr><tr><td align=\"left\">K</td><td align=\"left\">31.3</td><td char=\".\" align=\"char\">0.11</td><td char=\".\" align=\"char\">1.46</td></tr></tbody></table></table-wrap>", "<table-wrap id=\"Tab2\"><label>Table 2</label><caption><p>Technical index of coal and gangue identification system.</p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th align=\"left\">Detector range</th><th align=\"left\">Background number</th><th align=\"left\">Sensitivity</th><th align=\"left\">Energy threshold</th><th align=\"left\">Relative error</th><th align=\"left\">Minimum sampling period</th></tr></thead><tbody><tr><td align=\"left\">0.001–100uGy/h</td><td align=\"left\"> ≥ 100cps</td><td align=\"left\">1uGy/h ≥ 1000cps</td><td align=\"left\">35 keV</td><td align=\"left\">≤  ± 5%</td><td align=\"left\">50 ms</td></tr></tbody></table></table-wrap>" ]

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[ "<title>Introduction</title>", "<p id=\"Par2\">For many years, laser surgery has been an accepted tool in various surgical fields^{##REF##21500720##1##}. The use of lasers in hospitals is also increasing^{##UREF##0##2##}. The advantage of lasers is that they can achieve results similar to those of conventional surgery while being minimally invasive^{##REF##23729304##3##,##REF##12102411##4##}. At the same time, laser surgery usually has a high healing potential, with less post-operative inflammation and swelling^{##REF##6239644##5##}. The coagulation effect of the laser radiation improves visibility by coagulating small blood vessels^{##REF##565230##6##}. While these benefits are widely accepted, the practical advantage is still debated. For example, Seifi and Matini^{##REF##29263776##7##} showed in a small meta-study that there was no benefit from cutting soft dental tissue with a laser. They conclude: “Introducing an appropriate laser with suitable wavelength, input power and other properties for mentioned indications needs more research and clinical trials”.</p>", "<p id=\"Par3\">This shows that there is still a lot of research to be done. Even in recent studies, the types of lasers used are still under investigation^{##REF##34409842##8##}. For a more general view, it is important to look more closely at the laser-material interaction to differentiate the regimes of material ablation. Boulnois^{##UREF##1##9##} distinguishes between vaporisation, photoablation and photodisruption. These mechanisms play an important role in the efficiency of laser surgery. For example, Werner et al.^{##UREF##2##10##} showed that a laser could achieve more than ten times the ablation rate of the frequency doubled laser at 532 nm.</p>", "<p id=\"Par4\">Sánchez et al.^{##UREF##3##11##} compared a laser with an Er,CR:YSGG laser on gingiva. While the laser cut faster and without bleeding, the Er,CR:YSGG allowed a faster healing time. A slower healing ability of the laser was also found in later studies^{##REF##33105594##12##}. However, a meta-analysis from 2019^{##REF##31624426##13##} shows that a subgroup analysis for the type of laser cannot be done because the amount of data is too small. Furthermore, Protásio et al.^{##REF##31624426##13##} conclude “... that labial frenectomies performed with high-intensity surgical lasers are faster and offer a better prognosis in terms of pain and discomfort during speech and chewing than those performed with conventional scalpels”, if publication bias is not taken into account. A later randomized double-blinded and controlled pediatric clinical study in 2021 by Fioravanti et al.^{##REF##34204017##14##} with obstructive sleep apnea syndrome (OSAS) was performed. There, a milli second pulsed laser with a wavelength of 980 nm strongly decreased the OSAS compared to the control group contrasting the result from the review by Protásio et al.^{##REF##31624426##13##}. Also, a case report in 2023 showed a good outcome for healing of osteonecrosis of the jaw^{##UREF##4##15##} by laser treatment with a 980 nm laser. Moreover, a review from Lesniewski et al.^{##REF##35904936##16##} reveals that “... diode lasers and LEDs are equally effective tools for the phototherapy in periodontology and oral surgery”. Therefore, currently no firm conclusion can be drawn regarding whether laser surgery and which laser type result in a better surgery performance in the oral cavity. However, lasers in the near infrared range seem to be advantageous^{##REF##35904936##16##}.</p>", "<p id=\"Par5\">While this was an example of labial frenectomies, one of the main drawbacks of some studies is the statistics with too few samples and examined parameters. For example, the highly cited study by Cercadillo-Ibarguren et al.^{##UREF##5##17##} varied the four parameters laser power, laser type, air spray and pulsed laser operation with 117 samples. It was concluded that Er,CR:YSGG lasers performed well, while diode and lasers did not perform well. However, other authors state that a laser rarely causes any unwanted tissue damage when used correctly^{##UREF##6##18##–##REF##19715451##20##}.</p>", "<p id=\"Par6\">While this is obviously a contradiction, the important question is: why are there differences in the results? To investigate this question, various parameters are collected from the literature. A study by El-Sherif and King^{##REF##14505197##21##} for a laser with a wavelength of 2 m showed that the pulsed mode results in less damage of soft tissue than the CW mode of the laser. In addition, the heat affected zone ranges from 120 to 160 m for the pulsed laser mode and 400–800 m for the CW mode^{##REF##14505197##21##}. For bone tissue, the heat affected zone is for a laser only 6 m in comparison^{##REF##12417980##22##}. Similar results were found for an Er:YAG laser with a wavelength of 2.94 m. While the heat affected zone for the q-switched laser was 5–10 m, it increased to 10–50 m for the spiking mode with longer pulse durations^{##REF##2761327##23##}. Furthermore, the results did not change for different tissue types including soft and hard tissue^{##REF##2761327##23##}. It can also be concluded from the results of Krapchev et al.^{##UREF##8##24##} that the pulse frequency and duration should be below the thermal relaxation time to prevent unwanted tissue damage. This leads to the first two potential influencing parameters: wavelength and pulse duration.</p>", "<p id=\"Par7\">Another factor to consider is cooling of the tissue. Ivanenko et al.^{##REF##12417980##22##} were able to show that small amounts of water prevented carbonisation, while larger amounts of water showed no further improvement. This is due to the fact that the water evaporates when too much heat is present, leading to a cooling effect^{##REF##2345475##25##}. More specifically, the additional water creates a temperature gradient from the water to the tissue, causing the heat to flow to the water rather than the surrounding tissue^{##REF##8915947##26##}. For air cooling, the results are contradictory: while Ivanenko and Hering^{##UREF##9##27##} claim that pure gas cooling leads to more thermal damage, Afilal^{##UREF##10##28##} claims the opposite. This leads to the next two potential influencing parameters: water cooling and air cooling.</p>", "<p id=\"Par8\">Finally, the laser parameters should be considered. While the laser power is an obvious influencing parameter, the effect of scan speed could be shown by Afilal^{##UREF##10##28##}. He was able to show that carbonisation could be prevented by using the correct scanning parameters. In another case, the temperature increase could be limited to 30 K instead of 400 K by using a scanning technology^{##UREF##11##29##}. It is also known from the field of laser material processing (e.g. welding, cutting and additive manufacturing with lasers) that the line energy is an important parameter. This leads to the final potential influencing parameters: laser power, laser scan speed and laser line energy.</p>", "<p id=\"Par9\">In total, seven potential influencing parameters that affect the laser surgery process can be identified from the literature: wavelength, pulse duration, water cooling, air cooling, laser power, laser scan speed and laser line energy. To quantify the quality of the cut, the previously published scoring system^{##REF##36042339##30##} was used. With the help of the scoring system, all previously mentioned parameters, except wavelength and type of cooling, will be investigated in this study because studying the influence of wavelength would require too many different lasers and water cooling is known to result in a significant improvement in cut quality at the cost of increased complexity^{##REF##33408994##31##,##UREF##12##32##} and longer cutting times. In addition, the number of breaks between laser scans and the effect of an exhaust system are studied. The number of breaks is used to study the effect of heat dissipation on the cut. The exhaust parameter is taken into account as it may be important to remove the plumes from the laser surgery process. All these parameters are compared to the effect of the inter animal variation as a benchmark for their importance.</p>" ]

[ "<title>Methods</title>", "<p id=\"Par10\">The method section consists of four sections. In the first two sections, experimental procedure including the origin of the tissue and the statistical analysis is explained. In the following two sections, the varied parameters are discussed in detail.</p>", "<title>Experimental procedure</title>", "<p id=\"Par11\">\n\n</p>", "<p id=\"Par12\">In this section, the set-up and the experimental parameters that are not altered in this study are explained. The experimental setup is shown in Fig. ##FIG##0##1##. The used laser is from MICROSTORM (FEHA LasterTec GmbH, Germany) with a wavelength of 10.6 m. It is a diffusion cooled -laser with an acousto-optical modulator. The modulator is driven by an external pulse generator (DG1032Z, Rigol Technologies).</p>", "<p id=\"Par13\">The experimental parameters are divided into constant and variable parameters. In this section only the constant parameters are explained and summarized in Table ##TAB##0##1##. The temporal shape of the laser is rectangular. The focal distance is 13.7 cm from the lens to the tissue sample. The focal diameter is 258 m. Furthermore, the deflection of the laser beam is realised by a scan head (Scanlab GmbH). This allows the laser beam to be moved at a controlled speed of the sample. The temporal laser profile is set to be rectangular to ensure that the same light intensity is always applied to the tissue surface. A total of 1 cm long incisions were cut. There are 20 scans for each incision with a scan time of 10 ms. This leads to a cutting depth of up to 4 mm. If breaks are made between scans, the break time is 2 s. If the number of breaks is one, a break is done after 10 scans, if it is 3, a break is done after each 5 scans and if the number is 19 a break is done after each scan. The laser comes perpendicular to the surface of the table, However in practice, the laser may not be entirely perpendicular to the tissue as it is not flat. This can be seen in Fig. ##FIG##0##1##. Hence, the incident angle of the laser is seen as to be approximately zero degree.</p>", "<p id=\"Par14\">For all experiments, bisected pork tissues of food quality were purchased from the local butcher. Therefore, an ethical proposal for animal experimentation is not required. The samples used were 1 cm thick pieces of pork from the topside. The laser power of 235 W for 0.2 s produced the highest energy input observed in all experiments − 47 <italic>J</italic>. As a result, the sample size was always sufficiently large compared to the heat affected zone so that the size of the tissue had no effect on the heat dissipation.</p>", "<p id=\"Par15\">Table ##TAB##1##2## provides an overview about the number of samples. There are five repetitions for each animal sample and parameter setting for the non-laser parameters, resulting in a total of 30 data points (n = 30) per parameter setting. The experiments for the laser parameters include 3 different animal samples with 10 repetitions each, so there are 30 data points (n = 30) per parameter setting.</p>", "<title>Statistical analysis</title>", "<title>Experiment A</title>", "<p id=\"Par16\">For the quantitative analysis all cuts scored on the scoring system presented in a previous paper^{##REF##36042339##30##} from our group (especially Fig. ##FIG##1##2## provides an good overview). In short, by looking at tissue damage at the rim or the cutting front of the cut, a score from 1 to 5 is given, where 5 is the best possible score. The scoring is performed based on the presence/amount of carbonization and the colour of the tissue. Figure ##FIG##1##2## shows examplary scores for different cuts. It should be noted that a score of 3–4 already denotes a pretty good cut in comparison to cuts from actual procedures such as in Fig. ##FIG##3##4##b from Vanderhem et al.^{##REF##15598583##33##}. The cut shown there would be scored as 2. As it could be shown that the scoring of the cutting front is more reliable^{##REF##36042339##30##}, the complete analysis in the study is based on the scores of the cutting front.</p>", "<p id=\"Par17\">For each data set, the effect of the variable is presented and the samples are compared for significant difference. All commands are taken from SciPy^{##REF##32015543##34##} and the names in parentheses indicate the corresponding SciPy command. The difference between the samples is tested with the Wilcoxon–Mann–Whitney-Test (“scipy.stats.mannwhitneyu”) as the scoring is a ordinal scale. As multiple comparisons are performed, the significance levels are reduced to 0.01 (*), 0.001 (**), 0.0001 (***) and 0.00001 (****). Afterwards, an analysis of variance (ANOVA) is performed with the help of the statsmodel framework^{##UREF##13##35##}.</p>", "<title>Influence non-laser parameters</title>", "<p id=\"Par18\">\n\n</p>", "<p id=\"Par19\">The aim of this part is to evaluate the influencing parameters considered relevant in the literature, except the laser parameters. An overview of the varied parameters is given in Table ##TAB##2##3##. In order to reduce the number of experiments required, two sets of experiments have to be performed: In experiment A, the effect of air cooling, number of breaks and pulse duration is investigated. In experiment B, the effect of exhausting the plumes is investigated. In both data sets, each parameter combination is repeated five times on six different animals. To conduct the analysis, a benchmark is required to determine the relevance of a given parameter. This can be established by examining the variance in the effects of cutting different animals as for a practical applications, the effect of cutting different animals has to be lower than the effect of a given parameter. By attributing how much variance of the different animals is affecting the scoring, a benchmark is generated that takes into account the tissue’s storage and pre-treatment.</p>", "<p id=\"Par20\">The experiments in experiment A are performed with two laser powers (<italic>P</italic>):  W (irradiance = 160 ) and P = 129 W (irradiance = 247 ), in order to evaluate all non-laser parameters for a more optimal case (83 W) and a case with more tissue damage (129 W). For the first intensity, it is tested if the parameters can worsen the cut and in the second case, if the tested parameters can lead to optimal laser cuts. For all experiments, the scan speed is kept constant at 1 , resulting in an illumination time of 10 ms. Furthermore, for both laser powers a separate ANOVA analysis is performed.</p>", "<p id=\"Par21\">For experiment B, only the laser power of 129 W and the pulse duration of 964 ns are used. The parameters number of breaks and air cooling are varied as in A. The number of parameters is reduced to decrease the number of experiments required. The pulse duration is excluded as it is a laser parameter, and the laser power of 129 W is chosen to evaluate if the exhaustion of the plumes can lead to an improvement of a non-optimal laser cut.</p>", "<title>Influence laser parameters</title>", "<p id=\"Par22\">The aim of this part is to evaluate the influencing laser parameters. An overview of the parameters is provided in Table ##TAB##3##4##. For parameters, the line energies () are chosen to be: 83 , 172  and 235 . The line energy is calculated as followed:where is the scan speed. A total of 270 cuts are evaluated for three animals: 90 for each line energy. For each line energy, 3 laser powers/scanning speeds are selected. Thus, each parameter combination is performed ten times.</p>" ]

[ "<title>Results</title>", "<p id=\"Par23\">Each section of the results is divided into two parts. The first part presents the means and compares the significance of the data sets. The second part presents the results of the ANOVA analysis.</p>", "<title>Influence non-laser parameters</title>", "<title>Experiment A</title>", "<p id=\"Par24\">\n\n</p>", "<p id=\"Par25\">Figures ##FIG##2##3## and ##FIG##3##4## show the average effect and the different parameters with standard deviation and significance levels. For the laser power of 83 W, the cutting quality is overall higher as the cuts with a laser power of 129 W. This is due to the fact that the laser power of the cuts with 129 W is chosen to be too high to see the possible improvements by the different parameters. In both cases, a higher number of breaks reduces the unwanted tissue damage. Also for both pump powers, air cooling and different pulse durations show no or little significance and variation.</p>", "<p id=\"Par26\">For 83 W, a shorter pulse duration seems to reduce the cutting quality. However, the significance is small. The air pressure seems to reduce the cutting quality, but it is not a significant influence. The already good results could be improved by giving the laser more breaks.</p>", "<p id=\"Par27\">For 129 W, there is no significant effect of pulse duration. As for the 83 W, the air pressure seems to reduce the cutting quality a little. This effect is slightly significant. As the laser power was too high, the increase in the number of breaks shows a strong increase in the cutting quality. This could be due to the fact that the heat has more time to diffuse into the surrounding tissue, resulting in less heat accumulation. As more cuts reach five points, even the coagulated tissue is decreased.</p>", "<p id=\"Par28\">Tables ##TAB##4##5## and ##TAB##5##6## show the results of the ANOVA for a laser power of 83 and 129 W, respectively. Overall, the results are similar to the previous analysis. The effect of breaks is comparably large and highly significant. In addition, the effect of inter-animal variation is highly significant and can explain more of the variance than pulse duration and air cooling. Therefore, in addition to the conflicting significance, the effect of these two parameters can be discarded as they explain less variance. It should also be noted that and especially the adjusted is relatively low. Thus, most of the variance cannot be explained.</p>", "<p id=\"Par29\">For 83 W, air cooling shows no significant effect and the explainable variance is low. While the pulse duration shows a significant effect, the explainable variance is lower than for the inter-animal variation. Thus, the normal inter-animal variation masks the effect of both parameters. It should also be noted that the adjusted is only 0.38.</p>", "<p id=\"Par30\">For 129 W, the effect of the air cooling and the pulse duration are the opposite than for 83 W. The pulse duration shows no significant effect and the explainable variance is low. While air cooling shows a significant effect, the explainable variance is comparable to the inter-animal variation. For this case, the adjusted is 0.45.</p>", "<p id=\"Par31\">In summary, it can be concluded that, contrary to the literature, pulse duration has no or a contradictory effect on cutting quality. Furthermore, its effect on the final result is less than the inter-animal variation. Therefore, its effect can be ignored. Air cooling also has little or no effect on cutting quality. Therefore, air cooling should not be used, also in the interest of reducing the complexity of the set-up. The number of breaks between cuts shows a strong effect and significantly improves the results. In addition, the number of breaks leads to much stronger effects than the inter-animal variation. This parameter should therefore be taken into account. For practical application, however, there is a conflict of objectives. On the one hand, tissue damage should be low. This is favoured by a high number of breaks. On the other hand, the operation should be fast. This is favoured by a low number of breaks.</p>", "<title>Experiment B</title>", "<p id=\"Par32\">\n\n</p>", "<p id=\"Par33\">Figure ##FIG##4##5## shows the average effect and the different parameters with standard deviation and significance levels. The results of the breaks and the cooling by air are almost identical to the results for 129 W in experiment A. Hence, the results of this study are reproducible. The plume exhaust parameter shows no significant effect. These results are also supported by the ANOVA in Table ##TAB##6##7##. The significance of the number of breaks is high and it explains a large amount of variance. At the same time, exhaust and air cooling have a minimal effect on the explainable variance and the significance level is much lower than for the number of breaks. Therefore, both air cooling and exhaust do not play a major role. Both parameters can be adjusted according to other requirements of laser surgery.</p>", "<title>Influence laser parameters</title>", "<p id=\"Par34\">\n\n</p>", "<p id=\"Par35\">Figure ##FIG##5##6## shows the effect of line energy and laser power or scan speed, respectively. It can be seen that the line energy has a strong influence on the resulting cut quality. The laser line energy at smallest value 83 , gives the best cutting quality. The scan speed of 1  gives the best results regardless of the laser power and line energy. Thus, it appears that for a given line energy, the scan speed has a more significant effect than the laser power. For the comparable optimal line energy of 83 , the scan speed has only a small effect. For less optimal parameters, a non-optimal scan speed can further reduce the quality of the cut. The strongest effect of the scan speed occurs when the scan speed is low, whereas at higher scan speeds, the influence of the scan speed becomes insignificant. In summary, line energy is the most important parameter and scan speeds at 1  or higher are preferred.</p>", "<p id=\"Par36\">These overall results are supported by the ANOVA analysis which is shown in Table ##TAB##7##8##: all two laser parameters are significant and the line energy explains most of the variance. The explained variance is higher than for any of the non-laser parameters. As both parameters are also highly significant, it can be concluded that the laser parameters are the most important parameters. The effect of laser power/scan speed is analysed again separately for each line energy to clearly show whether laser power/scan speed or line energy is the more important parameter. It should be noted that the parameters of scan speed and laser power cannot be separated as they are indirectly proportional. The importance has to be concluded from Fig. ##FIG##5##6##.</p>", "<p id=\"Par37\">Tables ##TAB##8##9##, ##TAB##9##10## and ##TAB##10##11## show the effect of laser power/scan speed for line energies of 83 , 172  and 235 . For an optimal line energy of 83  (Table ##TAB##8##9##), the effect of laser power/scanning speed is hardly significant and only 11% of the variance can be explained. This effect is supported by the low value of . Therefore, for an optimal line energy, laser power and scan speed are not important.</p>", "<p id=\"Par38\">These results change when a non-optimal line energy is used, as shown in Tables ##TAB##9##10## and ##TAB##10##11##. In this case, the effect of laser power and scan speed becomes significant and explains a higher percentage of the variance. The value increases in this case. All three effects increase when the line energy is further away from the optimal value. Nevertheless, the optimal choice of laser power or scan speed can only improve the cut quality to a certain extent, which is limited by the non-optimal line energy. In other words: the line energy determines the maximum achievable cut quality.</p>", "<title>Limitations</title>", "<p id=\"Par39\">The main limitation of this study is the fact that all experiments were performed in an ex-vivo setting. This clearly limits the generalisability of the results presented as important parameters such as time to heal cannot be assessed. However, the chosen ex-vivo setting allows the study of a large number of laser cuts and, due to the lack of perfusion, tissue damage may appear easier. Therefore, it is possible that parameters from the ex-vivo setting can be transferred to in-vivo.</p>", "<p id=\"Par40\">The second limitation is that only one type of tissue is used for all laser cuts. While this is a requirement of this study to really understand the effect of different parameters, it does limit the generalisability. It is likely that the results can be transferred to at least some types of soft tissue, but generalisation to hard tissue such as bone is not possible without further experiments. However, with the current approach demonstrated, it may be possible to find a parameter setting that allows bone cutting with a laser.</p>", "<p id=\"Par41\">A third limitation is that pig tissue is used instead of human tissue. Thus, the transferability is limited. However, the parameters tested show a much stronger effect than the inter-animal variation. Therefore, it is expected that the presented results are at least partially transferable to other species.</p>", "<p id=\"Par42\">The fourth limitation is the size of the cuts and fixed laser focus. All cuts had a length of 1 cm and a depth of up to 4 mm. This leads to two conclusions. Deeper cuts are expected to result in more tissue damage. Therefore, the optimal parameters may be different or even have to be adjusted during laser surgery or in other words the optimal laser parameters might be depth dependent. Secondly, the fixed laser focus might have the effect of limiting the cutting depth. Hence, the cutting depth could be greater with an adjusted focus position.</p>" ]

[]

[ "<title>Conclusion</title>", "<p id=\"Par43\">In this study, the parameters of pulse duration, laser power, laser scan speed, line energy, air cooling, exhaust system and number of breaks between cuts were investigated. Among these parameters, line energy is the most important. It determines the maximum cutting quality that can be achieved. If the correct line energy is known, the scan speed and the number of breaks are similarly important parameters. Interestingly, the scan speed of about 1  is optimal for all line energies tested. For the number of breaks, it can be said that the more breaks there are, the more time the heat has to dissipate and the less unwanted tissue damage there will be. However, the choice of the break numbers should consider the practical requirement. A lower number of breaks is required to speed up the laser surgery process. As the line energy has to be optimized and the scan should be around 1 , the laser power is fixed. All the other parameters (pulse duration, air cooling, exhaust system) are not relevant for the laser surgery with a .</p>", "<p id=\"Par44\">This leads to the following conclusions for the use of the laser in soft tissue. Because of the importance of line energy and scan speed, the laser is not suitable for a manual handheld device. The laser should be used with a robot or remote system to achieve high quality cuts. Nevertheless, it is possible to use a laser for tissue ablation. Therefore, the laser is still an attractive laser for some applications^{##REF##15598583##33##,##REF##6415252##36##,##REF##31188758##37##}. However, the laser shines under remote operation conditions. As the pulse duration of the laser has no effect on the cutting quality, any pulse duration can be used. However, this result cannot be transferred to near infrared laser types, as the laser is superficially absorbed, unlike e.g. Nd:YAG lasers operating at 1064 nm. However, for frequency doubled or tripled Nd:YAG lasers or Er:YAG lasers, the absorption is also comparably high which is caused by hemoglobin for the frequency doubled or tripled Nd:YAG lasers and water for the Er:YAG lasers. Hence, it makes sense to investigate if the pulse duration may be irrelevant for the cutting quality for these lasers. The small effect of the air cooling and exhaust system parameters can probably be generalised to most other laser types. Therefore, air cooling is not required and the exhaust system can be adapted to other requirements of the laser surgery system.</p>" ]

[ "<p id=\"Par1\">In recent years, the laser has become an important tool in hospitals. Laser surgery in particular has many advantages. However, there is still a lack of the understanding of the influence of the relevant parameters for laser surgery. In order to fill this gap, the parameters pulse frequency, use of an exhaustion system, air cooling, laser power, laser scan speed, laser line energy and waiting time between cuts were analysed by ANOVA using inter-animal variation as a benchmark. The quality of the cuts was quantized by a previously published scoring system. A total of 1710 cuts were performed with a laser. Of the parameters investigated, laser power and scan speed have the strongest influence. Only the right combination of these two parameters allows good results. Other effects, such as the use of pulsed or continuous wave (CW) laser operation, or air cooling, show a small or negligible influence. By modulating only the laser power and scan speed, an almost perfect cut can be achieved with a laser, regardless of the external cooling used or the laser pulse duration or repetition rate from CW to nanosecond pulses.</p>", "<title>Subject terms</title>", "<p>Open Access funding enabled and organized by Projekt DEAL.</p>" ]

[ "<title>Supplementary Information</title>", "<p>\n</p>" ]

[ "<title>Supplementary Information</title>", "<p>The online version contains supplementary material available at 10.1038/s41598-024-51449-1.</p>", "<title>Acknowledgements</title>", "<p>The authors gratefully acknowledge funding of the Erlangen Graduate School in Advanced Optical Technologies (SAOT) by the Bavarian State Ministry for Science and Art.</p>", "<title>Disclosure</title>", "<p id=\"Par45\">In the writing process during the preparation of this work, the authors used “DeepL Write” in order to improve language and readability. After using this tool, the authors reviewed and edited the content as necessary and they take full responsibility for the content of the publication.</p>", "<title>Author contributions</title>", "<p>M.H. conceptualized the research, did the data analysis, prepared the manuscript and supported D.K. in the lab work. D.K. did the experimental work did part of the data analysis. D.N., M.S. and A.G. supported the data analysis. M.R. specified the work from the medical point of view. F.S., F.K. and M.S. guided the general research strategy. All authors reviewed the manuscript.</p>", "<title>Funding</title>", "<p>Open Access funding enabled and organized by Projekt DEAL.</p>", "<title>Data availability</title>", "<p>All data analysed during this study is included in the supplementary information files.</p>", "<title>Competing interests</title>", "<p id=\"Par46\">The authors declare no competing interests.</p>" ]

[ "<fig id=\"Fig1\"><label>Figure 1</label><caption><p>Setup with tissue, the compressed air supply, the exhaustion system and the laser source.</p></caption></fig>", "<fig id=\"Fig2\"><label>Figure 2</label><caption><p>Exemplary scoring for the five different cuts (taken from our previous publication^{##REF##36042339##30##}).</p></caption></fig>", "<fig id=\"Fig3\"><label>Figure 3</label><caption><p>Effect of the parameters number of breaks, pulse duration and cooling by air for the laser power of 83 W. The errorbars show the standard deviation. The significance levels are 0.01 (*), 0.001 (**), 0.0001 (***) and 0.00001 (****) by a Wilcoxon–Mann–Whitney-test.</p></caption></fig>", "<fig id=\"Fig4\"><label>Figure 4</label><caption><p>Effect of the parameters number of breaks, pulse duration and cooling by air for the laser power of 129 W. The errorbars show the standard deviation. The significance levels are 0.01 (*), 0.001 (**), 0.0001 (***) and 0.00001 (****) by a Wilcoxon–Mann–Whitney-test.</p></caption></fig>", "<fig id=\"Fig5\"><label>Figure 5</label><caption><p>Effect of the parameters number of breaks, exhaustion and cooling by air for the laser power of 129 W. The errorbars show the standard deviation. The significance levels are 0.01 (*), 0.001 (**), 0.0001 (***) and 0.00001 (****) by a Wilcoxon–Mann–Whitney-test.</p></caption></fig>", "<fig id=\"Fig6\"><label>Figure 6</label><caption><p>Effect of the line energy and the laser power/scan speed. The errorbars show the standard deviation. The significance levels are 0.01 (*), 0.001 (**), 0.0001 (***) and 0.00001 (****).</p></caption></fig>" ]

[ "<table-wrap id=\"Tab1\"><label>Table 1</label><caption><p>Constant parameters in all experiments.</p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th align=\"left\">Constant parameter</th><th align=\"left\">Value</th></tr></thead><tbody><tr><td align=\"left\">Laser wavelength</td><td align=\"left\">10.6 m</td></tr><tr><td align=\"left\">Temporal beam shape</td><td align=\"left\">Rectangular</td></tr><tr><td align=\"left\">Focus distance</td><td align=\"left\">13.7 cm</td></tr><tr><td align=\"left\">Spot size</td><td align=\"left\">258 m</td></tr><tr><td align=\"left\">Cutting length</td><td align=\"left\">1 cm</td></tr><tr><td align=\"left\">Number of scans</td><td align=\"left\">20</td></tr><tr><td align=\"left\">Break duration</td><td align=\"left\">2 s</td></tr><tr><td align=\"left\">Incident angle of the laser</td><td align=\"left\"> 0 </td></tr></tbody></table></table-wrap>", "<table-wrap id=\"Tab2\"><label>Table 2</label><caption><p>Samples and repetions for each experiment.</p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th align=\"left\"/><th align=\"left\">Non-laser parameter</th><th align=\"left\">Laser parameters</th></tr></thead><tbody><tr><td align=\"left\">Total number of samples</td><td align=\"left\">1440</td><td align=\"left\">270</td></tr><tr><td align=\"left\">Number of animals</td><td align=\"left\">6</td><td align=\"left\">3</td></tr><tr><td align=\"left\">Number of samples per parameter</td><td align=\"left\">30</td><td align=\"left\">30</td></tr></tbody></table></table-wrap>", "<table-wrap id=\"Tab3\"><label>Table 3</label><caption><p>Potential influence parameters known from literature which are varied.</p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th align=\"left\">Parameter</th><th align=\"left\">Tested values</th><th align=\"left\">Experiment Nr. (# samples)</th></tr></thead><tbody><tr><td align=\"left\">Cooling by air</td><td align=\"left\">True, false</td><td align=\"left\">A (2 540), B (540)</td></tr><tr><td align=\"left\">Number of breaks</td><td align=\"left\">1, 3, 19</td><td align=\"left\">A (2 540), B (540)</td></tr><tr><td align=\"left\">Pulse duration</td><td align=\"left\">cw (10 ms), 100 s, 964 ns</td><td align=\"left\">A (2 540)</td></tr><tr><td align=\"left\">Exhaustion direction</td><td align=\"left\">off, 90  (perpendicular), 45 </td><td align=\"left\">B (540; 180 same as A)</td></tr></tbody></table></table-wrap>", "<table-wrap id=\"Tab4\"><label>Table 4</label><caption><p>Laser parameters which are varied.</p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th align=\"left\">Parameter</th><th align=\"left\">Tested values</th></tr></thead><tbody><tr><td align=\"left\">Line energy</td><td align=\"left\">83 , 172 , 235 </td></tr><tr><td align=\"left\">Scan speed</td><td align=\"left\">0.04–2.83 </td></tr><tr><td align=\"left\">Laser power</td><td align=\"left\">10.4–235 W</td></tr></tbody></table></table-wrap>", "<table-wrap id=\"Tab5\"><label>Table 5</label><caption><p>ANOVA analysis of the non laser parameter for a laser power of 83 W.</p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th align=\"left\">Parameter</th><th align=\"left\">df</th><th align=\"left\">Sum sq.</th><th align=\"left\">Mean sq</th><th align=\"left\">F</th><th align=\"left\">p</th><th align=\"left\"></th></tr></thead><tbody><tr><td align=\"left\">Breaks</td><td align=\"left\">2</td><td align=\"left\">26</td><td align=\"left\">12.8</td><td align=\"left\">39</td><td align=\"left\">3e-23</td><td align=\"left\">0.19</td></tr><tr><td align=\"left\">Cooling by air</td><td align=\"left\">1</td><td align=\"left\">0.98</td><td align=\"left\">0.98</td><td align=\"left\">4.5</td><td align=\"left\">0.04</td><td align=\"left\">0.01</td></tr><tr><td align=\"left\">Pulse duration</td><td align=\"left\">2</td><td align=\"left\">4.9</td><td align=\"left\">2.5</td><td align=\"left\">11</td><td align=\"left\">2e-05</td><td align=\"left\">0.04</td></tr><tr><td align=\"left\">Animal</td><td align=\"left\">5</td><td align=\"left\">12</td><td align=\"left\">2.4</td><td align=\"left\">11</td><td align=\"left\">6e-10</td><td align=\"left\">0.09</td></tr><tr><td align=\"left\">:</td><td align=\"left\">0.50</td><td align=\"left\" colspan=\"2\">Adjusted :</td><td align=\"left\" colspan=\"3\">0.38</td></tr></tbody></table></table-wrap>", "<table-wrap id=\"Tab6\"><label>Table 6</label><caption><p>ANOVA analysis of the non laser parameter for a laser power of 129 W.</p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th align=\"left\">Parameter</th><th align=\"left\">df</th><th align=\"left\">Sum sq.</th><th align=\"left\">Mean sq</th><th align=\"left\"> F</th><th align=\"left\">p</th><th align=\"left\"></th></tr></thead><tbody><tr><td align=\"left\">Breaks</td><td align=\"left\">2</td><td align=\"left\">64</td><td align=\"left\">32</td><td align=\"left\">120</td><td align=\"left\">4e-41</td><td align=\"left\">0.32</td></tr><tr><td align=\"left\">Cooling by air</td><td align=\"left\">1</td><td align=\"left\">5.0</td><td align=\"left\">5.0</td><td align=\"left\">18</td><td align=\"left\">3e-5</td><td align=\"left\">0.03</td></tr><tr><td align=\"left\">Pulse duration</td><td align=\"left\">2</td><td align=\"left\">2.5</td><td align=\"left\">1.3</td><td align=\"left\">4.6</td><td align=\"left\">0.01</td><td align=\"left\">0.01</td></tr><tr><td align=\"left\">Animal</td><td align=\"left\">5</td><td align=\"left\">8.3</td><td align=\"left\">1.7</td><td align=\"left\">6.0</td><td align=\"left\">2e-5</td><td align=\"left\">0.04</td></tr><tr><td align=\"left\">:</td><td align=\"left\">0.56</td><td align=\"left\" colspan=\"2\">Adjusted :</td><td align=\"left\" colspan=\"3\">0.45</td></tr></tbody></table></table-wrap>", "<table-wrap id=\"Tab7\"><label>Table 7</label><caption><p>ANOVA analysis of the non laser parameter with the exhaustion of plumes for a laser power of 129 W.</p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th align=\"left\">Parameter</th><th align=\"left\">df</th><th align=\"left\">Sum sq.</th><th align=\"left\">Mean sq</th><th align=\"left\">F</th><th align=\"left\">p</th><th align=\"left\"></th></tr></thead><tbody><tr><td align=\"left\">Breaks</td><td align=\"left\">2</td><td align=\"left\">65</td><td align=\"left\">33</td><td align=\"left\">110</td><td align=\"left\">3e-39</td><td align=\"left\">0.30</td></tr><tr><td align=\"left\">Cooling by air</td><td align=\"left\">1</td><td align=\"left\">3.8</td><td align=\"left\">3.8</td><td align=\"left\">13</td><td align=\"left\">4e-4</td><td align=\"left\">0.02</td></tr><tr><td align=\"left\">Exhaustion of plumes</td><td align=\"left\">2</td><td align=\"left\">3.2</td><td align=\"left\">1.6</td><td align=\"left\">5.4</td><td align=\"left\">0.005</td><td align=\"left\">0.02</td></tr><tr><td align=\"left\">Animal</td><td align=\"left\">5</td><td align=\"left\">16</td><td align=\"left\">3.2</td><td align=\"left\">11</td><td align=\"left\">1e-9</td><td align=\"left\">0.07</td></tr><tr><td align=\"left\">:</td><td align=\"left\">0.54</td><td align=\"left\" colspan=\"2\">Adjusted :</td><td align=\"left\" colspan=\"3\">0.43</td></tr></tbody></table></table-wrap>", "<table-wrap id=\"Tab8\"><label>Table 8</label><caption><p>ANOVA analysis of the laser parameters for all measurements.</p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th align=\"left\">Parameter</th><th align=\"left\">df</th><th align=\"left\">Sum sq.</th><th align=\"left\">Mean sq</th><th align=\"left\">F</th><th align=\"left\">p</th><th align=\"left\"></th></tr></thead><tbody><tr><td align=\"left\">Line energy</td><td align=\"left\">2</td><td align=\"left\">160</td><td align=\"left\">80</td><td align=\"left\">290</td><td align=\"left\">3e-65</td><td align=\"left\">0.54</td></tr><tr><td align=\"left\">Laser power/scan speed</td><td align=\"left\">6</td><td align=\"left\">65</td><td align=\"left\">11</td><td align=\"left\">39</td><td align=\"left\">3e-33</td><td align=\"left\">0.22</td></tr><tr><td align=\"left\">Animal</td><td align=\"left\">2</td><td align=\"left\">3.3</td><td align=\"left\">1.6</td><td align=\"left\">5.9</td><td align=\"left\">0.003</td><td align=\"left\">0.11</td></tr><tr><td align=\"left\">:</td><td align=\"left\">0.78</td><td align=\"left\" colspan=\"2\">Adjusted :</td><td align=\"left\" colspan=\"3\">0.76</td></tr></tbody></table></table-wrap>", "<table-wrap id=\"Tab9\"><label>Table 9</label><caption><p>ANOVA analysis of the laser power/scan speed for 83 .</p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th align=\"left\">Parameter</th><th align=\"left\">df</th><th align=\"left\">Sum sq.</th><th align=\"left\">Mean sq</th><th align=\"left\">F</th><th align=\"left\">p</th><th align=\"left\"></th></tr></thead><tbody><tr><td align=\"left\">Laser power/scan speed</td><td align=\"left\">2</td><td align=\"left\">3.5</td><td align=\"left\">1.7</td><td align=\"left\">5.7</td><td align=\"left\">0.005</td><td align=\"left\">0.11</td></tr><tr><td align=\"left\">Animal</td><td align=\"left\">2</td><td align=\"left\">2.9</td><td align=\"left\">1.4</td><td align=\"left\">4.7</td><td align=\"left\">0.01</td><td align=\"left\">0.09</td></tr><tr><td align=\"left\">:</td><td align=\"left\">0.24</td><td align=\"left\" colspan=\"2\">Adjusted :</td><td align=\"left\" colspan=\"3\">0.16</td></tr></tbody></table></table-wrap>", "<table-wrap id=\"Tab10\"><label>Table 10</label><caption><p>ANOVA analysis of the laser power/ scan speed for 172 .</p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th align=\"left\">Parameter</th><th align=\"left\">df</th><th align=\"left\">Sum sq.</th><th align=\"left\">Mean sq</th><th align=\"left\">F</th><th align=\"left\">p</th><th align=\"left\"></th></tr></thead><tbody><tr><td align=\"left\">Laser power/scan speed</td><td align=\"left\">2</td><td align=\"left\">12</td><td align=\"left\">6.1</td><td align=\"left\">21</td><td align=\"left\">5e-5</td><td align=\"left\">0.32</td></tr><tr><td align=\"left\">Animal</td><td align=\"left\">2</td><td align=\"left\">2.2</td><td align=\"left\">1.1</td><td align=\"left\">3.7</td><td align=\"left\">0.03</td><td align=\"left\">0.06</td></tr><tr><td align=\"left\">:</td><td align=\"left\">0.44</td><td align=\"left\" colspan=\"2\">Adjusted :</td><td align=\"left\" colspan=\"3\">0.38</td></tr></tbody></table></table-wrap>", "<table-wrap id=\"Tab11\"><label>Table 11</label><caption><p>ANOVA analysis of the laser power/scan speed for 235 .</p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th align=\"left\">Parameter</th><th align=\"left\">df</th><th align=\"left\">Sum sq.</th><th align=\"left\">Mean sq</th><th align=\"left\">F</th><th align=\"left\">p</th><th align=\"left\"></th></tr></thead><tbody><tr><td align=\"left\">Laser power/scan speed</td><td align=\"left\">2</td><td align=\"left\">49</td><td align=\"left\">25</td><td align=\"left\">109</td><td align=\"left\">1e-23</td><td align=\"left\">0.73</td></tr><tr><td align=\"left\">Animal</td><td align=\"left\">2</td><td align=\"left\">0.089</td><td align=\"left\">0.044</td><td align=\"left\">0.20</td><td align=\"left\">0.82</td><td align=\"left\">0.00</td></tr><tr><td align=\"left\">:</td><td align=\"left\">0.73</td><td align=\"left\" colspan=\"2\">Adjusted :</td><td align=\"left\" colspan=\"3\">0.71</td></tr></tbody></table></table-wrap>" ]

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mathvariant=\"normal\">μ</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq13\"><alternatives><tex-math id=\"M25\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\text {CO}_2$$\\end{document}</tex-math><mml:math id=\"M26\"><mml:msub><mml:mtext>CO</mml:mtext><mml:mn>2</mml:mn></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq14\"><alternatives><tex-math id=\"M27\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\upmu$$\\end{document}</tex-math><mml:math id=\"M28\"><mml:mi mathvariant=\"normal\">μ</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq15\"><alternatives><tex-math id=\"M29\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\upmu$$\\end{document}</tex-math><mml:math id=\"M30\"><mml:mi mathvariant=\"normal\">μ</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq16\"><alternatives><tex-math id=\"M31\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\upmu$$\\end{document}</tex-math><mml:math id=\"M32\"><mml:mi mathvariant=\"normal\">μ</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq17\"><alternatives><tex-math id=\"M33\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\upmu$$\\end{document}</tex-math><mml:math id=\"M34\"><mml:mi mathvariant=\"normal\">μ</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq18\"><alternatives><tex-math id=\"M35\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\upmu$$\\end{document}</tex-math><mml:math id=\"M36\"><mml:mi mathvariant=\"normal\">μ</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq19\"><alternatives><tex-math id=\"M37\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\text {CO}_2$$\\end{document}</tex-math><mml:math id=\"M38\"><mml:msub><mml:mtext>CO</mml:mtext><mml:mn>2</mml:mn></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq20\"><alternatives><tex-math id=\"M39\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\upmu$$\\end{document}</tex-math><mml:math id=\"M40\"><mml:mi mathvariant=\"normal\">μ</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq21\"><alternatives><tex-math id=\"M41\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\upmu$$\\end{document}</tex-math><mml:math id=\"M42\"><mml:mi mathvariant=\"normal\">μ</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq22\"><alternatives><tex-math id=\"M43\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\approx$$\\end{document}</tex-math><mml:math id=\"M44\"><mml:mo>≈</mml:mo></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq23\"><alternatives><tex-math id=\"M45\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$^\\circ$$\\end{document}</tex-math><mml:math id=\"M46\"><mml:msup><mml:mrow/><mml:mo>∘</mml:mo></mml:msup></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq24\"><alternatives><tex-math id=\"M47\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\upmu$$\\end{document}</tex-math><mml:math id=\"M48\"><mml:mi mathvariant=\"normal\">μ</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq25\"><alternatives><tex-math id=\"M49\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\times$$\\end{document}</tex-math><mml:math id=\"M50\"><mml:mo>×</mml:mo></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq26\"><alternatives><tex-math id=\"M51\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\times$$\\end{document}</tex-math><mml:math id=\"M52\"><mml:mo>×</mml:mo></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq27\"><alternatives><tex-math id=\"M53\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\upmu$$\\end{document}</tex-math><mml:math id=\"M54\"><mml:mi mathvariant=\"normal\">μ</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq28\"><alternatives><tex-math id=\"M55\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\times$$\\end{document}</tex-math><mml:math id=\"M56\"><mml:mo>×</mml:mo></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq29\"><alternatives><tex-math id=\"M57\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$^\\circ$$\\end{document}</tex-math><mml:math id=\"M58\"><mml:msup><mml:mrow/><mml:mo>∘</mml:mo></mml:msup></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq30\"><alternatives><tex-math id=\"M59\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$^\\circ$$\\end{document}</tex-math><mml:math id=\"M60\"><mml:msup><mml:mrow/><mml:mo>∘</mml:mo></mml:msup></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq31\"><alternatives><tex-math id=\"M61\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$P=83$$\\end{document}</tex-math><mml:math id=\"M62\"><mml:mrow><mml:mi>P</mml:mi><mml:mo>=</mml:mo><mml:mn>83</mml:mn></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq32\"><alternatives><tex-math id=\"M63\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\text {kW}/\\text {cm}^2$$\\end{document}</tex-math><mml:math id=\"M64\"><mml:mrow><mml:mtext>kW</mml:mtext><mml:mo stretchy=\"false\">/</mml:mo><mml:msup><mml:mtext>cm</mml:mtext><mml:mn>2</mml:mn></mml:msup></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq33\"><alternatives><tex-math id=\"M65\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\text {kW}/\\text {cm}^2$$\\end{document}</tex-math><mml:math id=\"M66\"><mml:mrow><mml:mtext>kW</mml:mtext><mml:mo stretchy=\"false\">/</mml:mo><mml:msup><mml:mtext>cm</mml:mtext><mml:mn>2</mml:mn></mml:msup></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq34\"><alternatives><tex-math id=\"M67\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\text {m}/\\text {s}$$\\end{document}</tex-math><mml:math id=\"M68\"><mml:mrow><mml:mtext>m</mml:mtext><mml:mo stretchy=\"false\">/</mml:mo><mml:mtext>s</mml:mtext></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq35\"><alternatives><tex-math id=\"M69\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$E_l$$\\end{document}</tex-math><mml:math id=\"M70\"><mml:msub><mml:mi>E</mml:mi><mml:mi>l</mml:mi></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq36\"><alternatives><tex-math id=\"M71\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\text {J}/\\text {m}$$\\end{document}</tex-math><mml:math id=\"M72\"><mml:mrow><mml:mtext>J</mml:mtext><mml:mo stretchy=\"false\">/</mml:mo><mml:mtext>m</mml:mtext></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq37\"><alternatives><tex-math id=\"M73\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\text {J}/\\text {m}$$\\end{document}</tex-math><mml:math id=\"M74\"><mml:mrow><mml:mtext>J</mml:mtext><mml:mo stretchy=\"false\">/</mml:mo><mml:mtext>m</mml:mtext></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq38\"><alternatives><tex-math id=\"M75\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\text {J}/\\text {m}$$\\end{document}</tex-math><mml:math id=\"M76\"><mml:mrow><mml:mtext>J</mml:mtext><mml:mo stretchy=\"false\">/</mml:mo><mml:mtext>m</mml:mtext></mml:mrow></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ1\"><label>1</label><alternatives><tex-math id=\"M77\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} E_l=\\frac{P}{v_{scan}} \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M78\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:msub><mml:mi>E</mml:mi><mml:mi>l</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mfrac><mml:mi>P</mml:mi><mml:msub><mml:mi>v</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">scan</mml:mi></mml:mrow></mml:msub></mml:mfrac></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq39\"><alternatives><tex-math id=\"M79\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$v_{scan}$$\\end{document}</tex-math><mml:math id=\"M80\"><mml:msub><mml:mi>v</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">scan</mml:mi></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq40\"><alternatives><tex-math id=\"M81\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\text {J}/\\text {m}$$\\end{document}</tex-math><mml:math id=\"M82\"><mml:mrow><mml:mtext>J</mml:mtext><mml:mo stretchy=\"false\">/</mml:mo><mml:mtext>m</mml:mtext></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq41\"><alternatives><tex-math id=\"M83\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\text {J}/\\text {m}$$\\end{document}</tex-math><mml:math id=\"M84\"><mml:mrow><mml:mtext>J</mml:mtext><mml:mo stretchy=\"false\">/</mml:mo><mml:mtext>m</mml:mtext></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq42\"><alternatives><tex-math id=\"M85\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\text {J}/\\text {m}$$\\end{document}</tex-math><mml:math id=\"M86\"><mml:mrow><mml:mtext>J</mml:mtext><mml:mo stretchy=\"false\">/</mml:mo><mml:mtext>m</mml:mtext></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq43\"><alternatives><tex-math id=\"M87\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\text {m}/\\text {s}$$\\end{document}</tex-math><mml:math id=\"M88\"><mml:mrow><mml:mtext>m</mml:mtext><mml:mo stretchy=\"false\">/</mml:mo><mml:mtext>s</mml:mtext></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq44\"><alternatives><tex-math id=\"M89\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$R^2$$\\end{document}</tex-math><mml:math id=\"M90\"><mml:msup><mml:mi>R</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq45\"><alternatives><tex-math id=\"M91\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$R^2$$\\end{document}</tex-math><mml:math id=\"M92\"><mml:msup><mml:mi>R</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq46\"><alternatives><tex-math id=\"M93\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\partial \\eta ^2$$\\end{document}</tex-math><mml:math id=\"M94\"><mml:mrow><mml:mi>∂</mml:mi><mml:msup><mml:mi>η</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq47\"><alternatives><tex-math id=\"M95\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$R^2$$\\end{document}</tex-math><mml:math id=\"M96\"><mml:msup><mml:mi>R</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq48\"><alternatives><tex-math id=\"M97\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$R^2$$\\end{document}</tex-math><mml:math id=\"M98\"><mml:msup><mml:mi>R</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq49\"><alternatives><tex-math id=\"M99\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$R^2$$\\end{document}</tex-math><mml:math id=\"M100\"><mml:msup><mml:mi>R</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq50\"><alternatives><tex-math id=\"M101\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\partial \\eta ^2$$\\end{document}</tex-math><mml:math id=\"M102\"><mml:mrow><mml:mi>∂</mml:mi><mml:msup><mml:mi>η</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq51\"><alternatives><tex-math id=\"M103\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$R^2$$\\end{document}</tex-math><mml:math id=\"M104\"><mml:msup><mml:mi>R</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq52\"><alternatives><tex-math id=\"M105\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$R^2$$\\end{document}</tex-math><mml:math id=\"M106\"><mml:msup><mml:mi>R</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq53\"><alternatives><tex-math id=\"M107\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$R^2$$\\end{document}</tex-math><mml:math id=\"M108\"><mml:msup><mml:mi>R</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq54\"><alternatives><tex-math id=\"M109\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\partial \\eta ^2$$\\end{document}</tex-math><mml:math id=\"M110\"><mml:mrow><mml:mi>∂</mml:mi><mml:msup><mml:mi>η</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq55\"><alternatives><tex-math id=\"M111\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$R^2$$\\end{document}</tex-math><mml:math id=\"M112\"><mml:msup><mml:mi>R</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq56\"><alternatives><tex-math id=\"M113\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$R^2$$\\end{document}</tex-math><mml:math id=\"M114\"><mml:msup><mml:mi>R</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq57\"><alternatives><tex-math id=\"M115\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\text {J}/\\text {m}$$\\end{document}</tex-math><mml:math id=\"M116\"><mml:mrow><mml:mtext>J</mml:mtext><mml:mo stretchy=\"false\">/</mml:mo><mml:mtext>m</mml:mtext></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq58\"><alternatives><tex-math id=\"M117\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\text {m}/\\text {s}$$\\end{document}</tex-math><mml:math id=\"M118\"><mml:mrow><mml:mtext>m</mml:mtext><mml:mo stretchy=\"false\">/</mml:mo><mml:mtext>s</mml:mtext></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq59\"><alternatives><tex-math id=\"M119\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\text {J}/\\text {m}$$\\end{document}</tex-math><mml:math id=\"M120\"><mml:mrow><mml:mtext>J</mml:mtext><mml:mo stretchy=\"false\">/</mml:mo><mml:mtext>m</mml:mtext></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq60\"><alternatives><tex-math id=\"M121\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\text {m}/\\text {s}$$\\end{document}</tex-math><mml:math id=\"M122\"><mml:mrow><mml:mtext>m</mml:mtext><mml:mo stretchy=\"false\">/</mml:mo><mml:mtext>s</mml:mtext></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq61\"><alternatives><tex-math id=\"M123\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\partial \\eta ^2$$\\end{document}</tex-math><mml:math id=\"M124\"><mml:mrow><mml:mi>∂</mml:mi><mml:msup><mml:mi>η</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq62\"><alternatives><tex-math id=\"M125\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$R^2$$\\end{document}</tex-math><mml:math id=\"M126\"><mml:msup><mml:mi>R</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq63\"><alternatives><tex-math id=\"M127\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$R^2$$\\end{document}</tex-math><mml:math id=\"M128\"><mml:msup><mml:mi>R</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq64\"><alternatives><tex-math id=\"M129\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\text {J}/\\text {m}$$\\end{document}</tex-math><mml:math id=\"M130\"><mml:mrow><mml:mtext>J</mml:mtext><mml:mo stretchy=\"false\">/</mml:mo><mml:mtext>m</mml:mtext></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq65\"><alternatives><tex-math id=\"M131\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\partial \\eta ^2$$\\end{document}</tex-math><mml:math id=\"M132\"><mml:mrow><mml:mi>∂</mml:mi><mml:msup><mml:mi>η</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq66\"><alternatives><tex-math id=\"M133\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$R^2$$\\end{document}</tex-math><mml:math id=\"M134\"><mml:msup><mml:mi>R</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq67\"><alternatives><tex-math id=\"M135\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$R^2$$\\end{document}</tex-math><mml:math id=\"M136\"><mml:msup><mml:mi>R</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq68\"><alternatives><tex-math id=\"M137\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\text {J}/\\text {m}$$\\end{document}</tex-math><mml:math id=\"M138\"><mml:mrow><mml:mtext>J</mml:mtext><mml:mo stretchy=\"false\">/</mml:mo><mml:mtext>m</mml:mtext></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq69\"><alternatives><tex-math id=\"M139\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\text {J}/\\text 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[ "<supplementary-material content-type=\"local-data\" id=\"MOESM1\"></supplementary-material>" ]

[ "<table-wrap-foot><p>For experiment B, 540 cuts are evaluated. From these, 180 cuts are the same as in experiment A.</p></table-wrap-foot>", "<table-wrap-foot><p>Three line energies are taken and for each line energy three scan speeds and corresponding laser powers are selected.</p></table-wrap-foot>", "<fn-group><fn><p><bold>Publisher's note</bold></p><p>Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.</p></fn></fn-group>" ]

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[ "<media xlink:href=\"41598_2024_51449_MOESM1_ESM.zip\"><caption><p>Supplementary Information.</p></caption></media>" ]

{ "acronym": [], "definition": [] }

[ "<title>Introduction</title>", "<p id=\"Par2\">Pilot-wave theory (also called de Broglie-Bohm theory or Bohmian mechanics) is a realist, nonlocal formulation of quantum mechanics originally presented in the 1927 Solvay conference by de Brogile^{##UREF##0##1##,##UREF##1##2##}. In 1952, Bohm showed how the theory solves the vexed measurement problem in orthodox quantum mechanics by describing the measurement apparatus within the theory^{##UREF##2##3##,##UREF##3##4##}. The theory has been extended to relativistic domain^{##UREF##4##5##–##UREF##8##9##}, applied to astrophysical and cosmological scenarios^{##UREF##9##10##–##UREF##12##13##}, and provides a counter-example to the claim that quantum phenomena imply a denial of realism.</p>", "<p id=\"Par3\">In his description of the theory, Bohm pointed out that certain assumptions are necessary to reproduce orthodox quantum mechanics. Further, he opined that these assumptions may need modifications in regimes not yet experimentally accessible, so that the theory may either supersede or depart from orthodox quantum mechanics in the future^{##UREF##2##3##–##UREF##4##5##,##UREF##13##14##}. One of these assumptions is that the initial density of configurations equals the Born rule density. This assumption has been criticised on the grounds that, since there is no logical relation between the initial configuration density and the quantum state in the theory, it is ad hoc^{##UREF##14##15##,##UREF##15##16##}. Bohm was able to show that adding random collisions^{##UREF##13##14##} or random fluid fluctuations^{##UREF##16##17##} to the dynamics of the theory leads to relaxation from an arbitrary density to the Born rule density. Later, Valentini showed that the original dynamics alone is sufficient for relaxation to occur at a coarse grained level^{##UREF##17##18##,##UREF##18##19##}. Numerous computational studies have since been conducted that have furthered our understanding of the relaxation process in various scenarios (see^{##UREF##12##13##} for a review).</p>", "<p id=\"Par4\">However, a simple but important conceptual point has remained largely unnoticed in the literature: if there is no logical relationship between the configuration density and the quantum state in pilot-wave theory, then why should the quantum state be normalizable? In orthodox quantum mechanics, normalizability is necessary as statistical predictions are extracted from the quantum state according to the Born rule. On the other hand, in pilot-wave theory the quantum state serves as a physical field that determines the evolution of the configuration. To extract statistical predictions from the theory, one only needs to define an <italic>ensemble with a normalized density of configurations</italic> – normalizability of the quantum state is unnecessary. This opens up the possibility of physically interpreting non-normalizable quantum states that occur as solutions to physical constraints in quantum gravity, such as the Kodama state^{##UREF##19##20##–##UREF##21##22##}.</p>", "<p id=\"Par5\">However, to the best of our knowledge, the behaviour of non-normalizable solutions to the Schrodinger equation has not been studied from a pilot-wave perspective. In this article, we make a first step in this direction by studying the non-normalizable solutions of the harmonic-oscillator potential. We choose the harmonic oscillator as it is widely found in nature, and because the normalizability constraint leads to the important discretization of energy levels. The article is structured as follows. We first study the non-normalizable solutions of the harmonic oscillator, using both the analytic approach and the ladder operator approach. We then study the pilot-wave theory of the non-normalizable states. We show that the pilot-wave velocity field for the non-normalizable states at large . We discuss the relaxation behaviour for these states. We then introduce the notion of pilot-wave equilibrium and define the new <italic>H</italic>-function . We prove an <italic>H</italic>-theorem applicable to non-normalizable states using a coarse-grained , analogous to the <italic>H</italic>-theorem for quantum equilibrium. We study the relationship between relaxation to pilot-wave equilibrium and relaxation to quantum equilibrium. Lastly, we discuss the theoretical and experimental implications of our work. In particular, we show that non-normalizable states are unstable in the presence of perturbations and environmental interactions, and thereby give an explanation of quantization in pilot-wave theory.</p>" ]

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[ "<title>Discussion</title>", "<p id=\"Par54\">We have discussed some of the implications of our work in the previous section. However, the list of implications is necessarily inexhaustive as the normalizability constraint is ubiquitous in orthodox quantum mechanics. It would, for example, be interesting to study non-normalization solutions to the Schrodinger equation for other systems, say the Hydrogen atom, or to the Dirac equation. An important result of our work is that the non-normalizable harmonic-oscillator solutions are bound states, in the sense that the pilot-wave velocity field at large . It is important to figure out the general conditions under which the pilot-wave velocity field has this behaviour. Another important result is that perturbations and interactions make non-normalizable states unstable, in the sense that the system configuration becomes overwhelmingly likely with time to be in a normalizable branch of the total quantum state. Lastly, it remains unclear how to construct a well-defined basis for such states.</p>", "<p id=\"Par55\">We note that, according to our work, the explanation for quantization given by pilot-wave theory is drastically different from that of quantum mechanics. Quantization in quantum mechanics arises from the axiom of Born rule, whereas in pilot-wave theory quantization is an emergent phenomenon that arises from the instability of non-normalizable states due to perturbations and environmental interactions. In this sense, the status of non-normalizable states in the theory may be said to be analogous to that of non-equilibrium ensembles as (<bold>a</bold>) the conceptual structure of the theory allows the logical possibility of both non-normalizable states and non-equilibrium densities, and (<bold>b</bold>) the theory also possesses the internal logic necessary to explain why we do not observe either of them in present-day laboratories.</p>", "<p id=\"Par56\">We note that the <italic>H</italic>-theorem does not by itself prove that relaxation to pilot-wave equilibrium occurs, but provides a general mechanism to understand how equilibrium is approached, similar to the status of the generalized <italic>H</italic>-theorem in classical statistical mechanics^{##UREF##24##25##}. Whether relaxation in fact occurs in finite time, if it is monotonic etc. significantly depend on whether the velocity field yields sufficient mixing. It is well-known in the literature on relaxation in pilot-wave theory^{##UREF##12##13##,##UREF##18##19##,##UREF##33##34##} that the velocity field varies rapidly around nodes (if they exist) and thereby causes efficient relaxation in general. Therefore, future numerical simulations using superpositions of non-normalizable eigenstates can provide evidence whether relaxation to pilot-wave equilibrium indeed occurs, similar to relaxation to quantum equilibrium for normalizable states. It is useful to note here that the boundedness of the solutions ensures that the support does not necessarily become filamentous with time. For example, if is sufficiently large to cover the region around the origin and is very large near its boundary , then will remain effectively static as the radial velocity field will be very small in that region. Lastly, we note that the coarse-graining cells do not become filamentous as they do not evolve with time, unlike the configuration density.</p>", "<p id=\"Par57\">From a historical perspective, we know that the initial conditions of pilot-wave theory have usually been so restricted as to reproduce orthodox quantum mechanics. An important departure was made when nonequilibrium densities were taken seriously in the theory, and the notion of quantum equilibrium was defined^{##UREF##17##18##,##UREF##27##28##}. But the notion of quantum equilibrium is still restrictive as it assumes that a density in equilibrium always reproduces orthodox quantum mechanics. The notion of pilot-wave equilibrium makes one further step, in which this restriction is jettisoned. Therefore, generalising the notion of quantum equilibrium to pilot-wave equilibrium may be seen as a logical step towards treating pilot-wave theory as a theory in its own right, instead of as a hidden-variable reformulation of orthodox quantum mechanics.</p>", "<p id=\"Par58\">It may appear that the restriction of the configuration density to compact supports limits the physical applicability of pilot-wave equilibrium. However, this is incorrect as we can always approximate a density with global support up to arbitrary accuracy using a density with compact support. This can be done by defining an arbitrarily small but finite cut-off parameter so that if the global density at a particular point on the configuration space, we define the compact density , where (up to normalization) at all other . Further, global supports imply arbitrarily small probabilities that cannot be empirically verified and are, therefore, mathematical idealisations. For example, a Hydrogen atom in a lab on Earth has a finite but arbitrarily small probability of being found, in a position measurement, arbitrarily far away from the Earth. But observing such an extremely tiny probability trillions of light years away would take many times more than the current age of the universe in any realistic experimental setup.</p>", "<p id=\"Par59\">There are several implications of our work for pilot-wave theory. First, our work suggests a constraint on the pilot-wave velocity field. We know that the pilot-wave velocity is not uniquely defined as one can always add a divergence-free term to the current. In the context of non-normalizable states, the velocity field plays the important role of determining whether a given state is bounded. Therefore, it seems reasonable to impose the constraint that the addition of divergence-free term to the current does not affect the boundedness of the state. That is, if the (usually defined) pilot-wave velocity field goes to 0 at , then this behaviour must be preserved on modifying . It would be interesting to figure out the class of possible that satisfy this property. Second, our work may help in distinguishing pilot-wave theory from orthodox quantum mechanics and other realist interpretations of quantum mechanics. For example, some authors have claimed that the system configuration in pilot-wave theory is superflous and the theory is actually a many-worlds theory in disguise^{##UREF##34##35##–##UREF##36##37##}. As we have seen, however, the existence of a configuration density in the theory makes it possible to extract statistical predictions from non-normalizable quantum states. Therefore, the interpretation of non-normalizable states may turn out to be a crucial difference between the two theories. Third, we note that the notion of pilot-wave equilibrium, although introduced in the context of non-normalizable quantum states, is equally applicable to normalizable quantum states. It would be of interest to figure out whether densities partially relaxed to quantum equilibrium in previous numerical simulations have in fact relaxed to pilot-wave equilibrium. Lastly, our results imply that a unitary evolution involving non-normalizable states is dynamically equivalent to a corresponding non-unitary evolution involving appropriate normalizable states. This suggests that non-unitary evolution in some applications of orthodox quantum mechanics may in fact be an artefact of insistence on state normalizability. This also implies that, for normalizable states, unitary evolution is not necessary for relaxation to pilot-wave equilibrium.</p>", "<p id=\"Par60\">Our work also has implications for the -ontic versus -epistemic debate^{##UREF##37##38##–##UREF##39##40##}. Non-normalizable quantum states do not make sense from a -epistemic viewpoint, in which the role of the quantum state is to define probabilities. If the existence of non-normalizable quantum states is proved experimentally, or if such states are found to be crucial in fields like quantum cosmology or quantum gravity, then it would be difficult to argue in favour of -epistemicity. We note that, once pilot-wave equilibrium is reached at a coarse-grained level, then the relation on suggests how a -epistemic interpretation may emerge at an effective level from an underlying -ontic theory.</p>", "<p id=\"Par61\">We conclude that pilot-wave theory naturally suggests consideration of the possibility of non-normalizable quantum states, which we have studied for the case of harmonic oscillator. Such states have a physically-meaningful notion of an equilibrium density. We have argued that quantization emerges in pilot-wave theory due to the instability of non-normalizable states to perturbations and environmental interactions. Further work is needed to determine whether such states actually exist in nature.</p>" ]

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[ "<p id=\"Par1\">Non-normalizable states are difficult to interpret in the orthodox quantum formalism but often occur as solutions to physical constraints in quantum gravity. We argue that pilot-wave theory gives a straightforward physical interpretation of non-normalizable quantum states, as the theory requires only a normalized density of configurations to generate statistical predictions. In order to better understand such states, we conduct the first study of non-normalizable solutions of the harmonic oscillator from a pilot-wave perspective. We show that, contrary to intuitions from orthodox quantum mechanics, the non-normalizable eigenstates and their superpositions are bound states in the sense that the velocity field at large . We argue that defining a physically meaningful equilibrium density for such states requires a new notion of equilibrium, named pilot-wave equilibrium, which is a generalisation of the notion of quantum equilibrium. We define a new <italic>H</italic>-function , and prove that a density in pilot-wave equilibrium minimises , is equivariant, and remains in equilibrium with time. We prove an <italic>H</italic>-theorem for the coarse-grained , under assumptions similar to those for relaxation to quantum equilibrium. We give an explanation of the emergence of quantization in pilot-wave theory in terms of instability of non-normalizable states due to perturbations and environmental interactions. Lastly, we discuss applications in quantum field theory and quantum gravity, and implications for pilot-wave theory and quantum foundations in general.</p>", "<title>Subject terms</title>" ]

[ "<title>Non-normalizable solutions of the harmonic oscillator</title>", "<p id=\"Par6\">We start by noting that several elementary theorems in orthodox quantum mechanics are no longer applicable once the normalizability constraint on quantum state is dropped. In the non-normalizable scenario, eigenstates in one dimension are generally degenerate and complex as relevant theorems on degeneracy and reality of eigenstates no longer apply. Furthermore, a non-normalizable quantum state does not have a Fourier transform, and therefore a momentum representation, in general. This is because Fourier transform exists only if the concerned function does not diverge faster than a polynomial at large values of its argument. Therefore, we are restricted to the position representation of the quantum state in general. This makes sense from a pilot-wave perspective, as the position basis is the preferred basis in the theory. We also note that the momentum operator is in general non-Hermitian in this scenario.</p>", "<p id=\"Par7\">For the harmonic-oscillator potential, the energy eigenvalues are not quantized and can also take negative values in this scenario. Mathematically, the eigenvalues can also be complex in this scenario, but this is not physically meaningful from a pilot-wave perspective. Consider a von-Neumann energy measurement, which leads to apparatus wavefunctions of the form , where <italic>E</italic> is the energy eigenvalue and <italic>g</italic> is the strength of interaction between the system and apparatus. The wavefunction is not defined on configuration space if <italic>E</italic> is complex. Therefore, allowing complex eigenvalues is only possible if one abandons the configuration space as the fundamental arena of pilot-wave theory. Lastly, we restrict the initial wavefunction to only eigenstates and finite superpositions, as the time-evolution operator may not be not well-defined for an arbitrary initial wavefunction^{##UREF##22##23##}. With these facts in mind, let us study the non-normalizable solutions to the harmonic oscillator from a pilot-wave perspective.</p>", "<p id=\"Par8\">The time-independent Schrodinger equation for the harmonic-oscillator potential can be written aswhere and . The equation is traditionally solved by using the ansatz . Substituting the ansatz into Eq. (##FORMU##12##1##), we getEquation (##FORMU##16##2##) is known as the Hermite differential equation. It contains both normalizable and non-normalizable solutions to (##FORMU##12##1##). Using the Frobenius method, the general solution to (##FORMU##16##2##) can be written aswhere and are two arbitrary complex constants and the recurrence relation between ’s can be obtained to be . It is useful for us to rewrite Eq. (##FORMU##17##3##) aswhere and . Clearly, the term consists only of even powers of <italic>y</italic>, whereas the term consists only of odd powers.</p>", "<p id=\"Par9\">It is useful to note that , can be expressed in closed form as follows:whereis the confluent hypergeometric function of the first kind and is the Pochhammer symbol.</p>", "<p id=\"Par10\">The general solution to the time-independent Schrodinger Eq. (##FORMU##12##1##) can be written aswhere and . Equation (##FORMU##35##10##) is a valid solution to the Schrodinger Eq. (##FORMU##12##1##) for all (real) values of <italic>K</italic>. It can be shown that the series () terminates only if for an even (odd) <italic>n</italic>. In that case, () has a dependence at large and is normalizable. If for an even (odd) <italic>n</italic>, then () has a dependence at large and is non-normalizable.</p>", "<p id=\"Par11\">The complex coefficients , contain a total of 4 real parameters. We can eliminate 2 of the parameters by <italic>a)</italic> normalizing the coefficients so that (note that the quantum state is itself non normalizable in general) and <italic>b)</italic> eliminating the global phase. Both steps <italic>a</italic>) and <italic>b</italic>) make sense from a pilot-wave theory perspective as the pilot-wave velocity field , where <italic>j</italic>(<italic>y</italic>) is the quantum probability current (see Eq. (##FORMU##90##17##) below), does not depend on the global magnitude or the global phase of the quantum state. That is, a transformation of the form , where is a complex constant, does not change <italic>v</italic>(<italic>y</italic>). Therefore, we may further simplify Eq. (##FORMU##35##10##) towhere , , and , . In this form, it is clear that and act as basis vectors of the doubly degenerate subspace corresponding to <italic>K</italic>. We note that, in orthodox quantum mechanics, steps (a) and (b) are justified (for normalizable states) on the grounds that is a probability density. Clearly, cannot be interpreted as a probability density in our case but <italic>a)</italic>, <italic>b)</italic> are still valid from a pilot-wave perspective.</p>", "<p id=\"Par12\">We can connect the general solution (##FORMU##56##11##) to the allowed solutions in orthodox quantum mechanics as follows. We know that the allowed energy levels in orthodox quantum mechanics are given by , where <italic>n</italic> is a non-negative integer. Furthermore, we know from the preceding discussion that for all even <italic>n</italic>, is normalizable and is non-normalizable. Similarly, for odd <italic>n</italic>, is normalizable and is non-normalizable. Therefore,where is the harmonic-oscillator eigenstate in orthodox quantum mechanics, and is the relevant normalization constant.</p>", "<p id=\"Par13\">Let us consider a superposition of eigenstates corresponding to different values of <italic>K</italic>. Suppose . As before, we normalize the coefficients () and eliminate the global phase of , as the velocity field is unaffected by these changes. We also know, from the time-dependent Schrodinger equation, that will evolve asLastly, it is straightforward to extend the discussion to a system of <italic>N</italic> particles, each in a harmonic oscillator potential. Consider the quantum stateWe normalize the coefficients and eliminate the global phase of . The time evolution of can be easily calculated by the time-dependent Schrodinger equation. We discuss the action of ladder operators on non-normalizable states in the ##SUPPL##0##Supplementary Information##.</p>", "<title>Bound-state interpretation of non-normalizable harmonic oscillator states</title>", "<p id=\"Par14\">In pilot-wave theory, the quantum state serves to define the velocity field for the evolution of the system configuration. This can be a configuration of particles, as in pilot-wave theory of non-relativistic quantum mechanics, or a configuration of fields, as in pilot-wave theory of quantum field theory. Let us consider a system of <italic>N</italic> particles in the harmonic oscillator potential with the quantum state (##FORMU##80##14##). Without loss of generality, we suppose that all the particles have the same mass <italic>m</italic> for simplicity. The time-dependent Schrodinger equation implies the continuity equationwhere is a point on the configuration space, and the currentis defined in terms of and which is the complex conjugate of . From Eq. (##FORMU##84##15##), the quantityis defined as the pilot-wave velocity field. Let us consider an ensemble of the <italic>N</italic>-particle harmonic oscillator systems. As there is no <italic>a priori</italic> relationship between the quantum state and the configuration density in pilot-wave theory, we can define an initial normalized density for the ensemble. Equation (##FORMU##90##17##) supplies the velocity field to evolve :Clearly, experimental probabilities are well-defined as is normalized. However, there remains the question whether the velocity field (##FORMU##90##17##) behaves physically for non-normalizable states. One example of an unphysical behaviour would be if increases with as () for . In that case, the system configuration will escape to in finite time. In orthodox quantum mechanics, we know that such behaviour cannot occur as the normalizability constraint ensures that the probability density as . For this reason, the normalizable states are referred to as <italic>bound states</italic> in orthodox quantum mechanics.</p>", "<p id=\"Par15\">We can straightforwardly generalise the definition of bound state to the non-normalizable scenario: if the velocity field (##FORMU##90##17##) defined by is such that in the limit for all , then is a bound state. Such a velocity field ensures that any initial normalized configuration density will evolve to such that as for all . That is, the system configuration remains bounded at all (finite) times.</p>", "<p id=\"Par16\">Below, we prove that the non-normalizable solutions of the harmonic oscillator are bound states in this sense. We begin with the simplest case, that of an eigenstate in one dimension.</p>", "<title>Velocity field of an eigenstate in one-dimension</title>", "<p id=\"Par17\">Let us consider the velocity field of a harmonic oscillator eigenstate . We know from orthodox quantum mechanics that the normalizable eigenstates defined by (##FORMU##71##12##) are real. This implies that, for these states, the velocity field is zero everywhere and the particle is stationary. However, is complex in general. This implies that the velocity field for non-normalizable eigenstates is non-zero in general. Let us then calculate this velocity field.</p>", "<p id=\"Par18\">We first note the general result that, if is an eigenstate of the Hamiltonian, thenIn orthodox quantum mechanics, as as . In our case, on the other hand, as so that the left-hand side of Eq. (##FORMU##118##19##) becomes indeterminate at . However, it is convenient to evaluate the left-hand side of (##FORMU##118##19##) for at . This is because the following readily verifiable calculationsimply thatso that the current <italic>j</italic>(<italic>y</italic>) is constant and independent of <italic>K</italic>.</p>", "<p id=\"Par19\">Therefore, the velocity field iswhere, in Eq. (##FORMU##139##28##), we have used and (##FORMU##131##24##).</p>", "<p id=\"Par20\">Let us discuss the velocity field (##FORMU##139##28##). First, Eq. (##FORMU##139##28##) tells us that, for an eigenstate corresponding to <italic>K</italic>, the velocity field is constant with time. Second, it tells us that the velocity field depends on the angles , , so that degenerate eigenstates corresponding to the same <italic>K</italic> will, in general, have velocity fields that are different but proportional to each other at every <italic>y</italic>. Third, the velocity field does not change sign with <italic>y</italic>. Fourth, we note that the velocity field for an eigenstate corresponding to () has no apparent connection with the velocity field for an eigenstate corresponding to . Lastly, and most importantly, Eq. (##FORMU##139##28##) tells us that the velocity fields are inversely proportional to . This implies that, for as we know that diverges like at large . Therefore, the velocity field decreases very quickly to 0 as becomes large at (see Fig. ##FIG##0##1##). This implies that is a <italic>bound state</italic>, according to our definition, although it is non-normalizable. This is a surprising behaviour from the viewpoint of orthodox quantum mechanics, as a naive application of the Born rule would imply an infinitely large probability of the particle being found at large .</p>", "<title>Velocity field of a superposition of eigenstates</title>", "<p id=\"Par21\">Let us consider a quantum state that is a superposition of eigenstates corresponding to various <italic>K</italic>’s. We know from Eq. (##FORMU##90##17##) that the velocity field isTo study the asymptotic behaviour of (##FORMU##157##30##) as , we first need an asymptotic expression for as . We derive such an expression in the supplementary material, using the approach given in ref.^{##UREF##23##24##}.</p>", "<title>Asymptotic behaviour of the velocity field</title>", "<p id=\"Par22\">Using the expansion , we can express the current asUsing the asymptotic form derived in the Supplementary Information, we write at large , Eq. (##FORMU##167##32##) becomeswhere we have retained only the leading order of <italic>y</italic>. Similarly, we can prove thatTherefore, the velocity fieldEquation (##FORMU##172##35##) implies that (see Fig. ##FIG##1##2##). Therefore, a superposition of eigenstates corresponding to different <italic>K</italic>’s is a bound state. Let us proceed next to the case of multiple particles.</p>", "<title>Velocity field for multiple particles</title>", "<p id=\"Par23\">We want to check whether the asymptotic behaviour of the velocity field discussed in the previous subsections also hold in the case of multiple particles, each in a harmonic oscillator potential. Consider an <italic>N</italic>-particle quantum statewhere is an eigenstate of the g-th particle corresponding to the eigenvalue in the j-th term of the superposition. We know that the current in the r-th direction isSimilar to the previous subsection, we can express at large , and then simplify (##FORMU##188##37##) asOn the other hand,which implies thatEquation (##FORMU##193##40##) confirms that the velocity field is such that as \n. Therefore, the system configuration remains bounded at all times and is a bound state (see Fig. ##FIG##2##3##).</p>", "<title>Relaxation to equilibrium</title>", "<p id=\"Par24\">In pilot-wave theory for normalizable quantum states, it is well known that an arbitrary initial density of configurations relaxes to the Born rule density (called the equilibrium density) at a coarse-grained level, subject to standard statistical mechanical assumptions^{##UREF##12##13##,##UREF##17##18##,##UREF##18##19##}. In this section, we look at whether such a relaxation occurs to a well-defined equilibrium density when is non-normalizable.</p>", "<title>Pilot-wave equilibrium: a generalisation of quantum equilibrium</title>", "<p id=\"Par25\">Consider an ensemble of systems described by a non-normalizable quantum state with a normalized density of configurations . We want to understand if a physically-meaningful equilibrium density can be defined for the ensemble. In the case of normalizable quantum states, we know that the equilibrium density satisfies the following conditions: <list list-type=\"order\"><list-item><p id=\"Par26\">Entropy maximization: The equilibrium density minimises an appropriately defined <italic>H</italic>-function (the negative of which is maximised).</p></list-item><list-item><p id=\"Par27\">Equilibrium stability: The equilibrium density continues to be in equilibrium with time.</p></list-item><list-item><p id=\"Par28\">Equivariance: The functional form of the equilibrium density in terms of the quantum state is preserved with time.</p></list-item><list-item><p id=\"Par29\">Quantum-mechanical equivalence: The statistical predictions made by the equilibrium density is equal to that predicted by orthodox quantum mechanics for the same quantum state.</p></list-item></list>Let us check whether these conditions can be met in our scenario. Consider the first condition: we typically seek a density that minimises the <italic>H</italic>-function^{##UREF##17##18##}where the integral is defined over all of configuration space and is the set of all reals. Equation (##FORMU##204##41##) immediately lands us in trouble as it is formally the relative entropy from to – but , being non normalizable, is <italic>not</italic> a probability density over . Therefore, is not a mathematically well-defined relative entropy.</p>", "<p id=\"Par30\">Fortunately, it is straightforward to rectify the definition of <italic>H</italic> for our scenario. We note that, in general, the density may have support only over a proper subset of . Let us assume that is a proper subset of , that is, has a compact support. We can then treat as a probability density over once appropriately normalized. We define a candidate equilibrium densitywhere . We then replace byNote that, since is a valid probability density over , is a well-defined relative entropy from to . Equation (##FORMU##223##43##) can be written asso that the integrand is always non-negative, which implies that the lower bound , which is achieved when . Therefore, the newly-defined quantities and together satisfy the first condition set out at the beginning of the subsection.</p>", "<p id=\"Par31\">Let us next consider the second condition: does the initial density evolve to a that minimises ? We know that^{##UREF##13##14##}, since both and satisfy the same continuity equation, we havewhere . Equation (##FORMU##239##45##) implies that, given an initial density , we havewhere is the support of . We note that Eq. (##FORMU##242##46##) impliesThe time-dependent <italic>H</italic>-functionremains constant at its lower bound for the density . Thus, an initial density that minimises will evolve in time so as to minimise at all times.</p>", "<p id=\"Par32\">The third condition, of equivariance, is not directly met as the support is not determined by the quantum state. However, it is clear from (##FORMU##242##46##) that the functional form of in terms of over is invariant with time. We may therefore define the following condition to be pilot-wave invariance: the functional form of the density in terms of the quantum state over its support is invariant with time. Pilot-wave invariance is motivated by the notion of equivariance, and reduces to it in the special case that is normalizable and \n.</p>", "<p id=\"Par33\">Is the fourth condition also met? This condition ceases to make sense in our case, as we are dealing with quantum states that are non-normalizable. Such states are considered unphysical in orthodox quantum mechanics, and the theory provides no experimental probabilities for ensembles with such states. In view of the fact that conditions 1, 2 and 3 (suitably modified) are satisfied, and condition 4 is inapplicable, we may define a density that satisfies only the first three conditions to be in <italic>pilot-wave</italic> equilibrium (as opposed to quantum equilibrium). The terminology makes explicit the fact that quantifies relaxation to an equilibrium density in pilot-wave theory regardless of whether that density reproduces orthodox quantum mechanics, whereas quantifies relaxation to the equilibrium density that reproduces orthodox quantum mechanics. For normalizable states, the notion of pilot-wave equilibrium reduces to quantum equilibrium for the special case when .</p>", "<p id=\"Par34\">To conclude, we define a density with support to be in pilot-wave equilibrium if and only ifClearly, there are infinitely many that can be in pilot-wave equilibrium, as there are infinitely many subsets of . The density minimises the <italic>H</italic>-functionat all times. If does not satisfy condition (##FORMU##263##49##), then we define it to be in pilot-wave nonequilibrium. Note that a rescaling , where is a complex constant, does not change the equilibrium condition (##FORMU##263##49##), similar to the definition of the velocity field (##FORMU##90##17##). Lastly, we also note that although the concept of pilot-wave equilibrium has been motivated by a consideration of non-normalizable quantum states, it is applicable to normalizable quantum states as well.</p>", "<title><italic>H</italic>-theorem for relaxation to pilot-wave equilibrium</title>", "<p id=\"Par35\">We now turn to the question whether an arbitrary ensemble density will relax to pilot-wave equilibrium at a coarse-grained level, analogous to relaxation to quantum equilibrium for normalizable states. We show this is indeed the case by proving an <italic>H</italic>-theorem for .</p>", "<p id=\"Par36\">In the proof for relaxation to classical statistical equilibrium^{##UREF##24##25##} or quantum equilibrium^{##UREF##17##18##}, an important role is played by the fact that the exact <italic>H</italic>-function is constant with time. To build an analogous <italic>H</italic>-theorem for pilot-wave equilibrium, our first task then, is to ascertain if is constant with time. From Eqs. (##FORMU##204##41##), (##FORMU##220##42##) and (##FORMU##223##43##), the relationship between the two <italic>H</italic>-functions isClearly, it is sufficient to prove the constancy of to prove that is constant with time. We know, from Eq. (##FORMU##245##47##), that is constant with time if the initial density is in pilot-wave equilibrium. Let us consider an arbitrary initial density with support in pilot-wave nonequilibrium, piloted by a non-normalizable state . We also consider the pilot-wave equilibrium density over , where . As both and are piloted by , they will obey similar continuity equationswhere is determined by according to (##FORMU##90##17##). The velocity field provides the mapping from . We also know from Eq. (##FORMU##242##46##) thatTherefore, the quantityis in fact constant with time, and we can label it by . This implies that an arbitrary initial density with defined over a region of low (high) will ‘shrink’ (‘expand’) if it moves to a region of high (low) . Lastly, Eqs. (##FORMU##274##51##) and (##FORMU##294##55##) imply thatWe are now ready to prove the subquantum <italic>H</italic>-theorem for . We first subdivide the configuration space into small cells of volume . We then define the coarse-grained quantitieswhere the integral is performed over the cell which contains . Clearly, and are constant in each cell. We define the quantityand its coarse-grained version if , where of . Subtracting (##FORMU##288##53##) from (##FORMU##287##52##) and using the definition of , we havewhich is analogous to Eq. (##FORMU##239##45##). We define the coarse-grained version of to beAnalogous to the <italic>H</italic>-theorems for classical statistical equilibrium^{##UREF##24##25##} and for quantum equilibrium^{##UREF##17##18##}, we assume that there is no initial fine-grained structure, that is,Let us considerUsing the initial conditions (##FORMU##320##63##) and (##FORMU##321##64##), and the fact that is constant with time, we can simplify the first term in RHS of (##FORMU##322##65##) asThe second term in RHS of (##FORMU##322##65##) can be written aswhere the integral over has been broken up into integrals over each cell of volume . As and are constant over these cells, we can write , and if belongs to the cell. It then follows thatwhere, in Eq. (##FORMU##337##70##), we have used the relation (##FORMU##320##63##). Using (##FORMU##325##67##) and (##FORMU##338##71##), we can rewrite (##FORMU##322##65##) asWe note thatUsing (##FORMU##344##77##), we can rewrite Eq. (##FORMU##340##73##) asUsing the identity for all real <italic>x</italic>, <italic>y</italic>, it is then clear from Eq. (##FORMU##345##78##) that . We have, therefore, proven an <italic>H</italic>-theorem for , subject to assumptions similar to those assumed for relaxation to quantum equilibrium.</p>", "<title>Relationship between relaxation to pilot-wave equilibrium and to quantum equilibrium</title>", "<p id=\"Par37\">Although the <italic>H</italic>-theorem for gives the theoretical basis for relaxation to pilot-wave equilibrium, we need numerical evidence to determine whether relaxation in fact occurs. There exists a large body of results in the literature on the numerical evidence for relaxation to quantum equilibrium for normalizable states. It is, therefore, of interest to understand the relation between relaxation to pilot-wave equilibrium for non-normalizable states and relaxation to quantum equilibrium for normalizable states, if any.</p>", "<p id=\"Par38\">We begin by noting that Eq. (##FORMU##305##58##) can be written aswhere and . From Eqs. (##FORMU##318##61##) and (##FORMU##351##80##), we can then derivewhereIt is clear from (##FORMU##354##81##) that the lower bound of is , corresponding to pilot-wave equilibrium . The relationship (##FORMU##354##81##) implies that a study of the behaviour of is equivalent to that of . It now remains to recast this study in terms of normalizable states.</p>", "<p id=\"Par39\">Consider the non-normalizable quantum state from Eq. (##FORMU##185##36##). We know that the velocity field at large . Suppose a number <italic>L</italic> sufficiently large such that is very small at , then an initial distribution localised in the region cannot escape to for an arbitrarily long time (depending on the value of <italic>L</italic> chosen). This implies that we effectively need only for to know how evolves in the direction. We can utilise this feature of the velocity field to define a normalizable quantum state with the same velocity field in the region as that of the non-normalizable quantum state.</p>", "<p id=\"Par40\">Let us define the normalizable quantum statewhere is the Heaviside-step function, <italic>m</italic> is a positive integer and <italic>L</italic> is a very large constant such that is very small at for all . We know that is normalizable as at large for all . Clearly, we can replace by to evolve if has an initial support . The evolution of itself is non-unitary as . This is because is numerically, but not <italic>functionally</italic>, equal to in the subset . Therefore, we can study relaxation to pilot-wave equilibrium using normalizable states, but doing so would require non-unitary dynamics. A complete relaxation to pilot-wave equilibrium would correspond to a partial relaxation to quantum equilibrium (see Fig. ##FIG##3##4##).</p>", "<title>Theoretical and experimental implications</title>", "<p id=\"Par41\">In this section, we sketch the theoretical and experimental implications of our work. Although we have focused on the harmonic oscillator, the general approach adopted in this paper and the notion of pilot-wave equilibrium introduced are not exclusive to the harmonic oscillator. Therefore, where applicable, we discuss the implications in the broader context of non-normalizable quantum states with a normalized density of configurations.</p>", "<title>Non-relativistic quantum theory</title>", "<title>Experimental observation of continuous-energy eigenstates</title>", "<p id=\"Par42\">We have seen that pilot-wave theory gives a physical interpretation for non-normalizable harmonic oscillator states as bound states. However, such states have continuous energies and have never been experimentally observed. Does this directly falsify pilot-wave theory in favour of orthodox quantum mechanics?</p>", "<p id=\"Par43\">We first note that unitarity imposes restrictions on preparation of non-normalizable states in a laboratory. This is because, if the initial joint quantum state of the preparation apparatus (including all the atoms of all the equipments etc.) is normalizable, then the joint quantum state will remain normalizable after the preparation is completed. The argument can be repeated to conclude that non-normalizable states can be potentially detected today only if there existed non-normalizable states in the early universe.</p>", "<p id=\"Par44\">Consider an atom in the early universe in a non-normalizable eigenstate , where <italic>K</italic> is continuous. The atom will, in general, be subject to small perturbations across the universe. It can be shown, from time-dependent perturbation theory, that the quantum state will evolve asup to first order in , where is the unperturbed Hamiltonian of the atom. Note that, as the Dyson series does not assume state normalizability^{##UREF##25##26##}, Eq. (##FORMU##401##84##) is valid for . Let us consider realistic perturbations that are small and localised in space. That is, suppose the perturbations are of the approximate formso that they rapidly fall off around . Then, using the fact that is an eigenstate, we can write the integrand in (##FORMU##401##84##) asas is square integrable (although is not) and can be expanded in terms of the normalizable eigenstates of . Note that a perturbation arbitrarily distant from the atom is sufficient to make square integrable, given that falls off rapidly. Therefore, for realistic perturbations Eq. (##FORMU##401##84##) becomesso that the quantum state becomes a superposition of the non-normalizable and the normalizable ’s. If the atom now interacts strongly with the environment to cause an effective energy-measurement, then the possible eigenvalues are the discrete energies as well as the continuous energy . Using the von-Neumann measurement^{##UREF##26##27##} Hamiltonian , we can represent the combined state of the atom and an idealised pointer variable after such a measurement to bewhere <italic>g</italic> is the interaction constant, is the pointer state, and is used to represent both and in the superposition (##FORMU##417##87##). The probabilities will not be given by the Born rule as is non-normalizable, but will have to be computed from the normalized probability density . Note that decoherence will effectively occur as long as the pointer wavefunction is normalizable. Further interactions with macroscopic bodies will cause further decoherence^{##UREF##3##4##}, so that the measurement will be effectively irreversible as for normalizable quantum states.</p>", "<p id=\"Par45\">Therefore the atom, on account of perturbations and interactions with environment, may transition to a normalizable energy eigenstate. In that case, the total quantum state will remain non normalizable but the system configuration will enter an effectively-decohered normalizable branch. After <italic>N</italic> such measurements, the fraction that remains in the non-normalizable branch will be given bywhere the fraction lost to the normalizable branches in the j-th measurement is labelled by . Clearly, as unless \n where is some positive integer. The condition \n is possible if the initial density, the initial joint quantum state of the atom and the idealised measurement apparatus, and the perturbations are so finely tuned that the configuration density remains completely in the non-normalizable branch for all . Without such fine tuning, the probability of the atom remaining in becomes tiny after a sufficiently long time corresponding to a large <italic>N</italic>. Note the key role played by perturbations here as they continuously add superpositions of normalizable eigenstates to the total quantum state. Therefore, we would not in general expect non-normalizable states in the early universe to have survived to the present time. Further technical work is required to ascertain the survival timescales for various non-normalizable states and perturbations.</p>", "<title>Signalling and pilot-wave equilibrium</title>", "<p id=\"Par46\">We know that no-signalling is generally violated in quantum nonequilibrium^{##UREF##27##28##}. Given that quantum equilibrium (when applicable) is a special case of pilot-wave equilibrium, it is of interest to understand the signalling behaviour of ensembles in pilot-wave equilibrium. This is important to understand whether non-normalizable states in pilot-wave equilibrium are no-signalling. Below, we show that no-signalling is violated generally in pilot-wave equilibrium.</p>", "<p id=\"Par47\">Consider an initial two-particle entangled quantum state , where the two particles are located in space-like separated wings. Suppose an initial density with the support where are very small. Then . The density on is in pilot-wave equilibrium by definition.</p>", "<p id=\"Par48\">Suppose evolves under the Hamiltonian . The question is whether the marginal density of is affected by the distant local Hamiltonian under the control of the experimenter at the second wing. We know that, since is entangled, the velocity of the first particle will depend on and thereby on . Furthermore, in the limit , and the initial marginal density of the first particle will be . It is then clear that, since depends on and contains only the point , will depend on . The statistics of a position measurement performed at the first wing at time <italic>t</italic> will then depend on the Hamiltonian chosen by the experimenter at the second wing. We conclude that, in general, correlations generated by an ensemble in pilot-wave equilibrium are signalling, unless the ensemble is also in quantum equilibrium. As there is no notion of quantum equilibrium for non-normalizable states, we conclude that non-normalizable states generate signalling correlations in general.</p>", "<title>Quantum field theory</title>", "<p id=\"Par49\">We know that quantum fields can often be treated as a collection of harmonic oscillators^{##UREF##28##29##}. For illustration, let us consider the pilot-wave treatment^{##UREF##29##30##} of a free, massless real scalar field on a flat expanding space-time, with the Lagrangian density , where is the scale factor and for simplicity. The functional Schrodinger equation for this system iswhere is the quantum state defined over the configuration space , \n are real variables related to the Fourier-transform of byand is the canonical momentum. Here <italic>V</italic> is the box-normalization volume. Note that Eq. (##FORMU##470##90##) assumes a regularization so that a finite (but arbitrarily large) number of can be considered.</p>", "<p id=\"Par50\">Equation (##FORMU##470##90##) clearly shows that can be treated as a collection of independent harmonic oscillators in the Fourier space. Notably, although the field is assumed to have a Fourier-transform, we need not make the same assumption about which is piloting . Therefore, we can consider the non-normalizable solutions to (##FORMU##470##90##) explored in this paper. Such solutions may have implications in cosmological settings^{##UREF##12##13##,##UREF##29##30##}.</p>", "<title>Quantum gravity</title>", "<p id=\"Par51\">It is well known that non-normalizable quantum states are often encountered in quantum gravity^{##UREF##20##21##,##UREF##21##22##,##UREF##30##31##}. Such states are also encountered when pilot-wave dynamics is formulated on shape space, where a different approach to the problem of non-normalizability from a pilot-wave perspective has been explored^{##UREF##31##32##}. Recently, Valentini has argued for a pilot-wave approach to quantum gravity where statistical predictions are derived from a normalized configuration density^{##UREF##32##33##}. This is close to the approach adopted in our work, but there are several important differences. It is useful to discuss the implications of our work for quantum gravity in the context of ref.^{##UREF##32##33##}.</p>", "<p id=\"Par52\">First, ref.^{##UREF##32##33##} argues that there is no physical equilibrium density for non-normalizable quantum states, on the basis that the lower bound of diverges to . However, this argument has multiple flaws. Firstly, the lower bound of diverges only in the particular case where the support of the configuration density is the entire configuration space , that is . For all other cases the lower bound of is , as can be seen from Eq. (##FORMU##274##51##). Secondly, we have argued that, for non-normalizable quantum states, the notion of quantum equilibrium must be replaced by the more general notion of pilot-wave equilibrium. Correspondingly, must be replaced by to define a physical equilibrium density. Therefore, our results imply that some form of the Born rule arises as a physical equilibrium density for non-normalizable states.</p>", "<p id=\"Par53\">Second, ref.^{##UREF##32##33##} has emphasised that non-normalizability of the quantum state is due to the “deep physical reason” that the Wheeler-DeWitt equation on configuration space has a Klein-Gordon-like structure. In our approach on the other hand, there is no special role played by the structure of any particular equation. We have argued that non-normalizability is intrinsic to pilot-wave theory – only a normalized configuration density is needed to obtain statistical predictions. The quantum state, which defines the evolution of the configuration, need not be normalizable. Therefore, non-normalizable quantum states naturally follow from the first principles of the theory and the structure of the Wheeler-Dewitt equation can only play a technical role. This implies that non-normalizable solutions to the Schrodinger equation or Dirac equation are as valid from a pilot-wave perspective, where applicable, as that to the Wheeler-DeWitt equation.</p>", "<title>Supplementary Information</title>", "<p>\n</p>" ]

[ "<title>Supplementary Information</title>", "<p>The online version contains supplementary material available at 10.1038/s41598-023-50814-w.</p>", "<title>Acknowledgements</title>", "<p>I am thankful to Matt Leifer for encouragement and several helpful discussions. I am also thankful to Siddhant Das and Tathagata Karmakar for helpful discussions. The author was supported by a fellowship from the Grand Challenges Initiative at Chapman University.</p>", "<title>Author contributions</title>", "<p>I.S. is the sole author.</p>", "<title>Data availability</title>", "<p>The datasets used and/or analysed during the current study available from the corresponding author on reasonable request.</p>", "<title>Competing interests</title>", "<p id=\"Par62\">The author declares no competing interests.</p>" ]

[ "<fig id=\"Fig1\"><label>Figure 1</label><caption><p>Schematic illustration of (<bold>a</bold>) and (<bold>b</bold>) <italic>v</italic>(<italic>y</italic>) for the sample non-normalizable eigenstate . Note that at large .</p></caption></fig>", "<fig id=\"Fig2\"><label>Figure 2</label><caption><p>Schematic illustration of (<bold>a</bold>) and (<bold>b</bold>) <italic>v</italic>(<italic>y</italic>) for a sample superposition . Note that at large .</p></caption></fig>", "<fig id=\"Fig3\"><label>Figure 3</label><caption><p>Schematic illustration of (<bold>a</bold>) density plot for , (<bold>b</bold>) velocity plot for , (<bold>c</bold>) -velocity field and (<bold>d</bold>) -velocity field for a sample superposition . Note, from figures (<bold>b</bold>), (<bold>c</bold>) and (<bold>d</bold>), that at large and at large .</p></caption></fig>", "<fig id=\"Fig4\"><label>Figure 4</label><caption><p>Schematic illustration of the relationship between quantum equilibrium (Q eq) and the notion of pilot-wave equilibrium (PW eq) introduced in this paper. Given a normalizable quantum state , there is only a single density that is defined to be in quantum equilibrium (depicted as the dark red dot). On the other hand, there is an infinite number of densities that are in pilot-wave equilibrium (depicted as the light red region), corresponding to different subsets of the configuration space. Quantum equilibrium is a special case of pilot-wave equilibrium as depicted. For non-normalizable states, there is no density in quantum equilibrium (there is no red dot) but there are densities in pilot-wave equilibrium.</p></caption></fig>" ]

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\n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\pm y$$\\end{document}</tex-math><mml:math id=\"M4\"><mml:mrow><mml:mo>±</mml:mo><mml:mi>y</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq3\"><alternatives><tex-math id=\"M5\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$H_{pw}$$\\end{document}</tex-math><mml:math id=\"M6\"><mml:msub><mml:mi>H</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">pw</mml:mi></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq4\"><alternatives><tex-math id=\"M7\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$H_{pw}$$\\end{document}</tex-math><mml:math id=\"M8\"><mml:msub><mml:mi>H</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">pw</mml:mi></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq5\"><alternatives><tex-math id=\"M9\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$H_{pw}$$\\end{document}</tex-math><mml:math id=\"M10\"><mml:msub><mml:mi>H</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">pw</mml:mi></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq6\"><alternatives><tex-math id=\"M11\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$v_y \\rightarrow 0$$\\end{document}</tex-math><mml:math id=\"M12\"><mml:mrow><mml:msub><mml:mi>v</mml:mi><mml:mi>y</mml:mi></mml:msub><mml:mo stretchy=\"false\">→</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq7\"><alternatives><tex-math id=\"M13\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\pm y$$\\end{document}</tex-math><mml:math id=\"M14\"><mml:mrow><mml:mo>±</mml:mo><mml:mi>y</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq8\"><alternatives><tex-math id=\"M15\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$H_{pw}$$\\end{document}</tex-math><mml:math id=\"M16\"><mml:msub><mml:mi>H</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">pw</mml:mi></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq9\"><alternatives><tex-math id=\"M17\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$H_{pw}$$\\end{document}</tex-math><mml:math id=\"M18\"><mml:msub><mml:mi>H</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">pw</mml:mi></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq10\"><alternatives><tex-math id=\"M19\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\psi (y- gEt, 0)$$\\end{document}</tex-math><mml:math id=\"M20\"><mml:mrow><mml:mi>ψ</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>y</mml:mi><mml:mo>-</mml:mo><mml:mi>g</mml:mi><mml:mi>E</mml:mi><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:mn>0</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq11\"><alternatives><tex-math id=\"M21\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\psi (y- gEt, 0)$$\\end{document}</tex-math><mml:math id=\"M22\"><mml:mrow><mml:mi>ψ</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>y</mml:mi><mml:mo>-</mml:mo><mml:mi>g</mml:mi><mml:mi>E</mml:mi><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:mn>0</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq12\"><alternatives><tex-math id=\"M23\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$e^{-i{\\hat{H}}t/\\hbar }$$\\end{document}</tex-math><mml:math id=\"M24\"><mml:msup><mml:mi>e</mml:mi><mml:mrow><mml:mo>-</mml:mo><mml:mi>i</mml:mi><mml:mover accent=\"true\"><mml:mi>H</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">/</mml:mo><mml:mi>ħ</mml:mi></mml:mrow></mml:msup></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ1\"><label>1</label><alternatives><tex-math id=\"M25\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} {}-\\frac{d^2\\psi }{dy^2} + y^2\\psi = K\\psi \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M26\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:mrow/><mml:mo>-</mml:mo><mml:mfrac><mml:mrow><mml:msup><mml:mi>d</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mi>ψ</mml:mi></mml:mrow><mml:mrow><mml:mi>d</mml:mi><mml:msup><mml:mi>y</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mrow></mml:mfrac><mml:mo>+</mml:mo><mml:msup><mml:mi>y</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mi>ψ</mml:mi><mml:mo>=</mml:mo><mml:mi>K</mml:mi><mml:mi>ψ</mml:mi></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq13\"><alternatives><tex-math id=\"M27\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$y\\equiv \\sqrt{m\\omega /\\hbar }x$$\\end{document}</tex-math><mml:math id=\"M28\"><mml:mrow><mml:mi>y</mml:mi><mml:mo>≡</mml:mo><mml:msqrt><mml:mrow><mml:mi>m</mml:mi><mml:mi>ω</mml:mi><mml:mo stretchy=\"false\">/</mml:mo><mml:mi>ħ</mml:mi></mml:mrow></mml:msqrt><mml:mi>x</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq14\"><alternatives><tex-math id=\"M29\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$K \\equiv 2E/\\hbar \\omega$$\\end{document}</tex-math><mml:math id=\"M30\"><mml:mrow><mml:mi>K</mml:mi><mml:mo>≡</mml:mo><mml:mn>2</mml:mn><mml:mi>E</mml:mi><mml:mo stretchy=\"false\">/</mml:mo><mml:mi>ħ</mml:mi><mml:mi>ω</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq15\"><alternatives><tex-math id=\"M31\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$e^{-y^2/2}h^K(y)$$\\end{document}</tex-math><mml:math id=\"M32\"><mml:mrow><mml:msup><mml:mi>e</mml:mi><mml:mrow><mml:mo>-</mml:mo><mml:msup><mml:mi>y</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo stretchy=\"false\">/</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:msup><mml:mi>h</mml:mi><mml:mi>K</mml:mi></mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ2\"><label>2</label><alternatives><tex-math id=\"M33\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} \\frac{d^2 h^K}{dy^2} -2y\\frac{dh^K}{dy} + (K-1)h^K = 0 \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M34\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:mfrac><mml:mrow><mml:msup><mml:mi>d</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:msup><mml:mi>h</mml:mi><mml:mi>K</mml:mi></mml:msup></mml:mrow><mml:mrow><mml:mi>d</mml:mi><mml:msup><mml:mi>y</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mrow></mml:mfrac><mml:mo>-</mml:mo><mml:mn>2</mml:mn><mml:mi>y</mml:mi><mml:mfrac><mml:mrow><mml:mi>d</mml:mi><mml:msup><mml:mi>h</mml:mi><mml:mi>K</mml:mi></mml:msup></mml:mrow><mml:mrow><mml:mi mathvariant=\"italic\">dy</mml:mi></mml:mrow></mml:mfrac><mml:mo>+</mml:mo><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>K</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:msup><mml:mi>h</mml:mi><mml:mi>K</mml:mi></mml:msup><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ3\"><label>3</label><alternatives><tex-math id=\"M35\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} h^K(y)&amp;= \\sum _{n=0}^{\\infty } a_n y^n \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M36\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:msup><mml:mi>h</mml:mi><mml:mi>K</mml:mi></mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:mtd><mml:mtd columnalign=\"left\"><mml:mrow><mml:mo>=</mml:mo><mml:munderover><mml:mo>∑</mml:mo><mml:mrow><mml:mi>n</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow><mml:mi>∞</mml:mi></mml:munderover><mml:msub><mml:mi>a</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:msup><mml:mi>y</mml:mi><mml:mi>n</mml:mi></mml:msup></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq16\"><alternatives><tex-math id=\"M37\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$a_0$$\\end{document}</tex-math><mml:math id=\"M38\"><mml:msub><mml:mi>a</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq17\"><alternatives><tex-math id=\"M39\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$a_1$$\\end{document}</tex-math><mml:math id=\"M40\"><mml:msub><mml:mi>a</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq18\"><alternatives><tex-math id=\"M41\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$a_n$$\\end{document}</tex-math><mml:math id=\"M42\"><mml:msub><mml:mi>a</mml:mi><mml:mi>n</mml:mi></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq19\"><alternatives><tex-math id=\"M43\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$a_{n+2} = (2n+1-K)a_n/(n+1)(n+2)$$\\end{document}</tex-math><mml:math id=\"M44\"><mml:mrow><mml:msub><mml:mi>a</mml:mi><mml:mrow><mml:mi>n</mml:mi><mml:mo>+</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>2</mml:mn><mml:mi>n</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo>-</mml:mo><mml:mi>K</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:msub><mml:mi>a</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mo stretchy=\"false\">/</mml:mo><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>n</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>n</mml:mi><mml:mo>+</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ4\"><label>4</label><alternatives><tex-math id=\"M45\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} h^K(y)&amp;= a_0 \\sum _{n=0}^{\\infty } \\frac{a_{2n}}{a_0} y^{2n} + a_1 \\sum _{n=0}^{\\infty } \\frac{a_{2n+1}}{a_1} y^{2n+1} \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M46\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:msup><mml:mi>h</mml:mi><mml:mi>K</mml:mi></mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:mtd><mml:mtd columnalign=\"left\"><mml:mrow><mml:mo>=</mml:mo><mml:msub><mml:mi>a</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:munderover><mml:mo>∑</mml:mo><mml:mrow><mml:mi>n</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow><mml:mi>∞</mml:mi></mml:munderover><mml:mfrac><mml:msub><mml:mi>a</mml:mi><mml:mrow><mml:mn>2</mml:mn><mml:mi>n</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mi>a</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:mfrac><mml:msup><mml:mi>y</mml:mi><mml:mrow><mml:mn>2</mml:mn><mml:mi>n</mml:mi></mml:mrow></mml:msup><mml:mo>+</mml:mo><mml:msub><mml:mi>a</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:munderover><mml:mo>∑</mml:mo><mml:mrow><mml:mi>n</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow><mml:mi>∞</mml:mi></mml:munderover><mml:mfrac><mml:msub><mml:mi>a</mml:mi><mml:mrow><mml:mn>2</mml:mn><mml:mi>n</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mi>a</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:mfrac><mml:msup><mml:mi>y</mml:mi><mml:mrow><mml:mn>2</mml:mn><mml:mi>n</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ5\"><label>5</label><alternatives><tex-math id=\"M47\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned}&amp;= a_0 (1+ \\sum _{n=1}^{\\infty } \\frac{\\prod _{j=0}^{n-1} (4j+1-K)}{(2n)!} y^{2n}) + a_1 (y + \\sum _{n=1}^{\\infty } \\frac{\\prod _{j=0}^{n-1} (4j+3-K)}{(2n+1)!} y^{2n+1}) \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M48\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd/><mml:mtd columnalign=\"left\"><mml:mrow><mml:mo>=</mml:mo><mml:msub><mml:mi>a</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>1</mml:mn><mml:mo>+</mml:mo><mml:munderover><mml:mo>∑</mml:mo><mml:mrow><mml:mi>n</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mi>∞</mml:mi></mml:munderover><mml:mfrac><mml:mrow><mml:msubsup><mml:mo>∏</mml:mo><mml:mrow><mml:mi>j</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>4</mml:mn><mml:mi>j</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo>-</mml:mo><mml:mi>K</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>2</mml:mn><mml:mi>n</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>!</mml:mo></mml:mrow></mml:mfrac><mml:msup><mml:mi>y</mml:mi><mml:mrow><mml:mn>2</mml:mn><mml:mi>n</mml:mi></mml:mrow></mml:msup><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>+</mml:mo><mml:msub><mml:mi>a</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>y</mml:mi><mml:mo>+</mml:mo><mml:munderover><mml:mo>∑</mml:mo><mml:mrow><mml:mi>n</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mi>∞</mml:mi></mml:munderover><mml:mfrac><mml:mrow><mml:msubsup><mml:mo>∏</mml:mo><mml:mrow><mml:mi>j</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>4</mml:mn><mml:mi>j</mml:mi><mml:mo>+</mml:mo><mml:mn>3</mml:mn><mml:mo>-</mml:mo><mml:mi>K</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>2</mml:mn><mml:mi>n</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>!</mml:mo></mml:mrow></mml:mfrac><mml:msup><mml:mi>y</mml:mi><mml:mrow><mml:mn>2</mml:mn><mml:mi>n</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ6\"><label>6</label><alternatives><tex-math id=\"M49\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned}&amp;= a_0 h_0^K(y) + a_1 h_1^K(y) \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M50\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd/><mml:mtd columnalign=\"left\"><mml:mrow><mml:mo>=</mml:mo><mml:msub><mml:mi>a</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:msubsup><mml:mi>h</mml:mi><mml:mn>0</mml:mn><mml:mi>K</mml:mi></mml:msubsup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>+</mml:mo><mml:msub><mml:mi>a</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:msubsup><mml:mi>h</mml:mi><mml:mn>1</mml:mn><mml:mi>K</mml:mi></mml:msubsup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq20\"><alternatives><tex-math id=\"M51\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$h_0^K = (1+ \\sum _{n=1}^{\\infty } \\frac{\\prod _{j=0}^{n-1} (4j+1-K)}{(2n)!} y^{2n})$$\\end{document}</tex-math><mml:math id=\"M52\"><mml:mrow><mml:msubsup><mml:mi>h</mml:mi><mml:mn>0</mml:mn><mml:mi>K</mml:mi></mml:msubsup><mml:mo>=</mml:mo><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>1</mml:mn><mml:mo>+</mml:mo><mml:msubsup><mml:mo>∑</mml:mo><mml:mrow><mml:mi>n</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mi>∞</mml:mi></mml:msubsup><mml:mfrac><mml:mrow><mml:msubsup><mml:mo>∏</mml:mo><mml:mrow><mml:mi>j</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>4</mml:mn><mml:mi>j</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo>-</mml:mo><mml:mi>K</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>2</mml:mn><mml:mi>n</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>!</mml:mo></mml:mrow></mml:mfrac><mml:msup><mml:mi>y</mml:mi><mml:mrow><mml:mn>2</mml:mn><mml:mi>n</mml:mi></mml:mrow></mml:msup><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq21\"><alternatives><tex-math id=\"M53\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$h_1^K = (y + \\sum _{n=1}^{\\infty } \\frac{\\prod _{j=0}^{n-1} (4j+3-K)}{(2n+1)!} y^{2n+1})$$\\end{document}</tex-math><mml:math id=\"M54\"><mml:mrow><mml:msubsup><mml:mi>h</mml:mi><mml:mn>1</mml:mn><mml:mi>K</mml:mi></mml:msubsup><mml:mo>=</mml:mo><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>y</mml:mi><mml:mo>+</mml:mo><mml:msubsup><mml:mo>∑</mml:mo><mml:mrow><mml:mi>n</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mi>∞</mml:mi></mml:msubsup><mml:mfrac><mml:mrow><mml:msubsup><mml:mo>∏</mml:mo><mml:mrow><mml:mi>j</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>4</mml:mn><mml:mi>j</mml:mi><mml:mo>+</mml:mo><mml:mn>3</mml:mn><mml:mo>-</mml:mo><mml:mi>K</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>2</mml:mn><mml:mi>n</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>!</mml:mo></mml:mrow></mml:mfrac><mml:msup><mml:mi>y</mml:mi><mml:mrow><mml:mn>2</mml:mn><mml:mi>n</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq22\"><alternatives><tex-math id=\"M55\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$h_0^K$$\\end{document}</tex-math><mml:math id=\"M56\"><mml:msubsup><mml:mi>h</mml:mi><mml:mn>0</mml:mn><mml:mi>K</mml:mi></mml:msubsup></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq23\"><alternatives><tex-math id=\"M57\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$h_1^K$$\\end{document}</tex-math><mml:math id=\"M58\"><mml:msubsup><mml:mi>h</mml:mi><mml:mn>1</mml:mn><mml:mi>K</mml:mi></mml:msubsup></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq24\"><alternatives><tex-math id=\"M59\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$h_0^K(y)$$\\end{document}</tex-math><mml:math id=\"M60\"><mml:mrow><mml:msubsup><mml:mi>h</mml:mi><mml:mn>0</mml:mn><mml:mi>K</mml:mi></mml:msubsup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq25\"><alternatives><tex-math id=\"M61\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$h_1^K(y)$$\\end{document}</tex-math><mml:math id=\"M62\"><mml:mrow><mml:msubsup><mml:mi>h</mml:mi><mml:mn>1</mml:mn><mml:mi>K</mml:mi></mml:msubsup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ7\"><label>7</label><alternatives><tex-math id=\"M63\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} h_0^K(y)&amp;= M(\\frac{1}{4}(1-K), \\frac{1}{2}, y^2) \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M64\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:msubsup><mml:mi>h</mml:mi><mml:mn>0</mml:mn><mml:mi>K</mml:mi></mml:msubsup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:mtd><mml:mtd columnalign=\"left\"><mml:mrow><mml:mo>=</mml:mo><mml:mi>M</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:mn>4</mml:mn></mml:mfrac><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>1</mml:mn><mml:mo>-</mml:mo><mml:mi>K</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>,</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:mn>2</mml:mn></mml:mfrac><mml:mo>,</mml:mo><mml:msup><mml:mi>y</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ8\"><label>8</label><alternatives><tex-math id=\"M65\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} h_1^K(y)&amp;= y M(\\frac{1}{4}(3-K), \\frac{3}{2}, y^2) \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M66\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:msubsup><mml:mi>h</mml:mi><mml:mn>1</mml:mn><mml:mi>K</mml:mi></mml:msubsup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:mtd><mml:mtd columnalign=\"left\"><mml:mrow><mml:mo>=</mml:mo><mml:mi>y</mml:mi><mml:mi>M</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:mn>4</mml:mn></mml:mfrac><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>3</mml:mn><mml:mo>-</mml:mo><mml:mi>K</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>,</mml:mo><mml:mfrac><mml:mn>3</mml:mn><mml:mn>2</mml:mn></mml:mfrac><mml:mo>,</mml:mo><mml:msup><mml:mi>y</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ9\"><label>9</label><alternatives><tex-math id=\"M67\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} M(c, d, y) \\equiv \\sum _{j=0}^{\\infty } \\frac{(c)_j}{(d)_j} \\frac{y^j}{j!} \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M68\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:mi>M</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>c</mml:mi><mml:mo>,</mml:mo><mml:mi>d</mml:mi><mml:mo>,</mml:mo><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>≡</mml:mo><mml:munderover><mml:mo>∑</mml:mo><mml:mrow><mml:mi>j</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow><mml:mi>∞</mml:mi></mml:munderover><mml:mfrac><mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>c</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mi>j</mml:mi></mml:msub><mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>d</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mi>j</mml:mi></mml:msub></mml:mfrac><mml:mfrac><mml:msup><mml:mi>y</mml:mi><mml:mi>j</mml:mi></mml:msup><mml:mrow><mml:mi>j</mml:mi><mml:mo>!</mml:mo></mml:mrow></mml:mfrac></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq26\"><alternatives><tex-math id=\"M69\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$(t)_j \\equiv \\Gamma (t+j)/\\Gamma (t)$$\\end{document}</tex-math><mml:math id=\"M70\"><mml:mrow><mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mi>j</mml:mi></mml:msub><mml:mo>≡</mml:mo><mml:mi mathvariant=\"normal\">Γ</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo>+</mml:mo><mml:mi>j</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo stretchy=\"false\">/</mml:mo><mml:mi mathvariant=\"normal\">Γ</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ10\"><label>10</label><alternatives><tex-math id=\"M71\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} \\psi ^K(y) = a_0 \\varphi _0^K(y) + a_1 \\varphi _1^K(y) \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M72\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:msup><mml:mi>ψ</mml:mi><mml:mi>K</mml:mi></mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:msub><mml:mi>a</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:msubsup><mml:mi>φ</mml:mi><mml:mn>0</mml:mn><mml:mi>K</mml:mi></mml:msubsup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>+</mml:mo><mml:msub><mml:mi>a</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:msubsup><mml:mi>φ</mml:mi><mml:mn>1</mml:mn><mml:mi>K</mml:mi></mml:msubsup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq27\"><alternatives><tex-math id=\"M73\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\varphi _0^K \\equiv e^{-y^2/2} h_0^K(y)$$\\end{document}</tex-math><mml:math id=\"M74\"><mml:mrow><mml:msubsup><mml:mi>φ</mml:mi><mml:mn>0</mml:mn><mml:mi>K</mml:mi></mml:msubsup><mml:mo>≡</mml:mo><mml:msup><mml:mi>e</mml:mi><mml:mrow><mml:mo>-</mml:mo><mml:msup><mml:mi>y</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo stretchy=\"false\">/</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:msubsup><mml:mi>h</mml:mi><mml:mn>0</mml:mn><mml:mi>K</mml:mi></mml:msubsup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq28\"><alternatives><tex-math id=\"M75\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\varphi _1^K \\equiv e^{-y^2/2} h_1^K(y)$$\\end{document}</tex-math><mml:math id=\"M76\"><mml:mrow><mml:msubsup><mml:mi>φ</mml:mi><mml:mn>1</mml:mn><mml:mi>K</mml:mi></mml:msubsup><mml:mo>≡</mml:mo><mml:msup><mml:mi>e</mml:mi><mml:mrow><mml:mo>-</mml:mo><mml:msup><mml:mi>y</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo stretchy=\"false\">/</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:msubsup><mml:mi>h</mml:mi><mml:mn>1</mml:mn><mml:mi>K</mml:mi></mml:msubsup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq29\"><alternatives><tex-math id=\"M77\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$h_0^K(y)$$\\end{document}</tex-math><mml:math id=\"M78\"><mml:mrow><mml:msubsup><mml:mi>h</mml:mi><mml:mn>0</mml:mn><mml:mi>K</mml:mi></mml:msubsup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq30\"><alternatives><tex-math id=\"M79\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$h_1^K(y)$$\\end{document}</tex-math><mml:math id=\"M80\"><mml:mrow><mml:msubsup><mml:mi>h</mml:mi><mml:mn>1</mml:mn><mml:mi>K</mml:mi></mml:msubsup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq31\"><alternatives><tex-math id=\"M81\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$K = (2n+1)$$\\end{document}</tex-math><mml:math id=\"M82\"><mml:mrow><mml:mi>K</mml:mi><mml:mo>=</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>2</mml:mn><mml:mi>n</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq32\"><alternatives><tex-math id=\"M83\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\varphi _0^K(y)$$\\end{document}</tex-math><mml:math id=\"M84\"><mml:mrow><mml:msubsup><mml:mi>φ</mml:mi><mml:mn>0</mml:mn><mml:mi>K</mml:mi></mml:msubsup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq33\"><alternatives><tex-math id=\"M85\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\varphi _1^K(y)$$\\end{document}</tex-math><mml:math id=\"M86\"><mml:mrow><mml:msubsup><mml:mi>φ</mml:mi><mml:mn>1</mml:mn><mml:mi>K</mml:mi></mml:msubsup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq34\"><alternatives><tex-math id=\"M87\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$e^{-y^2/2}$$\\end{document}</tex-math><mml:math id=\"M88\"><mml:msup><mml:mi>e</mml:mi><mml:mrow><mml:mo>-</mml:mo><mml:msup><mml:mi>y</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo stretchy=\"false\">/</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq35\"><alternatives><tex-math id=\"M89\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\pm y$$\\end{document}</tex-math><mml:math id=\"M90\"><mml:mrow><mml:mo>±</mml:mo><mml:mi>y</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq36\"><alternatives><tex-math id=\"M91\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$K \\ne (2n+1)$$\\end{document}</tex-math><mml:math id=\"M92\"><mml:mrow><mml:mi>K</mml:mi><mml:mo>≠</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>2</mml:mn><mml:mi>n</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq37\"><alternatives><tex-math id=\"M93\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\varphi _0^K(y)$$\\end{document}</tex-math><mml:math id=\"M94\"><mml:mrow><mml:msubsup><mml:mi>φ</mml:mi><mml:mn>0</mml:mn><mml:mi>K</mml:mi></mml:msubsup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq38\"><alternatives><tex-math id=\"M95\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\varphi _1^K(y)$$\\end{document}</tex-math><mml:math id=\"M96\"><mml:mrow><mml:msubsup><mml:mi>φ</mml:mi><mml:mn>1</mml:mn><mml:mi>K</mml:mi></mml:msubsup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq39\"><alternatives><tex-math id=\"M97\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$e^{y^2/2}$$\\end{document}</tex-math><mml:math id=\"M98\"><mml:msup><mml:mi>e</mml:mi><mml:mrow><mml:msup><mml:mi>y</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo stretchy=\"false\">/</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq40\"><alternatives><tex-math id=\"M99\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\pm y$$\\end{document}</tex-math><mml:math id=\"M100\"><mml:mrow><mml:mo>±</mml:mo><mml:mi>y</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq41\"><alternatives><tex-math id=\"M101\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$a_0$$\\end{document}</tex-math><mml:math id=\"M102\"><mml:msub><mml:mi>a</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq42\"><alternatives><tex-math id=\"M103\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$a_1$$\\end{document}</tex-math><mml:math id=\"M104\"><mml:msub><mml:mi>a</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq43\"><alternatives><tex-math id=\"M105\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$|a_0|^2 + |a_1|^2 = 1$$\\end{document}</tex-math><mml:math id=\"M106\"><mml:mrow><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:msub><mml:mi>a</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:msup><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:msup><mml:mo>+</mml:mo><mml:msup><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo><mml:msub><mml:mi>a</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:msup><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq44\"><alternatives><tex-math id=\"M107\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$v(y) = j(y)/|\\psi (y)|^2$$\\end{document}</tex-math><mml:math id=\"M108\"><mml:mrow><mml:mi>v</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mi>j</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo stretchy=\"false\">/</mml:mo><mml:msup><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo><mml:mi>ψ</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:msup></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq45\"><alternatives><tex-math id=\"M109\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\psi (y) \\rightarrow \\alpha \\psi (y)$$\\end{document}</tex-math><mml:math id=\"M110\"><mml:mrow><mml:mi>ψ</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"false\">→</mml:mo><mml:mi>α</mml:mi><mml:mi>ψ</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq46\"><alternatives><tex-math id=\"M111\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\alpha$$\\end{document}</tex-math><mml:math id=\"M112\"><mml:mi>α</mml:mi></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ11\"><label>11</label><alternatives><tex-math id=\"M113\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} \\psi ^K_{\\theta , \\phi }(y) = \\cos {\\theta } \\varphi _0^K(y) + \\sin {\\theta } e^{i\\phi } \\varphi _1^K(y) \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M114\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:msubsup><mml:mi>ψ</mml:mi><mml:mrow><mml:mi>θ</mml:mi><mml:mo>,</mml:mo><mml:mi>ϕ</mml:mi></mml:mrow><mml:mi>K</mml:mi></mml:msubsup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mo>cos</mml:mo><mml:mi>θ</mml:mi><mml:msubsup><mml:mi>φ</mml:mi><mml:mn>0</mml:mn><mml:mi>K</mml:mi></mml:msubsup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>+</mml:mo><mml:mo>sin</mml:mo><mml:mi>θ</mml:mi><mml:msup><mml:mi>e</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>ϕ</mml:mi></mml:mrow></mml:msup><mml:msubsup><mml:mi>φ</mml:mi><mml:mn>1</mml:mn><mml:mi>K</mml:mi></mml:msubsup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq47\"><alternatives><tex-math id=\"M115\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\cos {\\theta } = |a_0|/\\sqrt{|a_0|^2 +|a_1|^2}$$\\end{document}</tex-math><mml:math id=\"M116\"><mml:mrow><mml:mrow><mml:mo>cos</mml:mo><mml:mi>θ</mml:mi><mml:mo>=</mml:mo><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:msub><mml:mi>a</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo><mml:mo stretchy=\"false\">/</mml:mo></mml:mrow><mml:msqrt><mml:mrow><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:msub><mml:mi>a</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:msup><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:msup><mml:mo>+</mml:mo><mml:msup><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo><mml:msub><mml:mi>a</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:msup></mml:mrow></mml:msqrt></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq48\"><alternatives><tex-math id=\"M117\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\sin {\\theta } = |a_1|/\\sqrt{|a_0|^2 +|a_1|^2}$$\\end{document}</tex-math><mml:math id=\"M118\"><mml:mrow><mml:mrow><mml:mo>sin</mml:mo><mml:mi>θ</mml:mi><mml:mo>=</mml:mo><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:msub><mml:mi>a</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo><mml:mo stretchy=\"false\">/</mml:mo></mml:mrow><mml:msqrt><mml:mrow><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:msub><mml:mi>a</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:msup><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:msup><mml:mo>+</mml:mo><mml:msup><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo><mml:msub><mml:mi>a</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:msup></mml:mrow></mml:msqrt></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq49\"><alternatives><tex-math id=\"M119\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\phi = -i\\ln (a_1 |a_0|/a_0|a_1|)$$\\end{document}</tex-math><mml:math id=\"M120\"><mml:mrow><mml:mi>ϕ</mml:mi><mml:mo>=</mml:mo><mml:mo>-</mml:mo><mml:mi>i</mml:mi><mml:mo>ln</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>a</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo stretchy=\"false\">|</mml:mo><mml:msub><mml:mi>a</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo stretchy=\"false\">|</mml:mo><mml:mo stretchy=\"false\">/</mml:mo><mml:msub><mml:mi>a</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo stretchy=\"false\">|</mml:mo><mml:msub><mml:mi>a</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo stretchy=\"false\">|</mml:mo><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq50\"><alternatives><tex-math id=\"M121\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\theta \\in [0,\\pi ]$$\\end{document}</tex-math><mml:math id=\"M122\"><mml:mrow><mml:mi>θ</mml:mi><mml:mo>∈</mml:mo><mml:mo stretchy=\"false\">[</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mi>π</mml:mi><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq51\"><alternatives><tex-math id=\"M123\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\phi \\in [0,2\\pi )$$\\end{document}</tex-math><mml:math id=\"M124\"><mml:mrow><mml:mi>ϕ</mml:mi><mml:mo>∈</mml:mo><mml:mo stretchy=\"false\">[</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mn>2</mml:mn><mml:mi>π</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq52\"><alternatives><tex-math id=\"M125\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\varphi _0^K(y)$$\\end{document}</tex-math><mml:math id=\"M126\"><mml:mrow><mml:msubsup><mml:mi>φ</mml:mi><mml:mn>0</mml:mn><mml:mi>K</mml:mi></mml:msubsup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq53\"><alternatives><tex-math id=\"M127\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\varphi _1^K(y)$$\\end{document}</tex-math><mml:math id=\"M128\"><mml:mrow><mml:msubsup><mml:mi>φ</mml:mi><mml:mn>1</mml:mn><mml:mi>K</mml:mi></mml:msubsup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq54\"><alternatives><tex-math id=\"M129\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$|\\psi (y)|^2$$\\end{document}</tex-math><mml:math id=\"M130\"><mml:msup><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo><mml:mi>ψ</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:msup></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq55\"><alternatives><tex-math id=\"M131\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$|\\psi (y)|^2$$\\end{document}</tex-math><mml:math id=\"M132\"><mml:msup><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo><mml:mi>ψ</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:msup></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq56\"><alternatives><tex-math id=\"M133\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$K(n) = (2n + 1)$$\\end{document}</tex-math><mml:math id=\"M134\"><mml:mrow><mml:mi>K</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>n</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>=</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>2</mml:mn><mml:mi>n</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq57\"><alternatives><tex-math id=\"M135\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\varphi _0^{K(n)}(y)$$\\end{document}</tex-math><mml:math id=\"M136\"><mml:mrow><mml:msubsup><mml:mi>φ</mml:mi><mml:mn>0</mml:mn><mml:mrow><mml:mi>K</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>n</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:msubsup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq58\"><alternatives><tex-math id=\"M137\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\varphi _1^{K(n)}(y)$$\\end{document}</tex-math><mml:math id=\"M138\"><mml:mrow><mml:msubsup><mml:mi>φ</mml:mi><mml:mn>1</mml:mn><mml:mrow><mml:mi>K</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>n</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:msubsup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq59\"><alternatives><tex-math id=\"M139\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\varphi _1^{K(n)}(y)$$\\end{document}</tex-math><mml:math id=\"M140\"><mml:mrow><mml:msubsup><mml:mi>φ</mml:mi><mml:mn>1</mml:mn><mml:mrow><mml:mi>K</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>n</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:msubsup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq60\"><alternatives><tex-math id=\"M141\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\varphi _0^{K(n)}(y)$$\\end{document}</tex-math><mml:math id=\"M142\"><mml:mrow><mml:msubsup><mml:mi>φ</mml:mi><mml:mn>0</mml:mn><mml:mrow><mml:mi>K</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>n</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:msubsup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ12\"><label>12</label><alternatives><tex-math id=\"M143\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} \\Psi _n(y) = {\\left\\{ \\begin{array}{ll} N_n \\varphi _0^{K(n)} &amp;{} \\text {, if}\\, n \\,\\text {even}\\\\ N_n \\varphi _1^{K(n)} &amp;{} \\text {, if}\\, n\\, \\text {odd} \\end{array}\\right. } \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M144\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:msub><mml:mi mathvariant=\"normal\">Ψ</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mfenced open=\"{\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"left\"><mml:mrow><mml:msub><mml:mi>N</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:msubsup><mml:mi>φ</mml:mi><mml:mn>0</mml:mn><mml:mrow><mml:mi>K</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>n</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:msubsup></mml:mrow></mml:mtd><mml:mtd columnalign=\"left\"><mml:mrow><mml:mrow/><mml:mtext>, if</mml:mtext><mml:mspace width=\"0.166667em\"/><mml:mi>n</mml:mi><mml:mspace width=\"0.166667em\"/><mml:mtext>even</mml:mtext></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign=\"left\"><mml:mrow><mml:mrow/><mml:msub><mml:mi>N</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:msubsup><mml:mi>φ</mml:mi><mml:mn>1</mml:mn><mml:mrow><mml:mi>K</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>n</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:msubsup></mml:mrow></mml:mtd><mml:mtd columnalign=\"left\"><mml:mrow><mml:mrow/><mml:mtext>, if</mml:mtext><mml:mspace width=\"0.166667em\"/><mml:mi>n</mml:mi><mml:mspace width=\"0.166667em\"/><mml:mtext>odd</mml:mtext></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq61\"><alternatives><tex-math id=\"M145\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\Psi _n(y)$$\\end{document}</tex-math><mml:math id=\"M146\"><mml:mrow><mml:msub><mml:mi mathvariant=\"normal\">Ψ</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq62\"><alternatives><tex-math id=\"M147\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$n^{th}$$\\end{document}</tex-math><mml:math id=\"M148\"><mml:msup><mml:mi>n</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">th</mml:mi></mml:mrow></mml:msup></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq63\"><alternatives><tex-math id=\"M149\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$N_n$$\\end{document}</tex-math><mml:math id=\"M150\"><mml:msub><mml:mi>N</mml:mi><mml:mi>n</mml:mi></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq64\"><alternatives><tex-math id=\"M151\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\psi (y) = \\sum _n c_n \\psi ^{K_n}_{\\theta _n, \\phi _n}(y)$$\\end{document}</tex-math><mml:math id=\"M152\"><mml:mrow><mml:mi>ψ</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:msub><mml:mo>∑</mml:mo><mml:mi>n</mml:mi></mml:msub><mml:msub><mml:mi>c</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:msubsup><mml:mi>ψ</mml:mi><mml:mrow><mml:msub><mml:mi>θ</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>ϕ</mml:mi><mml:mi>n</mml:mi></mml:msub></mml:mrow><mml:msub><mml:mi>K</mml:mi><mml:mi>n</mml:mi></mml:msub></mml:msubsup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq65\"><alternatives><tex-math id=\"M153\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\sum _n |c_n|^2 = 1$$\\end{document}</tex-math><mml:math id=\"M154\"><mml:mrow><mml:msub><mml:mo>∑</mml:mo><mml:mi>n</mml:mi></mml:msub><mml:msup><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo><mml:msub><mml:mi>c</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:msup><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq66\"><alternatives><tex-math id=\"M155\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\psi (y)$$\\end{document}</tex-math><mml:math id=\"M156\"><mml:mrow><mml:mi>ψ</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq67\"><alternatives><tex-math id=\"M157\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\psi (y)$$\\end{document}</tex-math><mml:math id=\"M158\"><mml:mrow><mml:mi>ψ</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ13\"><label>13</label><alternatives><tex-math id=\"M159\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} \\psi (y,t) = \\sum _n c_n e^{-i K_n\\omega t/2} \\psi ^{K_n}_{\\theta _n, \\phi _n}(y) \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M160\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:mi>ψ</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>y</mml:mi><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:munder><mml:mo>∑</mml:mo><mml:mi>n</mml:mi></mml:munder><mml:msub><mml:mi>c</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:msup><mml:mi>e</mml:mi><mml:mrow><mml:mo>-</mml:mo><mml:mi>i</mml:mi><mml:msub><mml:mi>K</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mi>ω</mml:mi><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">/</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:msubsup><mml:mi>ψ</mml:mi><mml:mrow><mml:msub><mml:mi>θ</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>ϕ</mml:mi><mml:mi>n</mml:mi></mml:msub></mml:mrow><mml:msub><mml:mi>K</mml:mi><mml:mi>n</mml:mi></mml:msub></mml:msubsup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ14\"><label>14</label><alternatives><tex-math id=\"M161\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} \\psi (y_1, y_2,...y_N) = \\sum _m c_m \\prod _{j=1}^N \\psi _{\\theta _j^m, \\phi _j^m}^{K_j^m}(y_j) \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M162\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:mi>ψ</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>y</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>y</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mo>,</mml:mo><mml:mo>.</mml:mo><mml:mo>.</mml:mo><mml:mo>.</mml:mo><mml:msub><mml:mi>y</mml:mi><mml:mi>N</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:munder><mml:mo>∑</mml:mo><mml:mi>m</mml:mi></mml:munder><mml:msub><mml:mi>c</mml:mi><mml:mi>m</mml:mi></mml:msub><mml:munderover><mml:mo>∏</mml:mo><mml:mrow><mml:mi>j</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mi>N</mml:mi></mml:munderover><mml:msubsup><mml:mi>ψ</mml:mi><mml:mrow><mml:msubsup><mml:mi>θ</mml:mi><mml:mi>j</mml:mi><mml:mi>m</mml:mi></mml:msubsup><mml:mo>,</mml:mo><mml:msubsup><mml:mi>ϕ</mml:mi><mml:mi>j</mml:mi><mml:mi>m</mml:mi></mml:msubsup></mml:mrow><mml:msubsup><mml:mi>K</mml:mi><mml:mi>j</mml:mi><mml:mi>m</mml:mi></mml:msubsup></mml:msubsup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>y</mml:mi><mml:mi>j</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq68\"><alternatives><tex-math id=\"M163\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$c_m$$\\end{document}</tex-math><mml:math id=\"M164\"><mml:msub><mml:mi>c</mml:mi><mml:mi>m</mml:mi></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq69\"><alternatives><tex-math id=\"M165\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\psi (y_1, y_2,...y_N)$$\\end{document}</tex-math><mml:math id=\"M166\"><mml:mrow><mml:mi>ψ</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>y</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>y</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mo>,</mml:mo><mml:mo>.</mml:mo><mml:mo>.</mml:mo><mml:mo>.</mml:mo><mml:msub><mml:mi>y</mml:mi><mml:mi>N</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq70\"><alternatives><tex-math id=\"M167\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\psi (y_1, y_2,...y_N)$$\\end{document}</tex-math><mml:math id=\"M168\"><mml:mrow><mml:mi>ψ</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>y</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>y</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mo>,</mml:mo><mml:mo>.</mml:mo><mml:mo>.</mml:mo><mml:mo>.</mml:mo><mml:msub><mml:mi>y</mml:mi><mml:mi>N</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ15\"><label>15</label><alternatives><tex-math id=\"M169\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} \\partial _t |\\psi ({\\overrightarrow{y}}, t)|^2 + \\vec {\\nabla }\\cdot \\vec{j}({\\overrightarrow{y}}, t) =0 \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M170\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:msub><mml:mi>∂</mml:mi><mml:mi>t</mml:mi></mml:msub><mml:msup><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo><mml:mi>ψ</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:msup><mml:mo>+</mml:mo><mml:mover accent=\"true\"><mml:mi mathvariant=\"normal\">∇</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>·</mml:mo><mml:mover accent=\"true\"><mml:mi>j</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq71\"><alternatives><tex-math id=\"M171\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\overrightarrow{y}} = (y_1, y_2,... y_N)$$\\end{document}</tex-math><mml:math id=\"M172\"><mml:mrow><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>=</mml:mo><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>y</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>y</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mo>,</mml:mo><mml:mo>.</mml:mo><mml:mo>.</mml:mo><mml:mo>.</mml:mo><mml:msub><mml:mi>y</mml:mi><mml:mi>N</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ16\"><label>16</label><alternatives><tex-math id=\"M173\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} \\vec{j}({\\overrightarrow{y}}, t) = \\frac{\\hbar }{2mi}\\big [{\\bar{\\psi }}({\\overrightarrow{y}}, t)\\vec {\\nabla } \\psi ({\\overrightarrow{y}}, t) - \\psi ({\\overrightarrow{y}}, t)\\vec {\\nabla } {\\bar{\\psi }}({\\overrightarrow{y}}, t) \\big ] \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M174\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:mover accent=\"true\"><mml:mi>j</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mfrac><mml:mi>ħ</mml:mi><mml:mrow><mml:mn>2</mml:mn><mml:mi>m</mml:mi><mml:mi>i</mml:mi></mml:mrow></mml:mfrac><mml:mrow><mml:mo maxsize=\"1.2em\" minsize=\"1.2em\" stretchy=\"true\">[</mml:mo></mml:mrow><mml:mover accent=\"true\"><mml:mrow><mml:mi>ψ</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">¯</mml:mo></mml:mrow></mml:mover><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mover accent=\"true\"><mml:mi mathvariant=\"normal\">∇</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mi>ψ</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>-</mml:mo><mml:mi>ψ</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mover accent=\"true\"><mml:mi mathvariant=\"normal\">∇</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mover accent=\"true\"><mml:mrow><mml:mi>ψ</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">¯</mml:mo></mml:mrow></mml:mover><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mrow><mml:mo maxsize=\"1.2em\" minsize=\"1.2em\" stretchy=\"true\">]</mml:mo></mml:mrow></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq72\"><alternatives><tex-math id=\"M175\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\vec {\\nabla } = \\sum _{i=1}^N {\\hat{y}}_i \\partial /\\partial y_i$$\\end{document}</tex-math><mml:math id=\"M176\"><mml:mrow><mml:mover accent=\"true\"><mml:mi mathvariant=\"normal\">∇</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>=</mml:mo><mml:msubsup><mml:mo>∑</mml:mo><mml:mrow><mml:mi>i</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mi>N</mml:mi></mml:msubsup><mml:msub><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover><mml:mi>i</mml:mi></mml:msub><mml:mi>∂</mml:mi><mml:mo stretchy=\"false\">/</mml:mo><mml:mi>∂</mml:mi><mml:msub><mml:mi>y</mml:mi><mml:mi>i</mml:mi></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq73\"><alternatives><tex-math id=\"M177\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\bar{\\psi }}({\\overrightarrow{y}}, t)$$\\end{document}</tex-math><mml:math id=\"M178\"><mml:mrow><mml:mover accent=\"true\"><mml:mrow><mml:mi>ψ</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">¯</mml:mo></mml:mrow></mml:mover><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq74\"><alternatives><tex-math id=\"M179\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\psi ({\\overrightarrow{y}}, t)$$\\end{document}</tex-math><mml:math id=\"M180\"><mml:mrow><mml:mi>ψ</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ17\"><label>17</label><alternatives><tex-math id=\"M181\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} \\vec{v}({\\overrightarrow{y}}, t) \\equiv \\frac{\\vec{j}({\\overrightarrow{y}}, t)}{|\\psi ({\\overrightarrow{y}}, t)|^2} \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M182\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:mover accent=\"true\"><mml:mi>v</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>≡</mml:mo><mml:mfrac><mml:mrow><mml:mover accent=\"true\"><mml:mi>j</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow><mml:mrow><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo><mml:mi>ψ</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:msup><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:msup></mml:mrow></mml:mfrac></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq75\"><alternatives><tex-math id=\"M183\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\rho ({\\overrightarrow{y}}, 0)$$\\end{document}</tex-math><mml:math id=\"M184\"><mml:mrow><mml:mi>ρ</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mn>0</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq76\"><alternatives><tex-math id=\"M185\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\rho ({\\overrightarrow{y}}, t)$$\\end{document}</tex-math><mml:math id=\"M186\"><mml:mrow><mml:mi>ρ</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ18\"><label>18</label><alternatives><tex-math id=\"M187\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} \\partial _t \\rho ({\\overrightarrow{y}}, t) + \\vec {\\nabla }\\cdot \\big (\\rho ({\\overrightarrow{y}}, t)\\vec{v}({\\overrightarrow{y}}, t)\\big ) =0 \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M188\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:msub><mml:mi>∂</mml:mi><mml:mi>t</mml:mi></mml:msub><mml:mi>ρ</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>+</mml:mo><mml:mover accent=\"true\"><mml:mi mathvariant=\"normal\">∇</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>·</mml:mo><mml:mrow><mml:mo maxsize=\"1.2em\" minsize=\"1.2em\" stretchy=\"true\">(</mml:mo></mml:mrow><mml:mi>ρ</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mover accent=\"true\"><mml:mi>v</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mrow><mml:mo maxsize=\"1.2em\" minsize=\"1.2em\" stretchy=\"true\">)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq77\"><alternatives><tex-math id=\"M189\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\rho ({\\overrightarrow{y}}, t)$$\\end{document}</tex-math><mml:math id=\"M190\"><mml:mrow><mml:mi>ρ</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq78\"><alternatives><tex-math id=\"M191\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$v_{y_i}({\\overrightarrow{y}}, t)$$\\end{document}</tex-math><mml:math id=\"M192\"><mml:mrow><mml:msub><mml:mi>v</mml:mi><mml:msub><mml:mi>y</mml:mi><mml:mi>i</mml:mi></mml:msub></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq79\"><alternatives><tex-math id=\"M193\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$y_i$$\\end{document}</tex-math><mml:math id=\"M194\"><mml:msub><mml:mi>y</mml:mi><mml:mi>i</mml:mi></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq80\"><alternatives><tex-math id=\"M195\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\sim y_i^{1+\\epsilon }$$\\end{document}</tex-math><mml:math id=\"M196\"><mml:mrow><mml:mo>∼</mml:mo><mml:msubsup><mml:mi>y</mml:mi><mml:mi>i</mml:mi><mml:mrow><mml:mn>1</mml:mn><mml:mo>+</mml:mo><mml:mi>ϵ</mml:mi></mml:mrow></mml:msubsup></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq81\"><alternatives><tex-math id=\"M197\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\epsilon &gt;0$$\\end{document}</tex-math><mml:math id=\"M198\"><mml:mrow><mml:mi>ϵ</mml:mi><mml:mo>&gt;</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq82\"><alternatives><tex-math id=\"M199\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$i \\in \\{1, 2,...N\\}$$\\end{document}</tex-math><mml:math id=\"M200\"><mml:mrow><mml:mi>i</mml:mi><mml:mo>∈</mml:mo><mml:mo stretchy=\"false\">{</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mo>.</mml:mo><mml:mo>.</mml:mo><mml:mo>.</mml:mo><mml:mi>N</mml:mi><mml:mo stretchy=\"false\">}</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq83\"><alternatives><tex-math id=\"M201\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$y_i \\rightarrow \\infty$$\\end{document}</tex-math><mml:math id=\"M202\"><mml:mrow><mml:msub><mml:mi>y</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo stretchy=\"false\">→</mml:mo><mml:mi>∞</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq84\"><alternatives><tex-math id=\"M203\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$|\\psi ({\\overrightarrow{y}}, t)|^2 \\rightarrow 0$$\\end{document}</tex-math><mml:math id=\"M204\"><mml:mrow><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo><mml:mi>ψ</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:msup><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:msup><mml:mo stretchy=\"false\">→</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq85\"><alternatives><tex-math id=\"M205\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$y_i \\rightarrow \\pm \\infty$$\\end{document}</tex-math><mml:math id=\"M206\"><mml:mrow><mml:msub><mml:mi>y</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo stretchy=\"false\">→</mml:mo><mml:mo>±</mml:mo><mml:mi>∞</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq86\"><alternatives><tex-math id=\"M207\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\psi ({\\overrightarrow{y}}, t)$$\\end{document}</tex-math><mml:math id=\"M208\"><mml:mrow><mml:mi>ψ</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq87\"><alternatives><tex-math id=\"M209\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$v_{y_i}({\\overrightarrow{y}}, t) \\rightarrow 0$$\\end{document}</tex-math><mml:math id=\"M210\"><mml:mrow><mml:msub><mml:mi>v</mml:mi><mml:msub><mml:mi>y</mml:mi><mml:mi>i</mml:mi></mml:msub></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo stretchy=\"false\">→</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq88\"><alternatives><tex-math id=\"M211\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$y_i \\rightarrow \\pm \\infty$$\\end{document}</tex-math><mml:math id=\"M212\"><mml:mrow><mml:msub><mml:mi>y</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo stretchy=\"false\">→</mml:mo><mml:mo>±</mml:mo><mml:mi>∞</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq89\"><alternatives><tex-math id=\"M213\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$i \\in \\{1, 2,...N\\}$$\\end{document}</tex-math><mml:math id=\"M214\"><mml:mrow><mml:mi>i</mml:mi><mml:mo>∈</mml:mo><mml:mo stretchy=\"false\">{</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mo>.</mml:mo><mml:mo>.</mml:mo><mml:mo>.</mml:mo><mml:mi>N</mml:mi><mml:mo stretchy=\"false\">}</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq90\"><alternatives><tex-math id=\"M215\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\psi ({\\overrightarrow{y}}, t)$$\\end{document}</tex-math><mml:math id=\"M216\"><mml:mrow><mml:mi>ψ</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq91\"><alternatives><tex-math id=\"M217\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\rho ({\\overrightarrow{y}}, 0)$$\\end{document}</tex-math><mml:math id=\"M218\"><mml:mrow><mml:mi>ρ</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mn>0</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq92\"><alternatives><tex-math id=\"M219\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\rho ({\\overrightarrow{y}}, t)$$\\end{document}</tex-math><mml:math id=\"M220\"><mml:mrow><mml:mi>ρ</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq93\"><alternatives><tex-math id=\"M221\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\rho ({\\overrightarrow{y}}, t) \\rightarrow 0$$\\end{document}</tex-math><mml:math id=\"M222\"><mml:mrow><mml:mi>ρ</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"false\">→</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq94\"><alternatives><tex-math id=\"M223\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$y_i \\rightarrow \\pm \\infty$$\\end{document}</tex-math><mml:math id=\"M224\"><mml:mrow><mml:msub><mml:mi>y</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo stretchy=\"false\">→</mml:mo><mml:mo>±</mml:mo><mml:mi>∞</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq95\"><alternatives><tex-math id=\"M225\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$i \\in \\{1, 2,...N\\}$$\\end{document}</tex-math><mml:math id=\"M226\"><mml:mrow><mml:mi>i</mml:mi><mml:mo>∈</mml:mo><mml:mo stretchy=\"false\">{</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mo>.</mml:mo><mml:mo>.</mml:mo><mml:mo>.</mml:mo><mml:mi>N</mml:mi><mml:mo stretchy=\"false\">}</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq96\"><alternatives><tex-math id=\"M227\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\overrightarrow{y}}$$\\end{document}</tex-math><mml:math id=\"M228\"><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq97\"><alternatives><tex-math id=\"M229\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\psi ^K_{\\theta , \\phi }(y)$$\\end{document}</tex-math><mml:math id=\"M230\"><mml:mrow><mml:msubsup><mml:mi>ψ</mml:mi><mml:mrow><mml:mi>θ</mml:mi><mml:mo>,</mml:mo><mml:mi>ϕ</mml:mi></mml:mrow><mml:mi>K</mml:mi></mml:msubsup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq98\"><alternatives><tex-math id=\"M231\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\Psi _n(y)$$\\end{document}</tex-math><mml:math id=\"M232\"><mml:mrow><mml:msub><mml:mi mathvariant=\"normal\">Ψ</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq99\"><alternatives><tex-math id=\"M233\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\psi ^K_{\\theta , \\phi }(y) = \\cos {\\theta } \\varphi _0^K(y) + \\sin {\\theta } e^{i\\phi } \\varphi _1^K(y)$$\\end{document}</tex-math><mml:math id=\"M234\"><mml:mrow><mml:msubsup><mml:mi>ψ</mml:mi><mml:mrow><mml:mi>θ</mml:mi><mml:mo>,</mml:mo><mml:mi>ϕ</mml:mi></mml:mrow><mml:mi>K</mml:mi></mml:msubsup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mo>cos</mml:mo><mml:mi>θ</mml:mi><mml:msubsup><mml:mi>φ</mml:mi><mml:mn>0</mml:mn><mml:mi>K</mml:mi></mml:msubsup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>+</mml:mo><mml:mo>sin</mml:mo><mml:mi>θ</mml:mi><mml:msup><mml:mi>e</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>ϕ</mml:mi></mml:mrow></mml:msup><mml:msubsup><mml:mi>φ</mml:mi><mml:mn>1</mml:mn><mml:mi>K</mml:mi></mml:msubsup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq100\"><alternatives><tex-math id=\"M235\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\psi (y)$$\\end{document}</tex-math><mml:math id=\"M236\"><mml:mrow><mml:mi>ψ</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ19\"><label>19</label><alternatives><tex-math id=\"M237\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} \\bar{\\psi} (y)\\psi '(y) -\\psi (y){\\bar{\\psi }}'(y) = c \\text { (constant)} \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M238\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:mover accent=\"true\"><mml:mrow><mml:mi>ψ</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">¯</mml:mo></mml:mrow></mml:mover><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:msup><mml:mi>ψ</mml:mi><mml:mo>′</mml:mo></mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>-</mml:mo><mml:mi>ψ</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:msup><mml:mrow><mml:mover accent=\"true\"><mml:mrow><mml:mi>ψ</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">¯</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mo>′</mml:mo></mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mi>c</mml:mi><mml:mspace width=\"0.333333em\"/><mml:mtext>(constant)</mml:mtext></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq101\"><alternatives><tex-math id=\"M239\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$c=0$$\\end{document}</tex-math><mml:math id=\"M240\"><mml:mrow><mml:mi>c</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq102\"><alternatives><tex-math id=\"M241\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\psi (y) \\rightarrow 0$$\\end{document}</tex-math><mml:math id=\"M242\"><mml:mrow><mml:mi>ψ</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"false\">→</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq103\"><alternatives><tex-math id=\"M243\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$y \\rightarrow \\infty$$\\end{document}</tex-math><mml:math id=\"M244\"><mml:mrow><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo><mml:mi>∞</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq104\"><alternatives><tex-math id=\"M245\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\psi (y) \\rightarrow \\infty$$\\end{document}</tex-math><mml:math id=\"M246\"><mml:mrow><mml:mi>ψ</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"false\">→</mml:mo><mml:mi>∞</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq105\"><alternatives><tex-math id=\"M247\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$y \\rightarrow \\infty$$\\end{document}</tex-math><mml:math id=\"M248\"><mml:mrow><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo><mml:mi>∞</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq106\"><alternatives><tex-math id=\"M249\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$y \\rightarrow \\infty$$\\end{document}</tex-math><mml:math id=\"M250\"><mml:mrow><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo><mml:mi>∞</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq107\"><alternatives><tex-math id=\"M251\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\psi ^K_{\\theta , \\phi }(y)$$\\end{document}</tex-math><mml:math id=\"M252\"><mml:mrow><mml:msubsup><mml:mi>ψ</mml:mi><mml:mrow><mml:mi>θ</mml:mi><mml:mo>,</mml:mo><mml:mi>ϕ</mml:mi></mml:mrow><mml:mi>K</mml:mi></mml:msubsup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq108\"><alternatives><tex-math id=\"M253\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$y=0$$\\end{document}</tex-math><mml:math id=\"M254\"><mml:mrow><mml:mi>y</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ20\"><label>20</label><alternatives><tex-math id=\"M255\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} \\varphi _0^K(0)&amp;= 1 \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M256\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:msubsup><mml:mi>φ</mml:mi><mml:mn>0</mml:mn><mml:mi>K</mml:mi></mml:msubsup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>0</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:mtd><mml:mtd columnalign=\"left\"><mml:mrow><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ21\"><label>21</label><alternatives><tex-math id=\"M257\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} \\varphi _1^K(0)&amp;= 0 \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M258\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:msubsup><mml:mi>φ</mml:mi><mml:mn>1</mml:mn><mml:mi>K</mml:mi></mml:msubsup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>0</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:mtd><mml:mtd columnalign=\"left\"><mml:mrow><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ22\"><label>22</label><alternatives><tex-math id=\"M259\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} \\frac{d\\varphi _0^K(0)}{dy}&amp;= 0 \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M260\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mfrac><mml:mrow><mml:mi>d</mml:mi><mml:msubsup><mml:mi>φ</mml:mi><mml:mn>0</mml:mn><mml:mi>K</mml:mi></mml:msubsup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>0</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow><mml:mrow><mml:mi mathvariant=\"italic\">dy</mml:mi></mml:mrow></mml:mfrac></mml:mtd><mml:mtd columnalign=\"left\"><mml:mrow><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ23\"><label>23</label><alternatives><tex-math id=\"M261\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} \\frac{d\\varphi _1^K(0)}{dy}&amp;= 1 \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M262\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mfrac><mml:mrow><mml:mi>d</mml:mi><mml:msubsup><mml:mi>φ</mml:mi><mml:mn>1</mml:mn><mml:mi>K</mml:mi></mml:msubsup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>0</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow><mml:mrow><mml:mi mathvariant=\"italic\">dy</mml:mi></mml:mrow></mml:mfrac></mml:mtd><mml:mtd columnalign=\"left\"><mml:mrow><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ24\"><label>24</label><alternatives><tex-math id=\"M263\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} {\\bar{\\psi }}^K_{\\theta , \\phi }(0)\\psi '^K_{\\theta , \\phi }(0) -\\psi ^K_{\\theta , \\phi }(0)\\bar{\\psi '}^K_{\\theta , \\phi }(0) = 2i\\cos {\\theta }\\sin {\\theta }\\sin \\phi = c \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M264\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:msubsup><mml:mrow><mml:mover accent=\"true\"><mml:mrow><mml:mi>ψ</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">¯</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mi>θ</mml:mi><mml:mo>,</mml:mo><mml:mi>ϕ</mml:mi></mml:mrow><mml:mi>K</mml:mi></mml:msubsup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>0</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:msubsup><mml:mi>ψ</mml:mi><mml:mrow><mml:mi>θ</mml:mi><mml:mo>,</mml:mo><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo><mml:mi>K</mml:mi></mml:mrow></mml:msubsup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>0</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>-</mml:mo><mml:msubsup><mml:mi>ψ</mml:mi><mml:mrow><mml:mi>θ</mml:mi><mml:mo>,</mml:mo><mml:mi>ϕ</mml:mi></mml:mrow><mml:mi>K</mml:mi></mml:msubsup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>0</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:msubsup><mml:mover accent=\"true\"><mml:mrow><mml:msup><mml:mi>ψ</mml:mi><mml:mo>′</mml:mo></mml:msup></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">¯</mml:mo></mml:mrow></mml:mover><mml:mrow><mml:mi>θ</mml:mi><mml:mo>,</mml:mo><mml:mi>ϕ</mml:mi></mml:mrow><mml:mi>K</mml:mi></mml:msubsup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>0</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mn>2</mml:mn><mml:mi>i</mml:mi><mml:mo>cos</mml:mo><mml:mi>θ</mml:mi><mml:mo>sin</mml:mo><mml:mi>θ</mml:mi><mml:mo>sin</mml:mo><mml:mi>ϕ</mml:mi><mml:mo>=</mml:mo><mml:mi>c</mml:mi></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq109\"><alternatives><tex-math id=\"M265\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$|\\psi (y)|^2$$\\end{document}</tex-math><mml:math id=\"M266\"><mml:msup><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo><mml:mi>ψ</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:msup></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq110\"><alternatives><tex-math id=\"M267\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\psi (y) = \\psi ^{14}_{16\\pi /5, 3\\pi /2}(y)$$\\end{document}</tex-math><mml:math id=\"M268\"><mml:mrow><mml:mi>ψ</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:msubsup><mml:mi>ψ</mml:mi><mml:mrow><mml:mn>16</mml:mn><mml:mi>π</mml:mi><mml:mo stretchy=\"false\">/</mml:mo><mml:mn>5</mml:mn><mml:mo>,</mml:mo><mml:mn>3</mml:mn><mml:mi>π</mml:mi><mml:mo stretchy=\"false\">/</mml:mo><mml:mn>2</mml:mn></mml:mrow><mml:mn>14</mml:mn></mml:msubsup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq111\"><alternatives><tex-math id=\"M269\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$v(y) \\rightarrow 0$$\\end{document}</tex-math><mml:math id=\"M270\"><mml:mrow><mml:mi>v</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"false\">→</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq112\"><alternatives><tex-math id=\"M271\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\pm y$$\\end{document}</tex-math><mml:math id=\"M272\"><mml:mrow><mml:mo>±</mml:mo><mml:mi>y</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ25\"><label>25</label><alternatives><tex-math id=\"M273\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} v(y, t)&amp;= \\frac{j(y)}{|\\psi ^K_{\\theta , \\phi }(y,t)|^2} \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M274\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:mi>v</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>y</mml:mi><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mtd><mml:mtd columnalign=\"left\"><mml:mrow><mml:mo>=</mml:mo><mml:mfrac><mml:mrow><mml:mi>j</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mrow><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:msubsup><mml:mi>ψ</mml:mi><mml:mrow><mml:mi>θ</mml:mi><mml:mo>,</mml:mo><mml:mi>ϕ</mml:mi></mml:mrow><mml:mi>K</mml:mi></mml:msubsup><mml:msup><mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>y</mml:mi><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:msup></mml:mrow></mml:mfrac></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ26\"><label>26</label><alternatives><tex-math id=\"M275\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned}&amp;= \\frac{\\hbar }{2mi} \\frac{{\\bar{\\psi }}^K_{\\theta , \\phi }(y,t)\\psi '^K_{\\theta , \\phi }(y,t) -\\psi ^K_{\\theta , \\phi }(y,t)\\bar{\\psi '}^K_{\\theta , \\phi }(y,t)}{|\\psi ^K_{\\theta , \\phi }(y,t)|^2} \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M276\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd/><mml:mtd columnalign=\"left\"><mml:mrow><mml:mo>=</mml:mo><mml:mfrac><mml:mi>ħ</mml:mi><mml:mrow><mml:mn>2</mml:mn><mml:mi>m</mml:mi><mml:mi>i</mml:mi></mml:mrow></mml:mfrac><mml:mfrac><mml:mrow><mml:msubsup><mml:mrow><mml:mover accent=\"true\"><mml:mrow><mml:mi>ψ</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">¯</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mi>θ</mml:mi><mml:mo>,</mml:mo><mml:mi>ϕ</mml:mi></mml:mrow><mml:mi>K</mml:mi></mml:msubsup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>y</mml:mi><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:msubsup><mml:mi>ψ</mml:mi><mml:mrow><mml:mi>θ</mml:mi><mml:mo>,</mml:mo><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo><mml:mi>K</mml:mi></mml:mrow></mml:msubsup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>y</mml:mi><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>-</mml:mo><mml:msubsup><mml:mi>ψ</mml:mi><mml:mrow><mml:mi>θ</mml:mi><mml:mo>,</mml:mo><mml:mi>ϕ</mml:mi></mml:mrow><mml:mi>K</mml:mi></mml:msubsup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>y</mml:mi><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:msubsup><mml:mover accent=\"true\"><mml:mrow><mml:msup><mml:mi>ψ</mml:mi><mml:mo>′</mml:mo></mml:msup></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">¯</mml:mo></mml:mrow></mml:mover><mml:mrow><mml:mi>θ</mml:mi><mml:mo>,</mml:mo><mml:mi>ϕ</mml:mi></mml:mrow><mml:mi>K</mml:mi></mml:msubsup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>y</mml:mi><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow><mml:mrow><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:msubsup><mml:mi>ψ</mml:mi><mml:mrow><mml:mi>θ</mml:mi><mml:mo>,</mml:mo><mml:mi>ϕ</mml:mi></mml:mrow><mml:mi>K</mml:mi></mml:msubsup><mml:msup><mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>y</mml:mi><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:msup></mml:mrow></mml:mfrac></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ27\"><label>27</label><alternatives><tex-math id=\"M277\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned}&amp;= \\frac{\\hbar }{2mi} \\frac{{\\bar{\\psi }}^K_{\\theta , \\phi }(0,t)\\psi '^K_{\\theta , \\phi }(0,t) -\\psi ^K_{\\theta , \\phi }(0,t)\\bar{\\psi '}^K_{\\theta , \\phi }(0,t)}{|\\psi ^K_{\\theta , \\phi }(y,t)|^2} \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M278\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd/><mml:mtd columnalign=\"left\"><mml:mrow><mml:mo>=</mml:mo><mml:mfrac><mml:mi>ħ</mml:mi><mml:mrow><mml:mn>2</mml:mn><mml:mi>m</mml:mi><mml:mi>i</mml:mi></mml:mrow></mml:mfrac><mml:mfrac><mml:mrow><mml:msubsup><mml:mrow><mml:mover accent=\"true\"><mml:mrow><mml:mi>ψ</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">¯</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mi>θ</mml:mi><mml:mo>,</mml:mo><mml:mi>ϕ</mml:mi></mml:mrow><mml:mi>K</mml:mi></mml:msubsup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:msubsup><mml:mi>ψ</mml:mi><mml:mrow><mml:mi>θ</mml:mi><mml:mo>,</mml:mo><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo><mml:mi>K</mml:mi></mml:mrow></mml:msubsup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>-</mml:mo><mml:msubsup><mml:mi>ψ</mml:mi><mml:mrow><mml:mi>θ</mml:mi><mml:mo>,</mml:mo><mml:mi>ϕ</mml:mi></mml:mrow><mml:mi>K</mml:mi></mml:msubsup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:msubsup><mml:mover accent=\"true\"><mml:mrow><mml:msup><mml:mi>ψ</mml:mi><mml:mo>′</mml:mo></mml:msup></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">¯</mml:mo></mml:mrow></mml:mover><mml:mrow><mml:mi>θ</mml:mi><mml:mo>,</mml:mo><mml:mi>ϕ</mml:mi></mml:mrow><mml:mi>K</mml:mi></mml:msubsup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow><mml:mrow><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:msubsup><mml:mi>ψ</mml:mi><mml:mrow><mml:mi>θ</mml:mi><mml:mo>,</mml:mo><mml:mi>ϕ</mml:mi></mml:mrow><mml:mi>K</mml:mi></mml:msubsup><mml:msup><mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>y</mml:mi><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:msup></mml:mrow></mml:mfrac></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ28\"><label>28</label><alternatives><tex-math id=\"M279\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned}&amp;= \\frac{\\hbar }{m}\\frac{\\cos {\\theta }\\sin {\\theta }\\sin \\phi }{|\\psi ^K_{\\theta , \\phi }(y,0)|^2} \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M280\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd/><mml:mtd columnalign=\"left\"><mml:mrow><mml:mo>=</mml:mo><mml:mfrac><mml:mi>ħ</mml:mi><mml:mi>m</mml:mi></mml:mfrac><mml:mfrac><mml:mrow><mml:mo>cos</mml:mo><mml:mi>θ</mml:mi><mml:mo>sin</mml:mo><mml:mi>θ</mml:mi><mml:mo>sin</mml:mo><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:msubsup><mml:mi>ψ</mml:mi><mml:mrow><mml:mi>θ</mml:mi><mml:mo>,</mml:mo><mml:mi>ϕ</mml:mi></mml:mrow><mml:mi>K</mml:mi></mml:msubsup><mml:msup><mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>y</mml:mi><mml:mo>,</mml:mo><mml:mn>0</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:msup></mml:mrow></mml:mfrac></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq113\"><alternatives><tex-math id=\"M281\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\psi ^K_{\\theta , \\phi }(y,t) = e^{-iKwt/2}\\psi ^K_{\\theta , \\phi }(y,0)$$\\end{document}</tex-math><mml:math id=\"M282\"><mml:mrow><mml:msubsup><mml:mi>ψ</mml:mi><mml:mrow><mml:mi>θ</mml:mi><mml:mo>,</mml:mo><mml:mi>ϕ</mml:mi></mml:mrow><mml:mi>K</mml:mi></mml:msubsup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>y</mml:mi><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:msup><mml:mi>e</mml:mi><mml:mrow><mml:mo>-</mml:mo><mml:mi>i</mml:mi><mml:mi>K</mml:mi><mml:mi>w</mml:mi><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">/</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:msubsup><mml:mi>ψ</mml:mi><mml:mrow><mml:mi>θ</mml:mi><mml:mo>,</mml:mo><mml:mi>ϕ</mml:mi></mml:mrow><mml:mi>K</mml:mi></mml:msubsup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>y</mml:mi><mml:mo>,</mml:mo><mml:mn>0</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq114\"><alternatives><tex-math id=\"M283\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\theta$$\\end{document}</tex-math><mml:math id=\"M284\"><mml:mi>θ</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq115\"><alternatives><tex-math id=\"M285\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\phi$$\\end{document}</tex-math><mml:math id=\"M286\"><mml:mi>ϕ</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq116\"><alternatives><tex-math id=\"M287\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$K = -K_0$$\\end{document}</tex-math><mml:math id=\"M288\"><mml:mrow><mml:mi>K</mml:mi><mml:mo>=</mml:mo><mml:mo>-</mml:mo><mml:msub><mml:mi>K</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq117\"><alternatives><tex-math id=\"M289\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$K_0 &gt;0$$\\end{document}</tex-math><mml:math id=\"M290\"><mml:mrow><mml:msub><mml:mi>K</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo>&gt;</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq118\"><alternatives><tex-math id=\"M291\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$K = +K_0$$\\end{document}</tex-math><mml:math id=\"M292\"><mml:mrow><mml:mi>K</mml:mi><mml:mo>=</mml:mo><mml:mo>+</mml:mo><mml:msub><mml:mi>K</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq119\"><alternatives><tex-math id=\"M293\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$|\\psi ^K_{\\theta , \\phi }(y,0)|^2$$\\end{document}</tex-math><mml:math id=\"M294\"><mml:mrow><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:msubsup><mml:mi>ψ</mml:mi><mml:mrow><mml:mi>θ</mml:mi><mml:mo>,</mml:mo><mml:mi>ϕ</mml:mi></mml:mrow><mml:mi>K</mml:mi></mml:msubsup><mml:msup><mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>y</mml:mi><mml:mo>,</mml:mo><mml:mn>0</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:msup></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq120\"><alternatives><tex-math id=\"M295\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$y \\rightarrow \\pm \\infty$$\\end{document}</tex-math><mml:math id=\"M296\"><mml:mrow><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo><mml:mo>±</mml:mo><mml:mi>∞</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ29\"><label>29</label><alternatives><tex-math id=\"M297\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} v(y, t) \\sim \\frac{\\hbar }{m}\\frac{\\cos {\\theta }\\sin {\\theta }\\sin \\phi }{e^{y^2}} \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M298\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:mi>v</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>y</mml:mi><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>∼</mml:mo><mml:mfrac><mml:mi>ħ</mml:mi><mml:mi>m</mml:mi></mml:mfrac><mml:mfrac><mml:mrow><mml:mo>cos</mml:mo><mml:mi>θ</mml:mi><mml:mo>sin</mml:mo><mml:mi>θ</mml:mi><mml:mo>sin</mml:mo><mml:mi>ϕ</mml:mi></mml:mrow><mml:msup><mml:mi>e</mml:mi><mml:msup><mml:mi>y</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:msup></mml:mfrac></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq121\"><alternatives><tex-math id=\"M299\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\psi ^K_{\\theta , \\phi }(y,t)$$\\end{document}</tex-math><mml:math id=\"M300\"><mml:mrow><mml:msubsup><mml:mi>ψ</mml:mi><mml:mrow><mml:mi>θ</mml:mi><mml:mo>,</mml:mo><mml:mi>ϕ</mml:mi></mml:mrow><mml:mi>K</mml:mi></mml:msubsup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>y</mml:mi><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq122\"><alternatives><tex-math id=\"M301\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\sim e^{y^2/2}$$\\end{document}</tex-math><mml:math id=\"M302\"><mml:mrow><mml:mo>∼</mml:mo><mml:msup><mml:mi>e</mml:mi><mml:mrow><mml:msup><mml:mi>y</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo stretchy=\"false\">/</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq123\"><alternatives><tex-math id=\"M303\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\pm y$$\\end{document}</tex-math><mml:math id=\"M304\"><mml:mrow><mml:mo>±</mml:mo><mml:mi>y</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq124\"><alternatives><tex-math id=\"M305\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$|\\psi ^K_{\\theta , \\phi }(y,0)|^2$$\\end{document}</tex-math><mml:math id=\"M306\"><mml:mrow><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:msubsup><mml:mi>ψ</mml:mi><mml:mrow><mml:mi>θ</mml:mi><mml:mo>,</mml:mo><mml:mi>ϕ</mml:mi></mml:mrow><mml:mi>K</mml:mi></mml:msubsup><mml:msup><mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>y</mml:mi><mml:mo>,</mml:mo><mml:mn>0</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:msup></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq125\"><alternatives><tex-math id=\"M307\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$y \\rightarrow \\pm \\infty$$\\end{document}</tex-math><mml:math id=\"M308\"><mml:mrow><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo><mml:mo>±</mml:mo><mml:mi>∞</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq126\"><alternatives><tex-math id=\"M309\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\psi ^K_{\\theta , \\phi }(y,0)$$\\end{document}</tex-math><mml:math id=\"M310\"><mml:mrow><mml:msubsup><mml:mi>ψ</mml:mi><mml:mrow><mml:mi>θ</mml:mi><mml:mo>,</mml:mo><mml:mi>ϕ</mml:mi></mml:mrow><mml:mi>K</mml:mi></mml:msubsup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>y</mml:mi><mml:mo>,</mml:mo><mml:mn>0</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq127\"><alternatives><tex-math id=\"M311\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\pm y$$\\end{document}</tex-math><mml:math id=\"M312\"><mml:mrow><mml:mo>±</mml:mo><mml:mi>y</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq128\"><alternatives><tex-math id=\"M313\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\psi (y, t) = \\sum _j c_j(t) \\psi ^{K_j}_{\\theta _j, \\phi _j}(y)$$\\end{document}</tex-math><mml:math id=\"M314\"><mml:mrow><mml:mi>ψ</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>y</mml:mi><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:msub><mml:mo>∑</mml:mo><mml:mi>j</mml:mi></mml:msub><mml:msub><mml:mi>c</mml:mi><mml:mi>j</mml:mi></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:msubsup><mml:mi>ψ</mml:mi><mml:mrow><mml:msub><mml:mi>θ</mml:mi><mml:mi>j</mml:mi></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>ϕ</mml:mi><mml:mi>j</mml:mi></mml:msub></mml:mrow><mml:msub><mml:mi>K</mml:mi><mml:mi>j</mml:mi></mml:msub></mml:msubsup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ30\"><label>30</label><alternatives><tex-math id=\"M315\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} v(y, t) = \\frac{\\hbar }{2mi} \\frac{{\\bar{\\psi }}(y,t)\\psi '(y,t) -\\psi (y,t)\\bar{\\psi '}(y,t)}{|\\psi (y,t)|^2} \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M316\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:mi>v</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>y</mml:mi><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mfrac><mml:mi>ħ</mml:mi><mml:mrow><mml:mn>2</mml:mn><mml:mi>m</mml:mi><mml:mi>i</mml:mi></mml:mrow></mml:mfrac><mml:mfrac><mml:mrow><mml:mover accent=\"true\"><mml:mrow><mml:mi>ψ</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">¯</mml:mo></mml:mrow></mml:mover><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>y</mml:mi><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:msup><mml:mi>ψ</mml:mi><mml:mo>′</mml:mo></mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>y</mml:mi><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>-</mml:mo><mml:mi>ψ</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>y</mml:mi><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mover accent=\"true\"><mml:mrow><mml:msup><mml:mi>ψ</mml:mi><mml:mo>′</mml:mo></mml:msup></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">¯</mml:mo></mml:mrow></mml:mover><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>y</mml:mi><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow><mml:msup><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo><mml:mi>ψ</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>y</mml:mi><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:msup></mml:mfrac></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq129\"><alternatives><tex-math id=\"M317\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$y \\rightarrow \\pm \\infty$$\\end{document}</tex-math><mml:math id=\"M318\"><mml:mrow><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo><mml:mo>±</mml:mo><mml:mi>∞</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq130\"><alternatives><tex-math id=\"M319\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\psi (y)$$\\end{document}</tex-math><mml:math id=\"M320\"><mml:mrow><mml:mi>ψ</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq131\"><alternatives><tex-math id=\"M321\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$y \\rightarrow \\pm \\infty$$\\end{document}</tex-math><mml:math id=\"M322\"><mml:mrow><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo><mml:mo>±</mml:mo><mml:mi>∞</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq132\"><alternatives><tex-math id=\"M323\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$|\\psi (y)|^2$$\\end{document}</tex-math><mml:math id=\"M324\"><mml:msup><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo><mml:mi>ψ</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:msup></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq133\"><alternatives><tex-math id=\"M325\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\psi (y) = 1/\\sqrt{6} \\psi ^{15.2}_{\\pi /3, \\pi /4} (y) + \\sqrt{2/3} e^{i\\pi /5} \\psi ^{5.8}_{\\pi /2, \\pi } (y) + 1/\\sqrt{6} e^{i\\pi /8} \\psi ^{10.2}_{\\pi /7, \\pi /5} (y)$$\\end{document}</tex-math><mml:math id=\"M326\"><mml:mrow><mml:mi>ψ</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">/</mml:mo><mml:msqrt><mml:mn>6</mml:mn></mml:msqrt><mml:msubsup><mml:mi>ψ</mml:mi><mml:mrow><mml:mi>π</mml:mi><mml:mo stretchy=\"false\">/</mml:mo><mml:mn>3</mml:mn><mml:mo>,</mml:mo><mml:mi>π</mml:mi><mml:mo stretchy=\"false\">/</mml:mo><mml:mn>4</mml:mn></mml:mrow><mml:mrow><mml:mn>15.2</mml:mn></mml:mrow></mml:msubsup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>+</mml:mo><mml:msqrt><mml:mrow><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">/</mml:mo><mml:mn>3</mml:mn></mml:mrow></mml:msqrt><mml:msup><mml:mi>e</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>π</mml:mi><mml:mo stretchy=\"false\">/</mml:mo><mml:mn>5</mml:mn></mml:mrow></mml:msup><mml:msubsup><mml:mi>ψ</mml:mi><mml:mrow><mml:mi>π</mml:mi><mml:mo stretchy=\"false\">/</mml:mo><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mi>π</mml:mi></mml:mrow><mml:mrow><mml:mn>5.8</mml:mn></mml:mrow></mml:msubsup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">/</mml:mo><mml:msqrt><mml:mn>6</mml:mn></mml:msqrt><mml:msup><mml:mi>e</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>π</mml:mi><mml:mo stretchy=\"false\">/</mml:mo><mml:mn>8</mml:mn></mml:mrow></mml:msup><mml:msubsup><mml:mi>ψ</mml:mi><mml:mrow><mml:mi>π</mml:mi><mml:mo stretchy=\"false\">/</mml:mo><mml:mn>7</mml:mn><mml:mo>,</mml:mo><mml:mi>π</mml:mi><mml:mo stretchy=\"false\">/</mml:mo><mml:mn>5</mml:mn></mml:mrow><mml:mrow><mml:mn>10.2</mml:mn></mml:mrow></mml:msubsup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq134\"><alternatives><tex-math id=\"M327\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$v(y) \\rightarrow 0$$\\end{document}</tex-math><mml:math id=\"M328\"><mml:mrow><mml:mi>v</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"false\">→</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq135\"><alternatives><tex-math id=\"M329\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\pm y$$\\end{document}</tex-math><mml:math id=\"M330\"><mml:mrow><mml:mo>±</mml:mo><mml:mi>y</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq136\"><alternatives><tex-math id=\"M331\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\psi (y,t) = \\sum _j c_j(t) \\psi ^{K_j}(y)$$\\end{document}</tex-math><mml:math id=\"M332\"><mml:mrow><mml:mi>ψ</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>y</mml:mi><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:msub><mml:mo>∑</mml:mo><mml:mi>j</mml:mi></mml:msub><mml:msub><mml:mi>c</mml:mi><mml:mi>j</mml:mi></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:msup><mml:mi>ψ</mml:mi><mml:msub><mml:mi>K</mml:mi><mml:mi>j</mml:mi></mml:msub></mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ31\"><label>31</label><alternatives><tex-math id=\"M333\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} j(y, t)&amp;= \\frac{\\hbar }{2mi}{\\bar{\\psi }}(y,t)\\psi '(y,t) -\\psi (y,t)\\bar{\\psi '}(y,t) \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M334\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:mi>j</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>y</mml:mi><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mtd><mml:mtd columnalign=\"left\"><mml:mrow><mml:mo>=</mml:mo><mml:mfrac><mml:mi>ħ</mml:mi><mml:mrow><mml:mn>2</mml:mn><mml:mi>m</mml:mi><mml:mi>i</mml:mi></mml:mrow></mml:mfrac><mml:mover accent=\"true\"><mml:mrow><mml:mi>ψ</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">¯</mml:mo></mml:mrow></mml:mover><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>y</mml:mi><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:msup><mml:mi>ψ</mml:mi><mml:mo>′</mml:mo></mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>y</mml:mi><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>-</mml:mo><mml:mi>ψ</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>y</mml:mi><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mover accent=\"true\"><mml:mrow><mml:msup><mml:mi>ψ</mml:mi><mml:mo>′</mml:mo></mml:msup></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">¯</mml:mo></mml:mrow></mml:mover><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>y</mml:mi><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ32\"><label>32</label><alternatives><tex-math id=\"M335\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned}{}&amp;= \\frac{\\hbar }{2mi} \\sum _{l, j} {\\bar{c}}_l c_j \\big [{\\bar{\\psi }}^{K_l}(y)\\psi '^{K_j}(y) -\\psi ^{K_j}(y)\\bar{\\psi '}^{K_l}(y) \\big ] \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M336\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow/></mml:mtd><mml:mtd columnalign=\"left\"><mml:mrow><mml:mo>=</mml:mo><mml:mfrac><mml:mi>ħ</mml:mi><mml:mrow><mml:mn>2</mml:mn><mml:mi>m</mml:mi><mml:mi>i</mml:mi></mml:mrow></mml:mfrac><mml:munder><mml:mo>∑</mml:mo><mml:mrow><mml:mi>l</mml:mi><mml:mo>,</mml:mo><mml:mi>j</mml:mi></mml:mrow></mml:munder><mml:msub><mml:mover accent=\"true\"><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">¯</mml:mo></mml:mrow></mml:mover><mml:mi>l</mml:mi></mml:msub><mml:msub><mml:mi>c</mml:mi><mml:mi>j</mml:mi></mml:msub><mml:mrow><mml:mo maxsize=\"1.2em\" minsize=\"1.2em\" stretchy=\"true\">[</mml:mo></mml:mrow><mml:msup><mml:mrow><mml:mover accent=\"true\"><mml:mrow><mml:mi>ψ</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">¯</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:msub><mml:mi>K</mml:mi><mml:mi>l</mml:mi></mml:msub></mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:msup><mml:mi>ψ</mml:mi><mml:mrow><mml:mo>′</mml:mo><mml:msub><mml:mi>K</mml:mi><mml:mi>j</mml:mi></mml:msub></mml:mrow></mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>-</mml:mo><mml:msup><mml:mi>ψ</mml:mi><mml:msub><mml:mi>K</mml:mi><mml:mi>j</mml:mi></mml:msub></mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:msup><mml:mover accent=\"true\"><mml:mrow><mml:msup><mml:mi>ψ</mml:mi><mml:mo>′</mml:mo></mml:msup></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">¯</mml:mo></mml:mrow></mml:mover><mml:msub><mml:mi>K</mml:mi><mml:mi>l</mml:mi></mml:msub></mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mrow><mml:mo maxsize=\"1.2em\" minsize=\"1.2em\" stretchy=\"true\">]</mml:mo></mml:mrow></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq137\"><alternatives><tex-math id=\"M337\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\psi ^{K_j}(y) \\approx e^{\\frac{y^2}{2}}y^{-\\frac{1+K}{2}}[1 + \\frac{(3+K)(1+K)}{16y^2}]$$\\end{document}</tex-math><mml:math id=\"M338\"><mml:mrow><mml:msup><mml:mi>ψ</mml:mi><mml:msub><mml:mi>K</mml:mi><mml:mi>j</mml:mi></mml:msub></mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>≈</mml:mo><mml:msup><mml:mi>e</mml:mi><mml:mfrac><mml:msup><mml:mi>y</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mn>2</mml:mn></mml:mfrac></mml:msup><mml:msup><mml:mi>y</mml:mi><mml:mrow><mml:mo>-</mml:mo><mml:mfrac><mml:mrow><mml:mn>1</mml:mn><mml:mo>+</mml:mo><mml:mi>K</mml:mi></mml:mrow><mml:mn>2</mml:mn></mml:mfrac></mml:mrow></mml:msup><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mn>1</mml:mn><mml:mo>+</mml:mo><mml:mfrac><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>3</mml:mn><mml:mo>+</mml:mo><mml:mi>K</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>1</mml:mn><mml:mo>+</mml:mo><mml:mi>K</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mrow><mml:mn>16</mml:mn><mml:msup><mml:mi>y</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mrow></mml:mfrac><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq138\"><alternatives><tex-math id=\"M339\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\pm y$$\\end{document}</tex-math><mml:math id=\"M340\"><mml:mrow><mml:mo>±</mml:mo><mml:mi>y</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ33\"><label>33</label><alternatives><tex-math id=\"M341\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} j(y, t)&amp;\\approx \\frac{\\hbar }{2mi} \\sum _{l, j} {\\bar{c}}_l(t) c_j(t) \\frac{e^{y^2}(K_l - K_j)}{2y^2\\sqrt{y^{K_j}}\\sqrt{y^{K_l}}} \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M342\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:mi>j</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>y</mml:mi><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mtd><mml:mtd columnalign=\"left\"><mml:mrow><mml:mo>≈</mml:mo><mml:mfrac><mml:mi>ħ</mml:mi><mml:mrow><mml:mn>2</mml:mn><mml:mi>m</mml:mi><mml:mi>i</mml:mi></mml:mrow></mml:mfrac><mml:munder><mml:mo>∑</mml:mo><mml:mrow><mml:mi>l</mml:mi><mml:mo>,</mml:mo><mml:mi>j</mml:mi></mml:mrow></mml:munder><mml:msub><mml:mover accent=\"true\"><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">¯</mml:mo></mml:mrow></mml:mover><mml:mi>l</mml:mi></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:msub><mml:mi>c</mml:mi><mml:mi>j</mml:mi></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mfrac><mml:mrow><mml:msup><mml:mi>e</mml:mi><mml:msup><mml:mi>y</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>K</mml:mi><mml:mi>l</mml:mi></mml:msub><mml:mo>-</mml:mo><mml:msub><mml:mi>K</mml:mi><mml:mi>j</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:msup><mml:mi>y</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:msqrt><mml:msup><mml:mi>y</mml:mi><mml:msub><mml:mi>K</mml:mi><mml:mi>j</mml:mi></mml:msub></mml:msup></mml:msqrt><mml:msqrt><mml:msup><mml:mi>y</mml:mi><mml:msub><mml:mi>K</mml:mi><mml:mi>l</mml:mi></mml:msub></mml:msup></mml:msqrt></mml:mrow></mml:mfrac></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ34\"><label>34</label><alternatives><tex-math id=\"M343\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} |\\psi (y,t)|^2 \\approx \\sum _{l, j} {\\bar{c}}_l(t) c_j(t) \\frac{e^{y^2}}{y\\sqrt{y^{K_j}}\\sqrt{y^{K_l}}} \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M344\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:msup><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo><mml:mi>ψ</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>y</mml:mi><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:msup><mml:mo>≈</mml:mo><mml:munder><mml:mo>∑</mml:mo><mml:mrow><mml:mi>l</mml:mi><mml:mo>,</mml:mo><mml:mi>j</mml:mi></mml:mrow></mml:munder><mml:msub><mml:mover accent=\"true\"><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">¯</mml:mo></mml:mrow></mml:mover><mml:mi>l</mml:mi></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:msub><mml:mi>c</mml:mi><mml:mi>j</mml:mi></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mfrac><mml:msup><mml:mi>e</mml:mi><mml:msup><mml:mi>y</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:msup><mml:mrow><mml:mi>y</mml:mi><mml:msqrt><mml:msup><mml:mi>y</mml:mi><mml:msub><mml:mi>K</mml:mi><mml:mi>j</mml:mi></mml:msub></mml:msup></mml:msqrt><mml:msqrt><mml:msup><mml:mi>y</mml:mi><mml:msub><mml:mi>K</mml:mi><mml:mi>l</mml:mi></mml:msub></mml:msup></mml:msqrt></mml:mrow></mml:mfrac></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ35\"><label>35</label><alternatives><tex-math id=\"M345\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} v(y, t)= \\frac{j(y, t)}{|\\psi (y,t)|^2} \\sim \\frac{1}{y} \\text { at large} \\pm y \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M346\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:mi>v</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>y</mml:mi><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mfrac><mml:mrow><mml:mi>j</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>y</mml:mi><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:msup><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo><mml:mi>ψ</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>y</mml:mi><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:msup></mml:mfrac><mml:mo>∼</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:mi>y</mml:mi></mml:mfrac><mml:mspace width=\"0.333333em\"/><mml:mtext>at large</mml:mtext><mml:mo>±</mml:mo><mml:mi>y</mml:mi></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq139\"><alternatives><tex-math id=\"M347\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\lim _{y \\rightarrow \\pm \\infty } v(y, t) = 0$$\\end{document}</tex-math><mml:math id=\"M348\"><mml:mrow><mml:msub><mml:mo movablelimits=\"true\">lim</mml:mo><mml:mrow><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo><mml:mo>±</mml:mo><mml:mi>∞</mml:mi></mml:mrow></mml:msub><mml:mi>v</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>y</mml:mi><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq140\"><alternatives><tex-math id=\"M349\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$|\\psi (y_1, y_2)|^2$$\\end{document}</tex-math><mml:math id=\"M350\"><mml:mrow><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo><mml:mi>ψ</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>y</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>y</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:msup><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:msup></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq141\"><alternatives><tex-math id=\"M351\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\vec{v}(y_1, y_2)$$\\end{document}</tex-math><mml:math id=\"M352\"><mml:mrow><mml:mover accent=\"true\"><mml:mi>v</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>y</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>y</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq142\"><alternatives><tex-math id=\"M353\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$y_1$$\\end{document}</tex-math><mml:math id=\"M354\"><mml:msub><mml:mi>y</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq143\"><alternatives><tex-math id=\"M355\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$v_{y_1}(y_1, y_2)$$\\end{document}</tex-math><mml:math id=\"M356\"><mml:mrow><mml:msub><mml:mi>v</mml:mi><mml:msub><mml:mi>y</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>y</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>y</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq144\"><alternatives><tex-math id=\"M357\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$y_2$$\\end{document}</tex-math><mml:math id=\"M358\"><mml:msub><mml:mi>y</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq145\"><alternatives><tex-math id=\"M359\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$v_{y_2}(y_1, y_2)$$\\end{document}</tex-math><mml:math id=\"M360\"><mml:mrow><mml:msub><mml:mi>v</mml:mi><mml:msub><mml:mi>y</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>y</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>y</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq146\"><alternatives><tex-math id=\"M361\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\psi (y_1, y_2) = \\sqrt{2}/3 \\psi ^{1.4}_{3\\pi /4, 4\\pi /3}(y_1) \\psi ^8_{2.2\\pi , 4.1\\pi }(y_2) + \\sqrt{2}/3 e^{i\\pi /5}\\psi ^{5}_{8\\pi /5, 5.8\\pi }(y_1) \\psi ^{15.6}_{2\\pi /5, 9\\pi /16}(y_2) + 1/3 e^{i\\pi /8}\\psi ^{9}_{\\pi /5, \\pi /7}(y_1) \\psi ^{0.75}_{\\pi /6, \\pi /9}(y_2) + 2/3 e^{i\\pi /9}\\psi ^{11.4}_{5\\pi /3, 6\\pi /7}(y_1) \\psi ^{12.6}_{2\\pi /5, 7\\pi /16}(y_2)$$\\end{document}</tex-math><mml:math id=\"M362\"><mml:mrow><mml:mi>ψ</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>y</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>y</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:msqrt><mml:mn>2</mml:mn></mml:msqrt><mml:mo stretchy=\"false\">/</mml:mo><mml:mn>3</mml:mn><mml:msubsup><mml:mi>ψ</mml:mi><mml:mrow><mml:mn>3</mml:mn><mml:mi>π</mml:mi><mml:mo stretchy=\"false\">/</mml:mo><mml:mn>4</mml:mn><mml:mo>,</mml:mo><mml:mn>4</mml:mn><mml:mi>π</mml:mi><mml:mo stretchy=\"false\">/</mml:mo><mml:mn>3</mml:mn></mml:mrow><mml:mrow><mml:mn>1.4</mml:mn></mml:mrow></mml:msubsup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>y</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:msubsup><mml:mi>ψ</mml:mi><mml:mrow><mml:mn>2.2</mml:mn><mml:mi>π</mml:mi><mml:mo>,</mml:mo><mml:mn>4.1</mml:mn><mml:mi>π</mml:mi></mml:mrow><mml:mn>8</mml:mn></mml:msubsup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>y</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>+</mml:mo><mml:msqrt><mml:mn>2</mml:mn></mml:msqrt><mml:mo stretchy=\"false\">/</mml:mo><mml:mn>3</mml:mn><mml:msup><mml:mi>e</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>π</mml:mi><mml:mo stretchy=\"false\">/</mml:mo><mml:mn>5</mml:mn></mml:mrow></mml:msup><mml:msubsup><mml:mi>ψ</mml:mi><mml:mrow><mml:mn>8</mml:mn><mml:mi>π</mml:mi><mml:mo stretchy=\"false\">/</mml:mo><mml:mn>5</mml:mn><mml:mo>,</mml:mo><mml:mn>5.8</mml:mn><mml:mi>π</mml:mi></mml:mrow><mml:mn>5</mml:mn></mml:msubsup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>y</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:msubsup><mml:mi>ψ</mml:mi><mml:mrow><mml:mn>2</mml:mn><mml:mi>π</mml:mi><mml:mo stretchy=\"false\">/</mml:mo><mml:mn>5</mml:mn><mml:mo>,</mml:mo><mml:mn>9</mml:mn><mml:mi>π</mml:mi><mml:mo stretchy=\"false\">/</mml:mo><mml:mn>16</mml:mn></mml:mrow><mml:mrow><mml:mn>15.6</mml:mn></mml:mrow></mml:msubsup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>y</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">/</mml:mo><mml:mn>3</mml:mn><mml:msup><mml:mi>e</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>π</mml:mi><mml:mo stretchy=\"false\">/</mml:mo><mml:mn>8</mml:mn></mml:mrow></mml:msup><mml:msubsup><mml:mi>ψ</mml:mi><mml:mrow><mml:mi>π</mml:mi><mml:mo stretchy=\"false\">/</mml:mo><mml:mn>5</mml:mn><mml:mo>,</mml:mo><mml:mi>π</mml:mi><mml:mo stretchy=\"false\">/</mml:mo><mml:mn>7</mml:mn></mml:mrow><mml:mn>9</mml:mn></mml:msubsup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>y</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:msubsup><mml:mi>ψ</mml:mi><mml:mrow><mml:mi>π</mml:mi><mml:mo stretchy=\"false\">/</mml:mo><mml:mn>6</mml:mn><mml:mo>,</mml:mo><mml:mi>π</mml:mi><mml:mo stretchy=\"false\">/</mml:mo><mml:mn>9</mml:mn></mml:mrow><mml:mrow><mml:mn>0.75</mml:mn></mml:mrow></mml:msubsup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>y</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>+</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">/</mml:mo><mml:mn>3</mml:mn><mml:msup><mml:mi>e</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>π</mml:mi><mml:mo stretchy=\"false\">/</mml:mo><mml:mn>9</mml:mn></mml:mrow></mml:msup><mml:msubsup><mml:mi>ψ</mml:mi><mml:mrow><mml:mn>5</mml:mn><mml:mi>π</mml:mi><mml:mo stretchy=\"false\">/</mml:mo><mml:mn>3</mml:mn><mml:mo>,</mml:mo><mml:mn>6</mml:mn><mml:mi>π</mml:mi><mml:mo stretchy=\"false\">/</mml:mo><mml:mn>7</mml:mn></mml:mrow><mml:mrow><mml:mn>11.4</mml:mn></mml:mrow></mml:msubsup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>y</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:msubsup><mml:mi>ψ</mml:mi><mml:mrow><mml:mn>2</mml:mn><mml:mi>π</mml:mi><mml:mo stretchy=\"false\">/</mml:mo><mml:mn>5</mml:mn><mml:mo>,</mml:mo><mml:mn>7</mml:mn><mml:mi>π</mml:mi><mml:mo stretchy=\"false\">/</mml:mo><mml:mn>16</mml:mn></mml:mrow><mml:mrow><mml:mn>12.6</mml:mn></mml:mrow></mml:msubsup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>y</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq147\"><alternatives><tex-math id=\"M363\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$v_{y_1}(y_1, y_2) \\rightarrow 0$$\\end{document}</tex-math><mml:math id=\"M364\"><mml:mrow><mml:msub><mml:mi>v</mml:mi><mml:msub><mml:mi>y</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>y</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>y</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo stretchy=\"false\">→</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq148\"><alternatives><tex-math id=\"M365\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\pm y_1$$\\end{document}</tex-math><mml:math id=\"M366\"><mml:mrow><mml:mo>±</mml:mo><mml:msub><mml:mi>y</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq149\"><alternatives><tex-math id=\"M367\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$v_{y_2}(y_1, y_2) \\rightarrow 0$$\\end{document}</tex-math><mml:math id=\"M368\"><mml:mrow><mml:msub><mml:mi>v</mml:mi><mml:msub><mml:mi>y</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>y</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>y</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo stretchy=\"false\">→</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq150\"><alternatives><tex-math id=\"M369\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\pm y_2$$\\end{document}</tex-math><mml:math id=\"M370\"><mml:mrow><mml:mo>±</mml:mo><mml:msub><mml:mi>y</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ36\"><label>36</label><alternatives><tex-math id=\"M371\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} \\psi ({\\overrightarrow{y}}, t) = \\sum _{j=1}^n c_j(t) \\prod _{g=1}^N \\psi ^{K_j^g}(y_g) \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M372\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:mi>ψ</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:munderover><mml:mo>∑</mml:mo><mml:mrow><mml:mi>j</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mi>n</mml:mi></mml:munderover><mml:msub><mml:mi>c</mml:mi><mml:mi>j</mml:mi></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:munderover><mml:mo>∏</mml:mo><mml:mrow><mml:mi>g</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mi>N</mml:mi></mml:munderover><mml:msup><mml:mi>ψ</mml:mi><mml:msubsup><mml:mi>K</mml:mi><mml:mi>j</mml:mi><mml:mi>g</mml:mi></mml:msubsup></mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>y</mml:mi><mml:mi>g</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq151\"><alternatives><tex-math id=\"M373\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\psi ^{K_j^g}(y_g)$$\\end{document}</tex-math><mml:math id=\"M374\"><mml:mrow><mml:msup><mml:mi>ψ</mml:mi><mml:msubsup><mml:mi>K</mml:mi><mml:mi>j</mml:mi><mml:mi>g</mml:mi></mml:msubsup></mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>y</mml:mi><mml:mi>g</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq153\"><alternatives><tex-math id=\"M375\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${K_j^g}$$\\end{document}</tex-math><mml:math id=\"M376\"><mml:msubsup><mml:mi>K</mml:mi><mml:mi>j</mml:mi><mml:mi>g</mml:mi></mml:msubsup></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ37\"><label>37</label><alternatives><tex-math id=\"M377\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} j_r({\\overrightarrow{y}}, t)&amp;= \\frac{\\hbar }{2mi} \\sum _{l, j} {\\bar{c}}_l(t) c_j(t) \\prod _{g \\ne r}^N \\psi ^{K_j^g}(y_g) \\prod _{w\\ne r}^N {\\bar{\\psi }}^{K_l^w}(y_w) \\big [{\\bar{\\psi }}^{K_l^r}(y_r)\\psi '^{K_j^r}(y_r) -\\psi ^{K_j^r}(y_r)\\bar{\\psi '}^{K_l^r}(y_r) \\big ] \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M378\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:msub><mml:mi>j</mml:mi><mml:mi>r</mml:mi></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:mtd><mml:mtd columnalign=\"left\"><mml:mrow><mml:mo>=</mml:mo><mml:mfrac><mml:mi>ħ</mml:mi><mml:mrow><mml:mn>2</mml:mn><mml:mi>m</mml:mi><mml:mi>i</mml:mi></mml:mrow></mml:mfrac><mml:munder><mml:mo>∑</mml:mo><mml:mrow><mml:mi>l</mml:mi><mml:mo>,</mml:mo><mml:mi>j</mml:mi></mml:mrow></mml:munder><mml:msub><mml:mover accent=\"true\"><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">¯</mml:mo></mml:mrow></mml:mover><mml:mi>l</mml:mi></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:msub><mml:mi>c</mml:mi><mml:mi>j</mml:mi></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:munderover><mml:mo>∏</mml:mo><mml:mrow><mml:mi>g</mml:mi><mml:mo>≠</mml:mo><mml:mi>r</mml:mi></mml:mrow><mml:mi>N</mml:mi></mml:munderover><mml:msup><mml:mi>ψ</mml:mi><mml:msubsup><mml:mi>K</mml:mi><mml:mi>j</mml:mi><mml:mi>g</mml:mi></mml:msubsup></mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>y</mml:mi><mml:mi>g</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:munderover><mml:mo>∏</mml:mo><mml:mrow><mml:mi>w</mml:mi><mml:mo>≠</mml:mo><mml:mi>r</mml:mi></mml:mrow><mml:mi>N</mml:mi></mml:munderover><mml:msup><mml:mrow><mml:mover accent=\"true\"><mml:mrow><mml:mi>ψ</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">¯</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:msubsup><mml:mi>K</mml:mi><mml:mi>l</mml:mi><mml:mi>w</mml:mi></mml:msubsup></mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>y</mml:mi><mml:mi>w</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mrow><mml:mo maxsize=\"1.2em\" minsize=\"1.2em\" stretchy=\"true\">[</mml:mo></mml:mrow><mml:msup><mml:mrow><mml:mover accent=\"true\"><mml:mrow><mml:mi>ψ</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">¯</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:msubsup><mml:mi>K</mml:mi><mml:mi>l</mml:mi><mml:mi>r</mml:mi></mml:msubsup></mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>y</mml:mi><mml:mi>r</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:msup><mml:mi>ψ</mml:mi><mml:mrow><mml:mo>′</mml:mo><mml:msubsup><mml:mi>K</mml:mi><mml:mi>j</mml:mi><mml:mi>r</mml:mi></mml:msubsup></mml:mrow></mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>y</mml:mi><mml:mi>r</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>-</mml:mo><mml:msup><mml:mi>ψ</mml:mi><mml:msubsup><mml:mi>K</mml:mi><mml:mi>j</mml:mi><mml:mi>r</mml:mi></mml:msubsup></mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>y</mml:mi><mml:mi>r</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:msup><mml:mover accent=\"true\"><mml:mrow><mml:msup><mml:mi>ψ</mml:mi><mml:mo>′</mml:mo></mml:msup></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">¯</mml:mo></mml:mrow></mml:mover><mml:msubsup><mml:mi>K</mml:mi><mml:mi>l</mml:mi><mml:mi>r</mml:mi></mml:msubsup></mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>y</mml:mi><mml:mi>r</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mrow><mml:mo maxsize=\"1.2em\" minsize=\"1.2em\" stretchy=\"true\">]</mml:mo></mml:mrow></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq156\"><alternatives><tex-math id=\"M379\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\psi ^{K_j^r}(y_r) \\approx e^{y_r^2/2}y_r^{-\\frac{1+(K_j^r)^2}{2}}[1 + (3+K_j^r)(1+K_j^r)/(16y_r^2)]$$\\end{document}</tex-math><mml:math id=\"M380\"><mml:mrow><mml:msup><mml:mi>ψ</mml:mi><mml:msubsup><mml:mi>K</mml:mi><mml:mi>j</mml:mi><mml:mi>r</mml:mi></mml:msubsup></mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>y</mml:mi><mml:mi>r</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>≈</mml:mo><mml:msup><mml:mi>e</mml:mi><mml:mrow><mml:msubsup><mml:mi>y</mml:mi><mml:mi>r</mml:mi><mml:mn>2</mml:mn></mml:msubsup><mml:mo stretchy=\"false\">/</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:msubsup><mml:mi>y</mml:mi><mml:mi>r</mml:mi><mml:mrow><mml:mo>-</mml:mo><mml:mfrac><mml:mrow><mml:mn>1</mml:mn><mml:mo>+</mml:mo><mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msubsup><mml:mi>K</mml:mi><mml:mi>j</mml:mi><mml:mi>r</mml:mi></mml:msubsup><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:msup></mml:mrow><mml:mn>2</mml:mn></mml:mfrac></mml:mrow></mml:msubsup><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mn>1</mml:mn><mml:mo>+</mml:mo><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>3</mml:mn><mml:mo>+</mml:mo><mml:msubsup><mml:mi>K</mml:mi><mml:mi>j</mml:mi><mml:mi>r</mml:mi></mml:msubsup><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>1</mml:mn><mml:mo>+</mml:mo><mml:msubsup><mml:mi>K</mml:mi><mml:mi>j</mml:mi><mml:mi>r</mml:mi></mml:msubsup><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo stretchy=\"false\">/</mml:mo><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>16</mml:mn><mml:msubsup><mml:mi>y</mml:mi><mml:mi>r</mml:mi><mml:mn>2</mml:mn></mml:msubsup><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq157\"><alternatives><tex-math id=\"M381\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\pm y_r$$\\end{document}</tex-math><mml:math id=\"M382\"><mml:mrow><mml:mo>±</mml:mo><mml:msub><mml:mi>y</mml:mi><mml:mi>r</mml:mi></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ38\"><label>38</label><alternatives><tex-math id=\"M383\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} j_r({\\overrightarrow{y}}, t)&amp;\\approx \\frac{\\hbar }{2mi} \\sum _{l, j} {\\bar{c}}_l(t) c_j(t) \\prod _{g \\ne r}^N \\psi ^{K_j^g}(y_g) \\prod _{w\\ne r}^N {\\bar{\\psi }}^{K_l^w}(y_w) \\frac{e^{y_r^2}(K_l^r - K_j^r)}{2y_r^2 \\sqrt{y_r^{K_j^r}}\\sqrt{y_r^{K_l^r}}} \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M384\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:msub><mml:mi>j</mml:mi><mml:mi>r</mml:mi></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:mtd><mml:mtd columnalign=\"left\"><mml:mrow><mml:mo>≈</mml:mo><mml:mfrac><mml:mi>ħ</mml:mi><mml:mrow><mml:mn>2</mml:mn><mml:mi>m</mml:mi><mml:mi>i</mml:mi></mml:mrow></mml:mfrac><mml:munder><mml:mo>∑</mml:mo><mml:mrow><mml:mi>l</mml:mi><mml:mo>,</mml:mo><mml:mi>j</mml:mi></mml:mrow></mml:munder><mml:msub><mml:mover accent=\"true\"><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">¯</mml:mo></mml:mrow></mml:mover><mml:mi>l</mml:mi></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:msub><mml:mi>c</mml:mi><mml:mi>j</mml:mi></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:munderover><mml:mo>∏</mml:mo><mml:mrow><mml:mi>g</mml:mi><mml:mo>≠</mml:mo><mml:mi>r</mml:mi></mml:mrow><mml:mi>N</mml:mi></mml:munderover><mml:msup><mml:mi>ψ</mml:mi><mml:msubsup><mml:mi>K</mml:mi><mml:mi>j</mml:mi><mml:mi>g</mml:mi></mml:msubsup></mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>y</mml:mi><mml:mi>g</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:munderover><mml:mo>∏</mml:mo><mml:mrow><mml:mi>w</mml:mi><mml:mo>≠</mml:mo><mml:mi>r</mml:mi></mml:mrow><mml:mi>N</mml:mi></mml:munderover><mml:msup><mml:mrow><mml:mover accent=\"true\"><mml:mrow><mml:mi>ψ</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">¯</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:msubsup><mml:mi>K</mml:mi><mml:mi>l</mml:mi><mml:mi>w</mml:mi></mml:msubsup></mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>y</mml:mi><mml:mi>w</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mfrac><mml:mrow><mml:msup><mml:mi>e</mml:mi><mml:msubsup><mml:mi>y</mml:mi><mml:mi>r</mml:mi><mml:mn>2</mml:mn></mml:msubsup></mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msubsup><mml:mi>K</mml:mi><mml:mi>l</mml:mi><mml:mi>r</mml:mi></mml:msubsup><mml:mo>-</mml:mo><mml:msubsup><mml:mi>K</mml:mi><mml:mi>j</mml:mi><mml:mi>r</mml:mi></mml:msubsup><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:msubsup><mml:mi>y</mml:mi><mml:mi>r</mml:mi><mml:mn>2</mml:mn></mml:msubsup><mml:msqrt><mml:msubsup><mml:mi>y</mml:mi><mml:mi>r</mml:mi><mml:msubsup><mml:mi>K</mml:mi><mml:mi>j</mml:mi><mml:mi>r</mml:mi></mml:msubsup></mml:msubsup></mml:msqrt><mml:msqrt><mml:msubsup><mml:mi>y</mml:mi><mml:mi>r</mml:mi><mml:msubsup><mml:mi>K</mml:mi><mml:mi>l</mml:mi><mml:mi>r</mml:mi></mml:msubsup></mml:msubsup></mml:msqrt></mml:mrow></mml:mfrac></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ39\"><label>39</label><alternatives><tex-math id=\"M385\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} |\\psi ({\\overrightarrow{y}},t)|^2 \\approx \\sum _{l, j} {\\bar{c}}_l(t) c_j(t) \\prod _{g \\ne r}^N \\psi ^{K_j^g}(y_g) \\prod _{w\\ne r}^N {\\bar{\\psi }}^{K_l^w}(y_w) \\frac{e^{y_r^2}}{y_r\\sqrt{y_r^{K_j^r}}\\sqrt{y_r^{K_l^r}}} \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M386\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo><mml:mi>ψ</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:msup><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:msup><mml:mo>≈</mml:mo><mml:munder><mml:mo>∑</mml:mo><mml:mrow><mml:mi>l</mml:mi><mml:mo>,</mml:mo><mml:mi>j</mml:mi></mml:mrow></mml:munder><mml:msub><mml:mover accent=\"true\"><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">¯</mml:mo></mml:mrow></mml:mover><mml:mi>l</mml:mi></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:msub><mml:mi>c</mml:mi><mml:mi>j</mml:mi></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:munderover><mml:mo>∏</mml:mo><mml:mrow><mml:mi>g</mml:mi><mml:mo>≠</mml:mo><mml:mi>r</mml:mi></mml:mrow><mml:mi>N</mml:mi></mml:munderover><mml:msup><mml:mi>ψ</mml:mi><mml:msubsup><mml:mi>K</mml:mi><mml:mi>j</mml:mi><mml:mi>g</mml:mi></mml:msubsup></mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>y</mml:mi><mml:mi>g</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:munderover><mml:mo>∏</mml:mo><mml:mrow><mml:mi>w</mml:mi><mml:mo>≠</mml:mo><mml:mi>r</mml:mi></mml:mrow><mml:mi>N</mml:mi></mml:munderover><mml:msup><mml:mrow><mml:mover accent=\"true\"><mml:mrow><mml:mi>ψ</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">¯</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:msubsup><mml:mi>K</mml:mi><mml:mi>l</mml:mi><mml:mi>w</mml:mi></mml:msubsup></mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>y</mml:mi><mml:mi>w</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mfrac><mml:msup><mml:mi>e</mml:mi><mml:msubsup><mml:mi>y</mml:mi><mml:mi>r</mml:mi><mml:mn>2</mml:mn></mml:msubsup></mml:msup><mml:mrow><mml:msub><mml:mi>y</mml:mi><mml:mi>r</mml:mi></mml:msub><mml:msqrt><mml:msubsup><mml:mi>y</mml:mi><mml:mi>r</mml:mi><mml:msubsup><mml:mi>K</mml:mi><mml:mi>j</mml:mi><mml:mi>r</mml:mi></mml:msubsup></mml:msubsup></mml:msqrt><mml:msqrt><mml:msubsup><mml:mi>y</mml:mi><mml:mi>r</mml:mi><mml:msubsup><mml:mi>K</mml:mi><mml:mi>l</mml:mi><mml:mi>r</mml:mi></mml:msubsup></mml:msubsup></mml:msqrt></mml:mrow></mml:mfrac></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ40\"><label>40</label><alternatives><tex-math id=\"M387\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} v_r({\\overrightarrow{y}}, t)= \\frac{j_r({\\overrightarrow{y}}, t)}{|\\psi ({\\overrightarrow{y}},t)|^2} \\sim \\frac{1}{y_r} \\text {at large} \\pm y_r \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M388\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:msub><mml:mi>v</mml:mi><mml:mi>r</mml:mi></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mfrac><mml:mrow><mml:msub><mml:mi>j</mml:mi><mml:mi>r</mml:mi></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow><mml:mrow><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo><mml:mi>ψ</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:msup><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:msup></mml:mrow></mml:mfrac><mml:mo>∼</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:msub><mml:mi>y</mml:mi><mml:mi>r</mml:mi></mml:msub></mml:mfrac><mml:mtext>at large</mml:mtext><mml:mo>±</mml:mo><mml:msub><mml:mi>y</mml:mi><mml:mi>r</mml:mi></mml:msub></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq158\"><alternatives><tex-math id=\"M389\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$v_r({\\overrightarrow{y}}, t) \\rightarrow 0$$\\end{document}</tex-math><mml:math id=\"M390\"><mml:mrow><mml:msub><mml:mi>v</mml:mi><mml:mi>r</mml:mi></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo stretchy=\"false\">→</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq159\"><alternatives><tex-math id=\"M391\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$y_r \\rightarrow \\infty$$\\end{document}</tex-math><mml:math id=\"M392\"><mml:mrow><mml:msub><mml:mi>y</mml:mi><mml:mi>r</mml:mi></mml:msub><mml:mo stretchy=\"false\">→</mml:mo><mml:mi>∞</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq160\"><alternatives><tex-math id=\"M393\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\forall r \\in \\{1, 2, ...N\\}$$\\end{document}</tex-math><mml:math id=\"M394\"><mml:mrow><mml:mo>∀</mml:mo><mml:mi>r</mml:mi><mml:mo>∈</mml:mo><mml:mo stretchy=\"false\">{</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mo>.</mml:mo><mml:mo>.</mml:mo><mml:mo>.</mml:mo><mml:mi>N</mml:mi><mml:mo stretchy=\"false\">}</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq161\"><alternatives><tex-math id=\"M395\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\overrightarrow{y}}$$\\end{document}</tex-math><mml:math id=\"M396\"><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq162\"><alternatives><tex-math id=\"M397\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\psi ({\\overrightarrow{y}}, t)$$\\end{document}</tex-math><mml:math id=\"M398\"><mml:mrow><mml:mi>ψ</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq163\"><alternatives><tex-math id=\"M399\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$|\\psi ({\\overrightarrow{y}}, t)|^2$$\\end{document}</tex-math><mml:math id=\"M400\"><mml:mrow><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo><mml:mi>ψ</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:msup><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:msup></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq164\"><alternatives><tex-math id=\"M401\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\psi ({\\overrightarrow{y}}, t)$$\\end{document}</tex-math><mml:math id=\"M402\"><mml:mrow><mml:mi>ψ</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq165\"><alternatives><tex-math id=\"M403\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\psi ({\\overrightarrow{y}})$$\\end{document}</tex-math><mml:math id=\"M404\"><mml:mrow><mml:mi>ψ</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq166\"><alternatives><tex-math id=\"M405\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\rho ({\\overrightarrow{y}})$$\\end{document}</tex-math><mml:math id=\"M406\"><mml:mrow><mml:mi>ρ</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq167\"><alternatives><tex-math id=\"M407\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\rho ({\\overrightarrow{y}})$$\\end{document}</tex-math><mml:math id=\"M408\"><mml:mrow><mml:mi>ρ</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ41\"><label>41</label><alternatives><tex-math id=\"M409\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} H_q \\equiv \\int _{\\mathscr {C}} \\rho ({\\overrightarrow{y}}) \\ln \\frac{\\rho ({\\overrightarrow{y}})}{|\\psi ({\\overrightarrow{y}})|^2} d{\\overrightarrow{y}} \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M410\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:msub><mml:mi>H</mml:mi><mml:mi>q</mml:mi></mml:msub><mml:mo>≡</mml:mo><mml:msub><mml:mo>∫</mml:mo><mml:mi mathvariant=\"script\">C</mml:mi></mml:msub><mml:mi>ρ</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>ln</mml:mo><mml:mfrac><mml:mrow><mml:mi>ρ</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mrow><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo><mml:mi>ψ</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:msup><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:msup></mml:mrow></mml:mfrac><mml:mi>d</mml:mi><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq168\"><alternatives><tex-math id=\"M411\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\mathscr {C}} = \\{{\\overrightarrow{y}}| y_r \\in {\\mathcal {R}} {\\textbf { }} \\forall r \\}$$\\end{document}</tex-math><mml:math id=\"M412\"><mml:mrow><mml:mi mathvariant=\"script\">C</mml:mi><mml:mo>=</mml:mo><mml:mo stretchy=\"false\">{</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo stretchy=\"false\">|</mml:mo><mml:msub><mml:mi>y</mml:mi><mml:mi>r</mml:mi></mml:msub><mml:mo>∈</mml:mo><mml:mi mathvariant=\"script\">R</mml:mi><mml:mrow/><mml:mo>∀</mml:mo><mml:mi>r</mml:mi><mml:mo stretchy=\"false\">}</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq169\"><alternatives><tex-math id=\"M413\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\mathcal {R}}$$\\end{document}</tex-math><mml:math id=\"M414\"><mml:mi mathvariant=\"script\">R</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq170\"><alternatives><tex-math id=\"M415\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\rho ({\\overrightarrow{y}})$$\\end{document}</tex-math><mml:math id=\"M416\"><mml:mrow><mml:mi>ρ</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq171\"><alternatives><tex-math id=\"M417\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$|\\psi ({\\overrightarrow{y}})|^2$$\\end{document}</tex-math><mml:math id=\"M418\"><mml:mrow><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo><mml:mi>ψ</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:msup><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:msup></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq172\"><alternatives><tex-math id=\"M419\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$|\\psi ({\\overrightarrow{y}})|^2$$\\end{document}</tex-math><mml:math id=\"M420\"><mml:mrow><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo><mml:mi>ψ</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:msup><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:msup></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq173\"><alternatives><tex-math id=\"M421\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\mathscr {C}}$$\\end{document}</tex-math><mml:math id=\"M422\"><mml:mi mathvariant=\"script\">C</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq174\"><alternatives><tex-math id=\"M423\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$H_q$$\\end{document}</tex-math><mml:math id=\"M424\"><mml:msub><mml:mi>H</mml:mi><mml:mi>q</mml:mi></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq175\"><alternatives><tex-math id=\"M425\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\rho ({\\overrightarrow{y}})$$\\end{document}</tex-math><mml:math id=\"M426\"><mml:mrow><mml:mi>ρ</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq176\"><alternatives><tex-math id=\"M427\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\Omega \\equiv \\{{\\overrightarrow{y}}|\\rho ({\\overrightarrow{y}}) &gt;0\\}$$\\end{document}</tex-math><mml:math id=\"M428\"><mml:mrow><mml:mi mathvariant=\"normal\">Ω</mml:mi><mml:mo>≡</mml:mo><mml:mo stretchy=\"false\">{</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo stretchy=\"false\">|</mml:mo><mml:mi>ρ</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>&gt;</mml:mo><mml:mn>0</mml:mn><mml:mo stretchy=\"false\">}</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq177\"><alternatives><tex-math id=\"M429\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\mathscr {C}}$$\\end{document}</tex-math><mml:math id=\"M430\"><mml:mi mathvariant=\"script\">C</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq178\"><alternatives><tex-math id=\"M431\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\Omega$$\\end{document}</tex-math><mml:math id=\"M432\"><mml:mi mathvariant=\"normal\">Ω</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq179\"><alternatives><tex-math id=\"M433\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\mathscr {C}}$$\\end{document}</tex-math><mml:math id=\"M434\"><mml:mi mathvariant=\"script\">C</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq180\"><alternatives><tex-math id=\"M435\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\rho ({\\overrightarrow{y}})$$\\end{document}</tex-math><mml:math id=\"M436\"><mml:mrow><mml:mi>ρ</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq181\"><alternatives><tex-math id=\"M437\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$|\\psi ({\\overrightarrow{y}})|^2$$\\end{document}</tex-math><mml:math id=\"M438\"><mml:mrow><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo><mml:mi>ψ</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:msup><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:msup></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq182\"><alternatives><tex-math id=\"M439\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\Omega$$\\end{document}</tex-math><mml:math id=\"M440\"><mml:mi mathvariant=\"normal\">Ω</mml:mi></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ42\"><label>42</label><alternatives><tex-math id=\"M441\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} \\rho _{pw}({\\overrightarrow{y}}) \\equiv {\\left\\{ \\begin{array}{ll} |\\psi ({\\overrightarrow{y}})|^2/{\\mathcal {N}} &amp;{}\\text {, for}\\, {\\overrightarrow{y}} \\in \\Omega \\\\ 0 &amp;{}\\text {, for}\\, {\\overrightarrow{y}} \\in {\\mathscr {C}} \\setminus \\Omega \\end{array}\\right. } \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M442\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:msub><mml:mi>ρ</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">pw</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>≡</mml:mo><mml:mfenced open=\"{\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"left\"><mml:mrow><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo><mml:mi>ψ</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:msup><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:msup><mml:mo stretchy=\"false\">/</mml:mo><mml:mi mathvariant=\"script\">N</mml:mi></mml:mrow></mml:mtd><mml:mtd columnalign=\"left\"><mml:mrow><mml:mrow/><mml:mtext>, for</mml:mtext><mml:mspace width=\"0.166667em\"/><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>∈</mml:mo><mml:mi mathvariant=\"normal\">Ω</mml:mi></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign=\"left\"><mml:mrow><mml:mrow/><mml:mn>0</mml:mn></mml:mrow></mml:mtd><mml:mtd columnalign=\"left\"><mml:mrow><mml:mrow/><mml:mtext>, for</mml:mtext><mml:mspace width=\"0.166667em\"/><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>∈</mml:mo><mml:mi mathvariant=\"script\">C</mml:mi><mml:mo lspace=\"0.15em\" rspace=\"0.15em\" stretchy=\"false\">\\</mml:mo><mml:mi mathvariant=\"normal\">Ω</mml:mi></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq183\"><alternatives><tex-math id=\"M443\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\mathcal {N}} \\equiv \\int _\\Omega |\\psi ({\\overrightarrow{y}})|^2 d{\\overrightarrow{y}}$$\\end{document}</tex-math><mml:math id=\"M444\"><mml:mrow><mml:mi mathvariant=\"script\">N</mml:mi><mml:mo>≡</mml:mo><mml:msub><mml:mo>∫</mml:mo><mml:mi mathvariant=\"normal\">Ω</mml:mi></mml:msub><mml:msup><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo><mml:mi>ψ</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:msup><mml:mi>d</mml:mi><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq184\"><alternatives><tex-math id=\"M445\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$H_q$$\\end{document}</tex-math><mml:math id=\"M446\"><mml:msub><mml:mi>H</mml:mi><mml:mi>q</mml:mi></mml:msub></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ43\"><label>43</label><alternatives><tex-math id=\"M447\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} H_{pw} \\equiv \\int _{{\\mathscr {C}}} \\rho ({\\overrightarrow{y}}) \\ln \\frac{\\rho ({\\overrightarrow{y}})}{\\rho _{pw}({\\overrightarrow{y}})} d{\\overrightarrow{y}} \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M448\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:msub><mml:mi>H</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">pw</mml:mi></mml:mrow></mml:msub><mml:mo>≡</mml:mo><mml:msub><mml:mo>∫</mml:mo><mml:mi mathvariant=\"script\">C</mml:mi></mml:msub><mml:mi>ρ</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>ln</mml:mo><mml:mfrac><mml:mrow><mml:mi>ρ</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mrow><mml:msub><mml:mi>ρ</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">pw</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:mfrac><mml:mi>d</mml:mi><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq185\"><alternatives><tex-math id=\"M449\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\rho _{pw}({\\overrightarrow{y}})$$\\end{document}</tex-math><mml:math id=\"M450\"><mml:mrow><mml:msub><mml:mi>ρ</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">pw</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq186\"><alternatives><tex-math id=\"M451\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\mathscr {C}}$$\\end{document}</tex-math><mml:math id=\"M452\"><mml:mi mathvariant=\"script\">C</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq187\"><alternatives><tex-math id=\"M453\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$H_{pw}$$\\end{document}</tex-math><mml:math id=\"M454\"><mml:msub><mml:mi>H</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">pw</mml:mi></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq188\"><alternatives><tex-math id=\"M455\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\rho ({\\overrightarrow{y}})$$\\end{document}</tex-math><mml:math id=\"M456\"><mml:mrow><mml:mi>ρ</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq189\"><alternatives><tex-math id=\"M457\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\rho _{pw}({\\overrightarrow{y}})$$\\end{document}</tex-math><mml:math id=\"M458\"><mml:mrow><mml:msub><mml:mi>ρ</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">pw</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ44\"><label>44</label><alternatives><tex-math id=\"M459\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} H_{pw} = \\int _{\\mathscr {C}} \\bigg ( \\rho ({\\overrightarrow{y}}) \\ln \\frac{\\rho ({\\overrightarrow{y}})}{\\rho _{pw}({\\overrightarrow{y}})} - \\rho ({\\overrightarrow{y}}) + \\rho _{pw}({\\overrightarrow{y}}) \\bigg ) d{\\overrightarrow{y}} \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M460\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:msub><mml:mi>H</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">pw</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:msub><mml:mo>∫</mml:mo><mml:mi mathvariant=\"script\">C</mml:mi></mml:msub><mml:mrow><mml:mo maxsize=\"2.047em\" minsize=\"2.047em\" stretchy=\"true\">(</mml:mo></mml:mrow><mml:mi>ρ</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>ln</mml:mo><mml:mfrac><mml:mrow><mml:mi>ρ</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mrow><mml:msub><mml:mi>ρ</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">pw</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:mfrac><mml:mo>-</mml:mo><mml:mi>ρ</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>+</mml:mo><mml:msub><mml:mi>ρ</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">pw</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mrow><mml:mo maxsize=\"2.047em\" minsize=\"2.047em\" stretchy=\"true\">)</mml:mo></mml:mrow><mml:mi>d</mml:mi><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq190\"><alternatives><tex-math id=\"M461\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$H_{pw}^{min} = 0$$\\end{document}</tex-math><mml:math id=\"M462\"><mml:mrow><mml:msubsup><mml:mi>H</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">pw</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"italic\">min</mml:mi></mml:mrow></mml:msubsup><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq191\"><alternatives><tex-math id=\"M463\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\rho ({\\overrightarrow{y}}) = \\rho _{pw}({\\overrightarrow{y}})$$\\end{document}</tex-math><mml:math id=\"M464\"><mml:mrow><mml:mi>ρ</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:msub><mml:mi>ρ</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">pw</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq192\"><alternatives><tex-math id=\"M465\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\rho _{pw}({\\overrightarrow{y}})$$\\end{document}</tex-math><mml:math id=\"M466\"><mml:mrow><mml:msub><mml:mi>ρ</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">pw</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq193\"><alternatives><tex-math id=\"M467\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$H_{pw}$$\\end{document}</tex-math><mml:math id=\"M468\"><mml:msub><mml:mi>H</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">pw</mml:mi></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq194\"><alternatives><tex-math id=\"M469\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\rho ({\\overrightarrow{y}}, 0) = \\rho _{pw}({\\overrightarrow{y}}, 0)$$\\end{document}</tex-math><mml:math id=\"M470\"><mml:mrow><mml:mi>ρ</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mn>0</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:msub><mml:mi>ρ</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">pw</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mn>0</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq195\"><alternatives><tex-math id=\"M471\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\rho ({\\overrightarrow{y}}, t)$$\\end{document}</tex-math><mml:math id=\"M472\"><mml:mrow><mml:mi>ρ</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq196\"><alternatives><tex-math id=\"M473\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$H_{pw}(t)$$\\end{document}</tex-math><mml:math id=\"M474\"><mml:mrow><mml:msub><mml:mi>H</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">pw</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq197\"><alternatives><tex-math id=\"M475\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\rho ({\\overrightarrow{y}}, t)$$\\end{document}</tex-math><mml:math id=\"M476\"><mml:mrow><mml:mi>ρ</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq198\"><alternatives><tex-math id=\"M477\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$|\\psi ({\\overrightarrow{y}}, t)|^2$$\\end{document}</tex-math><mml:math id=\"M478\"><mml:mrow><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo><mml:mi>ψ</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:msup><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:msup></mml:mrow></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ45\"><label>45</label><alternatives><tex-math id=\"M479\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} \\frac{df({\\overrightarrow{y}}, t)}{dt} = \\partial _t f({\\overrightarrow{y}}, t) + \\vec {\\nabla }f({\\overrightarrow{y}}, t) \\cdot \\vec{v}({\\overrightarrow{y}}, t) = 0 \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M480\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:mfrac><mml:mrow><mml:mi>d</mml:mi><mml:mi>f</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mrow><mml:mi mathvariant=\"italic\">dt</mml:mi></mml:mrow></mml:mfrac><mml:mo>=</mml:mo><mml:msub><mml:mi>∂</mml:mi><mml:mi>t</mml:mi></mml:msub><mml:mi>f</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>+</mml:mo><mml:mover accent=\"true\"><mml:mi mathvariant=\"normal\">∇</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mi>f</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>·</mml:mo><mml:mover accent=\"true\"><mml:mi>v</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq199\"><alternatives><tex-math id=\"M481\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$f({\\overrightarrow{y}}, t) \\equiv \\rho ({\\overrightarrow{y}}, t)/|\\psi ({\\overrightarrow{y}}, t)|^2$$\\end{document}</tex-math><mml:math id=\"M482\"><mml:mrow><mml:mi>f</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>≡</mml:mo><mml:mi>ρ</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo stretchy=\"false\">/</mml:mo><mml:msup><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo><mml:mi>ψ</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:msup></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq200\"><alternatives><tex-math id=\"M483\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\rho _{pw}({\\overrightarrow{y}}, 0)$$\\end{document}</tex-math><mml:math id=\"M484\"><mml:mrow><mml:msub><mml:mi>ρ</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">pw</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mn>0</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ46\"><label>46</label><alternatives><tex-math id=\"M485\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} \\rho _{pw}({\\overrightarrow{y}}, t) = |\\psi ({\\overrightarrow{y}}, t)|^2/{\\mathcal {N}} \\text {, if} {\\overrightarrow{y}} \\in \\Omega _t \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M486\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:msub><mml:mi>ρ</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">pw</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:msup><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo><mml:mi>ψ</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:msup><mml:mo stretchy=\"false\">/</mml:mo><mml:mi mathvariant=\"script\">N</mml:mi><mml:mtext>, if</mml:mtext><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>∈</mml:mo><mml:msub><mml:mi mathvariant=\"normal\">Ω</mml:mi><mml:mi>t</mml:mi></mml:msub></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq201\"><alternatives><tex-math id=\"M487\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\Omega _t = \\{{\\overrightarrow{y}}|\\rho _{pw}({\\overrightarrow{y}}, t) &gt;0\\}$$\\end{document}</tex-math><mml:math id=\"M488\"><mml:mrow><mml:msub><mml:mi mathvariant=\"normal\">Ω</mml:mi><mml:mi>t</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mrow><mml:mo stretchy=\"false\">{</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo stretchy=\"false\">|</mml:mo><mml:msub><mml:mi>ρ</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">pw</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>&gt;</mml:mo><mml:mn>0</mml:mn><mml:mo stretchy=\"false\">}</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq202\"><alternatives><tex-math id=\"M489\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\rho _{pw}({\\overrightarrow{y}}, t)$$\\end{document}</tex-math><mml:math id=\"M490\"><mml:mrow><mml:msub><mml:mi>ρ</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">pw</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ47\"><label>47</label><alternatives><tex-math id=\"M491\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} \\int _{\\Omega _t} |\\psi ({\\overrightarrow{y}}, t)|^2 d{\\overrightarrow{y}} = {\\mathcal {N}} \\forall t \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M492\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:msub><mml:mo>∫</mml:mo><mml:msub><mml:mi mathvariant=\"normal\">Ω</mml:mi><mml:mi>t</mml:mi></mml:msub></mml:msub><mml:msup><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo><mml:mi>ψ</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:msup><mml:mi>d</mml:mi><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>=</mml:mo><mml:mi mathvariant=\"script\">N</mml:mi><mml:mo>∀</mml:mo><mml:mi>t</mml:mi></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ48\"><label>48</label><alternatives><tex-math id=\"M493\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} H_{pw}(t) = \\int _{{\\mathscr {C}}} \\bigg ( \\rho ({\\overrightarrow{y}}, t) \\ln \\frac{\\rho ({\\overrightarrow{y}}, t)}{\\rho _{pw}({\\overrightarrow{y}}, t)} - \\rho ({\\overrightarrow{y}}, t) + \\rho _{pw}({\\overrightarrow{y}}, t) \\bigg ) d{\\overrightarrow{y}} \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M494\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:msub><mml:mi>H</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">pw</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:msub><mml:mo>∫</mml:mo><mml:mi mathvariant=\"script\">C</mml:mi></mml:msub><mml:mrow><mml:mo maxsize=\"2.047em\" minsize=\"2.047em\" stretchy=\"true\">(</mml:mo></mml:mrow><mml:mi>ρ</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>ln</mml:mo><mml:mfrac><mml:mrow><mml:mi>ρ</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mrow><mml:msub><mml:mi>ρ</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">pw</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:mfrac><mml:mo>-</mml:mo><mml:mi>ρ</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>+</mml:mo><mml:msub><mml:mi>ρ</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">pw</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mrow><mml:mo maxsize=\"2.047em\" minsize=\"2.047em\" stretchy=\"true\">)</mml:mo></mml:mrow><mml:mi>d</mml:mi><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq203\"><alternatives><tex-math id=\"M495\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$H_{pw}^{min} (t)= 0$$\\end{document}</tex-math><mml:math id=\"M496\"><mml:mrow><mml:msubsup><mml:mi>H</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">pw</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"italic\">min</mml:mi></mml:mrow></mml:msubsup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq204\"><alternatives><tex-math id=\"M497\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\rho ({\\overrightarrow{y}}, t) = \\rho _{pw}({\\overrightarrow{y}}, t)$$\\end{document}</tex-math><mml:math id=\"M498\"><mml:mrow><mml:mi>ρ</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:msub><mml:mi>ρ</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">pw</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq205\"><alternatives><tex-math id=\"M499\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$H_{pw}(0)$$\\end{document}</tex-math><mml:math id=\"M500\"><mml:mrow><mml:msub><mml:mi>H</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">pw</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>0</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq206\"><alternatives><tex-math id=\"M501\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$H_{pw}(t)$$\\end{document}</tex-math><mml:math id=\"M502\"><mml:mrow><mml:msub><mml:mi>H</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">pw</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq207\"><alternatives><tex-math id=\"M503\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\Omega _t$$\\end{document}</tex-math><mml:math id=\"M504\"><mml:msub><mml:mi mathvariant=\"normal\">Ω</mml:mi><mml:mi>t</mml:mi></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq208\"><alternatives><tex-math id=\"M505\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\rho _{pw}({\\overrightarrow{y}}, t)$$\\end{document}</tex-math><mml:math id=\"M506\"><mml:mrow><mml:msub><mml:mi>ρ</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">pw</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq209\"><alternatives><tex-math id=\"M507\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\psi ({\\overrightarrow{y}}, t)$$\\end{document}</tex-math><mml:math id=\"M508\"><mml:mrow><mml:mi>ψ</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq210\"><alternatives><tex-math id=\"M509\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\Omega _t$$\\end{document}</tex-math><mml:math id=\"M510\"><mml:msub><mml:mi mathvariant=\"normal\">Ω</mml:mi><mml:mi>t</mml:mi></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq211\"><alternatives><tex-math id=\"M511\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\psi$$\\end{document}</tex-math><mml:math id=\"M512\"><mml:mi>ψ</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq212\"><alternatives><tex-math id=\"M513\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\Omega _t = {\\mathscr {C}}$$\\end{document}</tex-math><mml:math id=\"M514\"><mml:mrow><mml:msub><mml:mi mathvariant=\"normal\">Ω</mml:mi><mml:mi>t</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mi mathvariant=\"script\">C</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq213\"><alternatives><tex-math id=\"M515\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\forall t$$\\end{document}</tex-math><mml:math id=\"M516\"><mml:mrow><mml:mo>∀</mml:mo><mml:mi>t</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq214\"><alternatives><tex-math id=\"M517\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$H_{pw}$$\\end{document}</tex-math><mml:math id=\"M518\"><mml:msub><mml:mi>H</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">pw</mml:mi></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq215\"><alternatives><tex-math id=\"M519\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$H_q$$\\end{document}</tex-math><mml:math id=\"M520\"><mml:msub><mml:mi>H</mml:mi><mml:mi>q</mml:mi></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq216\"><alternatives><tex-math id=\"M521\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\Omega = {\\mathscr {C}}$$\\end{document}</tex-math><mml:math id=\"M522\"><mml:mrow><mml:mi mathvariant=\"normal\">Ω</mml:mi><mml:mo>=</mml:mo><mml:mi mathvariant=\"script\">C</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq217\"><alternatives><tex-math id=\"M523\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\rho ({\\overrightarrow{y}}, t)$$\\end{document}</tex-math><mml:math id=\"M524\"><mml:mrow><mml:mi>ρ</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq218\"><alternatives><tex-math id=\"M525\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\Omega$$\\end{document}</tex-math><mml:math id=\"M526\"><mml:mi mathvariant=\"normal\">Ω</mml:mi></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ49\"><label>49</label><alternatives><tex-math id=\"M527\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} \\rho ({\\overrightarrow{y}}, t) = \\rho _{pw}({\\overrightarrow{y}}, t) \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M528\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:mi>ρ</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:msub><mml:mi>ρ</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">pw</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq219\"><alternatives><tex-math id=\"M529\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\rho ({\\overrightarrow{y}}, t)$$\\end{document}</tex-math><mml:math id=\"M530\"><mml:mrow><mml:mi>ρ</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq220\"><alternatives><tex-math id=\"M531\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\Omega$$\\end{document}</tex-math><mml:math id=\"M532\"><mml:mi mathvariant=\"normal\">Ω</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq221\"><alternatives><tex-math id=\"M533\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\mathscr {C}}$$\\end{document}</tex-math><mml:math id=\"M534\"><mml:mi mathvariant=\"script\">C</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq222\"><alternatives><tex-math id=\"M535\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\rho _{pw}({\\overrightarrow{y}}, t)$$\\end{document}</tex-math><mml:math id=\"M536\"><mml:mrow><mml:msub><mml:mi>ρ</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">pw</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ50\"><label>50</label><alternatives><tex-math id=\"M537\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} H_{pw}(t) = \\int _{\\mathscr {C}} \\rho ({\\overrightarrow{y}}) \\ln \\frac{\\rho ({\\overrightarrow{y}})}{\\rho _{pw}({\\overrightarrow{y}}, t)} d{\\overrightarrow{y}} \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M538\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:msub><mml:mi>H</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">pw</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:msub><mml:mo>∫</mml:mo><mml:mi mathvariant=\"script\">C</mml:mi></mml:msub><mml:mi>ρ</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>ln</mml:mo><mml:mfrac><mml:mrow><mml:mi>ρ</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mrow><mml:msub><mml:mi>ρ</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">pw</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:mfrac><mml:mi>d</mml:mi><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq223\"><alternatives><tex-math id=\"M539\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\rho ({\\overrightarrow{y}}, t)$$\\end{document}</tex-math><mml:math id=\"M540\"><mml:mrow><mml:mi>ρ</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq224\"><alternatives><tex-math id=\"M541\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\psi ({\\overrightarrow{y}}, t) \\rightarrow \\alpha \\psi ({\\overrightarrow{y}}, t)$$\\end{document}</tex-math><mml:math id=\"M542\"><mml:mrow><mml:mi>ψ</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo stretchy=\"false\">→</mml:mo><mml:mi>α</mml:mi><mml:mi>ψ</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq225\"><alternatives><tex-math id=\"M543\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\alpha$$\\end{document}</tex-math><mml:math id=\"M544\"><mml:mi>α</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq226\"><alternatives><tex-math id=\"M545\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$H_{pw}$$\\end{document}</tex-math><mml:math id=\"M546\"><mml:msub><mml:mi>H</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">pw</mml:mi></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq227\"><alternatives><tex-math id=\"M547\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$H_{pw}(t)$$\\end{document}</tex-math><mml:math id=\"M548\"><mml:mrow><mml:msub><mml:mi>H</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">pw</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ51\"><label>51</label><alternatives><tex-math id=\"M549\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} H_{pw}(t) = \\ln {\\mathcal {N}}(t) + H_q(t) \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M550\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:msub><mml:mi>H</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">pw</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mo>ln</mml:mo><mml:mi mathvariant=\"script\">N</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>+</mml:mo><mml:msub><mml:mi>H</mml:mi><mml:mi>q</mml:mi></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq228\"><alternatives><tex-math id=\"M551\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\mathcal {N}}(t)$$\\end{document}</tex-math><mml:math id=\"M552\"><mml:mrow><mml:mi mathvariant=\"script\">N</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq229\"><alternatives><tex-math id=\"M553\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$H_{pw}(t)$$\\end{document}</tex-math><mml:math id=\"M554\"><mml:mrow><mml:msub><mml:mi>H</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">pw</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq230\"><alternatives><tex-math id=\"M555\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\mathcal {N}}(t)$$\\end{document}</tex-math><mml:math id=\"M556\"><mml:mrow><mml:mi mathvariant=\"script\">N</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq231\"><alternatives><tex-math id=\"M557\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\rho ({\\overrightarrow{y}}, 0)$$\\end{document}</tex-math><mml:math id=\"M558\"><mml:mrow><mml:mi>ρ</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mn>0</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq232\"><alternatives><tex-math id=\"M559\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\Omega _0$$\\end{document}</tex-math><mml:math id=\"M560\"><mml:msub><mml:mi mathvariant=\"normal\">Ω</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq233\"><alternatives><tex-math id=\"M561\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\psi ({\\overrightarrow{y}}, t)$$\\end{document}</tex-math><mml:math id=\"M562\"><mml:mrow><mml:mi>ψ</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq234\"><alternatives><tex-math id=\"M563\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\rho _{pw}({\\overrightarrow{y}}, 0) = |\\psi ({\\overrightarrow{y}}, 0)|^2/{\\mathcal {N}}(0)$$\\end{document}</tex-math><mml:math id=\"M564\"><mml:mrow><mml:msub><mml:mi>ρ</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">pw</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mn>0</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:msup><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo><mml:mi>ψ</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mn>0</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:msup><mml:mo stretchy=\"false\">/</mml:mo><mml:mi mathvariant=\"script\">N</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>0</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq235\"><alternatives><tex-math id=\"M565\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\Omega _0$$\\end{document}</tex-math><mml:math id=\"M566\"><mml:msub><mml:mi mathvariant=\"normal\">Ω</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq236\"><alternatives><tex-math id=\"M567\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\mathcal {N}}(0) = \\int _{\\Omega _0} |\\psi ({\\overrightarrow{y}}, 0)|^2 d{\\overrightarrow{y}}$$\\end{document}</tex-math><mml:math id=\"M568\"><mml:mrow><mml:mi mathvariant=\"script\">N</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>0</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:msub><mml:mo>∫</mml:mo><mml:msub><mml:mi mathvariant=\"normal\">Ω</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:msub><mml:msup><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo><mml:mi>ψ</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mn>0</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:msup><mml:mi>d</mml:mi><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq237\"><alternatives><tex-math id=\"M569\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\rho ({\\overrightarrow{y}}, 0)$$\\end{document}</tex-math><mml:math id=\"M570\"><mml:mrow><mml:mi>ρ</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mn>0</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq238\"><alternatives><tex-math id=\"M571\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\rho _{pw}({\\overrightarrow{y}}, 0)$$\\end{document}</tex-math><mml:math id=\"M572\"><mml:mrow><mml:msub><mml:mi>ρ</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">pw</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mn>0</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq239\"><alternatives><tex-math id=\"M573\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\psi ({\\overrightarrow{y}}, t)$$\\end{document}</tex-math><mml:math id=\"M574\"><mml:mrow><mml:mi>ψ</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ52\"><label>52</label><alternatives><tex-math id=\"M575\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} \\partial _t \\rho ({\\overrightarrow{y}}, t)&amp;+ \\vec {\\nabla } \\cdot \\big (\\rho ({\\overrightarrow{y}}, t) \\vec{v}({\\overrightarrow{y}}, t) \\big ) = 0 \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M576\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:msub><mml:mi>∂</mml:mi><mml:mi>t</mml:mi></mml:msub><mml:mi>ρ</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:mtd><mml:mtd columnalign=\"left\"><mml:mrow><mml:mo>+</mml:mo><mml:mover accent=\"true\"><mml:mi mathvariant=\"normal\">∇</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>·</mml:mo><mml:mrow><mml:mo maxsize=\"1.2em\" minsize=\"1.2em\" stretchy=\"true\">(</mml:mo></mml:mrow><mml:mi>ρ</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mover accent=\"true\"><mml:mi>v</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mrow><mml:mo maxsize=\"1.2em\" minsize=\"1.2em\" stretchy=\"true\">)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ53\"><label>53</label><alternatives><tex-math id=\"M577\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} \\partial _t \\rho _{pw}({\\overrightarrow{y}}, t)&amp;+ \\vec {\\nabla } \\cdot \\big (\\rho _{pw}({\\overrightarrow{y}}, t) \\vec{v}({\\overrightarrow{y}}, t) \\big ) = 0 \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M578\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:msub><mml:mi>∂</mml:mi><mml:mi>t</mml:mi></mml:msub><mml:msub><mml:mi>ρ</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">pw</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:mtd><mml:mtd columnalign=\"left\"><mml:mrow><mml:mo>+</mml:mo><mml:mover accent=\"true\"><mml:mi mathvariant=\"normal\">∇</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>·</mml:mo><mml:mrow><mml:mo maxsize=\"1.2em\" minsize=\"1.2em\" stretchy=\"true\">(</mml:mo></mml:mrow><mml:msub><mml:mi>ρ</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">pw</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mover accent=\"true\"><mml:mi>v</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mrow><mml:mo maxsize=\"1.2em\" minsize=\"1.2em\" stretchy=\"true\">)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq240\"><alternatives><tex-math id=\"M579\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\vec{v}({\\overrightarrow{y}}, t)$$\\end{document}</tex-math><mml:math id=\"M580\"><mml:mrow><mml:mover accent=\"true\"><mml:mi>v</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq241\"><alternatives><tex-math id=\"M581\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\psi ({\\overrightarrow{y}}, t)$$\\end{document}</tex-math><mml:math id=\"M582\"><mml:mrow><mml:mi>ψ</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq242\"><alternatives><tex-math id=\"M583\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\vec{v}({\\overrightarrow{y}}, t)$$\\end{document}</tex-math><mml:math id=\"M584\"><mml:mrow><mml:mover accent=\"true\"><mml:mi>v</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq243\"><alternatives><tex-math id=\"M585\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\Omega _0 \\rightarrow \\Omega _t$$\\end{document}</tex-math><mml:math id=\"M586\"><mml:mrow><mml:msub><mml:mi mathvariant=\"normal\">Ω</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo stretchy=\"false\">→</mml:mo><mml:msub><mml:mi mathvariant=\"normal\">Ω</mml:mi><mml:mi>t</mml:mi></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ54\"><label>54</label><alternatives><tex-math id=\"M587\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} \\rho _{pw}({\\overrightarrow{y}}, t) = |\\psi ({\\overrightarrow{y}}, t)|^2/{\\mathcal {N}}(0) \\text {, if}\\, {\\overrightarrow{y}} \\in \\Omega _t \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M588\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:msub><mml:mi>ρ</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">pw</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:msup><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo><mml:mi>ψ</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:msup><mml:mo stretchy=\"false\">/</mml:mo><mml:mi mathvariant=\"script\">N</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>0</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mtext>, if</mml:mtext><mml:mspace width=\"0.166667em\"/><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>∈</mml:mo><mml:msub><mml:mi mathvariant=\"normal\">Ω</mml:mi><mml:mi>t</mml:mi></mml:msub></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ55\"><label>55</label><alternatives><tex-math id=\"M589\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} {\\mathcal {N}}(t) = \\int _{\\Omega _t} |\\psi ({\\overrightarrow{y}}, t)|^2 d{\\overrightarrow{y}} = {\\mathcal {N}}(0) \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M590\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:mi mathvariant=\"script\">N</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:msub><mml:mo>∫</mml:mo><mml:msub><mml:mi mathvariant=\"normal\">Ω</mml:mi><mml:mi>t</mml:mi></mml:msub></mml:msub><mml:msup><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo><mml:mi>ψ</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:msup><mml:mi>d</mml:mi><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>=</mml:mo><mml:mi mathvariant=\"script\">N</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>0</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq244\"><alternatives><tex-math id=\"M591\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\mathcal {N}}$$\\end{document}</tex-math><mml:math id=\"M592\"><mml:mi mathvariant=\"script\">N</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq245\"><alternatives><tex-math id=\"M593\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\rho ({\\overrightarrow{y}}, 0)$$\\end{document}</tex-math><mml:math id=\"M594\"><mml:mrow><mml:mi>ρ</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mn>0</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq246\"><alternatives><tex-math id=\"M595\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\Omega _0$$\\end{document}</tex-math><mml:math id=\"M596\"><mml:msub><mml:mi mathvariant=\"normal\">Ω</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq247\"><alternatives><tex-math id=\"M597\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$|\\psi ({\\overrightarrow{y}}, 0)|^2$$\\end{document}</tex-math><mml:math id=\"M598\"><mml:mrow><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo><mml:mi>ψ</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mn>0</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:msup><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:msup></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq248\"><alternatives><tex-math id=\"M599\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$|\\psi ({\\overrightarrow{y}}, t)|^2$$\\end{document}</tex-math><mml:math id=\"M600\"><mml:mrow><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo><mml:mi>ψ</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:msup><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:msup></mml:mrow></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ56\"><label>56</label><alternatives><tex-math id=\"M601\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} \\frac{dH_{pw}(t)}{dt} = 0 \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M602\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:mfrac><mml:mrow><mml:mi>d</mml:mi><mml:msub><mml:mi>H</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">pw</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow><mml:mrow><mml:mi mathvariant=\"italic\">dt</mml:mi></mml:mrow></mml:mfrac><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq249\"><alternatives><tex-math id=\"M603\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$H_{pw}$$\\end{document}</tex-math><mml:math id=\"M604\"><mml:msub><mml:mi>H</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">pw</mml:mi></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq250\"><alternatives><tex-math id=\"M605\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\mathscr {C}}$$\\end{document}</tex-math><mml:math id=\"M606\"><mml:mi mathvariant=\"script\">C</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq251\"><alternatives><tex-math id=\"M607\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\delta V$$\\end{document}</tex-math><mml:math id=\"M608\"><mml:mrow><mml:mi>δ</mml:mi><mml:mi>V</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ57\"><label>57</label><alternatives><tex-math id=\"M609\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} \\overline{\\rho ({\\overrightarrow{y}}, t)}&amp;\\equiv \\frac{1}{\\delta V} \\int _{\\delta V} \\rho ({\\overrightarrow{y}}, t) d{\\overrightarrow{y}} \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M610\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mover><mml:mrow><mml:mi>ρ</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>¯</mml:mo></mml:mover></mml:mtd><mml:mtd columnalign=\"left\"><mml:mrow><mml:mo>≡</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:mrow><mml:mi>δ</mml:mi><mml:mi>V</mml:mi></mml:mrow></mml:mfrac><mml:msub><mml:mo>∫</mml:mo><mml:mrow><mml:mi>δ</mml:mi><mml:mi>V</mml:mi></mml:mrow></mml:msub><mml:mi>ρ</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mi>d</mml:mi><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ58\"><label>58</label><alternatives><tex-math id=\"M611\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} \\overline{\\rho _{pw}({\\overrightarrow{y}}, t)}&amp;\\equiv \\frac{1}{\\delta V} \\int _{\\delta V} \\rho _{pw}({\\overrightarrow{y}}, t) \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M612\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mover><mml:mrow><mml:msub><mml:mi>ρ</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">pw</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow><mml:mo>¯</mml:mo></mml:mover></mml:mtd><mml:mtd columnalign=\"left\"><mml:mrow><mml:mo>≡</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:mrow><mml:mi>δ</mml:mi><mml:mi>V</mml:mi></mml:mrow></mml:mfrac><mml:msub><mml:mo>∫</mml:mo><mml:mrow><mml:mi>δ</mml:mi><mml:mi>V</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mi>ρ</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">pw</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq252\"><alternatives><tex-math id=\"M613\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\int _{\\delta V} d{\\overrightarrow{y}}$$\\end{document}</tex-math><mml:math id=\"M614\"><mml:mrow><mml:msub><mml:mo>∫</mml:mo><mml:mrow><mml:mi>δ</mml:mi><mml:mi>V</mml:mi></mml:mrow></mml:msub><mml:mi>d</mml:mi><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq253\"><alternatives><tex-math id=\"M615\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\overrightarrow{y}}$$\\end{document}</tex-math><mml:math id=\"M616\"><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq254\"><alternatives><tex-math id=\"M617\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\overline{\\rho ({\\overrightarrow{y}}, t)}$$\\end{document}</tex-math><mml:math id=\"M618\"><mml:mover><mml:mrow><mml:mi>ρ</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>¯</mml:mo></mml:mover></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq255\"><alternatives><tex-math id=\"M619\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\overline{\\rho _{pw}({\\overrightarrow{y}}, t)}$$\\end{document}</tex-math><mml:math id=\"M620\"><mml:mover><mml:mrow><mml:msub><mml:mi>ρ</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">pw</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow><mml:mo>¯</mml:mo></mml:mover></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ59\"><label>59</label><alternatives><tex-math id=\"M621\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} g({\\overrightarrow{y}}, t) \\equiv {\\left\\{ \\begin{array}{ll} \\rho ({\\overrightarrow{y}}, t)/\\rho _{pw}({\\overrightarrow{y}}, t)&amp;{}, \\text {if }{\\overrightarrow{y}} \\in \\Omega _t \\\\ 0&amp;{}, \\text {if }{\\overrightarrow{y}} \\in {\\mathscr {C}}\\setminus \\Omega _t \\end{array}\\right. } \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M622\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:mi>g</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>≡</mml:mo><mml:mfenced open=\"{\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"left\"><mml:mrow><mml:mi>ρ</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo stretchy=\"false\">/</mml:mo><mml:msub><mml:mi>ρ</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">pw</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:mtd><mml:mtd columnalign=\"left\"><mml:mrow><mml:mrow/><mml:mo>,</mml:mo><mml:mtext>if</mml:mtext><mml:mspace width=\"0.333333em\"/><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>∈</mml:mo><mml:msub><mml:mi mathvariant=\"normal\">Ω</mml:mi><mml:mi>t</mml:mi></mml:msub></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign=\"left\"><mml:mrow><mml:mrow/><mml:mn>0</mml:mn></mml:mrow></mml:mtd><mml:mtd columnalign=\"left\"><mml:mrow><mml:mrow/><mml:mo>,</mml:mo><mml:mtext>if</mml:mtext><mml:mspace width=\"0.333333em\"/><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>∈</mml:mo><mml:mi mathvariant=\"script\">C</mml:mi><mml:mo lspace=\"0.15em\" rspace=\"0.15em\" stretchy=\"false\">\\</mml:mo><mml:msub><mml:mi mathvariant=\"normal\">Ω</mml:mi><mml:mi>t</mml:mi></mml:msub></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq256\"><alternatives><tex-math id=\"M623\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\overline{g({\\overrightarrow{y}}, t)} \\equiv \\overline{\\rho ({\\overrightarrow{y}}, t)}/\\overline{\\rho _{pw}({\\overrightarrow{y}}, t)}$$\\end{document}</tex-math><mml:math id=\"M624\"><mml:mrow><mml:mover><mml:mrow><mml:mi>g</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>¯</mml:mo></mml:mover><mml:mo>≡</mml:mo><mml:mover><mml:mrow><mml:mi>ρ</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>¯</mml:mo></mml:mover><mml:mo stretchy=\"false\">/</mml:mo><mml:mover><mml:mrow><mml:msub><mml:mi>ρ</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">pw</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow><mml:mo>¯</mml:mo></mml:mover></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq257\"><alternatives><tex-math id=\"M625\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\overrightarrow{y}} \\in \\overline{\\Omega _t}$$\\end{document}</tex-math><mml:math id=\"M626\"><mml:mrow><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>∈</mml:mo><mml:mover><mml:msub><mml:mi mathvariant=\"normal\">Ω</mml:mi><mml:mi>t</mml:mi></mml:msub><mml:mo>¯</mml:mo></mml:mover></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq258\"><alternatives><tex-math id=\"M627\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\overline{\\Omega _t} \\equiv \\{{\\overrightarrow{y}}|\\overline{\\rho ({\\overrightarrow{y}}, t)} &gt;0\\}$$\\end{document}</tex-math><mml:math id=\"M628\"><mml:mrow><mml:mover><mml:msub><mml:mi mathvariant=\"normal\">Ω</mml:mi><mml:mi>t</mml:mi></mml:msub><mml:mo>¯</mml:mo></mml:mover><mml:mo>≡</mml:mo><mml:mrow><mml:mo stretchy=\"false\">{</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo stretchy=\"false\">|</mml:mo><mml:mover><mml:mrow><mml:mi>ρ</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>¯</mml:mo></mml:mover><mml:mo>&gt;</mml:mo><mml:mn>0</mml:mn><mml:mo stretchy=\"false\">}</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq259\"><alternatives><tex-math id=\"M629\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\mathscr {C}}$$\\end{document}</tex-math><mml:math id=\"M630\"><mml:mi mathvariant=\"script\">C</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq260\"><alternatives><tex-math id=\"M631\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$g({\\overrightarrow{y}}, t)$$\\end{document}</tex-math><mml:math id=\"M632\"><mml:mrow><mml:mi>g</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ60\"><label>60</label><alternatives><tex-math id=\"M633\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} \\frac{dg({\\overrightarrow{y}}, t)}{dt} = \\partial _t g({\\overrightarrow{y}}, t) + \\vec {\\nabla }g({\\overrightarrow{y}}, t) \\cdot \\vec{v}({\\overrightarrow{y}}, t) = 0 \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M634\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:mfrac><mml:mrow><mml:mi>d</mml:mi><mml:mi>g</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mrow><mml:mi mathvariant=\"italic\">dt</mml:mi></mml:mrow></mml:mfrac><mml:mo>=</mml:mo><mml:msub><mml:mi>∂</mml:mi><mml:mi>t</mml:mi></mml:msub><mml:mi>g</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>+</mml:mo><mml:mover accent=\"true\"><mml:mi mathvariant=\"normal\">∇</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mi>g</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>·</mml:mo><mml:mover accent=\"true\"><mml:mi>v</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq261\"><alternatives><tex-math id=\"M635\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$H_{pw}$$\\end{document}</tex-math><mml:math id=\"M636\"><mml:msub><mml:mi>H</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">pw</mml:mi></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ61\"><label>61</label><alternatives><tex-math id=\"M637\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} \\overline{H_{pw}(t)}&amp;\\equiv \\int _{{\\mathscr {C}}} \\overline{\\rho ({\\overrightarrow{y}}, t)} \\ln \\frac{\\overline{\\rho ({\\overrightarrow{y}}, t)}}{\\overline{\\rho _{pw}({\\overrightarrow{y}}, t)}} d{\\overrightarrow{y}} \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M638\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mover><mml:mrow><mml:msub><mml:mi>H</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">pw</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow><mml:mo>¯</mml:mo></mml:mover></mml:mtd><mml:mtd columnalign=\"left\"><mml:mrow><mml:mo>≡</mml:mo><mml:msub><mml:mo>∫</mml:mo><mml:mi mathvariant=\"script\">C</mml:mi></mml:msub><mml:mover><mml:mrow><mml:mi>ρ</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>¯</mml:mo></mml:mover><mml:mo>ln</mml:mo><mml:mfrac><mml:mover><mml:mrow><mml:mi>ρ</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>¯</mml:mo></mml:mover><mml:mover><mml:mrow><mml:msub><mml:mi>ρ</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">pw</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow><mml:mo>¯</mml:mo></mml:mover></mml:mfrac><mml:mi>d</mml:mi><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ62\"><label>62</label><alternatives><tex-math id=\"M639\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned}{}&amp;= \\int _{{\\mathscr {C}}} \\overline{\\rho ({\\overrightarrow{y}}, t)} \\ln \\overline{g({\\overrightarrow{y}}, t)} d{\\overrightarrow{y}} \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M640\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow/></mml:mtd><mml:mtd columnalign=\"left\"><mml:mrow><mml:mo>=</mml:mo><mml:msub><mml:mo>∫</mml:mo><mml:mi mathvariant=\"script\">C</mml:mi></mml:msub><mml:mover><mml:mrow><mml:mi>ρ</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>¯</mml:mo></mml:mover><mml:mo>ln</mml:mo><mml:mover><mml:mrow><mml:mi>g</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>¯</mml:mo></mml:mover><mml:mi>d</mml:mi><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ63\"><label>63</label><alternatives><tex-math id=\"M641\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} \\rho ({\\overrightarrow{y}}, 0)&amp;= \\overline{\\rho ({\\overrightarrow{y}}, 0)} \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M642\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:mi>ρ</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mn>0</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mtd><mml:mtd columnalign=\"left\"><mml:mrow><mml:mo>=</mml:mo><mml:mover><mml:mrow><mml:mi>ρ</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mn>0</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>¯</mml:mo></mml:mover></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ64\"><label>64</label><alternatives><tex-math id=\"M643\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} \\rho _{pw}({\\overrightarrow{y}}, 0)&amp;= \\overline{\\rho _{pw}({\\overrightarrow{y}}, 0)} \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M644\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:msub><mml:mi>ρ</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">pw</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mn>0</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:mtd><mml:mtd columnalign=\"left\"><mml:mrow><mml:mo>=</mml:mo><mml:mover><mml:mrow><mml:msub><mml:mi>ρ</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">pw</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mn>0</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow><mml:mo>¯</mml:mo></mml:mover></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ65\"><label>65</label><alternatives><tex-math id=\"M645\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} \\overline{H_{pw}(0)} - \\overline{H_{pw}(t)} = \\int _{{\\mathscr {C}}} \\overline{\\rho ({\\overrightarrow{y}}, 0)} \\ln \\overline{g({\\overrightarrow{y}}, 0)} d{\\overrightarrow{y}} - \\int _{{\\mathscr {C}}} \\overline{\\rho ({\\overrightarrow{y}}, t)} \\ln \\overline{g({\\overrightarrow{y}}, t)} d{\\overrightarrow{y}} \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M646\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:mover><mml:mrow><mml:msub><mml:mi>H</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">pw</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>0</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow><mml:mo>¯</mml:mo></mml:mover><mml:mo>-</mml:mo><mml:mover><mml:mrow><mml:msub><mml:mi>H</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">pw</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow><mml:mo>¯</mml:mo></mml:mover><mml:mo>=</mml:mo><mml:msub><mml:mo>∫</mml:mo><mml:mi mathvariant=\"script\">C</mml:mi></mml:msub><mml:mover><mml:mrow><mml:mi>ρ</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mn>0</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>¯</mml:mo></mml:mover><mml:mo>ln</mml:mo><mml:mover><mml:mrow><mml:mi>g</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mn>0</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>¯</mml:mo></mml:mover><mml:mi>d</mml:mi><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>-</mml:mo><mml:msub><mml:mo>∫</mml:mo><mml:mi mathvariant=\"script\">C</mml:mi></mml:msub><mml:mover><mml:mrow><mml:mi>ρ</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>¯</mml:mo></mml:mover><mml:mo>ln</mml:mo><mml:mover><mml:mrow><mml:mi>g</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>¯</mml:mo></mml:mover><mml:mi>d</mml:mi><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq262\"><alternatives><tex-math id=\"M647\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$H_{pw}(t)$$\\end{document}</tex-math><mml:math id=\"M648\"><mml:mrow><mml:msub><mml:mi>H</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">pw</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ66\"><label>66</label><alternatives><tex-math id=\"M649\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} \\int _{{\\mathscr {C}}} \\overline{\\rho ({\\overrightarrow{y}}, 0)} \\ln \\overline{g({\\overrightarrow{y}}, 0)} d{\\overrightarrow{y}}&amp;= \\int _{{\\mathscr {C}}} \\rho ({\\overrightarrow{y}}, 0) \\ln g({\\overrightarrow{y}}, 0) d{\\overrightarrow{y}} \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M650\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:msub><mml:mo>∫</mml:mo><mml:mi mathvariant=\"script\">C</mml:mi></mml:msub><mml:mover><mml:mrow><mml:mi>ρ</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mn>0</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>¯</mml:mo></mml:mover><mml:mo>ln</mml:mo><mml:mover><mml:mrow><mml:mi>g</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mn>0</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>¯</mml:mo></mml:mover><mml:mi>d</mml:mi><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover></mml:mrow></mml:mtd><mml:mtd columnalign=\"left\"><mml:mrow><mml:mo>=</mml:mo><mml:msub><mml:mo>∫</mml:mo><mml:mi mathvariant=\"script\">C</mml:mi></mml:msub><mml:mi>ρ</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mn>0</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>ln</mml:mo><mml:mi>g</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mn>0</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mi>d</mml:mi><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ67\"><label>67</label><alternatives><tex-math id=\"M651\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned}{}&amp;= \\int _{{\\mathscr {C}}} \\rho ({\\overrightarrow{y}}, t) \\ln g({\\overrightarrow{y}}, t) d{\\overrightarrow{y}} \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M652\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow/></mml:mtd><mml:mtd columnalign=\"left\"><mml:mrow><mml:mo>=</mml:mo><mml:msub><mml:mo>∫</mml:mo><mml:mi mathvariant=\"script\">C</mml:mi></mml:msub><mml:mi>ρ</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>ln</mml:mo><mml:mi>g</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mi>d</mml:mi><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ68\"><label>68</label><alternatives><tex-math id=\"M653\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} \\int _{{\\mathscr {C}}} \\overline{\\rho ({\\overrightarrow{y}}, t)} \\ln \\overline{g({\\overrightarrow{y}}, t)} d{\\overrightarrow{y}} = \\sum _i \\int _{\\delta V_i} \\overline{\\rho ({\\overrightarrow{y}}, t)} \\ln \\overline{g({\\overrightarrow{y}}, t)} d{\\overrightarrow{y}} \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M654\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:msub><mml:mo>∫</mml:mo><mml:mi mathvariant=\"script\">C</mml:mi></mml:msub><mml:mover><mml:mrow><mml:mi>ρ</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>¯</mml:mo></mml:mover><mml:mo>ln</mml:mo><mml:mover><mml:mrow><mml:mi>g</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>¯</mml:mo></mml:mover><mml:mi>d</mml:mi><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>=</mml:mo><mml:munder><mml:mo>∑</mml:mo><mml:mi>i</mml:mi></mml:munder><mml:msub><mml:mo>∫</mml:mo><mml:mrow><mml:mi>δ</mml:mi><mml:msub><mml:mi>V</mml:mi><mml:mi>i</mml:mi></mml:msub></mml:mrow></mml:msub><mml:mover><mml:mrow><mml:mi>ρ</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>¯</mml:mo></mml:mover><mml:mo>ln</mml:mo><mml:mover><mml:mrow><mml:mi>g</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>¯</mml:mo></mml:mover><mml:mi>d</mml:mi><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq263\"><alternatives><tex-math id=\"M655\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\mathscr {C}}$$\\end{document}</tex-math><mml:math id=\"M656\"><mml:mi mathvariant=\"script\">C</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq264\"><alternatives><tex-math id=\"M657\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\delta V$$\\end{document}</tex-math><mml:math id=\"M658\"><mml:mrow><mml:mi>δ</mml:mi><mml:mi>V</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq265\"><alternatives><tex-math id=\"M659\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\overline{\\rho ({\\overrightarrow{y}}, t)}$$\\end{document}</tex-math><mml:math id=\"M660\"><mml:mover><mml:mrow><mml:mi>ρ</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>¯</mml:mo></mml:mover></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq266\"><alternatives><tex-math id=\"M661\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\overline{\\rho _{pw}({\\overrightarrow{y}}, t)}$$\\end{document}</tex-math><mml:math id=\"M662\"><mml:mover><mml:mrow><mml:msub><mml:mi>ρ</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">pw</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow><mml:mo>¯</mml:mo></mml:mover></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq267\"><alternatives><tex-math id=\"M663\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\overline{\\rho ({\\overrightarrow{y}}, t)} = \\overline{\\rho _i(t)}$$\\end{document}</tex-math><mml:math id=\"M664\"><mml:mrow><mml:mover><mml:mrow><mml:mi>ρ</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>¯</mml:mo></mml:mover><mml:mo>=</mml:mo><mml:mover><mml:mrow><mml:msub><mml:mi>ρ</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow><mml:mo>¯</mml:mo></mml:mover></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq268\"><alternatives><tex-math id=\"M665\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\overline{\\rho _{pw}({\\overrightarrow{y}}, t)} = \\overline{\\rho _{pwi}(t)}$$\\end{document}</tex-math><mml:math id=\"M666\"><mml:mrow><mml:mover><mml:mrow><mml:msub><mml:mi>ρ</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">pw</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow><mml:mo>¯</mml:mo></mml:mover><mml:mo>=</mml:mo><mml:mover><mml:mrow><mml:msub><mml:mi>ρ</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">pwi</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow><mml:mo>¯</mml:mo></mml:mover></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq269\"><alternatives><tex-math id=\"M667\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\overline{g({\\overrightarrow{y}}, t)} = \\overline{g_i(t)}$$\\end{document}</tex-math><mml:math id=\"M668\"><mml:mrow><mml:mover><mml:mrow><mml:mi>g</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>¯</mml:mo></mml:mover><mml:mo>=</mml:mo><mml:mover><mml:mrow><mml:msub><mml:mi>g</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow><mml:mo>¯</mml:mo></mml:mover></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq270\"><alternatives><tex-math id=\"M669\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\overrightarrow{y}}$$\\end{document}</tex-math><mml:math id=\"M670\"><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq271\"><alternatives><tex-math id=\"M671\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$i^{th}$$\\end{document}</tex-math><mml:math id=\"M672\"><mml:msup><mml:mi>i</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">th</mml:mi></mml:mrow></mml:msup></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ69\"><label>69</label><alternatives><tex-math id=\"M673\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} \\sum _i \\int _{\\delta V_i} \\overline{\\rho ({\\overrightarrow{y}}, t)} \\ln \\overline{g({\\overrightarrow{y}}, t)} d{\\overrightarrow{y}}&amp;= \\sum _i \\overline{\\rho _i(t)} \\ln \\overline{g_i(t)} \\delta V \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M674\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:munder><mml:mo>∑</mml:mo><mml:mi>i</mml:mi></mml:munder><mml:msub><mml:mo>∫</mml:mo><mml:mrow><mml:mi>δ</mml:mi><mml:msub><mml:mi>V</mml:mi><mml:mi>i</mml:mi></mml:msub></mml:mrow></mml:msub><mml:mover><mml:mrow><mml:mi>ρ</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>¯</mml:mo></mml:mover><mml:mo>ln</mml:mo><mml:mover><mml:mrow><mml:mi>g</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>¯</mml:mo></mml:mover><mml:mi>d</mml:mi><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover></mml:mrow></mml:mtd><mml:mtd columnalign=\"left\"><mml:mrow><mml:mo>=</mml:mo><mml:munder><mml:mo>∑</mml:mo><mml:mi>i</mml:mi></mml:munder><mml:mover><mml:mrow><mml:msub><mml:mi>ρ</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow><mml:mo>¯</mml:mo></mml:mover><mml:mo>ln</mml:mo><mml:mover><mml:mrow><mml:msub><mml:mi>g</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow><mml:mo>¯</mml:mo></mml:mover><mml:mi>δ</mml:mi><mml:mi>V</mml:mi></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ70\"><label>70</label><alternatives><tex-math id=\"M675\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned}{}&amp;= \\sum _i \\overline{\\rho _i(t)} \\ln \\overline{g_i(t)} \\frac{\\int _{\\delta V} \\rho ({\\overrightarrow{y}}, t) d{\\overrightarrow{y}}}{\\overline{\\rho _i(t)}} \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M676\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow/></mml:mtd><mml:mtd columnalign=\"left\"><mml:mrow><mml:mo>=</mml:mo><mml:munder><mml:mo>∑</mml:mo><mml:mi>i</mml:mi></mml:munder><mml:mover><mml:mrow><mml:msub><mml:mi>ρ</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow><mml:mo>¯</mml:mo></mml:mover><mml:mo>ln</mml:mo><mml:mover><mml:mrow><mml:msub><mml:mi>g</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow><mml:mo>¯</mml:mo></mml:mover><mml:mfrac><mml:mrow><mml:msub><mml:mo>∫</mml:mo><mml:mrow><mml:mi>δ</mml:mi><mml:mi>V</mml:mi></mml:mrow></mml:msub><mml:mi>ρ</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mi>d</mml:mi><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover></mml:mrow><mml:mover><mml:mrow><mml:msub><mml:mi>ρ</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow><mml:mo>¯</mml:mo></mml:mover></mml:mfrac></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ71\"><label>71</label><alternatives><tex-math id=\"M677\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned}{}&amp;= \\int _{{\\mathscr {C}}} \\rho ({\\overrightarrow{y}}, t) \\ln \\overline{g({\\overrightarrow{y}}, t)} d{\\overrightarrow{y}} \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M678\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow/></mml:mtd><mml:mtd columnalign=\"left\"><mml:mrow><mml:mo>=</mml:mo><mml:msub><mml:mo>∫</mml:mo><mml:mi mathvariant=\"script\">C</mml:mi></mml:msub><mml:mi>ρ</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>ln</mml:mo><mml:mover><mml:mrow><mml:mi>g</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>¯</mml:mo></mml:mover><mml:mi>d</mml:mi><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ72\"><label>72</label><alternatives><tex-math id=\"M679\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} \\overline{H_{pw}(0)} - \\overline{H_{pw}(t)}&amp;= \\int _{{\\mathscr {C}}} \\rho ({\\overrightarrow{y}}, t) \\ln \\frac{g({\\overrightarrow{y}}, t)}{\\overline{g({\\overrightarrow{y}}, t)}} d{\\overrightarrow{y}} \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M680\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:mover><mml:mrow><mml:msub><mml:mi>H</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">pw</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>0</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow><mml:mo>¯</mml:mo></mml:mover><mml:mo>-</mml:mo><mml:mover><mml:mrow><mml:msub><mml:mi>H</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">pw</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow><mml:mo>¯</mml:mo></mml:mover></mml:mrow></mml:mtd><mml:mtd columnalign=\"left\"><mml:mrow><mml:mo>=</mml:mo><mml:msub><mml:mo>∫</mml:mo><mml:mi mathvariant=\"script\">C</mml:mi></mml:msub><mml:mi>ρ</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>ln</mml:mo><mml:mfrac><mml:mrow><mml:mi>g</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mover><mml:mrow><mml:mi>g</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>¯</mml:mo></mml:mover></mml:mfrac><mml:mi>d</mml:mi><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ73\"><label>73</label><alternatives><tex-math id=\"M681\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned}{}&amp;= \\int _{{\\mathscr {C}}} \\rho _{pw}({\\overrightarrow{y}}, t) g({\\overrightarrow{y}}, t)\\ln \\frac{g({\\overrightarrow{y}}, t)}{\\overline{g({\\overrightarrow{y}}, t)}} d{\\overrightarrow{y}} \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M682\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow/></mml:mtd><mml:mtd columnalign=\"left\"><mml:mrow><mml:mo>=</mml:mo><mml:msub><mml:mo>∫</mml:mo><mml:mi mathvariant=\"script\">C</mml:mi></mml:msub><mml:msub><mml:mi>ρ</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">pw</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mi>g</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>ln</mml:mo><mml:mfrac><mml:mrow><mml:mi>g</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mover><mml:mrow><mml:mi>g</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>¯</mml:mo></mml:mover></mml:mfrac><mml:mi>d</mml:mi><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ74\"><label>74</label><alternatives><tex-math id=\"M683\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} \\int _{{\\mathscr {C}}} \\rho _{pw}({\\overrightarrow{y}}, t)\\overline{g({\\overrightarrow{y}}, t)}d{\\overrightarrow{y}}&amp;= \\sum _i \\int _{\\delta V_i} \\rho _{pw}({\\overrightarrow{y}}, t)\\frac{\\overline{\\rho ({\\overrightarrow{y}}, t)}}{\\overline{\\rho _{pw}({\\overrightarrow{y}}, t)}} d{\\overrightarrow{y}} \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M684\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:msub><mml:mo>∫</mml:mo><mml:mi mathvariant=\"script\">C</mml:mi></mml:msub><mml:msub><mml:mi>ρ</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">pw</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mover><mml:mrow><mml:mi>g</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>¯</mml:mo></mml:mover><mml:mi>d</mml:mi><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover></mml:mrow></mml:mtd><mml:mtd columnalign=\"left\"><mml:mrow><mml:mo>=</mml:mo><mml:munder><mml:mo>∑</mml:mo><mml:mi>i</mml:mi></mml:munder><mml:msub><mml:mo>∫</mml:mo><mml:mrow><mml:mi>δ</mml:mi><mml:msub><mml:mi>V</mml:mi><mml:mi>i</mml:mi></mml:msub></mml:mrow></mml:msub><mml:msub><mml:mi>ρ</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">pw</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mfrac><mml:mover><mml:mrow><mml:mi>ρ</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>¯</mml:mo></mml:mover><mml:mover><mml:mrow><mml:msub><mml:mi>ρ</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">pw</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow><mml:mo>¯</mml:mo></mml:mover></mml:mfrac><mml:mi>d</mml:mi><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ75\"><label>75</label><alternatives><tex-math id=\"M685\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned}{}&amp;= \\sum _i \\frac{\\overline{\\rho _i(t)}}{\\overline{\\rho _{pwi}(t)}} \\int _{\\delta V_i} \\rho _{pw}({\\overrightarrow{y}}, t) d{\\overrightarrow{y}} \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M686\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow/></mml:mtd><mml:mtd columnalign=\"left\"><mml:mrow><mml:mo>=</mml:mo><mml:munder><mml:mo>∑</mml:mo><mml:mi>i</mml:mi></mml:munder><mml:mfrac><mml:mover><mml:mrow><mml:msub><mml:mi>ρ</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow><mml:mo>¯</mml:mo></mml:mover><mml:mover><mml:mrow><mml:msub><mml:mi>ρ</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">pwi</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow><mml:mo>¯</mml:mo></mml:mover></mml:mfrac><mml:msub><mml:mo>∫</mml:mo><mml:mrow><mml:mi>δ</mml:mi><mml:msub><mml:mi>V</mml:mi><mml:mi>i</mml:mi></mml:msub></mml:mrow></mml:msub><mml:msub><mml:mi>ρ</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">pw</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mi>d</mml:mi><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ76\"><label>76</label><alternatives><tex-math id=\"M687\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned}{}&amp;= \\sum _i \\overline{\\rho _i(t)} \\delta V \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M688\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow/></mml:mtd><mml:mtd columnalign=\"left\"><mml:mrow><mml:mo>=</mml:mo><mml:munder><mml:mo>∑</mml:mo><mml:mi>i</mml:mi></mml:munder><mml:mover><mml:mrow><mml:msub><mml:mi>ρ</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow><mml:mo>¯</mml:mo></mml:mover><mml:mi>δ</mml:mi><mml:mi>V</mml:mi></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ77\"><label>77</label><alternatives><tex-math id=\"M689\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned}{}&amp;= \\int _{{\\mathscr {C}}} \\overline{\\rho ({\\overrightarrow{y}}, t)}d{\\overrightarrow{y}} = 1 \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M690\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow/></mml:mtd><mml:mtd columnalign=\"left\"><mml:mrow><mml:mo>=</mml:mo><mml:msub><mml:mo>∫</mml:mo><mml:mi mathvariant=\"script\">C</mml:mi></mml:msub><mml:mover><mml:mrow><mml:mi>ρ</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>¯</mml:mo></mml:mover><mml:mi>d</mml:mi><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ78\"><label>78</label><alternatives><tex-math id=\"M691\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} \\overline{H_{pw}(0)} - \\overline{H_{pw}(t)} = \\int _{{\\mathscr {C}}} \\rho _{pw}({\\overrightarrow{y}}, t) \\bigg ( g({\\overrightarrow{y}}, t)\\ln \\frac{g({\\overrightarrow{y}}, t)}{\\overline{g({\\overrightarrow{y}}, t)}} - g({\\overrightarrow{y}}, t) + \\overline{g({\\overrightarrow{y}}, t)} \\bigg ) d{\\overrightarrow{y}} \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M692\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:mover><mml:mrow><mml:msub><mml:mi>H</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">pw</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>0</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow><mml:mo>¯</mml:mo></mml:mover><mml:mo>-</mml:mo><mml:mover><mml:mrow><mml:msub><mml:mi>H</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">pw</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow><mml:mo>¯</mml:mo></mml:mover><mml:mo>=</mml:mo><mml:msub><mml:mo>∫</mml:mo><mml:mi mathvariant=\"script\">C</mml:mi></mml:msub><mml:msub><mml:mi>ρ</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">pw</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mrow><mml:mo maxsize=\"2.047em\" minsize=\"2.047em\" stretchy=\"true\">(</mml:mo></mml:mrow><mml:mi>g</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>ln</mml:mo><mml:mfrac><mml:mrow><mml:mi>g</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mover><mml:mrow><mml:mi>g</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>¯</mml:mo></mml:mover></mml:mfrac><mml:mo>-</mml:mo><mml:mi>g</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>+</mml:mo><mml:mover><mml:mrow><mml:mi>g</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>¯</mml:mo></mml:mover><mml:mrow><mml:mo maxsize=\"2.047em\" minsize=\"2.047em\" stretchy=\"true\">)</mml:mo></mml:mrow><mml:mi>d</mml:mi><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq272\"><alternatives><tex-math id=\"M693\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$x\\ln (x/y) -x + y \\ge 0$$\\end{document}</tex-math><mml:math id=\"M694\"><mml:mrow><mml:mi>x</mml:mi><mml:mo>ln</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>x</mml:mi><mml:mo stretchy=\"false\">/</mml:mo><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>-</mml:mo><mml:mi>x</mml:mi><mml:mo>+</mml:mo><mml:mi>y</mml:mi><mml:mo>≥</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq273\"><alternatives><tex-math id=\"M695\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\overline{H_{pw}(0)} - \\overline{H_{pw}(t)} \\ge 0$$\\end{document}</tex-math><mml:math id=\"M696\"><mml:mrow><mml:mover><mml:mrow><mml:msub><mml:mi>H</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">pw</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>0</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow><mml:mo>¯</mml:mo></mml:mover><mml:mo>-</mml:mo><mml:mover><mml:mrow><mml:msub><mml:mi>H</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">pw</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow><mml:mo>¯</mml:mo></mml:mover><mml:mo>≥</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq274\"><alternatives><tex-math id=\"M697\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\overline{H_{pw}(t)}$$\\end{document}</tex-math><mml:math id=\"M698\"><mml:mover><mml:mrow><mml:msub><mml:mi>H</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">pw</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow><mml:mo>¯</mml:mo></mml:mover></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq275\"><alternatives><tex-math id=\"M699\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$H_{pw}$$\\end{document}</tex-math><mml:math id=\"M700\"><mml:msub><mml:mi>H</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">pw</mml:mi></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ79\"><label>79</label><alternatives><tex-math id=\"M701\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} \\overline{\\rho _{pw}({\\overrightarrow{y}}, t)}&amp;= \\frac{1}{\\delta V} \\int _{\\delta V} \\frac{|\\psi ({\\overrightarrow{y}}, t)|^2}{{\\mathcal {N}}} \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M702\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mover><mml:mrow><mml:msub><mml:mi>ρ</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">pw</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow><mml:mo>¯</mml:mo></mml:mover></mml:mtd><mml:mtd columnalign=\"left\"><mml:mrow><mml:mo>=</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:mrow><mml:mi>δ</mml:mi><mml:mi>V</mml:mi></mml:mrow></mml:mfrac><mml:msub><mml:mo>∫</mml:mo><mml:mrow><mml:mi>δ</mml:mi><mml:mi>V</mml:mi></mml:mrow></mml:msub><mml:mfrac><mml:mrow><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo><mml:mi>ψ</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:msup><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:msup></mml:mrow><mml:mi mathvariant=\"script\">N</mml:mi></mml:mfrac></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ80\"><label>80</label><alternatives><tex-math id=\"M703\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned}{}&amp;= \\frac{\\overline{|\\psi ({\\overrightarrow{y}}, t)|^2}}{{\\mathcal {N}}} \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M704\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow/></mml:mtd><mml:mtd columnalign=\"left\"><mml:mrow><mml:mo>=</mml:mo><mml:mfrac><mml:mover><mml:mrow><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo><mml:mi>ψ</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:msup><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:msup></mml:mrow><mml:mo>¯</mml:mo></mml:mover><mml:mi mathvariant=\"script\">N</mml:mi></mml:mfrac></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq276\"><alternatives><tex-math id=\"M705\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\overline{|\\psi ({\\overrightarrow{y}}, t)|^2} \\equiv \\int _{\\delta V} |\\psi ({\\overrightarrow{y}}, t)|^2/\\delta V$$\\end{document}</tex-math><mml:math id=\"M706\"><mml:mrow><mml:mover><mml:mrow><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo><mml:mi>ψ</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:msup><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:msup></mml:mrow><mml:mo>¯</mml:mo></mml:mover><mml:mo>≡</mml:mo><mml:msub><mml:mo>∫</mml:mo><mml:mrow><mml:mi>δ</mml:mi><mml:mi>V</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo><mml:mi>ψ</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:msup><mml:mo stretchy=\"false\">/</mml:mo><mml:mi>δ</mml:mi><mml:mi>V</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq277\"><alternatives><tex-math id=\"M707\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\overrightarrow{y}} \\in \\Omega _t$$\\end{document}</tex-math><mml:math id=\"M708\"><mml:mrow><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>∈</mml:mo><mml:msub><mml:mi mathvariant=\"normal\">Ω</mml:mi><mml:mi>t</mml:mi></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ81\"><label>81</label><alternatives><tex-math id=\"M709\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} \\overline{H_{pw}(t)} = \\overline{H_{q}(t)} + \\ln {\\mathcal {N}} \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M710\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:mover><mml:mrow><mml:msub><mml:mi>H</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">pw</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow><mml:mo>¯</mml:mo></mml:mover><mml:mo>=</mml:mo><mml:mover><mml:mrow><mml:msub><mml:mi>H</mml:mi><mml:mi>q</mml:mi></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow><mml:mo>¯</mml:mo></mml:mover><mml:mo>+</mml:mo><mml:mo>ln</mml:mo><mml:mi mathvariant=\"script\">N</mml:mi></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ82\"><label>82</label><alternatives><tex-math id=\"M711\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} \\overline{H_{q}(t)} \\equiv \\int _{{\\mathscr {C}}} \\overline{\\rho ({\\overrightarrow{y}}, t)} \\ln \\frac{\\overline{\\rho ({\\overrightarrow{y}}, t)}}{\\overline{|\\psi ({\\overrightarrow{y}}, t)|^2}} d{\\overrightarrow{y}} \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M712\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:mover><mml:mrow><mml:msub><mml:mi>H</mml:mi><mml:mi>q</mml:mi></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow><mml:mo>¯</mml:mo></mml:mover><mml:mo>≡</mml:mo><mml:msub><mml:mo>∫</mml:mo><mml:mi mathvariant=\"script\">C</mml:mi></mml:msub><mml:mover><mml:mrow><mml:mi>ρ</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>¯</mml:mo></mml:mover><mml:mo>ln</mml:mo><mml:mfrac><mml:mover><mml:mrow><mml:mi>ρ</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>¯</mml:mo></mml:mover><mml:mover><mml:mrow><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo><mml:mi>ψ</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:msup><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:msup></mml:mrow><mml:mo>¯</mml:mo></mml:mover></mml:mfrac><mml:mi>d</mml:mi><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq278\"><alternatives><tex-math id=\"M713\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\overline{H_{q}(t)}$$\\end{document}</tex-math><mml:math id=\"M714\"><mml:mover><mml:mrow><mml:msub><mml:mi>H</mml:mi><mml:mi>q</mml:mi></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow><mml:mo>¯</mml:mo></mml:mover></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq279\"><alternatives><tex-math id=\"M715\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$H_{q}^{min} = -\\ln {\\mathcal {N}}$$\\end{document}</tex-math><mml:math id=\"M716\"><mml:mrow><mml:msubsup><mml:mi>H</mml:mi><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"italic\">min</mml:mi></mml:mrow></mml:msubsup><mml:mo>=</mml:mo><mml:mo>-</mml:mo><mml:mo>ln</mml:mo><mml:mi mathvariant=\"script\">N</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq280\"><alternatives><tex-math id=\"M717\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$H_{pw}^{min} = 0$$\\end{document}</tex-math><mml:math id=\"M718\"><mml:mrow><mml:msubsup><mml:mi>H</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">pw</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"italic\">min</mml:mi></mml:mrow></mml:msubsup><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq281\"><alternatives><tex-math id=\"M719\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\overline{H_{q}(t)}$$\\end{document}</tex-math><mml:math id=\"M720\"><mml:mover><mml:mrow><mml:msub><mml:mi>H</mml:mi><mml:mi>q</mml:mi></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow><mml:mo>¯</mml:mo></mml:mover></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq282\"><alternatives><tex-math id=\"M721\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\overline{H_{pw}(t)}$$\\end{document}</tex-math><mml:math id=\"M722\"><mml:mover><mml:mrow><mml:msub><mml:mi>H</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">pw</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow><mml:mo>¯</mml:mo></mml:mover></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq283\"><alternatives><tex-math id=\"M723\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\psi ({\\overrightarrow{y}}, t) = \\sum _{j=1}^n c_j(t) \\prod _{g=1}^N \\psi ^{K_j^g}(y_g)$$\\end{document}</tex-math><mml:math id=\"M724\"><mml:mrow><mml:mi>ψ</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:msubsup><mml:mo>∑</mml:mo><mml:mrow><mml:mi>j</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mi>n</mml:mi></mml:msubsup><mml:msub><mml:mi>c</mml:mi><mml:mi>j</mml:mi></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:msubsup><mml:mo>∏</mml:mo><mml:mrow><mml:mi>g</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mi>N</mml:mi></mml:msubsup><mml:msup><mml:mi>ψ</mml:mi><mml:msubsup><mml:mi>K</mml:mi><mml:mi>j</mml:mi><mml:mi>g</mml:mi></mml:msubsup></mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>y</mml:mi><mml:mi>g</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq284\"><alternatives><tex-math id=\"M725\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$v_r({\\overrightarrow{y}}) \\sim 1/y_r^2$$\\end{document}</tex-math><mml:math id=\"M726\"><mml:mrow><mml:msub><mml:mi>v</mml:mi><mml:mi>r</mml:mi></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>∼</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">/</mml:mo><mml:msubsup><mml:mi>y</mml:mi><mml:mi>r</mml:mi><mml:mn>2</mml:mn></mml:msubsup></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq285\"><alternatives><tex-math id=\"M727\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\pm y_r$$\\end{document}</tex-math><mml:math id=\"M728\"><mml:mrow><mml:mo>±</mml:mo><mml:msub><mml:mi>y</mml:mi><mml:mi>r</mml:mi></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq286\"><alternatives><tex-math id=\"M729\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$v_r({\\overrightarrow{y}})$$\\end{document}</tex-math><mml:math id=\"M730\"><mml:mrow><mml:msub><mml:mi>v</mml:mi><mml:mi>r</mml:mi></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq287\"><alternatives><tex-math id=\"M731\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$y_r = \\pm L$$\\end{document}</tex-math><mml:math id=\"M732\"><mml:mrow><mml:msub><mml:mi>y</mml:mi><mml:mi>r</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mo>±</mml:mo><mml:mi>L</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq288\"><alternatives><tex-math id=\"M733\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\rho ({\\overrightarrow{y}}, 0)$$\\end{document}</tex-math><mml:math id=\"M734\"><mml:mrow><mml:mi>ρ</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mn>0</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq289\"><alternatives><tex-math id=\"M735\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$|y_r| \\le L$$\\end{document}</tex-math><mml:math id=\"M736\"><mml:mrow><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:msub><mml:mi>y</mml:mi><mml:mi>r</mml:mi></mml:msub><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo><mml:mo>≤</mml:mo><mml:mi>L</mml:mi></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq290\"><alternatives><tex-math id=\"M737\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$|y_r| &gt; L$$\\end{document}</tex-math><mml:math id=\"M738\"><mml:mrow><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:msub><mml:mi>y</mml:mi><mml:mi>r</mml:mi></mml:msub><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo><mml:mo>&gt;</mml:mo><mml:mi>L</mml:mi></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq291\"><alternatives><tex-math id=\"M739\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$v_r({\\overrightarrow{y}})$$\\end{document}</tex-math><mml:math id=\"M740\"><mml:mrow><mml:msub><mml:mi>v</mml:mi><mml:mi>r</mml:mi></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq292\"><alternatives><tex-math id=\"M741\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$y_r \\in (-L, +L)$$\\end{document}</tex-math><mml:math id=\"M742\"><mml:mrow><mml:msub><mml:mi>y</mml:mi><mml:mi>r</mml:mi></mml:msub><mml:mo>∈</mml:mo><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mo>-</mml:mo><mml:mi>L</mml:mi><mml:mo>,</mml:mo><mml:mo>+</mml:mo><mml:mi>L</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq293\"><alternatives><tex-math id=\"M743\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\rho ({\\overrightarrow{y}}, t)$$\\end{document}</tex-math><mml:math id=\"M744\"><mml:mrow><mml:mi>ρ</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq294\"><alternatives><tex-math id=\"M745\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$y_r$$\\end{document}</tex-math><mml:math id=\"M746\"><mml:msub><mml:mi>y</mml:mi><mml:mi>r</mml:mi></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq295\"><alternatives><tex-math id=\"M747\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$y_r \\in (-L, +L)$$\\end{document}</tex-math><mml:math id=\"M748\"><mml:mrow><mml:msub><mml:mi>y</mml:mi><mml:mi>r</mml:mi></mml:msub><mml:mo>∈</mml:mo><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mo>-</mml:mo><mml:mi>L</mml:mi><mml:mo>,</mml:mo><mml:mo>+</mml:mo><mml:mi>L</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ83\"><label>83</label><alternatives><tex-math id=\"M749\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} \\psi _n({\\overrightarrow{y}}, t) \\equiv \\prod _{r=1}^N e^{-\\theta (y_r - L) (y_r - L)^{2m}} e^{-\\theta (-y_r - L) (y_r + L)^{2m}} \\psi ({\\overrightarrow{y}}, t) \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M750\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:msub><mml:mi>ψ</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>≡</mml:mo><mml:munderover><mml:mo>∏</mml:mo><mml:mrow><mml:mi>r</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mi>N</mml:mi></mml:munderover><mml:msup><mml:mi>e</mml:mi><mml:mrow><mml:mo>-</mml:mo><mml:mi>θ</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>y</mml:mi><mml:mi>r</mml:mi></mml:msub><mml:mo>-</mml:mo><mml:mi>L</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>y</mml:mi><mml:mi>r</mml:mi></mml:msub><mml:mo>-</mml:mo><mml:mi>L</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:mi>m</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:msup><mml:msup><mml:mi>e</mml:mi><mml:mrow><mml:mo>-</mml:mo><mml:mi>θ</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mo>-</mml:mo><mml:msub><mml:mi>y</mml:mi><mml:mi>r</mml:mi></mml:msub><mml:mo>-</mml:mo><mml:mi>L</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>y</mml:mi><mml:mi>r</mml:mi></mml:msub><mml:mo>+</mml:mo><mml:mi>L</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:mi>m</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:msup><mml:mi>ψ</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq296\"><alternatives><tex-math id=\"M751\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\theta (x)$$\\end{document}</tex-math><mml:math id=\"M752\"><mml:mrow><mml:mi>θ</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>x</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq297\"><alternatives><tex-math id=\"M753\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$v_r({\\overrightarrow{y}})$$\\end{document}</tex-math><mml:math id=\"M754\"><mml:mrow><mml:msub><mml:mi>v</mml:mi><mml:mi>r</mml:mi></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq298\"><alternatives><tex-math id=\"M755\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$y_r = \\pm L$$\\end{document}</tex-math><mml:math id=\"M756\"><mml:mrow><mml:msub><mml:mi>y</mml:mi><mml:mi>r</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mo>±</mml:mo><mml:mi>L</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq299\"><alternatives><tex-math id=\"M757\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$r \\in \\{1, 2, ...N\\}$$\\end{document}</tex-math><mml:math id=\"M758\"><mml:mrow><mml:mi>r</mml:mi><mml:mo>∈</mml:mo><mml:mo stretchy=\"false\">{</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mo>.</mml:mo><mml:mo>.</mml:mo><mml:mo>.</mml:mo><mml:mi>N</mml:mi><mml:mo stretchy=\"false\">}</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq300\"><alternatives><tex-math id=\"M759\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\psi _n({\\overrightarrow{y}}, t)$$\\end{document}</tex-math><mml:math id=\"M760\"><mml:mrow><mml:msub><mml:mi>ψ</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq301\"><alternatives><tex-math id=\"M761\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\psi ^{K_j^g}({\\overrightarrow{y}}) \\sim e^{y_r^2/2}$$\\end{document}</tex-math><mml:math id=\"M762\"><mml:mrow><mml:msup><mml:mi>ψ</mml:mi><mml:msubsup><mml:mi>K</mml:mi><mml:mi>j</mml:mi><mml:mi>g</mml:mi></mml:msubsup></mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>∼</mml:mo><mml:msup><mml:mi>e</mml:mi><mml:mrow><mml:msubsup><mml:mi>y</mml:mi><mml:mi>r</mml:mi><mml:mn>2</mml:mn></mml:msubsup><mml:mo stretchy=\"false\">/</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq302\"><alternatives><tex-math id=\"M763\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\pm y_r$$\\end{document}</tex-math><mml:math id=\"M764\"><mml:mrow><mml:mo>±</mml:mo><mml:msub><mml:mi>y</mml:mi><mml:mi>r</mml:mi></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq303\"><alternatives><tex-math id=\"M765\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$r \\in \\{1, 2, ...N\\}$$\\end{document}</tex-math><mml:math id=\"M766\"><mml:mrow><mml:mi>r</mml:mi><mml:mo>∈</mml:mo><mml:mo stretchy=\"false\">{</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mo>.</mml:mo><mml:mo>.</mml:mo><mml:mo>.</mml:mo><mml:mi>N</mml:mi><mml:mo stretchy=\"false\">}</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq304\"><alternatives><tex-math id=\"M767\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\psi ({\\overrightarrow{y}}, t)$$\\end{document}</tex-math><mml:math id=\"M768\"><mml:mrow><mml:mi>ψ</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq305\"><alternatives><tex-math id=\"M769\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\psi _n({\\overrightarrow{y}}, t)$$\\end{document}</tex-math><mml:math id=\"M770\"><mml:mrow><mml:msub><mml:mi>ψ</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq306\"><alternatives><tex-math id=\"M771\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\rho ({\\overrightarrow{y}}, t)$$\\end{document}</tex-math><mml:math id=\"M772\"><mml:mrow><mml:mi>ρ</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq307\"><alternatives><tex-math id=\"M773\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\rho ({\\overrightarrow{y}}, 0)$$\\end{document}</tex-math><mml:math id=\"M774\"><mml:mrow><mml:mi>ρ</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mn>0</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq308\"><alternatives><tex-math id=\"M775\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\Omega _0 \\subset \\Lambda \\equiv \\{{\\overrightarrow{y}}| y_r \\in (-L, +L) {\\textbf { }}\\forall {\\textbf { }} r\\}$$\\end{document}</tex-math><mml:math id=\"M776\"><mml:mrow><mml:msub><mml:mi mathvariant=\"normal\">Ω</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo>⊂</mml:mo><mml:mi mathvariant=\"normal\">Λ</mml:mi><mml:mo>≡</mml:mo><mml:mrow><mml:mo stretchy=\"false\">{</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo stretchy=\"false\">|</mml:mo><mml:msub><mml:mi>y</mml:mi><mml:mi>r</mml:mi></mml:msub><mml:mo>∈</mml:mo><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mo>-</mml:mo><mml:mi>L</mml:mi><mml:mo>,</mml:mo><mml:mo>+</mml:mo><mml:mi>L</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mrow/><mml:mo>∀</mml:mo><mml:mrow/><mml:mi>r</mml:mi><mml:mo stretchy=\"false\">}</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq309\"><alternatives><tex-math id=\"M777\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\psi _n({\\overrightarrow{y}}, t)$$\\end{document}</tex-math><mml:math id=\"M778\"><mml:mrow><mml:msub><mml:mi>ψ</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq310\"><alternatives><tex-math id=\"M779\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$e^{-i{\\hat{H}}t/\\hbar } \\psi _n({\\overrightarrow{y}}, 0) \\ne e^{-i{\\hat{H}}t/\\hbar } \\psi ({\\overrightarrow{y}}, 0)$$\\end{document}</tex-math><mml:math id=\"M780\"><mml:mrow><mml:msup><mml:mi>e</mml:mi><mml:mrow><mml:mo>-</mml:mo><mml:mi>i</mml:mi><mml:mover accent=\"true\"><mml:mi>H</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">/</mml:mo><mml:mi>ħ</mml:mi></mml:mrow></mml:msup><mml:msub><mml:mi>ψ</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mn>0</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>≠</mml:mo><mml:msup><mml:mi>e</mml:mi><mml:mrow><mml:mo>-</mml:mo><mml:mi>i</mml:mi><mml:mover accent=\"true\"><mml:mi>H</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">/</mml:mo><mml:mi>ħ</mml:mi></mml:mrow></mml:msup><mml:mi>ψ</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mn>0</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq311\"><alternatives><tex-math id=\"M781\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\psi _n({\\overrightarrow{y}}, 0)$$\\end{document}</tex-math><mml:math id=\"M782\"><mml:mrow><mml:msub><mml:mi>ψ</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mn>0</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq312\"><alternatives><tex-math id=\"M783\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\psi ({\\overrightarrow{y}}, 0)$$\\end{document}</tex-math><mml:math id=\"M784\"><mml:mrow><mml:mi>ψ</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mn>0</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq313\"><alternatives><tex-math id=\"M785\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\Lambda$$\\end{document}</tex-math><mml:math id=\"M786\"><mml:mi mathvariant=\"normal\">Λ</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq314\"><alternatives><tex-math id=\"M787\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$H_{pw}^{min} = 0$$\\end{document}</tex-math><mml:math id=\"M788\"><mml:mrow><mml:msubsup><mml:mi>H</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">pw</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"italic\">min</mml:mi></mml:mrow></mml:msubsup><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq315\"><alternatives><tex-math id=\"M789\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$H_{q}^{min} = -\\ln {\\mathcal {N}}$$\\end{document}</tex-math><mml:math id=\"M790\"><mml:mrow><mml:msubsup><mml:mi>H</mml:mi><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"italic\">min</mml:mi></mml:mrow></mml:msubsup><mml:mo>=</mml:mo><mml:mo>-</mml:mo><mml:mo>ln</mml:mo><mml:mi mathvariant=\"script\">N</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq316\"><alternatives><tex-math id=\"M791\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\psi$$\\end{document}</tex-math><mml:math id=\"M792\"><mml:mi>ψ</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq317\"><alternatives><tex-math id=\"M793\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\rho _q = |\\psi |^2$$\\end{document}</tex-math><mml:math id=\"M794\"><mml:mrow><mml:msub><mml:mi>ρ</mml:mi><mml:mi>q</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:msup><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo><mml:mi>ψ</mml:mi><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:msup></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq318\"><alternatives><tex-math id=\"M795\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\rho _{pw}$$\\end{document}</tex-math><mml:math id=\"M796\"><mml:msub><mml:mi>ρ</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">pw</mml:mi></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq319\"><alternatives><tex-math id=\"M797\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\Omega$$\\end{document}</tex-math><mml:math id=\"M798\"><mml:mi mathvariant=\"normal\">Ω</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq320\"><alternatives><tex-math id=\"M799\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\psi ^K({\\overrightarrow{y}})$$\\end{document}</tex-math><mml:math id=\"M800\"><mml:mrow><mml:msup><mml:mi>ψ</mml:mi><mml:mi>K</mml:mi></mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq321\"><alternatives><tex-math id=\"M801\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\delta V({\\overrightarrow{y}}, t)$$\\end{document}</tex-math><mml:math id=\"M802\"><mml:mrow><mml:mi>δ</mml:mi><mml:mi>V</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ84\"><label>84</label><alternatives><tex-math id=\"M803\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} \\psi ({\\overrightarrow{y}}, t) = e^{-i{\\hat{H}}_0 t/\\hbar } \\psi ^K({\\overrightarrow{y}}) - \\frac{ie^{-i{\\hat{H}}_0 t/\\hbar }}{\\hbar }\\int _0^t dt' e^{i{\\hat{H}}_0 t'/\\hbar } \\delta V({\\overrightarrow{y}}, t') e^{-i{\\hat{H}}_0 t'/\\hbar } \\psi ^K({\\overrightarrow{y}}) + {\\mathcal {O}}(\\delta V^2) \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M804\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:mi>ψ</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:msup><mml:mi>e</mml:mi><mml:mrow><mml:mo>-</mml:mo><mml:mi>i</mml:mi><mml:msub><mml:mover accent=\"true\"><mml:mi>H</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover><mml:mn>0</mml:mn></mml:msub><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">/</mml:mo><mml:mi>ħ</mml:mi></mml:mrow></mml:msup><mml:msup><mml:mi>ψ</mml:mi><mml:mi>K</mml:mi></mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>-</mml:mo><mml:mfrac><mml:mrow><mml:mi>i</mml:mi><mml:msup><mml:mi>e</mml:mi><mml:mrow><mml:mo>-</mml:mo><mml:mi>i</mml:mi><mml:msub><mml:mover accent=\"true\"><mml:mi>H</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover><mml:mn>0</mml:mn></mml:msub><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">/</mml:mo><mml:mi>ħ</mml:mi></mml:mrow></mml:msup></mml:mrow><mml:mi>ħ</mml:mi></mml:mfrac><mml:msubsup><mml:mo>∫</mml:mo><mml:mn>0</mml:mn><mml:mi>t</mml:mi></mml:msubsup><mml:mi>d</mml:mi><mml:msup><mml:mi>t</mml:mi><mml:mo>′</mml:mo></mml:msup><mml:msup><mml:mi>e</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:msub><mml:mover accent=\"true\"><mml:mi>H</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover><mml:mn>0</mml:mn></mml:msub><mml:msup><mml:mi>t</mml:mi><mml:mo>′</mml:mo></mml:msup><mml:mo stretchy=\"false\">/</mml:mo><mml:mi>ħ</mml:mi></mml:mrow></mml:msup><mml:mi>δ</mml:mi><mml:mi>V</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:msup><mml:mi>t</mml:mi><mml:mo>′</mml:mo></mml:msup><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:msup><mml:mi>e</mml:mi><mml:mrow><mml:mo>-</mml:mo><mml:mi>i</mml:mi><mml:msub><mml:mover accent=\"true\"><mml:mi>H</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover><mml:mn>0</mml:mn></mml:msub><mml:msup><mml:mi>t</mml:mi><mml:mo>′</mml:mo></mml:msup><mml:mo stretchy=\"false\">/</mml:mo><mml:mi>ħ</mml:mi></mml:mrow></mml:msup><mml:msup><mml:mi>ψ</mml:mi><mml:mi>K</mml:mi></mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>+</mml:mo><mml:mi mathvariant=\"script\">O</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>δ</mml:mi><mml:msup><mml:mi>V</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq322\"><alternatives><tex-math id=\"M805\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\delta V$$\\end{document}</tex-math><mml:math id=\"M806\"><mml:mrow><mml:mi>δ</mml:mi><mml:mi>V</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq323\"><alternatives><tex-math id=\"M807\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\hat{H}}_0$$\\end{document}</tex-math><mml:math id=\"M808\"><mml:msub><mml:mover accent=\"true\"><mml:mi>H</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover><mml:mn>0</mml:mn></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq324\"><alternatives><tex-math id=\"M809\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\psi ^K({\\overrightarrow{y}})$$\\end{document}</tex-math><mml:math id=\"M810\"><mml:mrow><mml:msup><mml:mi>ψ</mml:mi><mml:mi>K</mml:mi></mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq325\"><alternatives><tex-math id=\"M811\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\delta V({\\overrightarrow{y}}, t')$$\\end{document}</tex-math><mml:math id=\"M812\"><mml:mrow><mml:mi>δ</mml:mi><mml:mi>V</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:msup><mml:mi>t</mml:mi><mml:mo>′</mml:mo></mml:msup><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ85\"><label>85</label><alternatives><tex-math id=\"M813\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} \\delta V({\\overrightarrow{y}}, t') = \\sum _{n=1}^N e^{\\frac{-|{\\overrightarrow{y}}-{\\overrightarrow{y}}_n(t')|^4}{\\sigma_n}} \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M814\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:mi>δ</mml:mi><mml:mi>V</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:msup><mml:mi>t</mml:mi><mml:mo>′</mml:mo></mml:msup><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:munderover><mml:mo>∑</mml:mo><mml:mrow><mml:mi>n</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mi>N</mml:mi></mml:munderover><mml:msup><mml:mi>e</mml:mi><mml:mfrac><mml:mrow><mml:mrow><mml:mo>-</mml:mo><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>-</mml:mo><mml:msub><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mi>n</mml:mi></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msup><mml:mi>t</mml:mi><mml:mo>′</mml:mo></mml:msup><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:msup><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mn>4</mml:mn></mml:msup></mml:mrow><mml:msub><mml:mi>σ</mml:mi><mml:mi>n</mml:mi></mml:msub></mml:mfrac></mml:msup></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq326\"><alternatives><tex-math id=\"M815\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\overrightarrow{y}}_n(t')$$\\end{document}</tex-math><mml:math id=\"M816\"><mml:mrow><mml:msub><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mi>n</mml:mi></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msup><mml:mi>t</mml:mi><mml:mo>′</mml:mo></mml:msup><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq327\"><alternatives><tex-math id=\"M817\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\psi ^K({\\overrightarrow{y}})$$\\end{document}</tex-math><mml:math id=\"M818\"><mml:mrow><mml:msup><mml:mi>ψ</mml:mi><mml:mi>K</mml:mi></mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ86\"><label>86</label><alternatives><tex-math id=\"M819\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} e^{i{\\hat{H}}_0 t'/\\hbar } \\bigg ( e^{-iE_K t'/\\hbar } \\delta V({\\overrightarrow{y}}, t') \\psi ^K({\\overrightarrow{y}}) \\bigg ) = \\sum _j e^{i(E_j -E_K) t'/\\hbar } c_j(t') \\psi ^j({\\overrightarrow{y}}) \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M820\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:msup><mml:mi>e</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:msub><mml:mover accent=\"true\"><mml:mi>H</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover><mml:mn>0</mml:mn></mml:msub><mml:msup><mml:mi>t</mml:mi><mml:mo>′</mml:mo></mml:msup><mml:mo stretchy=\"false\">/</mml:mo><mml:mi>ħ</mml:mi></mml:mrow></mml:msup><mml:mrow><mml:mo maxsize=\"2.047em\" minsize=\"2.047em\" stretchy=\"true\">(</mml:mo></mml:mrow><mml:msup><mml:mi>e</mml:mi><mml:mrow><mml:mo>-</mml:mo><mml:mi>i</mml:mi><mml:msub><mml:mi>E</mml:mi><mml:mi>K</mml:mi></mml:msub><mml:msup><mml:mi>t</mml:mi><mml:mo>′</mml:mo></mml:msup><mml:mo stretchy=\"false\">/</mml:mo><mml:mi>ħ</mml:mi></mml:mrow></mml:msup><mml:mi>δ</mml:mi><mml:mi>V</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:msup><mml:mi>t</mml:mi><mml:mo>′</mml:mo></mml:msup><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:msup><mml:mi>ψ</mml:mi><mml:mi>K</mml:mi></mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mrow><mml:mo maxsize=\"2.047em\" minsize=\"2.047em\" stretchy=\"true\">)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:munder><mml:mo>∑</mml:mo><mml:mi>j</mml:mi></mml:munder><mml:msup><mml:mi>e</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>E</mml:mi><mml:mi>j</mml:mi></mml:msub><mml:mo>-</mml:mo><mml:msub><mml:mi>E</mml:mi><mml:mi>K</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:msup><mml:mi>t</mml:mi><mml:mo>′</mml:mo></mml:msup><mml:mo stretchy=\"false\">/</mml:mo><mml:mi>ħ</mml:mi></mml:mrow></mml:msup><mml:msub><mml:mi>c</mml:mi><mml:mi>j</mml:mi></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msup><mml:mi>t</mml:mi><mml:mo>′</mml:mo></mml:msup><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:msup><mml:mi>ψ</mml:mi><mml:mi>j</mml:mi></mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq328\"><alternatives><tex-math id=\"M821\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\delta V({\\overrightarrow{y}}, t')\\psi ^K({\\overrightarrow{y}})$$\\end{document}</tex-math><mml:math id=\"M822\"><mml:mrow><mml:mi>δ</mml:mi><mml:mi>V</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:msup><mml:mi>t</mml:mi><mml:mo>′</mml:mo></mml:msup><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:msup><mml:mi>ψ</mml:mi><mml:mi>K</mml:mi></mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq329\"><alternatives><tex-math id=\"M823\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\psi ^K({\\overrightarrow{y}})$$\\end{document}</tex-math><mml:math id=\"M824\"><mml:mrow><mml:msup><mml:mi>ψ</mml:mi><mml:mi>K</mml:mi></mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq330\"><alternatives><tex-math id=\"M825\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\psi ^j({\\overrightarrow{y}})$$\\end{document}</tex-math><mml:math id=\"M826\"><mml:mrow><mml:msup><mml:mi>ψ</mml:mi><mml:mi>j</mml:mi></mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq331\"><alternatives><tex-math id=\"M827\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$H_0$$\\end{document}</tex-math><mml:math id=\"M828\"><mml:msub><mml:mi>H</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq332\"><alternatives><tex-math id=\"M829\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\delta V({\\overrightarrow{y}}, t')$$\\end{document}</tex-math><mml:math id=\"M830\"><mml:mrow><mml:mi>δ</mml:mi><mml:mi>V</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:msup><mml:mi>t</mml:mi><mml:mo>′</mml:mo></mml:msup><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq333\"><alternatives><tex-math id=\"M831\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\delta V({\\overrightarrow{y}}, t')\\psi ^K({\\overrightarrow{y}})$$\\end{document}</tex-math><mml:math id=\"M832\"><mml:mrow><mml:mi>δ</mml:mi><mml:mi>V</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:msup><mml:mi>t</mml:mi><mml:mo>′</mml:mo></mml:msup><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:msup><mml:mi>ψ</mml:mi><mml:mi>K</mml:mi></mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq334\"><alternatives><tex-math id=\"M833\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\delta V({\\overrightarrow{y}}, t')$$\\end{document}</tex-math><mml:math id=\"M834\"><mml:mrow><mml:mi>δ</mml:mi><mml:mi>V</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:msup><mml:mi>t</mml:mi><mml:mo>′</mml:mo></mml:msup><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ87\"><label>87</label><alternatives><tex-math id=\"M835\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} \\psi ({\\overrightarrow{y}}, t) = e^{-iE_K t/\\hbar } \\psi ^K({\\overrightarrow{y}}) - \\frac{ie^{-i{\\hat{H}}_0 t/\\hbar }}{\\hbar }\\int _0^t dt' \\sum _j e^{i(E_j -E_K) t'/\\hbar } c_j(t') \\psi ^j({\\overrightarrow{y}}) + {\\mathcal {O}}(\\delta V^2) \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M836\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:mi>ψ</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:msup><mml:mi>e</mml:mi><mml:mrow><mml:mo>-</mml:mo><mml:mi>i</mml:mi><mml:msub><mml:mi>E</mml:mi><mml:mi>K</mml:mi></mml:msub><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">/</mml:mo><mml:mi>ħ</mml:mi></mml:mrow></mml:msup><mml:msup><mml:mi>ψ</mml:mi><mml:mi>K</mml:mi></mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>-</mml:mo><mml:mfrac><mml:mrow><mml:mi>i</mml:mi><mml:msup><mml:mi>e</mml:mi><mml:mrow><mml:mo>-</mml:mo><mml:mi>i</mml:mi><mml:msub><mml:mover accent=\"true\"><mml:mi>H</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover><mml:mn>0</mml:mn></mml:msub><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">/</mml:mo><mml:mi>ħ</mml:mi></mml:mrow></mml:msup></mml:mrow><mml:mi>ħ</mml:mi></mml:mfrac><mml:msubsup><mml:mo>∫</mml:mo><mml:mn>0</mml:mn><mml:mi>t</mml:mi></mml:msubsup><mml:mi>d</mml:mi><mml:msup><mml:mi>t</mml:mi><mml:mo>′</mml:mo></mml:msup><mml:munder><mml:mo>∑</mml:mo><mml:mi>j</mml:mi></mml:munder><mml:msup><mml:mi>e</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>E</mml:mi><mml:mi>j</mml:mi></mml:msub><mml:mo>-</mml:mo><mml:msub><mml:mi>E</mml:mi><mml:mi>K</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:msup><mml:mi>t</mml:mi><mml:mo>′</mml:mo></mml:msup><mml:mo stretchy=\"false\">/</mml:mo><mml:mi>ħ</mml:mi></mml:mrow></mml:msup><mml:msub><mml:mi>c</mml:mi><mml:mi>j</mml:mi></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msup><mml:mi>t</mml:mi><mml:mo>′</mml:mo></mml:msup><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:msup><mml:mi>ψ</mml:mi><mml:mi>j</mml:mi></mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>+</mml:mo><mml:mi mathvariant=\"script\">O</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>δ</mml:mi><mml:msup><mml:mi>V</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq335\"><alternatives><tex-math id=\"M837\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\psi ^K({\\overrightarrow{y}})$$\\end{document}</tex-math><mml:math id=\"M838\"><mml:mrow><mml:msup><mml:mi>ψ</mml:mi><mml:mi>K</mml:mi></mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq336\"><alternatives><tex-math id=\"M839\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\psi ^j({\\overrightarrow{y}})$$\\end{document}</tex-math><mml:math id=\"M840\"><mml:mrow><mml:msup><mml:mi>ψ</mml:mi><mml:mi>j</mml:mi></mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq337\"><alternatives><tex-math id=\"M841\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$E_j$$\\end{document}</tex-math><mml:math id=\"M842\"><mml:msub><mml:mi>E</mml:mi><mml:mi>j</mml:mi></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq338\"><alternatives><tex-math id=\"M843\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$E_K$$\\end{document}</tex-math><mml:math id=\"M844\"><mml:msub><mml:mi>E</mml:mi><mml:mi>K</mml:mi></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq339\"><alternatives><tex-math id=\"M845\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\hat{H}}_I = g {\\hat{E}}_{{\\overrightarrow{y}}}\\otimes {\\hat{p}}_x$$\\end{document}</tex-math><mml:math id=\"M846\"><mml:mrow><mml:msub><mml:mover accent=\"true\"><mml:mi>H</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover><mml:mi>I</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mi>g</mml:mi><mml:msub><mml:mover accent=\"true\"><mml:mi>E</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover></mml:msub><mml:mo>⊗</mml:mo><mml:msub><mml:mover accent=\"true\"><mml:mi>p</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover><mml:mi>x</mml:mi></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ88\"><label>88</label><alternatives><tex-math id=\"M847\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} \\Psi ({\\overrightarrow{y}}, x, t) = \\sum _n a_n \\phi (x - \\frac{gtE_n}{\\hbar ^2},0)\\psi ^n({\\overrightarrow{y}}) \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M848\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:mi mathvariant=\"normal\">Ψ</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:munder><mml:mo>∑</mml:mo><mml:mi>n</mml:mi></mml:munder><mml:msub><mml:mi>a</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mi>ϕ</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>x</mml:mi><mml:mo>-</mml:mo><mml:mfrac><mml:mrow><mml:mi>g</mml:mi><mml:mi>t</mml:mi><mml:msub><mml:mi>E</mml:mi><mml:mi>n</mml:mi></mml:msub></mml:mrow><mml:msup><mml:mi>ħ</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mfrac><mml:mo>,</mml:mo><mml:mn>0</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:msup><mml:mi>ψ</mml:mi><mml:mi>n</mml:mi></mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq340\"><alternatives><tex-math id=\"M849\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\phi (x - \\frac{gtE_n}{\\hbar ^2},0)$$\\end{document}</tex-math><mml:math id=\"M850\"><mml:mrow><mml:mi>ϕ</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>x</mml:mi><mml:mo>-</mml:mo><mml:mfrac><mml:mrow><mml:mi>g</mml:mi><mml:mi>t</mml:mi><mml:msub><mml:mi>E</mml:mi><mml:mi>n</mml:mi></mml:msub></mml:mrow><mml:msup><mml:mi>ħ</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mfrac><mml:mo>,</mml:mo><mml:mn>0</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq341\"><alternatives><tex-math id=\"M851\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\psi ^n({\\overrightarrow{y}})$$\\end{document}</tex-math><mml:math id=\"M852\"><mml:mrow><mml:msup><mml:mi>ψ</mml:mi><mml:mi>n</mml:mi></mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq342\"><alternatives><tex-math id=\"M853\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\psi ^j({\\overrightarrow{y}})$$\\end{document}</tex-math><mml:math id=\"M854\"><mml:mrow><mml:msup><mml:mi>ψ</mml:mi><mml:mi>j</mml:mi></mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq343\"><alternatives><tex-math id=\"M855\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\psi ^K({\\overrightarrow{y}})$$\\end{document}</tex-math><mml:math id=\"M856\"><mml:mrow><mml:msup><mml:mi>ψ</mml:mi><mml:mi>K</mml:mi></mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq344\"><alternatives><tex-math id=\"M857\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\Psi ({\\overrightarrow{y}}, x, t)$$\\end{document}</tex-math><mml:math id=\"M858\"><mml:mrow><mml:mi mathvariant=\"normal\">Ψ</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq345\"><alternatives><tex-math id=\"M859\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\rho ({\\overrightarrow{y}}, t)$$\\end{document}</tex-math><mml:math id=\"M860\"><mml:mrow><mml:mi>ρ</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq346\"><alternatives><tex-math id=\"M861\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\phi (x - \\frac{gtE_n}{\\hbar ^2},0)$$\\end{document}</tex-math><mml:math id=\"M862\"><mml:mrow><mml:mi>ϕ</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>x</mml:mi><mml:mo>-</mml:mo><mml:mfrac><mml:mrow><mml:mi>g</mml:mi><mml:mi>t</mml:mi><mml:msub><mml:mi>E</mml:mi><mml:mi>n</mml:mi></mml:msub></mml:mrow><mml:msup><mml:mi>ħ</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mfrac><mml:mo>,</mml:mo><mml:mn>0</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq347\"><alternatives><tex-math id=\"M863\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\Psi ({\\overrightarrow{y}}, x, t)$$\\end{document}</tex-math><mml:math id=\"M864\"><mml:mrow><mml:mi mathvariant=\"normal\">Ψ</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ89\"><label>89</label><alternatives><tex-math id=\"M865\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} f(N) = \\Pi _{j=1}^N (1-\\epsilon _j) \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M866\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:mi>f</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>N</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:msubsup><mml:mi mathvariant=\"normal\">Π</mml:mi><mml:mrow><mml:mi>j</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mi>N</mml:mi></mml:msubsup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>1</mml:mn><mml:mo>-</mml:mo><mml:msub><mml:mi>ϵ</mml:mi><mml:mi>j</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq349\"><alternatives><tex-math id=\"M867\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\epsilon _j$$\\end{document}</tex-math><mml:math id=\"M868\"><mml:msub><mml:mi>ϵ</mml:mi><mml:mi>j</mml:mi></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq350\"><alternatives><tex-math id=\"M869\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$f(N) \\rightarrow 0$$\\end{document}</tex-math><mml:math id=\"M870\"><mml:mrow><mml:mi>f</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>N</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"false\">→</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq351\"><alternatives><tex-math id=\"M871\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$N \\rightarrow \\infty$$\\end{document}</tex-math><mml:math id=\"M872\"><mml:mrow><mml:mi>N</mml:mi><mml:mo stretchy=\"false\">→</mml:mo><mml:mi>∞</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq352\"><alternatives><tex-math id=\"M873\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\epsilon _j = 0$$\\end{document}</tex-math><mml:math id=\"M874\"><mml:mrow><mml:msub><mml:mi>ϵ</mml:mi><mml:mi>j</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq353\"><alternatives><tex-math id=\"M875\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\forall j &gt; N_0$$\\end{document}</tex-math><mml:math id=\"M876\"><mml:mrow><mml:mo>∀</mml:mo><mml:mi>j</mml:mi><mml:mo>&gt;</mml:mo><mml:msub><mml:mi>N</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq354\"><alternatives><tex-math id=\"M877\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$N_0$$\\end{document}</tex-math><mml:math id=\"M878\"><mml:msub><mml:mi>N</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq355\"><alternatives><tex-math id=\"M879\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\epsilon _j = 0$$\\end{document}</tex-math><mml:math id=\"M880\"><mml:mrow><mml:msub><mml:mi>ϵ</mml:mi><mml:mi>j</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq356\"><alternatives><tex-math id=\"M881\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\forall j &gt; N_0$$\\end{document}</tex-math><mml:math id=\"M882\"><mml:mrow><mml:mo>∀</mml:mo><mml:mi>j</mml:mi><mml:mo>&gt;</mml:mo><mml:msub><mml:mi>N</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq357\"><alternatives><tex-math id=\"M883\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$j &gt; N_0$$\\end{document}</tex-math><mml:math id=\"M884\"><mml:mrow><mml:mi>j</mml:mi><mml:mo>&gt;</mml:mo><mml:msub><mml:mi>N</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq358\"><alternatives><tex-math id=\"M885\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\psi ^K({\\overrightarrow{y}})$$\\end{document}</tex-math><mml:math id=\"M886\"><mml:mrow><mml:msup><mml:mi>ψ</mml:mi><mml:mi>K</mml:mi></mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq359\"><alternatives><tex-math id=\"M887\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\psi (y_1, y_2, 0)$$\\end{document}</tex-math><mml:math id=\"M888\"><mml:mrow><mml:mi>ψ</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>y</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>y</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mo>,</mml:mo><mml:mn>0</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq360\"><alternatives><tex-math id=\"M889\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\Omega _0 \\equiv \\{(y_1, y_2)|y_1\\in (Y_1, Y_1 +\\delta y_1) , y_2\\in (Y_2, Y_2 +\\delta y_2) \\}$$\\end{document}</tex-math><mml:math id=\"M890\"><mml:mrow><mml:msub><mml:mi mathvariant=\"normal\">Ω</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo>≡</mml:mo><mml:mrow><mml:mo stretchy=\"false\">{</mml:mo><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>y</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>y</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo stretchy=\"false\">|</mml:mo><mml:msub><mml:mi>y</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>∈</mml:mo><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>Y</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>Y</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>+</mml:mo><mml:mi>δ</mml:mi><mml:msub><mml:mi>y</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>,</mml:mo><mml:msub><mml:mi>y</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mo>∈</mml:mo><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>Y</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>Y</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mo>+</mml:mo><mml:mi>δ</mml:mi><mml:msub><mml:mi>y</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo stretchy=\"false\">}</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq361\"><alternatives><tex-math id=\"M891\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\delta y_1, \\delta y_2$$\\end{document}</tex-math><mml:math id=\"M892\"><mml:mrow><mml:mi>δ</mml:mi><mml:msub><mml:mi>y</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>,</mml:mo><mml:mi>δ</mml:mi><mml:msub><mml:mi>y</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq362\"><alternatives><tex-math id=\"M893\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\mathcal {N}} = \\int _{\\Omega _0} |\\psi (y_1, y_2, 0)|^2 dy_1dy_2 \\approx |\\psi (Y_1, Y_2, 0)|^2 \\delta y_1\\delta y_2$$\\end{document}</tex-math><mml:math id=\"M894\"><mml:mrow><mml:mi mathvariant=\"script\">N</mml:mi><mml:mo>=</mml:mo><mml:msub><mml:mo>∫</mml:mo><mml:msub><mml:mi mathvariant=\"normal\">Ω</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:msub><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo><mml:mi>ψ</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>y</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>y</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mo>,</mml:mo><mml:mn>0</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:msup><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:msup><mml:mi>d</mml:mi><mml:msub><mml:mi>y</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mi>d</mml:mi><mml:msub><mml:mi>y</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mo>≈</mml:mo><mml:msup><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo><mml:mi>ψ</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>Y</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>Y</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mo>,</mml:mo><mml:mn>0</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:msup><mml:mi>δ</mml:mi><mml:msub><mml:mi>y</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mi>δ</mml:mi><mml:msub><mml:mi>y</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq363\"><alternatives><tex-math id=\"M895\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\rho (y_1, y_2, 0) \\equiv |\\psi (y_1, y_2, 0)|^2/{\\mathcal {N}} \\approx 1/ \\delta y_1\\delta y_2$$\\end{document}</tex-math><mml:math id=\"M896\"><mml:mrow><mml:mi>ρ</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>y</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>y</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mo>,</mml:mo><mml:mn>0</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>≡</mml:mo><mml:msup><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo><mml:mi>ψ</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>y</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>y</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mo>,</mml:mo><mml:mn>0</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:msup><mml:mo stretchy=\"false\">/</mml:mo><mml:mi mathvariant=\"script\">N</mml:mi><mml:mo>≈</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">/</mml:mo><mml:mi>δ</mml:mi><mml:msub><mml:mi>y</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mi>δ</mml:mi><mml:msub><mml:mi>y</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq364\"><alternatives><tex-math id=\"M897\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\Omega _0$$\\end{document}</tex-math><mml:math id=\"M898\"><mml:msub><mml:mi mathvariant=\"normal\">Ω</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq365\"><alternatives><tex-math id=\"M899\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\psi (y_1, y_2, t)$$\\end{document}</tex-math><mml:math id=\"M900\"><mml:mrow><mml:mi>ψ</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>y</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>y</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq366\"><alternatives><tex-math id=\"M901\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\hat{H}} = {\\hat{H}}_1\\otimes {\\hat{I}} + {\\hat{I}} \\otimes {\\hat{H}}_2$$\\end{document}</tex-math><mml:math id=\"M902\"><mml:mrow><mml:mover accent=\"true\"><mml:mi>H</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover><mml:mo>=</mml:mo><mml:msub><mml:mover accent=\"true\"><mml:mi>H</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover><mml:mn>1</mml:mn></mml:msub><mml:mo>⊗</mml:mo><mml:mover accent=\"true\"><mml:mi>I</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover><mml:mo>+</mml:mo><mml:mover accent=\"true\"><mml:mi>I</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover><mml:mo>⊗</mml:mo><mml:msub><mml:mover accent=\"true\"><mml:mi>H</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover><mml:mn>2</mml:mn></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq367\"><alternatives><tex-math id=\"M903\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$y_1$$\\end{document}</tex-math><mml:math id=\"M904\"><mml:msub><mml:mi>y</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq368\"><alternatives><tex-math id=\"M905\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\hat{H}}_2$$\\end{document}</tex-math><mml:math id=\"M906\"><mml:msub><mml:mover accent=\"true\"><mml:mi>H</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover><mml:mn>2</mml:mn></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq369\"><alternatives><tex-math id=\"M907\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\psi (y_1, y_2, t)$$\\end{document}</tex-math><mml:math id=\"M908\"><mml:mrow><mml:mi>ψ</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>y</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>y</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq370\"><alternatives><tex-math id=\"M909\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$v_1(y_1, y_2, t)$$\\end{document}</tex-math><mml:math id=\"M910\"><mml:mrow><mml:msub><mml:mi>v</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>y</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>y</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq371\"><alternatives><tex-math id=\"M911\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$y_2$$\\end{document}</tex-math><mml:math id=\"M912\"><mml:msub><mml:mi>y</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq372\"><alternatives><tex-math id=\"M913\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\hat{H}}_2$$\\end{document}</tex-math><mml:math id=\"M914\"><mml:msub><mml:mover accent=\"true\"><mml:mi>H</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover><mml:mn>2</mml:mn></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq373\"><alternatives><tex-math id=\"M915\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\delta y_1, \\delta y_2 \\rightarrow 0$$\\end{document}</tex-math><mml:math id=\"M916\"><mml:mrow><mml:mi>δ</mml:mi><mml:msub><mml:mi>y</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>,</mml:mo><mml:mi>δ</mml:mi><mml:msub><mml:mi>y</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mo stretchy=\"false\">→</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq374\"><alternatives><tex-math id=\"M917\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\rho (y_1, y_2, 0) = \\delta (y_1 - Y_1) \\delta (y_2 - Y_2)$$\\end{document}</tex-math><mml:math id=\"M918\"><mml:mrow><mml:mi>ρ</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>y</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>y</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mo>,</mml:mo><mml:mn>0</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mi>δ</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>y</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>-</mml:mo><mml:msub><mml:mi>Y</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mi>δ</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>y</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mo>-</mml:mo><mml:msub><mml:mi>Y</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq375\"><alternatives><tex-math id=\"M919\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\rho (y_1, 0) = \\delta (y_1 - Y_1)$$\\end{document}</tex-math><mml:math id=\"M920\"><mml:mrow><mml:mi>ρ</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>y</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>,</mml:mo><mml:mn>0</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mi>δ</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>y</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>-</mml:mo><mml:msub><mml:mi>Y</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq376\"><alternatives><tex-math id=\"M921\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$v_1(y_1, y_2, t)$$\\end{document}</tex-math><mml:math id=\"M922\"><mml:mrow><mml:msub><mml:mi>v</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>y</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>y</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq377\"><alternatives><tex-math id=\"M923\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\hat{H}}_2$$\\end{document}</tex-math><mml:math id=\"M924\"><mml:msub><mml:mover accent=\"true\"><mml:mi>H</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover><mml:mn>2</mml:mn></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq378\"><alternatives><tex-math id=\"M925\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\rho (y_1, 0)$$\\end{document}</tex-math><mml:math id=\"M926\"><mml:mrow><mml:mi>ρ</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>y</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>,</mml:mo><mml:mn>0</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq379\"><alternatives><tex-math id=\"M927\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$Y_1$$\\end{document}</tex-math><mml:math id=\"M928\"><mml:msub><mml:mi>Y</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq380\"><alternatives><tex-math id=\"M929\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\rho (y_1, t) = \\delta (y_1 - Y_1(t))$$\\end{document}</tex-math><mml:math id=\"M930\"><mml:mrow><mml:mi>ρ</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>y</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mi>δ</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>y</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>-</mml:mo><mml:msub><mml:mi>Y</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq381\"><alternatives><tex-math id=\"M931\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\hat{H}}_2$$\\end{document}</tex-math><mml:math id=\"M932\"><mml:msub><mml:mover accent=\"true\"><mml:mi>H</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover><mml:mn>2</mml:mn></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq382\"><alternatives><tex-math id=\"M933\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\phi (\\vec{x}, t)$$\\end{document}</tex-math><mml:math id=\"M934\"><mml:mrow><mml:mi>ϕ</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>x</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq383\"><alternatives><tex-math id=\"M935\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\mathcal {L}} = \\big (a^3 {\\dot{\\phi }}^2 - a (\\nabla \\phi )^2\\big )/2$$\\end{document}</tex-math><mml:math id=\"M936\"><mml:mrow><mml:mi mathvariant=\"script\">L</mml:mi><mml:mo>=</mml:mo><mml:mrow><mml:mo maxsize=\"1.2em\" minsize=\"1.2em\" stretchy=\"true\">(</mml:mo></mml:mrow><mml:msup><mml:mi>a</mml:mi><mml:mn>3</mml:mn></mml:msup><mml:msup><mml:mrow><mml:mover accent=\"true\"><mml:mi>ϕ</mml:mi><mml:mo>˙</mml:mo></mml:mover></mml:mrow><mml:mn>2</mml:mn></mml:msup><mml:mo>-</mml:mo><mml:mi>a</mml:mi><mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi mathvariant=\"normal\">∇</mml:mi><mml:mi>ϕ</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:msup><mml:mrow><mml:mo maxsize=\"1.2em\" minsize=\"1.2em\" stretchy=\"true\">)</mml:mo></mml:mrow><mml:mo stretchy=\"false\">/</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq384\"><alternatives><tex-math id=\"M937\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$a=a(t)$$\\end{document}</tex-math><mml:math id=\"M938\"><mml:mrow><mml:mi>a</mml:mi><mml:mo>=</mml:mo><mml:mi>a</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq385\"><alternatives><tex-math id=\"M939\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$c =1$$\\end{document}</tex-math><mml:math id=\"M940\"><mml:mrow><mml:mi>c</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ90\"><label>90</label><alternatives><tex-math id=\"M941\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} \\sum _{\\vec{k}, r} \\left(\\frac{1}{2a^3}\\pi _{\\vec{k}, r}^2 + \\frac{ak^2}{2}q_{\\vec{k},r}^2 \\right)\\psi = i\\frac{\\partial \\psi }{\\partial t} \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M942\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:munder><mml:mo>∑</mml:mo><mml:mrow><mml:mover accent=\"true\"><mml:mi>k</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mi>r</mml:mi></mml:mrow></mml:munder><mml:mfenced close=\")\" open=\"(\"><mml:mfrac><mml:mn>1</mml:mn><mml:mrow><mml:mn>2</mml:mn><mml:msup><mml:mi>a</mml:mi><mml:mn>3</mml:mn></mml:msup></mml:mrow></mml:mfrac><mml:msubsup><mml:mi>π</mml:mi><mml:mrow><mml:mover accent=\"true\"><mml:mi>k</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mi>r</mml:mi></mml:mrow><mml:mn>2</mml:mn></mml:msubsup><mml:mo>+</mml:mo><mml:mfrac><mml:mrow><mml:mi>a</mml:mi><mml:msup><mml:mi>k</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mrow><mml:mn>2</mml:mn></mml:mfrac><mml:msubsup><mml:mi>q</mml:mi><mml:mrow><mml:mover accent=\"true\"><mml:mi>k</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mi>r</mml:mi></mml:mrow><mml:mn>2</mml:mn></mml:msubsup></mml:mfenced><mml:mi>ψ</mml:mi><mml:mo>=</mml:mo><mml:mi>i</mml:mi><mml:mfrac><mml:mrow><mml:mi>∂</mml:mi><mml:mi>ψ</mml:mi></mml:mrow><mml:mrow><mml:mi>∂</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:mfrac></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq386\"><alternatives><tex-math id=\"M943\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\psi = \\psi (\\{q_{\\vec{k}, r}\\}, t)$$\\end{document}</tex-math><mml:math id=\"M944\"><mml:mrow><mml:mi>ψ</mml:mi><mml:mo>=</mml:mo><mml:mi>ψ</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mo stretchy=\"false\">{</mml:mo><mml:msub><mml:mi>q</mml:mi><mml:mrow><mml:mover accent=\"true\"><mml:mi>k</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mi>r</mml:mi></mml:mrow></mml:msub><mml:mo stretchy=\"false\">}</mml:mo></mml:mrow><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq387\"><alternatives><tex-math id=\"M945\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\{q_{\\vec{k}, r}\\} \\equiv (q_{\\vec{k}_1, r}, q_{\\vec{k}_2, r}, q_{\\vec{k}_3, r}...)$$\\end{document}</tex-math><mml:math id=\"M946\"><mml:mrow><mml:mrow><mml:mo stretchy=\"false\">{</mml:mo><mml:msub><mml:mi>q</mml:mi><mml:mrow><mml:mover accent=\"true\"><mml:mi>k</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mi>r</mml:mi></mml:mrow></mml:msub><mml:mo stretchy=\"false\">}</mml:mo></mml:mrow><mml:mo>≡</mml:mo><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>q</mml:mi><mml:mrow><mml:msub><mml:mover accent=\"true\"><mml:mi>k</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mn>1</mml:mn></mml:msub><mml:mo>,</mml:mo><mml:mi>r</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>q</mml:mi><mml:mrow><mml:msub><mml:mover accent=\"true\"><mml:mi>k</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mn>2</mml:mn></mml:msub><mml:mo>,</mml:mo><mml:mi>r</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>q</mml:mi><mml:mrow><mml:msub><mml:mover accent=\"true\"><mml:mi>k</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mn>3</mml:mn></mml:msub><mml:mo>,</mml:mo><mml:mi>r</mml:mi></mml:mrow></mml:msub><mml:mo>.</mml:mo><mml:mo>.</mml:mo><mml:mo>.</mml:mo><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq388\"><alternatives><tex-math id=\"M947\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$q_{\\vec{k}, r}$$\\end{document}</tex-math><mml:math id=\"M948\"><mml:msub><mml:mi>q</mml:mi><mml:mrow><mml:mover accent=\"true\"><mml:mi>k</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mi>r</mml:mi></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq389\"><alternatives><tex-math id=\"M949\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$(r = 1, 2)$$\\end{document}</tex-math><mml:math id=\"M950\"><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>r</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq390\"><alternatives><tex-math id=\"M951\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\phi (\\vec{x}, t)$$\\end{document}</tex-math><mml:math id=\"M952\"><mml:mrow><mml:mi>ϕ</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>x</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ91\"><label>91</label><alternatives><tex-math id=\"M953\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} \\phi (\\vec{k}, t) \\equiv \\frac{1}{(2\\pi )^{3/2}} \\int \\phi (\\vec{x}, t) e^{-i\\vec{k}\\cdot \\vec{x}} d\\vec{x} = \\frac{\\sqrt{V}}{(2\\pi )^{3/2}} \\big ( q_{\\vec{k}, 1}(t) + i q_{\\vec{k}, 2}(t)\\big ) \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M954\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:mi>ϕ</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>k</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>≡</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>2</mml:mn><mml:mi>π</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mrow><mml:mn>3</mml:mn><mml:mo stretchy=\"false\">/</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mfrac><mml:mo>∫</mml:mo><mml:mi>ϕ</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>x</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:msup><mml:mi>e</mml:mi><mml:mrow><mml:mo>-</mml:mo><mml:mi>i</mml:mi><mml:mover accent=\"true\"><mml:mi>k</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>·</mml:mo><mml:mover accent=\"true\"><mml:mi>x</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover></mml:mrow></mml:msup><mml:mi>d</mml:mi><mml:mover accent=\"true\"><mml:mi>x</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>=</mml:mo><mml:mfrac><mml:msqrt><mml:mi>V</mml:mi></mml:msqrt><mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>2</mml:mn><mml:mi>π</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mrow><mml:mn>3</mml:mn><mml:mo stretchy=\"false\">/</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mfrac><mml:mrow><mml:mo maxsize=\"1.2em\" minsize=\"1.2em\" stretchy=\"true\">(</mml:mo></mml:mrow><mml:msub><mml:mi>q</mml:mi><mml:mrow><mml:mover accent=\"true\"><mml:mi>k</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>+</mml:mo><mml:mi>i</mml:mi><mml:msub><mml:mi>q</mml:mi><mml:mrow><mml:mover accent=\"true\"><mml:mi>k</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mrow><mml:mo maxsize=\"1.2em\" minsize=\"1.2em\" stretchy=\"true\">)</mml:mo></mml:mrow></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq391\"><alternatives><tex-math id=\"M955\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\pi _{\\vec{k}, r} \\equiv \\partial (\\int {\\mathcal {L}} d\\vec{x})/\\partial {\\dot{q}}_{\\vec{k}, r} = a^3 {\\dot{q}}_{\\vec{k}, r}$$\\end{document}</tex-math><mml:math id=\"M956\"><mml:mrow><mml:msub><mml:mi>π</mml:mi><mml:mrow><mml:mover accent=\"true\"><mml:mi>k</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mi>r</mml:mi></mml:mrow></mml:msub><mml:mo>≡</mml:mo><mml:mi>∂</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mo>∫</mml:mo><mml:mi mathvariant=\"script\">L</mml:mi><mml:mi>d</mml:mi><mml:mover accent=\"true\"><mml:mi>x</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo stretchy=\"false\">/</mml:mo><mml:mi>∂</mml:mi><mml:msub><mml:mover accent=\"true\"><mml:mi>q</mml:mi><mml:mo>˙</mml:mo></mml:mover><mml:mrow><mml:mover accent=\"true\"><mml:mi>k</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mi>r</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:msup><mml:mi>a</mml:mi><mml:mn>3</mml:mn></mml:msup><mml:msub><mml:mover accent=\"true\"><mml:mi>q</mml:mi><mml:mo>˙</mml:mo></mml:mover><mml:mrow><mml:mover accent=\"true\"><mml:mi>k</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mi>r</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq392\"><alternatives><tex-math id=\"M957\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\vec{k}$$\\end{document}</tex-math><mml:math id=\"M958\"><mml:mover accent=\"true\"><mml:mi>k</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq393\"><alternatives><tex-math id=\"M959\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\phi (\\vec{x}, t)$$\\end{document}</tex-math><mml:math id=\"M960\"><mml:mrow><mml:mi>ϕ</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>x</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq394\"><alternatives><tex-math id=\"M961\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\phi (\\vec{x}, t)$$\\end{document}</tex-math><mml:math id=\"M962\"><mml:mrow><mml:mi>ϕ</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>x</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq395\"><alternatives><tex-math id=\"M963\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\psi (\\{q_{\\vec{k}, r}\\}, t)$$\\end{document}</tex-math><mml:math id=\"M964\"><mml:mrow><mml:mi>ψ</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mo stretchy=\"false\">{</mml:mo><mml:msub><mml:mi>q</mml:mi><mml:mrow><mml:mover accent=\"true\"><mml:mi>k</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mi>r</mml:mi></mml:mrow></mml:msub><mml:mo stretchy=\"false\">}</mml:mo></mml:mrow><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq396\"><alternatives><tex-math id=\"M965\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\phi (\\vec{x}, t)$$\\end{document}</tex-math><mml:math id=\"M966\"><mml:mrow><mml:mi>ϕ</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>x</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq397\"><alternatives><tex-math id=\"M967\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$H_q$$\\end{document}</tex-math><mml:math id=\"M968\"><mml:msub><mml:mi>H</mml:mi><mml:mi>q</mml:mi></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq398\"><alternatives><tex-math id=\"M969\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$-\\infty$$\\end{document}</tex-math><mml:math id=\"M970\"><mml:mrow><mml:mo>-</mml:mo><mml:mi>∞</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq399\"><alternatives><tex-math id=\"M971\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$H_q$$\\end{document}</tex-math><mml:math id=\"M972\"><mml:msub><mml:mi>H</mml:mi><mml:mi>q</mml:mi></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq400\"><alternatives><tex-math id=\"M973\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\Omega$$\\end{document}</tex-math><mml:math id=\"M974\"><mml:mi mathvariant=\"normal\">Ω</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq401\"><alternatives><tex-math id=\"M975\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\mathscr {C}}$$\\end{document}</tex-math><mml:math id=\"M976\"><mml:mi mathvariant=\"script\">C</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq402\"><alternatives><tex-math id=\"M977\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\Omega = {\\mathscr {C}}$$\\end{document}</tex-math><mml:math id=\"M978\"><mml:mrow><mml:mi mathvariant=\"normal\">Ω</mml:mi><mml:mo>=</mml:mo><mml:mi mathvariant=\"script\">C</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq403\"><alternatives><tex-math id=\"M979\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$H_q$$\\end{document}</tex-math><mml:math id=\"M980\"><mml:msub><mml:mi>H</mml:mi><mml:mi>q</mml:mi></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq404\"><alternatives><tex-math id=\"M981\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$H_q^{min} = -\\ln {\\mathcal {N}}$$\\end{document}</tex-math><mml:math id=\"M982\"><mml:mrow><mml:msubsup><mml:mi>H</mml:mi><mml:mi>q</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">min</mml:mi></mml:mrow></mml:msubsup><mml:mo>=</mml:mo><mml:mo>-</mml:mo><mml:mo>ln</mml:mo><mml:mi mathvariant=\"script\">N</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq405\"><alternatives><tex-math id=\"M983\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$H_q$$\\end{document}</tex-math><mml:math id=\"M984\"><mml:msub><mml:mi>H</mml:mi><mml:mi>q</mml:mi></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq406\"><alternatives><tex-math id=\"M985\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$H_{pw}$$\\end{document}</tex-math><mml:math id=\"M986\"><mml:msub><mml:mi>H</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">pw</mml:mi></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq407\"><alternatives><tex-math id=\"M987\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$v_y \\rightarrow 0$$\\end{document}</tex-math><mml:math id=\"M988\"><mml:mrow><mml:msub><mml:mi>v</mml:mi><mml:mi>y</mml:mi></mml:msub><mml:mo stretchy=\"false\">→</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq408\"><alternatives><tex-math id=\"M989\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\pm y$$\\end{document}</tex-math><mml:math id=\"M990\"><mml:mrow><mml:mo>±</mml:mo><mml:mi>y</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq409\"><alternatives><tex-math id=\"M991\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\Omega _t$$\\end{document}</tex-math><mml:math id=\"M992\"><mml:msub><mml:mi mathvariant=\"normal\">Ω</mml:mi><mml:mi>t</mml:mi></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq410\"><alternatives><tex-math id=\"M993\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\Omega _0$$\\end{document}</tex-math><mml:math id=\"M994\"><mml:msub><mml:mi mathvariant=\"normal\">Ω</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq411\"><alternatives><tex-math id=\"M995\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$|\\psi ({\\overrightarrow{y}},t)|^2$$\\end{document}</tex-math><mml:math id=\"M996\"><mml:mrow><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo><mml:mi>ψ</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:msup><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:msup></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq412\"><alternatives><tex-math id=\"M997\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\partial \\Omega _0$$\\end{document}</tex-math><mml:math id=\"M998\"><mml:mrow><mml:mi>∂</mml:mi><mml:msub><mml:mi mathvariant=\"normal\">Ω</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq413\"><alternatives><tex-math id=\"M999\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\partial \\Omega _t$$\\end{document}</tex-math><mml:math id=\"M1000\"><mml:mrow><mml:mi>∂</mml:mi><mml:msub><mml:mi mathvariant=\"normal\">Ω</mml:mi><mml:mi>t</mml:mi></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq414\"><alternatives><tex-math id=\"M1001\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\epsilon&lt;&lt; 1$$\\end{document}</tex-math><mml:math id=\"M1002\"><mml:mrow><mml:mi>ϵ</mml:mi><mml:mo>&lt;</mml:mo><mml:mo>&lt;</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq415\"><alternatives><tex-math id=\"M1003\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\rho _g ({\\overrightarrow{y}}) \\le \\epsilon$$\\end{document}</tex-math><mml:math id=\"M1004\"><mml:mrow><mml:msub><mml:mi>ρ</mml:mi><mml:mi>g</mml:mi></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>≤</mml:mo><mml:mi>ϵ</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq416\"><alternatives><tex-math id=\"M1005\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\overrightarrow{y}}$$\\end{document}</tex-math><mml:math id=\"M1006\"><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq417\"><alternatives><tex-math id=\"M1007\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\rho _c ({\\overrightarrow{y}}) \\equiv 0$$\\end{document}</tex-math><mml:math id=\"M1008\"><mml:mrow><mml:msub><mml:mi>ρ</mml:mi><mml:mi>c</mml:mi></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>≡</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq418\"><alternatives><tex-math id=\"M1009\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\rho _c ({\\overrightarrow{y}}) \\equiv \\rho _g ({\\overrightarrow{y}})$$\\end{document}</tex-math><mml:math id=\"M1010\"><mml:mrow><mml:msub><mml:mi>ρ</mml:mi><mml:mi>c</mml:mi></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>≡</mml:mo><mml:msub><mml:mi>ρ</mml:mi><mml:mi>g</mml:mi></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq419\"><alternatives><tex-math id=\"M1011\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\overrightarrow{y}}$$\\end{document}</tex-math><mml:math id=\"M1012\"><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq420\"><alternatives><tex-math id=\"M1013\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$v_y = j_y/|\\psi |^2$$\\end{document}</tex-math><mml:math id=\"M1014\"><mml:mrow><mml:msub><mml:mi>v</mml:mi><mml:mi>y</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:msub><mml:mi>j</mml:mi><mml:mi>y</mml:mi></mml:msub><mml:mo stretchy=\"false\">/</mml:mo><mml:msup><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo><mml:mi>ψ</mml:mi><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:msup></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq421\"><alternatives><tex-math id=\"M1015\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$y \\rightarrow \\pm \\infty$$\\end{document}</tex-math><mml:math id=\"M1016\"><mml:mrow><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo><mml:mo>±</mml:mo><mml:mi>∞</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq422\"><alternatives><tex-math id=\"M1017\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\vec{j} \\rightarrow \\vec{j} + \\vec{\\nabla } \\times \\vec{A}$$\\end{document}</tex-math><mml:math id=\"M1018\"><mml:mrow><mml:mover accent=\"true\"><mml:mi>j</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo stretchy=\"false\">→</mml:mo><mml:mover accent=\"true\"><mml:mi>j</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>+</mml:mo><mml:mover accent=\"true\"><mml:mi mathvariant=\"normal\">∇</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>×</mml:mo><mml:mover accent=\"true\"><mml:mi>A</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq423\"><alternatives><tex-math id=\"M1019\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\vec{A}$$\\end{document}</tex-math><mml:math id=\"M1020\"><mml:mover accent=\"true\"><mml:mi>A</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq424\"><alternatives><tex-math id=\"M1021\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\psi$$\\end{document}</tex-math><mml:math id=\"M1022\"><mml:mi>ψ</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq425\"><alternatives><tex-math id=\"M1023\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\psi$$\\end{document}</tex-math><mml:math id=\"M1024\"><mml:mi>ψ</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq426\"><alternatives><tex-math id=\"M1025\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\psi$$\\end{document}</tex-math><mml:math id=\"M1026\"><mml:mi>ψ</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq427\"><alternatives><tex-math id=\"M1027\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\psi$$\\end{document}</tex-math><mml:math id=\"M1028\"><mml:mi>ψ</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq428\"><alternatives><tex-math id=\"M1029\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\overline{\\rho ({\\overrightarrow{y}}, t)} = \\overline{|\\psi ({\\overrightarrow{y}}, t)|^2}/{\\mathcal {N}}$$\\end{document}</tex-math><mml:math id=\"M1030\"><mml:mrow><mml:mover><mml:mrow><mml:mi>ρ</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>¯</mml:mo></mml:mover><mml:mo>=</mml:mo><mml:mover><mml:mrow><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo><mml:mi>ψ</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:msup><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:msup></mml:mrow><mml:mo>¯</mml:mo></mml:mover><mml:mo stretchy=\"false\">/</mml:mo><mml:mi mathvariant=\"script\">N</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq429\"><alternatives><tex-math id=\"M1031\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\Omega _t$$\\end{document}</tex-math><mml:math id=\"M1032\"><mml:msub><mml:mi mathvariant=\"normal\">Ω</mml:mi><mml:mi>t</mml:mi></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq430\"><alternatives><tex-math id=\"M1033\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\psi$$\\end{document}</tex-math><mml:math id=\"M1034\"><mml:mi>ψ</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq431\"><alternatives><tex-math id=\"M1035\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\psi$$\\end{document}</tex-math><mml:math id=\"M1036\"><mml:mi>ψ</mml:mi></mml:math></alternatives></inline-formula>" ]

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[ "<supplementary-material content-type=\"local-data\" id=\"MOESM1\"></supplementary-material>" ]

[ "<fn-group><fn><p><bold>Publisher's note</bold></p><p>Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.</p></fn></fn-group>" ]

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[ "<media xlink:href=\"41598_2023_50814_MOESM1_ESM.pdf\"><caption><p>Supplementary Information 1.</p></caption></media>" ]

{ "acronym": [], "definition": [] }

[ "<title>Introduction</title>", "<p id=\"Par2\">Urinary Tract Infections (UTIs) are one of the most common bacterial infections in older adults, constituting around 25% of all infections^{##REF##12848468##1##–##UREF##1##5##}. Clinical presentation ranges from self-limited illness to severe sepsis. UTIs account for ~9–31% of cases of severe sepsis which itself has an estimated mortality of 20–40%^{##UREF##0##4##,##UREF##2##6##–##REF##23103175##9##}. To differentiate between asymptomatic bacteriuria and UTIs, clinicians rely on positive findings of bacteriuria and genitourinary symptoms. Diagnosis is further complicated by the presence of cognitive impairment or dementia since People Living with Dementia (PLWD) may find it challenging to report their symptoms, and this could result in further complications^{##REF##31638686##10##–##REF##24845761##12##}. As a result, acute infections might not be diagnosed until symptoms require hospitalisation^{##UREF##4##13##}. In the United Kingdom, over 20% of hospital beds are occupied by PLWD, with 9% of these attributed to UTIs^{##UREF##5##14##–##REF##30645599##17##}.</p>", "<p id=\"Par3\">Currently, a urine sample test and acute changes in baseline cognition are used to diagnose UTIs in PLWD^{##REF##34448496##18##}. However, samples can be difficult to obtain due to urinary incontinence, cognitive impairment, sample contamination or previous use of antibiotics^{##UREF##8##19##} and are taken on suspicion of an infection, which may be delayed. Additionally, although they can be used as rapid detectors, dipstick tests have a high false positive rate for older adults and require action from the PLWD or their carer which limits their effectiveness for diagnosis^{##REF##24570248##3##,##UREF##9##20##}. Highlighting UTI risk by identifying early symptoms would allow for prompt diagnosis, improved health outcomes and effective allocation of healthcare resources.</p>", "<p id=\"Par4\">Machine Learning (ML) offers opportunities for clinical diagnosis and decision-support and recent advances show promise for development of advanced predictive models that incorporate patient data to improve diagnostic performance. For UTI detection in PLWD, ML can improve diagnostic performance and timeliness. Several investigations have been conducted for UTI risk prediction on younger adult populations^{##UREF##10##21##,##UREF##11##22##}, which do not generalise to older adults. Existing methods developed for older adults also rely on typical symptoms as predictor variables, precluding their use in community-dwelling patients with dementia with atypical clinical manifestations and who may struggle expressing symptoms^{##UREF##12##23##}. In parallel, low-cost monitoring devices have been developed to offer complementary solutions to the typical diagnostic criteria^{##REF##29088123##24##,##UREF##13##25##}. Rantz et al.^{##UREF##14##26##} use activity data collected from in-home Passive Infra-Red (PIR) sensors to detect UTIs in older adults. However, their work is limited to 37 participants and does not utilise ML techniques. The study is also limited to the use of activity data and does not utilise physiological measurements. Work by Enshaeifar et al.^{##REF##30645599##17##} employed an unsupervised approach to predict UTIs based on in-home sensors and physiological measurements, however their work showed insufficient diagnostic performance and required the participant to record their own physiology measurements twice a day.</p>", "<p id=\"Par5\">This study presents a machine learning application to identifying the risk of UTI events in PLWD by analysing symptom-targeted features, engineered from continuous in-home activity and physiology data collected by low-cost and passive sensors (Fig. ##FIG##0##1## presents an overview). Then, through optimisation and consultations with clinicians, we determine thresholds for the stratification of the risk scores to improve the algorithm’s clinical applicability. The proposed approach has been evaluated in an observational clinical study consisting of 117 participants living with dementia within their own homes. We have worked closely with healthcare professionals to implement a reliable and non-intrusive UTI risk model. Our work will (1) aid clinicians in the early diagnosis of UTIs, and (2) enable a better understanding of in-home behaviour at the point of clinical decision-making. The use of high-resolution in-home observation and measurement data in conjunction with machine learning methods result in timely interventions that can have a significant impact on reducing preventable and unplanned hospital admissions in dementia patients. Such a tool allows for precise collection of urine samples for culture analysis, improved clinical outcomes, a reduction in the burden on healthcare services, and decreased antibiotic overuse and misuse in PLWD by reducing UTI detection time and providing practitioners with more complete pictures of their patients.</p>" ]

[ "<title>Methods</title>", "<title>Study design and population</title>", "<p id=\"Par18\">This study was performed in collaboration with Imperial College London and Surrey and Borders Partnership NHS Trust. Participants were recruited from the following: (1) health and social care partners within the primary care network and community NHS trusts, (2) urgent and acute care services within the NHS, (3) social services who oversee sheltered and extra care sheltered housing schemes, (4) NHS Community Mental Health Teams for older adults (CMHT-OP), and (5) specialist memory services at Surrey and Borders Partnership NHS Foundation Trust. All participants provided written informed consent. Capacity to consent was assessed according to Good Clinical Practice, as detailed in the Research Governance Framework for Health and Social Care (Department of Health 2005) and the Mental Capacity Act 2005. Participants were provided with a Participant Information Sheet (PIS) that includes information on how the study used their personal data collected in accordance with the GDPR requirements. If the participant was deemed to lack capacity, a personal or professional consultee was sought to provide written consent to the study. Additionally, capacity of both the participant and study partner is assessed at each research visit. Research staff conducting the assessment have completed the NIHR GCP training and Valid Informed Consent training. If a participant is deemed to lack capacity but is willing to take part in the research, a personal consultee is sought in the first instance to sign a declaration of consent. If no personal consultee can be found, a professional consultee, such as a key worker, is sought. This process is included in the study protocol and ethical panel approval is obtained.</p>", "<p id=\"Par19\">Eligible study participants included adults &gt;50 years with a clinically ascertained diagnosis of dementia or mild cognitive impairment and current or previous treatment at a psychiatric unit. Participants lacking capacity for informed consent were required to have a partner or caregiver who had known them for at least 6 months and was able to attend research assessments with them. Exclusion criteria were as follows: (1) patients receiving treatment for terminal illness (2) presence of severe mental health conditions including depression, anxiety, psychosis, and agitation (3) presence of active suicidal thoughts. In total, 117 participants were selected for participation using the above-mentioned recruitment process.</p>", "<p id=\"Par20\">The cohort characteristics can be seen in Table ##TAB##1##2## and a patient disposition is available in Supplementary Information Section ##SUPPL##1##1##.</p>", "<title>Data collection and definition of outcome</title>", "<p id=\"Par21\">Demographic data was collected during the baseline assessment, whilst psychometric scales were used to collect various physical and cognitive data during regular visits. In-home observation and measurement data was obtained using low-cost off-the-shelf monitoring technologies, including PIR sensors (for measuring activity) and sleep monitoring devices. Figure ##FIG##0##1## presents cohort-wide sleep and activity activations, and differences in sleep and activity for a participant with both UTI positive and negative days. PIR sensors can detect motion within 9 metres and with a maximum angle of 45^∘ and the sleep mat device can monitor breathing rates, heart rates, and sleep states. For an illustration of the layout of sensors see Supplementary Information Section ##SUPPL##1##2##.</p>", "<p id=\"Par22\">Urine samples were collected from several enrolled participants to be labelled by clinicians. Additionally, a baseline algorithm developed in our previous work^{##UREF##21##40##} suggested patients to the study monitoring team to check for additional symptoms of UTIs and arrange a sample collection and refer to the GP if needed. Once samples were collected, a urine sample analysis was performed and the results sent to clinicians, who with information from the monitoring team, determine a UTI. In total, we have 258 labelled urine samples from 64 participants, of which 81 were confirmed positive UTI cases. If a single day has been labelled, we assume the preceding and proceeding 3 days would also be labelled the same (see Supplementary Information Section ##SUPPL##1##6##). This extends the number of labelled days of data to 1752, consisting of 534 positives and 1218 negatives. For our experimentation, we used data collected between 2021/06/28 and 2022/12/01. The models were trained to predict whether a participant had a UTI on a given day (24 h time window). The distribution of labels can be found in Supplementary Information Section ##SUPPL##1##3##.</p>", "<title>Data pre-processing and feature selection</title>", "<p id=\"Par23\">In addition to sensor readings, we performed feature engineering inspired by well-known symptoms of UTIs such as incontinence, urgency and increased frequency of urination, and behavioural changes (<ext-link ext-link-type=\"uri\" xlink:href=\"https://www.nhs.uk/conditions/urinary-tract-infections-utis/\">https://www.nhs.uk/conditions/urinary-tract-infections-utis/</ext-link>) to allow clinical interpretability and improve model performance and generalisability.</p>", "<p id=\"Par24\">Raw features were: (1) frequency of bathroom, bedroom, hallway, kitchen, lounge activations; (2) mean and standard deviation of nocturnal heart rate and respiratory rate; (3) nocturnal awake occurrences. Engineered features were: (4) bathroom day and nocturnal frequencies, moving average, and percentage change; (5) mean and standard deviation of the movement time from any location within the house to bathroom; (6) daily entropy in PIR sensor activation; (7) number of previous UTIs to date. More information on the features selected can be found in Supplementary Information Section ##SUPPL##1##4##.</p>", "<p id=\"Par25\">Data collection occurred outside controlled environments using in-home devices so missing measurements inevitably occurred. To limit the effects of incomplete data^{##UREF##22##41##}, we imputed missing values based on strategies depending on the given features (see Supplementary Information Section ##SUPPL##1##5## for more information).</p>", "<title>Analysis platform</title>", "<p id=\"Par26\">All analyses were performed on a secure computing environment at Imperial College London using Python version 3.9. The Pandas^{##UREF##23##42##}, Numpy^{##REF##32939066##43##}, Scikit-Learn^{##UREF##24##44##}, and Pytorch^{##UREF##25##45##} packages formed much of our pipeline.</p>", "<title>Methodology</title>", "<p id=\"Par27\">To ensure generalisability, we evaluated our work in two different ways.</p>", "<p id=\"Par28\">Firstly, the dataset was split temporally into training and testing subsets in an 80:20 ratio. The data collected from 2021/06/28 to 2022/10/05 represented 80% (<italic>n</italic> = 1394 days in total, from <italic>p</italic> = 54 participants) of the dataset, whilst the data between 2022/10/05 and 2022/12/01 represented 20% (<italic>n</italic> = 358, <italic>p</italic> = 39).</p>", "<p id=\"Par29\">This formed the first analysis, evaluating the model at making predictions on future data from the same cohort as it was trained on. We will refer to this experimental setting as “Date Split\".</p>", "<p id=\"Par30\">In the second analysis, we used a leave-one-out cross-validation strategy^{##UREF##26##46##}. Here, data was split in the same way as in the first evaluation method. Then, training and testing of our machine learning models was performed using a leave-one-out strategy on data from each of the PLWD. This way, we are able to test the model performance on data from participants outside of the cohort it has been trained on. We will refer to this experimental setting as “Date-ID Split\".</p>", "<p id=\"Par31\">During model development and whilst optimising model parameters, validation sets were produced by splitting the training data on the date 2022/09/11. All experiments were performed multiple times, with each run using a bootstrap sample^{##UREF##26##46##} of the training set to ensure reproducibility. See Supplementary Information Section ##SUPPL##1##8## for a visualisation of this evaluation.</p>", "<p id=\"Par32\">We used sensitivity, specificity, and area under the precision-recall curve to measure model performance (for definitions of metrics, please see Supplementary Information Section ##SUPPL##1##7##).</p>", "<title>Model development</title>", "<p id=\"Par33\">We tested Logistic Regression (LR), Extreme Gradient Boosting Decision Tree (XGBoost)^{##UREF##15##27##}, Multilayer Perceptron (MLP), Self-Attention^{##UREF##16##28##}, Random Forest (RF)^{##UREF##17##29##} and Naive Bayes (NB) models at predicting the risk of UTIs. Hyper-parameters were tuned using Bayesian optimisation on train-validation splits, with the model producing the highest area under the precision-recall curve (on validation data) selected for the final analysis. The number of days of data used as input to the model was jointly tested, ranging from 1 day to 7 days. Supplementary Information Section ##SUPPL##1##9## contains information on the decisions made regarding each step of the UTI model pipeline.</p>", "<title>Stratification of risk scores for clinical reporting</title>", "<p id=\"Par34\">Risk scores from the model are stratified into three groups, used to inform clinical decisions in a concise way and provide precise control over the number of actionable alerts. Outputs are split into the groups Green, Amber, and Red; referring to minimal, medium, and high risk of a UTI respectively. By varying these thresholds, we can balance levels of sensitivity and specificity for the different groups with the number of alerts. This allows our process to be flexible to different clinical scenarios and resources.</p>", "<p id=\"Par35\">Within this work, the optimal thresholds used in our analysis of results were calculated using the algorithm’s predictions on the data collected between 2022/09/11 and 2022/10/05 (validation data), and with feedback from a clinical team. More information on the risk stratification is included in Supplementary Information Section ##SUPPL##1##12##.</p>", "<title>Ethics approval</title>", "<p id=\"Par36\">The study received ethical approval from the London-Surrey Borders Research Ethics Committee; TIHM 1.5 REC: 19/LO/0102. The study is registered with National Institute for Health and Care Research (NIHR) in the United Kingdom under Integrated Research Application System (IRAS) registration number 257561.</p>", "<title>Reporting summary</title>", "<p id=\"Par37\">Further information on research design is available in the ##SUPPL##2##Nature Research Reporting Summary## linked to this article.</p>" ]

[ "<title>Results</title>", "<title>Model performance</title>", "<p id=\"Par6\">We examined Logistic Regression (LR), Extreme Gradient Boosting Decision Trees (XGBoost)^{##UREF##15##27##}, Multilayer Perceptron (MLP), Self-Attention^{##UREF##16##28##}, Random Forest (RF)^{##UREF##17##29##}, and Naive Bayes (NB) in their effectiveness to predict UTI events and found the best-performing classification model was LR with L2 regularisation, acting over 3 consecutive days of data. Table ##TAB##0##1## presents this model performance on the different data splits. Results from the other models are included in Supplementary Information Section ##SUPPL##1##10##. Analysis of model reliability and calibration can be seen in Supplementary Information Section ##SUPPL##1##11##.</p>", "<title>Risk stratification</title>", "<p id=\"Par7\">To improve flexibility of the model to varying clinical settings, we calculate stratified risk scores as discussed in Section: Stratification of Risk Scores for Clinical Reporting. Figure ##FIG##1##2## shows the sensitivity and specificity that can be achieved on the validation set by stratifying the results. By varying the stratification thresholds, sensitivity and specificity can be balanced with the number of people given Green and Red alerts. In Supplementary Information Section ##SUPPL##1##12.1##, we present the performance variations when jointly changing the Red and Green thresholds. Here, we select thresholds , and (following interval notation) for groups Green, Amber, and Red respectively. Table ##TAB##0##1## shows the results of grouping the risk predictions on the Red and Green groups.</p>", "<title>Early detection</title>", "<p id=\"Par8\">We evaluated the model’s utility in correctly estimating the risk of UTIs prior to the recorded clinical urine tests. Figure ##FIG##2##3## demonstrates specificity, sensitivity and the area under precision-recall curve for days prior to the recorded UTI events. This shows that 2 days prior to a sample test, our model achieved a sensitivity of 64.4 (95% CI = 61.1–67.8), specificity of 68.9 (95% CI = 66.8–71.0), and area under the precision-recall curve of 64.5 (95% CI = 63.0–66.0), and 4 days prior, a sensitivity of 64.4 (95% CI = 61.1–67.8), specificity of 71.9 (95% CI = 67.9–75.8), and area under the precision-recall curve of 65.4 (95% CI = 60.8–70.0).</p>", "<title>Feature importance</title>", "<p id=\"Par9\">The most important features influencing predictions were identified using SHapley Additive exPlanations (SHAP)^{##REF##31001455##30##}, a method for producing explainable predictions and calculating contributions from individual features to risk scores. The results of this, on the test set, can be seen in Fig. ##FIG##3##4##a and reveal that the number of previous confirmed UTI events, the standard deviation of the nocturnal respiratory rate, the nocturnal average heart rate, and the number of nocturnal awake states were positively correlated with a higher risk score. We can also breakdown single predictions to understand contributions to a risk score, as show in Fig. ##FIG##3##4##b. Further examples can be seen in Supplementary Information Section ##SUPPL##1##13##.</p>", "<title>Frequency of generated alerts</title>", "<p id=\"Par10\">To understand the requirements of our model in a clinical setting, we calculated the risk groups of each day of data between 2022/10/05 and 2022/12/01, for each of the PLWD in our dataset. We find that on average, each of the PLWD will receive 0.25 Green alerts, 0.69 Amber alerts and 0.06 Red alerts each day. In Supplementary Information Section ##SUPPL##1##14##, we visualise how this risk score varies over time and in Supplementary Information Section ##SUPPL##1##15##, we present the model performance on subsets of the sensors. Additionally, in Supplementary Information Section ##SUPPL##1##16##, we compare the performance between those with recurrent and non-recurrent UTIs and in Supplementary Information Section ##SUPPL##1##17##, we compare the results between Male and Female participants.</p>" ]

[ "<title>Discussion</title>", "<p id=\"Par11\">The 2020 report of the Lancet Commission on dementia prevention, treatment, and care emphasises the significance of individualised interventions to address complex medical problems in dementia, which result in unnecessary hospital admissions, accelerated functional decline, and decreased quality of life^{##REF##32738937##31##}. An area of priority development is infection prevention and timely detection and treatment^{##UREF##18##32##}. By conducting preliminary experiments into early identification of possible UTIs in remote healthcare settings, we hope to contribute to directly addressing this priority by investigating more individualised, predictive, and preventative healthcare.</p>", "<p id=\"Par12\">We present a machine learning pipeline for continuous UTI risk screening via analysis of passively collected in-home activity and physiology data. We considered several models and found that LR acting over 3 days attained the top performance (sensitivity of 65.2% (95% CI = 64.3–66.2) and 54.5% (95% CI = 52.7–56.4), and specificity of 70.9% (95% CI = 68.6–73.1) and 73.0% (95% CI = 71.2–74.8) on “Date-ID Split\" and “Date Split\" respectively). The performance was higher on “Date-ID Split\" than “Date Split\", which we hypothesise is due to some PLWD who have opposing labels in the training and testing data. In this case, in “Date Split\", the model might over-fit to the training data from a PLWD. However, in “Date-ID Split\" all data seen by the model during testing is from participants not appearing in training. The ratios of positive to negatives in the test sets of the “Date Split\" and the “Date-ID Split\" are 0.31 and 0.32 respectively and 0.47 and 0.47 for the training set of the “Date Split\" and the “Date-ID Split\" respectively.</p>", "<p id=\"Par13\">Through stratification, risk scores were transformed into more accessible groups, allowing for the flexible management of actionable alerts within a time period. Following this, the performance on the Green and Red groups were significantly improved, achieving a sensitivity of 74.7% (95% CI = 67.9–81.5) and 69.0% (95% CI = 64.4–73.5), and specificity of 87.9% (95% CI = 85.0–90.9) and 94.1% (95% CI = 92.0–96.2) on the “Date-ID Split\" and “Date Split\" respectively.</p>", "<p id=\"Par14\">SHAP analysis then highlighted the features most strongly predictive of the risk score. Our analysis shows that an increase in the number of previously confirmed UTI events was associated with a positive UTI prediction, agreeing with the literature^{##REF##25410372##33##}. We also highlighted the frequency of the lounge and hallway activations as negatively correlated with risk score, whilst the bedroom frequency was positively correlated. We postulate that this results from participants spending more time in bed due to interrupted sleep, or due to the effects of comorbidities. Third, increases in the standard deviation of the night time respiratory rate and the night time average heart rate were correlated with a higher risk of a UTI. Nocturnal respiratory rate has been linked to stress, reflects physiologic and pathophysiologic determinants, and has been suggested as a biomarker for impending hospitalisation^{##UREF##19##34##–##REF##6872603##38##}. Increased nocturnal awake occurrences were associated with a higher UTI risk, suggesting PLWD with UTIs were having more disturbed nights of sleep; in agreement with the literature^{##UREF##20##39##}. This could additionally explain why increased standard deviation of the night time respiratory rate and the night time average heart rate were correlated with a higher risk of a UTI. Considering the clinical manifestations of UTIs in older adults, our feature importance results agree with the current understanding of UTIs in PLWD.</p>", "<p id=\"Par15\">This study contains a few limitations that would also allow for future research directions. Whilst this work was conducted using readily available and low-cost sensors (with preliminary analysis of sensor importance presented in Supplementary Information Section ##SUPPL##1##15##), further directions of work could improve the understanding of the balance between the cost and complexity of deployment and UTI risk prediction performance. The deployed PIR sensors allow us to collect data at low cost, but they do not allow for the distinction between data generated by the person of study and other members of the house. Further work could explore methods of passively collecting personalised data. We found that the sleep mat (which does collect personalised data) significantly improved the analysis performance (Supplementary Information Section ##SUPPL##1##15##). In Supplementary Information Section ##SUPPL##1##11##, we discuss the model reliability and calibration and find that our model overestimates UTI risk, likely because of the data imbalance in the training set. This motivates the applied risk stratification which allows the monitoring team to balance the sensitivity and specificity with the number of generated alerts; however, when deploying this system, work should be done to understand the trade-off between false positives and false negatives to ensure that the risk groups are well-calibrated. Finally, whilst this work focused on an important section of the population (People Living with Dementia), it would be helpful to apply these techniques in a larger cohort study or one containing older adults in community living environments such as care homes, assisted living, or skilled nursing facilities.</p>", "<p id=\"Par16\">Our feasibility study was conducted within real-world in-home settings on data collected in (near) real-time using off-the-shelf and low-cost sensory technologies and engineered, clinically meaningful, features for predicting UTIs determined by clinicians, urine sample analysis, and a clinical monitoring team. We provide preliminary evidence for the use of such an operation and model which, with further testing, could prove to reduce delays in detecting UTIs in PLWD, and potentially reduce the number of avoidable hospital admissions when used to support clinicians with care. The proposed approach can be scaled rapidly and enable human-in-the-loop decision support by taking advantage of technological advancements, cloud computing, and machine learning. Moreover, risk stratification allows for model calibration to improve patient outcomes and care delivery whilst balancing the cost associated with testing for UTIs. Within an ongoing study or in production, the group thresholds can be modified over time to account for care team resources. SHAP analysis will enable the presentation of explainable results (such as in Supplementary Information Section ##SUPPL##1##13)##, allowing clinicians to explore why the UTI algorithm has made a given prediction. Additional future work will involve continuing to investigate our in-home monitoring systems’ effects on clinical outcomes, as well as patients’ quality of life.</p>", "<p id=\"Par17\">When deployed, our model will be continually trained on new data as collected. To ensure the performance consistently meets a minimum standard, we will routinely evaluate the model on a test set and track its performance. Feature importance will also be monitored to confirm the algorithm is producing clinically founded results. This will enable rapid debugging of errors and maintain a high level of quality in predictions.</p>" ]

[]

[ "<p id=\"Par1\">Urinary Tract Infections (UTIs) are one of the most prevalent bacterial infections in older adults and a significant contributor to unplanned hospital admissions in People Living with Dementia (PLWD), with early detection being crucial due to the predicament of reporting symptoms and limited help-seeking behaviour. The most common diagnostic tool is urine sample analysis, which can be time-consuming and is only employed where UTI clinical suspicion exists. In this method development and proof-of-concept study, participants living with dementia were monitored via low-cost devices in the home that passively measure activity, sleep, and nocturnal physiology. Using 27828 person-days of remote monitoring data (from 117 participants), we engineered features representing symptoms used for diagnosing a UTI. We then evaluate explainable machine learning techniques in passively calculating UTI risk and perform stratification on scores to support clinical translation and allow control over the balance between alert rate and sensitivity and specificity. The proposed UTI algorithm achieves a sensitivity of 65.3% (95% Confidence Interval (CI) = 64.3–66.2) and specificity of 70.9% (68.6–73.1) when predicting UTIs on unseen participants and after risk stratification, a sensitivity of 74.7% (67.9–81.5) and specificity of 87.9% (85.0–90.9). In addition, feature importance methods reveal that the largest contributions to the predictions were bathroom visit statistics, night-time respiratory rate, and the number of previous UTI events, aligning with the literature. Our machine learning method alerts clinicians of UTI risk in subjects, enabling earlier detection and enhanced screening when considering treatment.</p>", "<title>Subject terms</title>" ]

[ "<title>Supplementary information</title>", "<p>\n\n\n\n</p>" ]

[ "<title>Supplementary information</title>", "<p>The online version contains supplementary material available at 10.1038/s41746-023-00995-5.</p>", "<title>Acknowledgements</title>", "<p>This study is funded by the UK Dementia Research Institute (UKDRI) Care Research and Technology Centre funded by the Medical Research Council (MRC), Alzheimer’s Research UK, Alzheimer’s Society (grant number: UKDRI-7002), and the UKRI Engineering and Physical Sciences Research Council (EPSRC) PROTECT Project (grant number: EP/W031892/1). Infrastructure support for this research was provided by the NIHR Imperial Biomedical Research Centre (BRC) and the UKRI Medical Research Council (MRC). The funders were not involved in the study design, data collection, data analysis or writing the manuscript.</p>", "<title>Author contributions</title>", "<p>AC, FP: Conceptualisation, Methodology, Software, Formal analysis, Investigation, Data Processing, Writing—Original Draft, Review and Editing, Visualisation; KZ, NFL: Writing—Original Draft, Review and Editing; CW: Writing—Original Draft, Review and Editing, Data Collection; TC: Methodology, Writing—Review and Editing; SK: Methodology, Writing—Original Draft, Review and Editing; RJ, MT, MC, KJ, RV, MK, SD: Reviewing, Data Collection; PF: Reviewing, Data Collection, Funding Acquisition JT: Data Collection; DW: Methodology, Data Collection; RN: Clinical Study Lead, Conceptualisation, Data Collection, Writing—Review and Editing, Funding Acquisition; PB: Conceptualisation, Methodology, Writing—Original Draft, Review and Editing, Supervision, Funding Acquisition; FP and KZ contributed equally to this work.</p>", "<title>Data availability</title>", "<p>The data that support the findings of this study are available from the corresponding author upon reasonable request.</p>", "<title>Code availability</title>", "<p>The code used in this study will be made available by the corresponding author upon reasonable request.</p>", "<title>Competing interests</title>", "<p id=\"Par38\">The authors declare no competing interests.</p>" ]

[ "<fig id=\"Fig1\"><label>Fig. 1</label><caption><title>The dataset.</title><p><bold>a</bold> All PIR activations measured and the corresponding time of day. <bold>b</bold> The average proportions of PIR activations in a time period for days containing a verified positive and negative UTI label for a selected participant. The difference in Bathroom activity is of note, showing a significant increase in use for those days corresponding to a positive UTI. <bold>c</bold> All sleep state measurements within the dataset and the corresponding time of day. <bold>d</bold> The first of the graphs show the percentage of a given time window a single participant spent in bed and awake; the second and third shows the difference in heart rate and respiratory rate distributions for days labelled as positive and negative UTI for a single participant. The elements of the boxplots correspond to: center line, median; box limits, upper and lower quartiles; whiskers, 1.5x interquartile range; points, outliers. Here HR corresponds to heart rate and RR corresponds to respiratory rate.</p></caption></fig>", "<fig id=\"Fig2\"><label>Fig. 2</label><caption><title>Sensitivity and specificity with different thresholds.</title><p>The variations in sensitivity and specificity that can be achieved on the validation set by changing the thresholds for defining Green and Red groups. The line colours represent different threshold values. Sensitivity and specificity are calculated on the data in the validation set corresponding to Red and Green alerts. Here, to calculate the metrics on the Green group, the Red threshold was set at &gt;50%, and when calculating the metrics on the Red group, the Green threshold was set at ≤50%. This figure shows the average results from the “Date Split\" and “Date-ID Split\".</p></caption></fig>", "<fig id=\"Fig3\"><label>Fig. 3</label><caption><title>Performance at days prior to UTI label.</title><p>The performance of the model when tasked with analysing the risk of a UTI from the test set, at different numbers of days prior to the verified recorded date. The error bands represent the 95% confidence interval (1000 bootstrap samples) of the mean of the values from the 10 bootstrap repeats.</p></caption></fig>", "<fig id=\"Fig4\"><label>Fig. 4</label><caption><title>SHAP values.</title><p><bold>a</bold> The feature importance for the top 10 most important features, as calculated by SHAP on the test set, and their corresponding feature values. The colour represents the normalised feature value, whilst the position in the <italic>x</italic>-axis represents the contribution that value made to the prediction. “MA\" refers to the moving average, whilst “Delta\" refers to the percentage change in the value from the previous day. These values are calculated on the test set of the “Date-ID Split\". <bold>b</bold> The breakdown of a single prediction shows how each feature contributed to a correct prediction of a positive UTI. Here, the values on the arrows correspond to the normalised feature value in units of standard deviations away from the mean.</p></caption></fig>" ]

[ "<table-wrap id=\"Tab1\"><label>Table 1</label><caption><p>Mean (95% CI) % of sensitivity, specificity, and area under the precision-recall curve of the UTI prediction model on the different data splits with 10 bootstrap repeats.</p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th colspan=\"5\">Before risk Stratification</th></tr><tr><th/><th/><th>Sensitivity</th><th>Specificity</th><th>AUC Precision-Recall</th></tr></thead><tbody><tr><td>Date</td><td>Validation</td><td>67.3 (65.9–68.6)</td><td>69.1 (66.9–71.4)</td><td>67.7 (66.4–69.0)</td></tr><tr><td>Date</td><td>Test</td><td>54.5 (52.7–56.4)</td><td>73.0 (71.2–74.8)</td><td>54.4 (53.4–55.4)</td></tr><tr><td>Date-ID</td><td>Validation</td><td>87.7 (85.2–90.2)</td><td>66.0 (64.4–67.7)</td><td>78.3 (76.8–79.7)</td></tr><tr><td>Date-ID</td><td>Test</td><td>65.2 (64.3–66.2)</td><td>70.9 (68.6–73.1)</td><td>63.5 (61.8–65.2)</td></tr></tbody></table><table frame=\"hsides\" rules=\"groups\"><thead><tr><th colspan=\"5\">After Risk Stratification</th></tr><tr><th/><th/><th>Sensitivity</th><th>Specificity</th><th>Precision</th></tr></thead><tbody><tr><td>Date</td><td>Validation</td><td>86.6 (80.9–92.3)</td><td>94.5 (91.7–97.3)</td><td>87.3 (82.2–92.4)</td></tr><tr><td>Date</td><td>Test</td><td>69.0 (64.4–73.5)</td><td>94.1 (92.0–96.2)</td><td>81.9 (75.5–88.2)</td></tr><tr><td>Date-ID</td><td>Validation</td><td>98.3 (95.5–101.1)</td><td>90.0 (85.5–94.5)</td><td>81.7 (74.4–89.1)</td></tr><tr><td>Date-ID</td><td>Test</td><td>74.7 (67.9–81.5)</td><td>87.9 (85.0–90.9)</td><td>77.0 (71.9–82.1)</td></tr></tbody></table></table-wrap>", "<table-wrap id=\"Tab2\"><label>Table 2</label><caption><p>Characteristics of the study cohort.</p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th>Characteristic</th><th colspan=\"2\">Entire cohort</th><th colspan=\"2\">Labelled cohort</th></tr><tr><th/><th>No.</th><th>%</th><th>No.</th><th>%</th></tr></thead><tbody><tr><td><bold>Sex</bold></td><td/><td/><td/><td/></tr><tr><td>Female</td><td>54</td><td>46</td><td>27</td><td>42</td></tr><tr><td>Male</td><td>63</td><td>54</td><td>37</td><td>58</td></tr><tr><td><bold>Birth year</bold></td><td/><td/><td/><td/></tr><tr><td>1920-1930</td><td>13</td><td>11</td><td>5</td><td>8</td></tr><tr><td>1930-1940</td><td>47</td><td>40</td><td>27</td><td>42</td></tr><tr><td>1940-1950</td><td>41</td><td>35</td><td>26</td><td>41</td></tr><tr><td>1950-1960</td><td>13</td><td>11</td><td>5</td><td>8</td></tr><tr><td>1960-1970</td><td>2</td><td>2</td><td>1</td><td>2</td></tr><tr><td>1970-1980</td><td>1</td><td>1</td><td>0</td><td>0</td></tr><tr><td><bold>Ethnicity</bold></td><td/><td/><td/><td/></tr><tr><td>White</td><td>95</td><td>81</td><td>60</td><td>94</td></tr><tr><td>Asian</td><td>8</td><td>7</td><td>3</td><td>5</td></tr><tr><td>Black/African/Caribbean</td><td>3</td><td>3</td><td>0</td><td>0</td></tr><tr><td>Mixed/Multiple Groups</td><td>1</td><td>1</td><td>0</td><td>0</td></tr><tr><td>N/A</td><td>10</td><td>9</td><td>1</td><td>2</td></tr><tr><td><bold>Household</bold></td><td/><td/><td/><td/></tr><tr><td>Lives Alone</td><td>45</td><td>38</td><td>16</td><td>25</td></tr><tr><td>Lives with Partner</td><td>60</td><td>47</td><td/><td>73</td></tr><tr><td>N/A</td><td>12</td><td>10</td><td>1</td><td>2</td></tr><tr><td><bold>Primary diagnosis</bold></td><td/><td/><td/><td/></tr><tr><td>Alzheimer’s</td><td>61</td><td>52</td><td>39</td><td>61</td></tr><tr><td>Vascular Dementia</td><td>10</td><td>9</td><td>5</td><td>8</td></tr><tr><td>Parkinson’s</td><td>5</td><td>4</td><td>2</td><td>3</td></tr><tr><td>Other and Mixed</td><td>40</td><td>34</td><td>18</td><td>28</td></tr><tr><td>N/A</td><td>1</td><td>1</td><td>0</td><td>0</td></tr></tbody></table></table-wrap>" ]

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[ "<supplementary-material content-type=\"local-data\" id=\"MOESM1\"></supplementary-material>", "<supplementary-material content-type=\"local-data\" id=\"MOESM2\"></supplementary-material>", "<supplementary-material content-type=\"local-data\" id=\"MOESM3\"></supplementary-material>" ]

[ "<table-wrap-foot><p>These results are reported before and after risk stratification is performed.</p></table-wrap-foot>", "<table-wrap-foot><p>Some participants requested not to share their information outside the study and correspond to the Not Available information.</p></table-wrap-foot>", "<fn-group><fn><p><bold>Publisher’s note</bold> Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.</p></fn><fn><p>These authors contributed equally: Francesca Palermo, Kimberley Zakka.</p></fn><fn><p>A list of authors and their affiliations appears at the end of the paper.</p></fn><fn><p>A full list of members and their affiliations appears in the Supplementary Information.</p></fn></fn-group>" ]

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[ "<media xlink:href=\"41746_2023_995_MOESM1_ESM.pdf\"><caption><p>CR&amp;T Group Members</p></caption></media>", "<media xlink:href=\"41746_2023_995_MOESM2_ESM.pdf\"><caption><p>Supplementary Materials</p></caption></media>", "<media xlink:href=\"41746_2023_995_MOESM3_ESM.pdf\"><caption><p>Reporting Summary</p></caption></media>" ]

{ "acronym": [], "definition": [] }

[ "<title>Introduction</title>", "<p id=\"Par2\">In the current era, ionizing radiation has been employed worldwide in many fields, such as medical diagnostics and treatments^{##UREF##0##1##}, nuclear facilities, agricultural and industrial applications, mining areas, and research^{##REF##35683923##2##}. In addition to these well-known fields, ionizing radiation is also used to screen people for non-medical reasons at border installations and military checkpoints, specifically to find bulk bombs or other illegal items concealed on the body, such as substances not picked up by metal detectors. Advanced imaging technology, such as accelerator-based container scanners and baggage inspection scanners with small, medium, and high energy, have been used in some countries for customs inspections against any threats that affect the countries’ border security. Some X-ray inspection systems, such as the Orion® 928DX inspection system, use X-rays with energies of nearly 168 keV in airports^{##UREF##1##3##}. Furthermore, the Eagle® G60 ZBx gantry inspection system uses X-rays with dual-energy 3 and 6 MeV in ports^{##REF##32390647##4##}. The introduction of X-ray inspection systems at the airline’s checkpoints and border control was met with social controversy in part connected to exposure dose to travelling people, operators and bystanders. Ionizing radiations employed in such systems may provide health risks if improperly handled or the equipment’s safety features are deficient. Well-known negative consequences of exposure to ionizing radiation on bone health include a reduction in mineral density, bone growth retardation in children, and spontaneous fractures in older women^{##UREF##2##5##}, and it primarily delivers harmful effects to damage DNA^{##REF##26406369##6##}.</p>", "<p id=\"Par3\">To reduce the drawbacks of ionizing radiation, appropriate safety precautions must be implemented to balance their potential benefits against their drawbacks. International radiation groups have proposed specific guidelines to lessen and reduce the risks of radiation exposure to human organs. ALARA (As Low As Reasonably Achievable) is the most popular radiation protection rule that is based on reducing radiation doses by minimizing the time of radiation exposure, putting as much distance as possible between the user and the source of radiation, and the usage of radiation shields must be applied^{##REF##35100213##7##}. Lead and concrete are the traditional and common shielding materials utilized as protective materials in radiation facilities. Concrete is an excellent shielding material that has attracted significant interest because of its affordability, environmental friendliness, optimum density for radiation attenuation, easily shaped, requires little upkeep, and strong mechanical properties^{##UREF##3##8##}. It is also widely used as an effective radiation shield for the X-ray inspection systems such as the EagleP60® ZBx and the Eagle M60® ZBx inspection systems located at the ports and border points of the country^{##UREF##4##9##}. However, due to flaws like cracking and immobility, it is useless for other applications. Lead is the most commonly utilized radiation shielding material due to its high density, low cost, and superior shielding performance^{##UREF##5##10##}. Even while lead seems like the perfect material for a shield, its toxicity^{##REF##27486361##11##} toward people and the environment makes it less valuable. The need for lead alternatives has increased recently, particularly in the medical industry^{##REF##32942196##12##}.</p>", "<p id=\"Par4\">Numerous materials’ characteristics have been created and enhanced for use as radiation shielding to overcome these drawbacks. The chemical stability, flexibility, lightweight, and low cost of polymer composites with inorganic fillers like micro and nanoparticles have led to extensive research into these materials as potential substitutes for conventional radiation shielding materials^{##REF##33192208##13##}. There are many polymers, including PMMA^{##UREF##6##14##}, polyethylene^{##UREF##7##15##}, polypropylene^{##UREF##8##16##}, polyvinyl chloride^{##UREF##9##17##}, epoxy^{##UREF##10##18##}, styrene-butadiene rubber^{##UREF##11##19##}, natural rubber^{##UREF##12##20##}, silicon rubber^{##REF##36604568##21##}, ethylene-propylene-dine monomer (EPDM)^{##REF##37173378##22##}, polystyrene^{##UREF##13##23##}, and recycled polymers^{##UREF##14##24##,##UREF##15##25##}, were studied as radiation-protective matrixes. Nagaraja et al. investigated the performance of radiation shielding for different types of polymers that are commonly used within the energy range 81–1332 keV. Among all the tested polymers, the lead tetragonal polymer is the most effective γ-rays absorber^{##UREF##16##26##}. The polymer matrix has been filled with metal oxides like PbO^{##REF##36127499##27##}, CdO^{##UREF##17##28##}, Bi₂O₃^{##UREF##18##29##}, ZnO^{##UREF##19##30##}, Gd₂O₃^{##UREF##20##31##}, and MgO^{##UREF##21##32##} to develop a radiation shield that can be used to attenuate X-rays and γ-rays.</p>", "<p id=\"Par5\">Metal oxide fillers reinforced in polymer composites at the nanoscale can significantly improve their mechanical, electrical, and optical properties. Additionally, their small size makes these materials extremely effective at attenuating radiation. For example, El-Khatib et al. compared the radiation-shielding abilities of different loadings of micro and nano CdO distributed HDPE matrix at photon energy ranging from 59.53 to 1408.01 keV and reported that nano-CdO/HDPE composites shielded γ-rays more effectively than micro-CdO/HDPE composites at the same weight fraction^{##REF##31690761##33##}. Another research investigated the radiation shielding ability of epoxy-based micro and nano WO₃ and Bi₂O₃ reinforced composites^{##UREF##22##34##}. The study demonstrated that nano dopant is more successful and effective in attenuating the photons. Abbas et al. studied the effect of Bi₂O₃ micro- and nano-particles content on silicon rubber’s (SR) γ-ray interaction parameters^{##REF##35267871##35##}. The attenuation coefficients of the obtained SR samples showed a clear advantage in lower energy levels compared to other energies. Furthermore, the SR’s nano-Bi₂O₃ was superior to the SR’s micro-Bi₂O₃. Additionally, the mechanical results revealed that the material’s flexibility decreased as the Bi₂O₃ filler was increased to 30%.</p>", "<p id=\"Par6\">Poly-methyl methacrylate (PMMA) is a significant kind of polymer among thermoplastics. PMMA is an optically transparent polymer with a refractive index of 1.49 and a density of 1.20 and is frequently used as an alternative to inorganic glass^{##UREF##23##36##,##REF##24873535##37##}. PMMA is an amorphous polymer that resists corrosion, abrasion, weather, and chemicals, as well as its ideal production conditions are lightweight and resistant to breaking. Numerous products, including coatings, additives, sealants, optical fibers, and neutron stoppers, have been made with PMMA. By incorporating filler into the PMMA matrix, this versatile material’s applications may be even more varied because well-dispersed filler could improve some of its physical characteristics^{##REF##33807421##38##}. Chen et al. evaluated the shielding properties of different samples, including pure PMMA, PMMA/MWCNT, and PMMA/MWCNT/Bi₂O₃, compared with aluminum (Al). According to the electron-beam attenuation properties, the PMMA/MWCNT/Bi₂O₃ nanocomposite was 37% lighter than Al while still providing the same level of radiation protection in the 9–20 MeV electron energy range^{##UREF##24##39##}. Cao et al. investigated the performance of γ-rays shielding and the physical and mechanical characteristics of PMMA composites doped with 0–44.0 wt% Bi₂O₃ prepared by the fast-curing technique. The results showed that for radiation energies up to 1000 keV, PMMA/Bi₂O₃ composites showed superior γ-rays shielding performance compared to pure PMMA. Additionally, the hardness measurement shows that mechanical hardness rises with increasing loading of Bi₂O₃^{##UREF##25##40##}. Another research reported the use of PMMA/Bi₂O₃ polymer composites as a replacement for concrete and gypsum in the construction of diagnostics radiation facilities^{##UREF##26##41##}. Furthermore, lightweight, environment-friendly, and cost-effective materials based on flexible Bi-PMMA composites are investigated as radiation shielding materials suitable for low energy γ-rays^{##UREF##27##42##}. Recently, PMMA polymer with different concentrations (0, 2, 5, 10, 15, and 20 wt%) of BaTiO₃ as a nanofiller was examined to be used as nuclear radiation shielding materials. The developed non-toxic, flexible, and transparent nanocomposite protective material is presented and the specimens containing 10–15 wt%, showed enhanced radiation attenuation^{##UREF##28##43##}.</p>", "<p id=\"Par7\">Zirconium oxide (ZrO₂), also known as Zirconia, is a material with significant technological importance due to its outstanding corrosion resistance, high strength, high chemical stability, chemical and microbiological resistance, and high mass attenuation coefficient^{##UREF##29##44##}. Furthermore, due to its excellent thermal stability and low neutron absorption cross-section, Zirconia is applied in nuclear reactor technology. Wahab et al. evaluated the effect of zirconia nanoparticles on the radiation shielding performance of the lead borate glass^{##UREF##30##45##}. Regarding the considerable collection of research indicated above, incorporating nanofillers into polymers is a promising method to develop novel radiation protective materials. There is also a strong need for more research to study how the filler size affects the shielding characteristics against γ-rays for various composite systems.</p>", "<p id=\"Par8\">The survey of the literature reveals that there are very limited studies that deal with the use of nano ZrO₂ as a filler in the polymeric matrix to attenuate γ-rays. Hence, the primary goal of this work is to investigate how the particle size and weight percentage of ZrO₂ affect the ability of ZrO₂/PMMA composites to shield against γ-rays. For this purpose, the mass attenuation coefficients of pure PMMA and PMMA loaded with 15, 30, and 45 wt% of micro and nano ZrO₂ were estimated experimentally and compared to results obtained theoretically from XCOM database. Additionally, other shielding parameters such as the linear attenuation coefficients (μ), half-value layer (HVL), tenth value layer (TVL), mean free pass (MFP), effective atomic number (Z_eff), effective electron density (N_eff), and equivalent atomic number (Z_eq), as well as exposure buildup factor (EBF) and energy absorption buildup factor (EABF) were calculated at various energies between 0.015 and 15 MeV to assess the γ-rays shielding ability of the prepared ZrO₂/PMMA composites.</p>" ]

[ "<title>Materials and methods</title>", "<title>Materials</title>", "<p id=\"Par9\">Self-cured acrylic resin (Acrostone Cold Cure Acrylic Resin), a commercial product with a density of 1.18 g/cm³, comes in two bottles containing powder (Poly methyl methacrylate, prepolymer (–(–CH2¼C(CH3) COOCH3-)n-PMMA) and (MMA) monomer liquid hardener (CH2¼C (CH3) COOCH3, MMA), were the matrix materials employed in this work and provided by the Acrostone Dental &amp; Medical Supplies Company in Cairo, Egypt. The physical properties of MMA liquid hardener and PMMA powder are summarized in Tables ##TAB##0##1## and ##TAB##1##2##, respectively. Zirconium oxide micro- and nanoparticles were employed as fillers and obtained from Nanoshel business Wilmington, DE 19808, USA. Micro Zirconium oxide particles were provided with a purity of 99.9% and an average size of about 1–2 µm. In comparison, the Zirconium oxide nanoparticles were supplied with a purity of 99.9% and an average size of about 80 nm. The physical properties of Zirconium oxide microparticles (MPs) and nanoparticles (NPs) are listed in Table ##TAB##2##3##.</p>", "<title>Cold (self)-cured PMMA</title>", "<p id=\"Par10\">In comparison to heat-cured PMMA, cold-cured PMMA possesses a different composition and polymerization technique, making it unnecessary to apply thermal energy. It is also known as chemically cured or auto-polymerized PMMA, indicating that the polymerization process initiates immediately after the powder and liquid components are combined. Consequently, no heat is required for the polymerization reaction to occur since the benzoyl peroxide initiator present in the pre-polymerized PMMA pellets can be chemically activated. The advantages of cold-cured PMMA over heat-cured PMMA are its superior adaptability and dimensional stability, resulting in reduced polymerization shrinkage^{##UREF##32##47##}.</p>", "<title>Preparation of ZrO₂/PMMA composites</title>", "<p id=\"Par11\">This study utilized the self-curing method to fabricate pure PMMA, micro- ZrO₂/PMMA, and nano- ZrO₂/PMMA composites. Table ##TAB##3##4## lists the sample codes, compositions, and densities of the produced composites. Three main groups of acrylic-resin specimens were fabricated; (a) the reference group (P-0Z) was prepared by blending self-cured PMMA powder and liquid MMA in a 3:1 by volume ratio as recommended by the manufacturer, (b) the modified micro ZrO₂/PMMA group (P-15mZ, P-30mZ, P-45mZ) which was fabricated at constant micro filler loadings of about 15, 30 and 45 wt%, and (C) the modified nano-ZrO₂/PMMA group (P-15nZ, P-30nZ, P-45nZ) which was fabricated with the same loadings of nanofiller.</p>", "<p id=\"Par12\">Before preparing the samples, each component was pre-weighed according to the weight fractions listed in Table ##TAB##3##4##, using a 0.0001 g sensitive electrical balance (Analytical Balance, GR200, Japan). The reference group (P-0Z) was prepared by mixing dry powder (PMMA) with liquid (MMA) in a clean and dry glass beaker and stirred continuously at room temperature for a maximum of 4.5 min (according to the manufacturer) to eliminate any gas bubbles from the specimens. The mixing process was performed using a mechanical mixer set to speed 20 rpm until the mixture reached the dough stage. Then, the mixture was poured into the center of an opening silicone rubber mold, shown in Fig. ##FIG##0##1##, until the mold was filled. The mold must be slowly shaken and pulsed from side to side. The mold was then kept standing on the workbench at room temperature for 20 min after the mixing process had started to allow the mixture to thicken and harden the surface of the casting. The modified groups were prepared by mixing ZrO₂ nanoparticles or ZrO₂ microparticles with the PMMA powder in a glass beaker and mixed with an electric mixer for 20 min to create a homogenous mixture. Then, as previously mentioned, the blended powder was added as one unit to the liquid monomer in a ratio of 3:1 by volume. When the mixed acrylic resin reached the dough stage, it was packed in a silicon rubber mold for 2 h at room temperature (25 ± 2 °C) to get the specimen’s final shape, as shown in Fig. ##FIG##1##2##.</p>", "<title>Morphology study and structural characterization</title>", "<p id=\"Par13\">The particle size of micro and nano ZrO₂ powder was analyzed using a transmission electron microscope (TEM) (JEM 1400 Plus, JEOL, Japan) at 200 kV. Furthermore, the cross-section morphologies of the prepared samples were examined using a scanning electron microscope (SEM) (JSM-6010LV, JEOL). To enhance the SEM image quality, a thin layer of gold (20 nm) was coated on the prepared samples using a low-vacuum sputtered coating system (JEOL-JFC-1100E). SEM images were acquired to compare the dispersion of ZrO₂ micro and nanoparticles within the PMMA matrix. TEM and SEM analyses were performed to study how the ZrO₂ particle size and its distribution affected the shielding properties of the investigated ZrO₂/PMMA composites. The present specimens were also analyzed using a Bruker Vertex 70 infrared spectrometer in the 4000–400 range, together with a Platinum ATR unit, Germany-Ray diffractometer (Schimadzu-7000), to collect FT-IR data and examine photon absorption and transmission in the IR region. FT-IR analysis was conducted to identify the functional groups in the ZrO₂/PMMA composites and the interaction mechanism of ZrO₂ particles and PMMA polymeric matrix.</p>", "<title>γ-ray spectroscopic setup</title>", "<p id=\"Par14\">The cylindrical high purity germanium detector (Model GC1520 from Canberra, United States) was employed in conjunction with a multichannel analyzer to conduct γ-ray spectroscopic measurements. The detector boasts a relative efficiency of 15% within the 50 keV to 10 MeV range, with a resolution of 1.85 keV at the 1.33 MeV γ-ray peak using Co-60^{##UREF##33##48##}. The detector was encased with a 15 cm thick lead shield to mitigate background radiation. The measurements were calibrated using standardized sources, including Am-241, Ba-133, Cs-137, Co-60, and Eu-152, in the energy range between 0.05953 and 1.4081 MeV. Table ##TAB##4##5## outlines the emitted energies and activities attributed to these sources. The experimental arrangement for the γ-ray measuring system is illustrated in Fig. ##FIG##2##3##.</p>", "<p id=\"Par15\">The γ-ray spectrum for each measurement was obtained based on the sample thickness, such that the statistical error was less than 1%. For example, Fig. ##FIG##3##4## depicts the obtained spectra using a Co-60 radioactive point source in the absence and presence of the P-0Z sample. The electrical signal generated by the detector was amplified and analyzed using Canberra’s Genie 2000 data acquisition and analysis software ISO 9001. The net area beneath the photo peak was calculated and divided by the acquisition time to determine the count rate. The counting rate was determined in the presence (I) and absence (I₀) of the specimen, respectively. Beer–Lambert’s law (Eq. ##FORMU##1##1##) was then utilized to determine the linear attenuation coefficient (cm⁻¹) of each sample at various γ-ray energies.</p>", "<title>Attenuation parameters calculations</title>", "<p id=\"Par16\">The linear attenuation coefficient (μ), a significant shielding parameter for determining the impact of γ-rays at proper energy on the materials under study, can be calculated from Beer–Lambert’s law as indicated in Eq. (##FORMU##1##1##)^{##UREF##34##49##}:where the initial (I₀) and transmitted (I intensities across the sample with a thickness (t) can be experimentally calculated as discussed in “<xref rid=\"Sec7\" ref-type=\"sec\">γ-ray spectroscopic setup</xref>” section. The MAC can be derived using Eq. (##FORMU##3##2##):where ρ_s is the density of the sample measured using Archimedes method^{##UREF##35##50##}:where ρ_s refers to the density of the samples, is the density of water at the test temperature, is the mass of the dry sample, and m₂ is the mass of the immersion sample.</p>", "<p id=\"Par17\">In developing an appropriate radiation shielding substance, two factors must be considered: the HVL and TVL. HVL and TVL are the material thicknesses required to reduce the γ-ray intensity to 50% and 10% of its original value, respectively. Equations (##FORMU##7##4##) and (##FORMU##8##5##) are typically used to estimate these values^{##UREF##36##51##}:</p>", "<p id=\"Par18\">The MFP, which is defined as the average distance moved by a photon between two subsequent reactions, this parameter was given in Eq. (##FORMU##9##6##)^{##REF##36837205##52##}:</p>", "<p id=\"Par19\">To determine a material’s radiation shielding capabilities, it is necessary to calculate its Z_eff and N_eff parameters. These values are obtained from the atomic cross section (σ_a) and electron cross section (σ_e) of the material. The σ_a and σ_e values directly relate to the number of atoms and electrons present in a unit volume of the material. Materials with higher σ_a and σ_e values are more effective as shielding materials. σ_a is calculated using Eq. (##FORMU##10##7##)^{##UREF##37##53##} and provides the probability of interaction per atom within the material’s unit volume.where N is Avogadro’s number, A_i and f_i are the atomic weight and fractional weight for each target element, respectively. σ_e provides the probability of interaction per electron in the specimen’s unit volume and is expressed by Eq. (##FORMU##11##8##)^{##UREF##38##54##}:where Z_i is the atomic number and f_i is the target element’s fractional abundance. Along with the use of σ_a and, σ_e. The effective atomic number is derived by applying Eq. (##FORMU##12##9##)^{##UREF##39##55##}:</p>", "<p id=\"Par20\">The N_eff values are referred to the electron numbers per unit mass of the interacting target and is given by Eq. (##FORMU##13##10##)^{##UREF##40##56##}:</p>", "<p id=\"Par21\">When constructing a shielding material, the two forms of buildup factor, the EABF and the EBF are crucial parameters that should be taken into account. Due to secondary γ-rays emissions^{##UREF##41##57##}, buildup factors are always more than one and correct the attenuation estimations in Beer–Lambert’s law. The three stages below were followed using the Geometric Progression (GP) fitting technique to compute the EABF and EBF for the produced ZrO₂/PMMA composites:<list list-type=\"simple\"><list-item><label>(i)</label><p id=\"Par22\">the composite’s Z_eq values, which is identical to the elemental atomic number, was initially calculated with the following Eq. (##FORMU##14##11##)^{##UREF##42##58##}: where R₁ and R₂ represent the (μ_Comp/μ_total) ratios for the elements with atomic numbers Z₁ and Z₂, and R represents the (μ_Comp/μ_total) ratio for the composite under investigation at a particular energy that falls between ratios R₁ and R₂.</p></list-item><list-item><label>(ii)</label><p id=\"Par23\">After getting the Z_eq values of the specified composites were then employed to estimate the GP fitting EABF and EBF coefficients [b, c, a, X_k, and d] in the range of energies (0.015–15 MeV) using the following formula^{##UREF##43##59##}:</p><p id=\"Par24\">C₁ and C₂ are the GP fitting parameters obtained from ANSI/ANS-6.4.3 standard data^{##UREF##44##60##}, equivalent to the atomic numbers Z₁ and Z₂ at which Z_eq of the produced ZrO₂/PMMA composites are located.</p></list-item><list-item><label>(iii)</label><p id=\"Par25\">Eventually, the resulting GP fitting parameters were used to compute the EABF and EBF using the following relationships^{##UREF##45##61##}:</p><p id=\"Par26\">and</p><p id=\"Par27\">where where <italic>x</italic> represents the penetration depth in terms of MFP and E represents the energy of the incident photon.</p></list-item></list></p>" ]

[ "<title>Result and discussion</title>", "<title>Characterization</title>", "<title>Transmission electron microscope (TEM) analysis</title>", "<p id=\"Par28\">The TEM micrographs of ZrO₂ MPs and NPs are depicted in Fig. ##FIG##4##5##. Figure ##FIG##4##5##a shows that ZrO₂ MPs are nearly spherical in shape and have an average particle size between 1.46 and 1.75 µm. On the other hand, Fig. ##FIG##4##5##b demonstrates the existence of ZrO₂ NPs, which have a consistent size distribution between 7.86 and 12 nm.</p>", "<title>Scanning electron microscope (SEM) analysis</title>", "<p id=\"Par29\">Figure ##FIG##5##6## displays the SEM images of P-0Z, P-15mZ, P-15nZ, P-45mZ, and P-45nZ composites. Figure ##FIG##5##6## describes how ZrO₂ particles impacted PMMA on a micro and nanoscale. Figure ##FIG##5##6##a depicts smooth and distinct variation compared to ZrO₂/PMMA composites. In other words, pure PMMA (Fig. ##FIG##5##6##a) and ZrO₂/PMMA composites (Fig. ##FIG##5##6##b–e) exhibit a distinct difference in morphology. The SEM images of micro and nano ZrO₂/PMMA composite samples with identical filler wt% are compared as indicated in Fig. ##FIG##5##6##b–d. It is clear that ZrO₂ Nps are uniformly scattered and thoroughly incorporated into the PMMA matrix in ZrO₂/PMMA composites, which may strengthen the interfacial adhesion between the PMMA matrix and ZrO₂ NPs and offer an interconnecting structure for shielding. While large ZrO₂ MPs are not fully covered by the PMMA matrix in micro ZrO₂/PMMA composites, and some of them peel off from the matrix due to insufficient interfacial adhesion, which behaves as voids for shielding. Additionally, it was evident that the porosity reduces and the distribution rises as the fraction of particles increases. This attribute proves that increasing the ZrO₂ content in PMMA will enhance structural, mechanical and shielding properties.</p>", "<title>Fourier transform-infrared (FT-IR)</title>", "<p id=\"Par30\">FT-IR spectroscopic analysis was performed on the produced samples using Bruker Vertex 70 infrared spectrometer to identify the functional groups in the ZrO₂/PMMA composites and the interaction mechanism of ZrO₂ particles and PMMA polymeric matrix. The FT-IR spectra of P-0Z, ZrO₂ MPs, P-15mZ, P-45mZ, ZrO₂ NPs, P-15nZ, and P-45nZ samples were collected in the wavelength region of the 400–4000 cm⁻¹ and displayed in Fig. ##FIG##6##7##. A discrete absorption band from 1142.19 to 1239 cm⁻¹ can be seen in the FT-IR spectrum P-0Z as depicted in Fig. ##FIG##6##7##a, which related to the C–O–C stretching vibration. The vibrations of the –methyl group can be identified to the pair of bands at 1386 cm⁻¹, and 750 cm⁻¹. The band at 980 cm⁻¹ is the characteristic absorption vibrations of PMMA, jointly with the bands at 1068 cm⁻¹ and 839 cm⁻¹. Due to the existence of ester carbonyl group stretching vibration (Acrylate carboxyl group), a sharp intensity peak at 1725.21 cm⁻¹ was conducted. The band at 1442.32 cm⁻¹ can be attributed to the bending vibration of the C–H bonds of the –CH₃ group. The band at 1442.32 cm⁻¹ can be allocated to the bending vibration of the C–H bonds of the –CH₃ group. The two bands at 2925.05 cm⁻¹ and 2854.34 cm⁻¹ can be linked to the C–H bond stretching vibrations of the –CH₃ and –CH₂– groups, respectively. In addition, two faint absorption bands at 3734 cm⁻¹ and 1644 cm⁻¹ result from the stretching and bending vibrations of the –OH group, respectively^{##REF##21787437##62##}.</p>", "<p id=\"Par31\">A broad peak can be seen in the FT-IR spectrum of ZrO₂ MPs in Fig. ##FIG##6##7##b at 3428.96 cm⁻¹, 1634.05 cm⁻¹, and (456.10, 604 cm⁻¹), which correspond to OH stretching, OH bending, and the Zr–O band, respectively. Peak locations, forms, and intensities have been determined in accordance with the fingerprint features, together with the material’s fundamental components^{##REF##28666841##63##}. As seen in Fig. ##FIG##6##7##c, the spectrum of the P-15mZ composite, compared to that of P-0Z, had new peaks at 453.94 cm⁻¹ and 599.12 cm⁻¹ referring to the metal–oxygen bond in ZrO₂. Furthermore, it is clear that when the weight fraction of ZrO₂ MPs was increased to 45 wt%, two peaks, as shown in Fig. ##FIG##6##7##d, were seen at 631 cm⁻¹ and 698 cm⁻¹, which are associated with stretching vibration modes of Zr–O and bending vibration of =C–H^{##UREF##46##64##}. The apparent presence of these peaks indicates that ZrO₂ MPs have been successfully embedded in the PMMA polymeric matrix.</p>", "<p id=\"Par32\">Figure ##FIG##6##7##e displays the FT-IR spectrum of ZrO₂ NPs. The peaks at 453.92 cm⁻¹ and 599.12 cm⁻¹ are related to the strong metal–oxygen conjunction in ZrO₂ NPs, and two peaks at 631 cm⁻¹ and 698 cm⁻¹ are attributed to the Zr–O bond and bending vibration of =C–H as previously mentioned. At 574 cm⁻¹ and 708 cm⁻¹, two additional bands have been observed that are connected to the ZrO₂ NPs. It is also clear from Fig. ##FIG##6##7##f,g that the spectra of P-15nZ and P-45nZ composites exhibit behavior that is comparable to that previously reported, in conjunction with the existence of two more peaks at 551 cm⁻¹ and 598.04 cm⁻¹ that were formed at greater concentrations of ZrO₂ NPs. Consequently, it can be deduced from Fig. ##FIG##6##7## that there was a chemical bond between the PMMA matrix and the ZrO₂ filler in all composites, which resulted in chemical interactions between them.</p>", "<title>γ-ray shielding results</title>", "<p id=\"Par33\">The MAC is a standard parameter utilized for measuring and comparing the performance of various shielding materials. Table ##TAB##5##6## displays the experimentally measured values of MACs for all the investigated composites (pure PMMA, micro ZrO₂/PMMA, and nano ZrO₂/PMMA composites) in the energy range between 0.05953 and 1.4081 MeV. Furthermore, using the XCOM software, MAC for the PMMA and micro ZrO₂/PMMA composites were generated theoretically, and the relative deviation (Δ%) between the experimental and theoretical results that were determined by using Eq. (##FORMU##19##16##):</p>", "<p id=\"Par34\">As can be seen from Table ##TAB##5##6##, both the MAC values obtained from XCOM and those achieved by laboratory measurement exhibit good comparability. This remark is accurate for all of the energies tested. However, some minor discrepancies were discovered between the two methods. These are acceptable because ordinarily, anyone can find a few minor errors in the experimental results, but generally, the experimental results are acceptable and agree with the XCOM results. This is a crucial and significant step since it clarifies the precision of the geometry used in the lab to calculate the MAC for the PMMA and ZrO₂/PMMA composites. According to Table ##TAB##5##6##, the Δ% for pure PMMA (free of ZrO₂ filler) is restricted to − 0.65 and 0.94%, whereas the Δ% for 15 wt% ZrO₂/PMMA is restricted to − 0.40 and 0.84%, and for 30 wt% ZrO₂/PMMA ranges between − 0.94 and 0.61%. However, it can only be between − 0.89 and 0.81% for 45 wt% ZrO₂/PMMA. These results support that the practical and theoretical results are compatible since the Δ% is less than 2%.</p>", "<p id=\"Par35\">The experimental results of μ values of pure PMMA, micro, and nano ZrO₂/PMMA composites filled with various concentrations (15 wt%, 30 wt%, and 45 wt%) as a function of γ-ray energies are shown in Fig. ##FIG##7##8##. As can be seen from Fig. ##FIG##7##8##, the energy of the incident photons and the compositions of the protective material have a significant impact on μ values. The μ values have been found to significantly increase with increasing micro- and nano-ZrO₂ concentrations in the composites and rapidly decline as photon energy increases. This trend may be demonstrated by focusing on the three primary processes by which energetic photons interact with matter; photoelectric effect, Compton scattering, and pair production, which all contribute to the energy loss of the incident photon. At energies less than 125 keV, the photoelectric effect is the primary mechanism that causes photons to be absorbed since the possibility of photoelectric absorption depends on Z³, where Z is the atomic number^{##UREF##47##65##}. Therefore, due to element Zr with atomic number (Z = 40), μ values increase as the concentration of ZrO₂ in the PMMA matrix increases. However, as photon energy increases beyond 125 keV, the likelihood of the photoelectric effect reduces roughly according to 1/E³^{##UREF##48##66##}, where E is the energy of the incident photons, which illustrates why μ for every composite decreases slightly as the photon energy goes above 125 keV. Meanwhile, in this energy range increasing the ZrO₂ filler wt% in the PMMA matrix is slightly affect the values of μ which have nearly the same value as the photon energy increase. This result is because, at this intermediate energy range, the effect of photoelectric absorption diminishes, and the Compton scattering mechanism takes over. Besides, the cross-section of the Compton scattering is practically independent of atomic number but depends on the number of electrons per unit mass.</p>", "<p id=\"Par36\">Additionally, the difference of μ between micro- and nano-sized ZrO₂/PMMA was also compared, as shown in Fig. ##FIG##7##8##. At all of the tested γ-ray energies, the nano ZrO₂/PMMA curves always are above the micro ZrO₂/PMMA curves for the same weight percent of ZrO₂ filler. As the size of ZrO₂ particles decreases from micro to nano size, the uniform distribution of ZrO₂ NPs over a greater surface area within the PMMA matrix would increase the likelihood of incident photons interacting with ZrO₂ NPs in nanocomposites as compared to micro composites and increase the probability of further scattering mechanisms for the photons until the photon’s energy is below 200 keV. Consequently, in PMMA-based radiation protective material, ZrO₂ NPs exhibit excellent attenuation performance than ZrO₂ MPs for the same chemical composition and weight percentage of the composite.</p>", "<p id=\"Par37\">To evaluate the superiority of the shielding ability of nanocomposites over micro composites, the relative increase rate (δ%) in μ values between nano and micro ZrO₂/PMMA composites was calculated according to Eq. (##FORMU##20##17##) and depicted in Fig. ##FIG##8##9## as a function of energy at various ZrO₂ loadings.</p>", "<p id=\"Par38\">As shown in Fig. ##FIG##8##9##, the relative increase rate (δ%) increases with an increase in ZrO₂ content; however, its value decreases as the photon’s energy increases from 0.05953 to 1.408 MeV. These findings suggest that, due to various photon interaction cross-sections at different photon energies, the size effect diminishes as photon energy increases. The absorption capability is Z-dependent when the photoelectric effect dominates at low photon energies. Due to the mid-high Z of the element zirconium in ZrO₂ particles and the low Z of the elements C, O, and H in the PMMA matrix, the photoelectric absorption of ZrO₂ particles is considerably higher than that of the PMMA matrix. As a result, these particles are extremely important for radiation shielding. At the energy of 0.05953 MeV, the δ% in the P-15nZ sample was 16.88%, compared to 17.29% in the P-30nZ sample and 17.84% in the P-45nZ sample. Since Compton scattering is more likely at higher energies and its cross-section can be regarded as the predominant interaction that does not rely on Z but rather on the free electrons, there is little difference in the ability of ZrO₂ particles compared to the PMMA matrix. As a result, the essential role of ZrO₂ particles diminishes, and the impact of particle size decreases. At the highest examined energy (1.408 MeV), the δ % was 6.67% for P-15nZ, whereas it was 7.23%and 8.60% for P-30nZ and P-45nZ samples, respectively. In summary, the δ % follows the general trend P-45nZ &gt; P-30nZ &gt; P-15nZ for all incident energies. So, the composite P-45nZ has superior shielding potentials over all the investigated samples.</p>", "<p id=\"Par39\">Three primary critical radiation shielding parameters, the HVL, the TVL, and the MFP have been researched in correlation to the radiation shielding capabilities of micro- and nano-structured composites^{##UREF##49##67##}. Figure ##FIG##9##10## displays the HVL, TVL, and MFP at different energies ranging from 0.0595 MeV to 1.408 MeV. The HVL is the thinnest sample at which 50% of the original γ-ray intensity passes through it. The findings of our calculation of the HVL for the chosen composites at the energies employed for the μ data are displayed in Fig. ##FIG##9##10##a. When analyzing the data in this Fig. ##FIG##9##10##a. One can see a gradual increase in HVL as energy is increased from 0.0595 to 1.408 MeV. This tendency indicates that the photons’ ability to penetrate samples rises along with their energy. The lowest HVL is found at 0.0595 MeV (in the range of 0.26111–3.2299 cm), and there is a significant increase across the upward energies (6.9944–10.1189 cm at 1.408 MeV), as shown in Fig. ##FIG##9##10##a. This fact emphasizes that as the radiation’s energy rises, more photons will be able to pass through the chosen samples. Figure ##FIG##9##10##a even further illustrates that the effective method for reducing the HVL and improving the ability of the chosen samples to attenuate γ-ray is the addition of ZrO₂ to the PMMA matrix. Comparing P-45nZ to the other materials, P-45nZ presents the lowest HVL at any energy. Our analysis shows that the HVL appears in the following sequence: P-0Z &gt; P-15mZ &gt; P-15nZ &gt; P-30mZ &gt; P-30nZ &gt; P-45mZ &gt; P-45nZ. This pattern emphasizes that the addition of more ZrO₂ improves photon shielding properties because ZrO₂ is denser than purified PMMA. Thus, it is clear that ZrO₂ can reduce the HVL, making the P-45nZ composite optimal.</p>", "<p id=\"Par40\">The TVL findings are shown as a function of energy in Fig. ##FIG##9##10##b. The TVL values of P-0Z and P-45nZ samples at the starting energy (i.e., 0.0595 MeV) show a dropping trend from 10.7269 cm to 0.8673 cm because the TVL highly relies on the sample density at all energies. It is evident that decreasing TVL results from the rising density of the composite. The TVL trend depicted in Fig. ##FIG##9##10##b is consistent with that in Fig. ##FIG##9##10##a for the HVL. The highest TVL values range from 23.2349 cm for P-45nZ to 33.6143 cm for the P-0Z sample at 1.408 MeV. The high ZrO₂ content of the P-45nZ sample contributed to its high density and showed the sample’s low TVL. The reverse of the μ values is the MFP values, which are depicted in a manner comparable to that of the HVL and TVL. The smaller the MFP of a composite, the superior the radiation shielding ability. Figure ##FIG##9##10##c depicts the relationship between the investigated composites’ MFP and the energy. At all energies, the MFP depends on the ZrO₂ content. Increasing the ZrO₂ insertion from 0 to 45 wt% in PMMA led to an increase in the density of the samples, from 1.176 for P-0Z to 1.8330 g/cm³ for P-45nZ. Consequently, the MFP values drop from 4.6598 for P-0Z sample to 0.3767 cm for P-45nZ at 0.0595 MeV. At higher energy of 1.408 MeV the MFP drops from 14.599 to 9.90 cm. Thus, we can deduce that the P-45nZ sample needs a thinner shielding layer than the other specimens in order to prevent the same radiation, and we can also infer that an increase in energy leads to a rise in the MFP. In conclusion, increasing the content and decreasing the size of ZrO₂ particles leads to lower values for the HVL, TVL, and MFP parameters, which optimize radiation shielding.</p>", "<p id=\"Par41\">For the examined pure PMMA and ZrO₂/PMMA micro composites, the change of Z_eff and N_eff with photon energy is shown in Figs. ##FIG##10##11## and ##FIG##11##12##, respectively. Evidently, at low energies, the Z_eff and N_eff reach their maximum values at 0.02 MeV and then decline as the energy increases. This trend can be attributed to the photoelectric process’s cross-section, which is inversely proportional to photon energy as E^3.5. However, as the photon energy exceeds 0.3 MeV, further increments of photon energy, the value of Z_eff becomes virtually independent of photon energy. This behavior might be because the Compton scattering mechanism predominates. At high energies above 1.5 MeV, the value of Z_eff slowly rises as the photon energy increases. The supremacy of pair production in this higher energy area can be used to explain this trend. Figure ##FIG##10##11## also reveals that, as the concentration of ZrO₂ filler increases in the PMMA matrix, the values of Z_eff increase. This increase is due to the density of ZrO₂, which increases the overall density of the PMMA-based composites. Therefore, P-45mZ with 45% ZrO₂ is discovered to have the highest value of Z_eff at all γ-ray energies. Eventually, the minimum Zeff corresponds effectively to pure PMMA with 0% of ZrO₂, which does not contain ZrO₂ filler. As shown in Fig. ##FIG##11##12##, N_eff exhibits approximately the same behavior as Z_eff since the two parameters are strongly linked.</p>", "<p id=\"Par42\">The Z_eq describes the shielding characteristics of the chosen polymers pertaining to equivalent elements and is also considered when determining the buildup factor. The composites having higher Z_eq is the best radiation-protective material. Figure ##FIG##12##13## depicts the Z_eq values for the micro ZrO₂/PMMA composites as a function of the photon energy in the range between 0.015 and 15 MeV. From Fig. ##FIG##12##13##, it is obvious that adding ZrO₂ in increasing amounts into the PMMA matrix causes the Z_eq to increase at the same γ-ray energy. Therefore, the P-0Z sample has the lowest Z_eq values, as seen in Fig. ##FIG##12##13##, whereas the P-45mZ sample has the highest values. Consequently, the P-45mZ composite has better shielding ability than other PMMA composites, which is consistent with the former results of MACs. Furthermore, it is also apparent that the Z_eq increases to reach its maximum value for all the ZrO₂/PMMA composites at 1 MeV due to the Compton scattering (CS) process. The higher observed rise in Z_eq values is related to the high rates of CS interaction in the mid-(γ) energy regions, where the Z_eq calculation largely depended on the ratio of (MAC_CS/MAC_total), implying substantial Compton scattering in the medium energy zone. Then, Z_eq drops rapidly as the γ-ray energy exceeds 1.22 MeV due to the pair production process dominating at the higher energy regions.</p>", "<p id=\"Par43\">Figure ##FIG##13##14## demonstrates the variations of EBF and EABF for P-0Z, P-15mZ, P-30mZ, and P-45mZ samples at various penetration depths as a function of photon energy. It is evident that the EBF and EABF values for the selected composites ascend to a maximal value at middle energies before beginning to fall. The predominant photon interaction mechanism in the low energy region is the photoelectric absorption, whose cross-section changes inversely with energy as E^3.5. Thus, in this low-energy region, the selected composites can absorb the most photons because of the predominance of this process. Therefore, it causes the EBF and EABF values in the lower energy regions to decrease. On the other hand, pair production, another photon absorption mechanism with a cross-section that is inversely proportional to energy as E², is also predominant in the higher energy area. Compton scattering, a predominant photon interaction process in the intermediate energy region, only reduces photon energy caused by scattering and cannot entirely remove the photon. Because the photon’s lifetime is longer in this energy range, it is more likely to escape from the polymer sample. The values of EBF and EABF are increased as a consequence of this process. Additionally, it is noted that repeated scattering events at large penetration depths cause an increase in the values of EBF and EABF to extremely high levels. It is essential to point out that the variance between EBF and EABF values at the same ZrO₂ concentration and the same energy is very close. Additionally, a significant decrease in the values of EBF and EABF, accompanied by a shift in their maximum values to higher energies, was observed as the ZrO₂ content increased.</p>", "<p id=\"Par44\">The variance of EBF and EABF with the radiant energy of all the chosen composites has also been plotted in Fig. ##FIG##13##14##a–d for certain penetrations depths up to 40 MFP to illustrate the effect of the chemical composition of the selected ZrO₂/PMMA composites on the EBF and EABF. It is evident that the equivalent atomic number of the chosen polymers has an inverse relationship with the EBF and EABF. Thus, P-0Z, the lowest Z_eq polymer, dominates EBF and EABF values at their maximums, while P-45mZ, the greatest Z_eq polymer, dominates EBF and EABF values at their minimums. Because P-0Z is a polymer with low-Z components, it could have the highest EBF. Additionally, according to Fig. ##FIG##13##14##a–d, increasing the thickness of the interacting substance, i.e. increasing the penetration depth of the chosen polymers, causes an increase in the scattering events inside the polymer. Consequently, the EBF and EABF values are incredibly high and display the highest values at the penetration depth of 40 MFP. In light of this, it can be said that P-45mZ has more vital X-ray and γ-ray shielding efficiency than P-0Z.</p>" ]

[ "<title>Result and discussion</title>", "<title>Characterization</title>", "<title>Transmission electron microscope (TEM) analysis</title>", "<p id=\"Par28\">The TEM micrographs of ZrO₂ MPs and NPs are depicted in Fig. ##FIG##4##5##. Figure ##FIG##4##5##a shows that ZrO₂ MPs are nearly spherical in shape and have an average particle size between 1.46 and 1.75 µm. On the other hand, Fig. ##FIG##4##5##b demonstrates the existence of ZrO₂ NPs, which have a consistent size distribution between 7.86 and 12 nm.</p>", "<title>Scanning electron microscope (SEM) analysis</title>", "<p id=\"Par29\">Figure ##FIG##5##6## displays the SEM images of P-0Z, P-15mZ, P-15nZ, P-45mZ, and P-45nZ composites. Figure ##FIG##5##6## describes how ZrO₂ particles impacted PMMA on a micro and nanoscale. Figure ##FIG##5##6##a depicts smooth and distinct variation compared to ZrO₂/PMMA composites. In other words, pure PMMA (Fig. ##FIG##5##6##a) and ZrO₂/PMMA composites (Fig. ##FIG##5##6##b–e) exhibit a distinct difference in morphology. The SEM images of micro and nano ZrO₂/PMMA composite samples with identical filler wt% are compared as indicated in Fig. ##FIG##5##6##b–d. It is clear that ZrO₂ Nps are uniformly scattered and thoroughly incorporated into the PMMA matrix in ZrO₂/PMMA composites, which may strengthen the interfacial adhesion between the PMMA matrix and ZrO₂ NPs and offer an interconnecting structure for shielding. While large ZrO₂ MPs are not fully covered by the PMMA matrix in micro ZrO₂/PMMA composites, and some of them peel off from the matrix due to insufficient interfacial adhesion, which behaves as voids for shielding. Additionally, it was evident that the porosity reduces and the distribution rises as the fraction of particles increases. This attribute proves that increasing the ZrO₂ content in PMMA will enhance structural, mechanical and shielding properties.</p>", "<title>Fourier transform-infrared (FT-IR)</title>", "<p id=\"Par30\">FT-IR spectroscopic analysis was performed on the produced samples using Bruker Vertex 70 infrared spectrometer to identify the functional groups in the ZrO₂/PMMA composites and the interaction mechanism of ZrO₂ particles and PMMA polymeric matrix. The FT-IR spectra of P-0Z, ZrO₂ MPs, P-15mZ, P-45mZ, ZrO₂ NPs, P-15nZ, and P-45nZ samples were collected in the wavelength region of the 400–4000 cm⁻¹ and displayed in Fig. ##FIG##6##7##. A discrete absorption band from 1142.19 to 1239 cm⁻¹ can be seen in the FT-IR spectrum P-0Z as depicted in Fig. ##FIG##6##7##a, which related to the C–O–C stretching vibration. The vibrations of the –methyl group can be identified to the pair of bands at 1386 cm⁻¹, and 750 cm⁻¹. The band at 980 cm⁻¹ is the characteristic absorption vibrations of PMMA, jointly with the bands at 1068 cm⁻¹ and 839 cm⁻¹. Due to the existence of ester carbonyl group stretching vibration (Acrylate carboxyl group), a sharp intensity peak at 1725.21 cm⁻¹ was conducted. The band at 1442.32 cm⁻¹ can be attributed to the bending vibration of the C–H bonds of the –CH₃ group. The band at 1442.32 cm⁻¹ can be allocated to the bending vibration of the C–H bonds of the –CH₃ group. The two bands at 2925.05 cm⁻¹ and 2854.34 cm⁻¹ can be linked to the C–H bond stretching vibrations of the –CH₃ and –CH₂– groups, respectively. In addition, two faint absorption bands at 3734 cm⁻¹ and 1644 cm⁻¹ result from the stretching and bending vibrations of the –OH group, respectively^{##REF##21787437##62##}.</p>", "<p id=\"Par31\">A broad peak can be seen in the FT-IR spectrum of ZrO₂ MPs in Fig. ##FIG##6##7##b at 3428.96 cm⁻¹, 1634.05 cm⁻¹, and (456.10, 604 cm⁻¹), which correspond to OH stretching, OH bending, and the Zr–O band, respectively. Peak locations, forms, and intensities have been determined in accordance with the fingerprint features, together with the material’s fundamental components^{##REF##28666841##63##}. As seen in Fig. ##FIG##6##7##c, the spectrum of the P-15mZ composite, compared to that of P-0Z, had new peaks at 453.94 cm⁻¹ and 599.12 cm⁻¹ referring to the metal–oxygen bond in ZrO₂. Furthermore, it is clear that when the weight fraction of ZrO₂ MPs was increased to 45 wt%, two peaks, as shown in Fig. ##FIG##6##7##d, were seen at 631 cm⁻¹ and 698 cm⁻¹, which are associated with stretching vibration modes of Zr–O and bending vibration of =C–H^{##UREF##46##64##}. The apparent presence of these peaks indicates that ZrO₂ MPs have been successfully embedded in the PMMA polymeric matrix.</p>", "<p id=\"Par32\">Figure ##FIG##6##7##e displays the FT-IR spectrum of ZrO₂ NPs. The peaks at 453.92 cm⁻¹ and 599.12 cm⁻¹ are related to the strong metal–oxygen conjunction in ZrO₂ NPs, and two peaks at 631 cm⁻¹ and 698 cm⁻¹ are attributed to the Zr–O bond and bending vibration of =C–H as previously mentioned. At 574 cm⁻¹ and 708 cm⁻¹, two additional bands have been observed that are connected to the ZrO₂ NPs. It is also clear from Fig. ##FIG##6##7##f,g that the spectra of P-15nZ and P-45nZ composites exhibit behavior that is comparable to that previously reported, in conjunction with the existence of two more peaks at 551 cm⁻¹ and 598.04 cm⁻¹ that were formed at greater concentrations of ZrO₂ NPs. Consequently, it can be deduced from Fig. ##FIG##6##7## that there was a chemical bond between the PMMA matrix and the ZrO₂ filler in all composites, which resulted in chemical interactions between them.</p>", "<title>γ-ray shielding results</title>", "<p id=\"Par33\">The MAC is a standard parameter utilized for measuring and comparing the performance of various shielding materials. Table ##TAB##5##6## displays the experimentally measured values of MACs for all the investigated composites (pure PMMA, micro ZrO₂/PMMA, and nano ZrO₂/PMMA composites) in the energy range between 0.05953 and 1.4081 MeV. Furthermore, using the XCOM software, MAC for the PMMA and micro ZrO₂/PMMA composites were generated theoretically, and the relative deviation (Δ%) between the experimental and theoretical results that were determined by using Eq. (##FORMU##19##16##):</p>", "<p id=\"Par34\">As can be seen from Table ##TAB##5##6##, both the MAC values obtained from XCOM and those achieved by laboratory measurement exhibit good comparability. This remark is accurate for all of the energies tested. However, some minor discrepancies were discovered between the two methods. These are acceptable because ordinarily, anyone can find a few minor errors in the experimental results, but generally, the experimental results are acceptable and agree with the XCOM results. This is a crucial and significant step since it clarifies the precision of the geometry used in the lab to calculate the MAC for the PMMA and ZrO₂/PMMA composites. According to Table ##TAB##5##6##, the Δ% for pure PMMA (free of ZrO₂ filler) is restricted to − 0.65 and 0.94%, whereas the Δ% for 15 wt% ZrO₂/PMMA is restricted to − 0.40 and 0.84%, and for 30 wt% ZrO₂/PMMA ranges between − 0.94 and 0.61%. However, it can only be between − 0.89 and 0.81% for 45 wt% ZrO₂/PMMA. These results support that the practical and theoretical results are compatible since the Δ% is less than 2%.</p>", "<p id=\"Par35\">The experimental results of μ values of pure PMMA, micro, and nano ZrO₂/PMMA composites filled with various concentrations (15 wt%, 30 wt%, and 45 wt%) as a function of γ-ray energies are shown in Fig. ##FIG##7##8##. As can be seen from Fig. ##FIG##7##8##, the energy of the incident photons and the compositions of the protective material have a significant impact on μ values. The μ values have been found to significantly increase with increasing micro- and nano-ZrO₂ concentrations in the composites and rapidly decline as photon energy increases. This trend may be demonstrated by focusing on the three primary processes by which energetic photons interact with matter; photoelectric effect, Compton scattering, and pair production, which all contribute to the energy loss of the incident photon. At energies less than 125 keV, the photoelectric effect is the primary mechanism that causes photons to be absorbed since the possibility of photoelectric absorption depends on Z³, where Z is the atomic number^{##UREF##47##65##}. Therefore, due to element Zr with atomic number (Z = 40), μ values increase as the concentration of ZrO₂ in the PMMA matrix increases. However, as photon energy increases beyond 125 keV, the likelihood of the photoelectric effect reduces roughly according to 1/E³^{##UREF##48##66##}, where E is the energy of the incident photons, which illustrates why μ for every composite decreases slightly as the photon energy goes above 125 keV. Meanwhile, in this energy range increasing the ZrO₂ filler wt% in the PMMA matrix is slightly affect the values of μ which have nearly the same value as the photon energy increase. This result is because, at this intermediate energy range, the effect of photoelectric absorption diminishes, and the Compton scattering mechanism takes over. Besides, the cross-section of the Compton scattering is practically independent of atomic number but depends on the number of electrons per unit mass.</p>", "<p id=\"Par36\">Additionally, the difference of μ between micro- and nano-sized ZrO₂/PMMA was also compared, as shown in Fig. ##FIG##7##8##. At all of the tested γ-ray energies, the nano ZrO₂/PMMA curves always are above the micro ZrO₂/PMMA curves for the same weight percent of ZrO₂ filler. As the size of ZrO₂ particles decreases from micro to nano size, the uniform distribution of ZrO₂ NPs over a greater surface area within the PMMA matrix would increase the likelihood of incident photons interacting with ZrO₂ NPs in nanocomposites as compared to micro composites and increase the probability of further scattering mechanisms for the photons until the photon’s energy is below 200 keV. Consequently, in PMMA-based radiation protective material, ZrO₂ NPs exhibit excellent attenuation performance than ZrO₂ MPs for the same chemical composition and weight percentage of the composite.</p>", "<p id=\"Par37\">To evaluate the superiority of the shielding ability of nanocomposites over micro composites, the relative increase rate (δ%) in μ values between nano and micro ZrO₂/PMMA composites was calculated according to Eq. (##FORMU##20##17##) and depicted in Fig. ##FIG##8##9## as a function of energy at various ZrO₂ loadings.</p>", "<p id=\"Par38\">As shown in Fig. ##FIG##8##9##, the relative increase rate (δ%) increases with an increase in ZrO₂ content; however, its value decreases as the photon’s energy increases from 0.05953 to 1.408 MeV. These findings suggest that, due to various photon interaction cross-sections at different photon energies, the size effect diminishes as photon energy increases. The absorption capability is Z-dependent when the photoelectric effect dominates at low photon energies. Due to the mid-high Z of the element zirconium in ZrO₂ particles and the low Z of the elements C, O, and H in the PMMA matrix, the photoelectric absorption of ZrO₂ particles is considerably higher than that of the PMMA matrix. As a result, these particles are extremely important for radiation shielding. At the energy of 0.05953 MeV, the δ% in the P-15nZ sample was 16.88%, compared to 17.29% in the P-30nZ sample and 17.84% in the P-45nZ sample. Since Compton scattering is more likely at higher energies and its cross-section can be regarded as the predominant interaction that does not rely on Z but rather on the free electrons, there is little difference in the ability of ZrO₂ particles compared to the PMMA matrix. As a result, the essential role of ZrO₂ particles diminishes, and the impact of particle size decreases. At the highest examined energy (1.408 MeV), the δ % was 6.67% for P-15nZ, whereas it was 7.23%and 8.60% for P-30nZ and P-45nZ samples, respectively. In summary, the δ % follows the general trend P-45nZ &gt; P-30nZ &gt; P-15nZ for all incident energies. So, the composite P-45nZ has superior shielding potentials over all the investigated samples.</p>", "<p id=\"Par39\">Three primary critical radiation shielding parameters, the HVL, the TVL, and the MFP have been researched in correlation to the radiation shielding capabilities of micro- and nano-structured composites^{##UREF##49##67##}. Figure ##FIG##9##10## displays the HVL, TVL, and MFP at different energies ranging from 0.0595 MeV to 1.408 MeV. The HVL is the thinnest sample at which 50% of the original γ-ray intensity passes through it. The findings of our calculation of the HVL for the chosen composites at the energies employed for the μ data are displayed in Fig. ##FIG##9##10##a. When analyzing the data in this Fig. ##FIG##9##10##a. One can see a gradual increase in HVL as energy is increased from 0.0595 to 1.408 MeV. This tendency indicates that the photons’ ability to penetrate samples rises along with their energy. The lowest HVL is found at 0.0595 MeV (in the range of 0.26111–3.2299 cm), and there is a significant increase across the upward energies (6.9944–10.1189 cm at 1.408 MeV), as shown in Fig. ##FIG##9##10##a. This fact emphasizes that as the radiation’s energy rises, more photons will be able to pass through the chosen samples. Figure ##FIG##9##10##a even further illustrates that the effective method for reducing the HVL and improving the ability of the chosen samples to attenuate γ-ray is the addition of ZrO₂ to the PMMA matrix. Comparing P-45nZ to the other materials, P-45nZ presents the lowest HVL at any energy. Our analysis shows that the HVL appears in the following sequence: P-0Z &gt; P-15mZ &gt; P-15nZ &gt; P-30mZ &gt; P-30nZ &gt; P-45mZ &gt; P-45nZ. This pattern emphasizes that the addition of more ZrO₂ improves photon shielding properties because ZrO₂ is denser than purified PMMA. Thus, it is clear that ZrO₂ can reduce the HVL, making the P-45nZ composite optimal.</p>", "<p id=\"Par40\">The TVL findings are shown as a function of energy in Fig. ##FIG##9##10##b. The TVL values of P-0Z and P-45nZ samples at the starting energy (i.e., 0.0595 MeV) show a dropping trend from 10.7269 cm to 0.8673 cm because the TVL highly relies on the sample density at all energies. It is evident that decreasing TVL results from the rising density of the composite. The TVL trend depicted in Fig. ##FIG##9##10##b is consistent with that in Fig. ##FIG##9##10##a for the HVL. The highest TVL values range from 23.2349 cm for P-45nZ to 33.6143 cm for the P-0Z sample at 1.408 MeV. The high ZrO₂ content of the P-45nZ sample contributed to its high density and showed the sample’s low TVL. The reverse of the μ values is the MFP values, which are depicted in a manner comparable to that of the HVL and TVL. The smaller the MFP of a composite, the superior the radiation shielding ability. Figure ##FIG##9##10##c depicts the relationship between the investigated composites’ MFP and the energy. At all energies, the MFP depends on the ZrO₂ content. Increasing the ZrO₂ insertion from 0 to 45 wt% in PMMA led to an increase in the density of the samples, from 1.176 for P-0Z to 1.8330 g/cm³ for P-45nZ. Consequently, the MFP values drop from 4.6598 for P-0Z sample to 0.3767 cm for P-45nZ at 0.0595 MeV. At higher energy of 1.408 MeV the MFP drops from 14.599 to 9.90 cm. Thus, we can deduce that the P-45nZ sample needs a thinner shielding layer than the other specimens in order to prevent the same radiation, and we can also infer that an increase in energy leads to a rise in the MFP. In conclusion, increasing the content and decreasing the size of ZrO₂ particles leads to lower values for the HVL, TVL, and MFP parameters, which optimize radiation shielding.</p>", "<p id=\"Par41\">For the examined pure PMMA and ZrO₂/PMMA micro composites, the change of Z_eff and N_eff with photon energy is shown in Figs. ##FIG##10##11## and ##FIG##11##12##, respectively. Evidently, at low energies, the Z_eff and N_eff reach their maximum values at 0.02 MeV and then decline as the energy increases. This trend can be attributed to the photoelectric process’s cross-section, which is inversely proportional to photon energy as E^3.5. However, as the photon energy exceeds 0.3 MeV, further increments of photon energy, the value of Z_eff becomes virtually independent of photon energy. This behavior might be because the Compton scattering mechanism predominates. At high energies above 1.5 MeV, the value of Z_eff slowly rises as the photon energy increases. The supremacy of pair production in this higher energy area can be used to explain this trend. Figure ##FIG##10##11## also reveals that, as the concentration of ZrO₂ filler increases in the PMMA matrix, the values of Z_eff increase. This increase is due to the density of ZrO₂, which increases the overall density of the PMMA-based composites. Therefore, P-45mZ with 45% ZrO₂ is discovered to have the highest value of Z_eff at all γ-ray energies. Eventually, the minimum Zeff corresponds effectively to pure PMMA with 0% of ZrO₂, which does not contain ZrO₂ filler. As shown in Fig. ##FIG##11##12##, N_eff exhibits approximately the same behavior as Z_eff since the two parameters are strongly linked.</p>", "<p id=\"Par42\">The Z_eq describes the shielding characteristics of the chosen polymers pertaining to equivalent elements and is also considered when determining the buildup factor. The composites having higher Z_eq is the best radiation-protective material. Figure ##FIG##12##13## depicts the Z_eq values for the micro ZrO₂/PMMA composites as a function of the photon energy in the range between 0.015 and 15 MeV. From Fig. ##FIG##12##13##, it is obvious that adding ZrO₂ in increasing amounts into the PMMA matrix causes the Z_eq to increase at the same γ-ray energy. Therefore, the P-0Z sample has the lowest Z_eq values, as seen in Fig. ##FIG##12##13##, whereas the P-45mZ sample has the highest values. Consequently, the P-45mZ composite has better shielding ability than other PMMA composites, which is consistent with the former results of MACs. Furthermore, it is also apparent that the Z_eq increases to reach its maximum value for all the ZrO₂/PMMA composites at 1 MeV due to the Compton scattering (CS) process. The higher observed rise in Z_eq values is related to the high rates of CS interaction in the mid-(γ) energy regions, where the Z_eq calculation largely depended on the ratio of (MAC_CS/MAC_total), implying substantial Compton scattering in the medium energy zone. Then, Z_eq drops rapidly as the γ-ray energy exceeds 1.22 MeV due to the pair production process dominating at the higher energy regions.</p>", "<p id=\"Par43\">Figure ##FIG##13##14## demonstrates the variations of EBF and EABF for P-0Z, P-15mZ, P-30mZ, and P-45mZ samples at various penetration depths as a function of photon energy. It is evident that the EBF and EABF values for the selected composites ascend to a maximal value at middle energies before beginning to fall. The predominant photon interaction mechanism in the low energy region is the photoelectric absorption, whose cross-section changes inversely with energy as E^3.5. Thus, in this low-energy region, the selected composites can absorb the most photons because of the predominance of this process. Therefore, it causes the EBF and EABF values in the lower energy regions to decrease. On the other hand, pair production, another photon absorption mechanism with a cross-section that is inversely proportional to energy as E², is also predominant in the higher energy area. Compton scattering, a predominant photon interaction process in the intermediate energy region, only reduces photon energy caused by scattering and cannot entirely remove the photon. Because the photon’s lifetime is longer in this energy range, it is more likely to escape from the polymer sample. The values of EBF and EABF are increased as a consequence of this process. Additionally, it is noted that repeated scattering events at large penetration depths cause an increase in the values of EBF and EABF to extremely high levels. It is essential to point out that the variance between EBF and EABF values at the same ZrO₂ concentration and the same energy is very close. Additionally, a significant decrease in the values of EBF and EABF, accompanied by a shift in their maximum values to higher energies, was observed as the ZrO₂ content increased.</p>", "<p id=\"Par44\">The variance of EBF and EABF with the radiant energy of all the chosen composites has also been plotted in Fig. ##FIG##13##14##a–d for certain penetrations depths up to 40 MFP to illustrate the effect of the chemical composition of the selected ZrO₂/PMMA composites on the EBF and EABF. It is evident that the equivalent atomic number of the chosen polymers has an inverse relationship with the EBF and EABF. Thus, P-0Z, the lowest Z_eq polymer, dominates EBF and EABF values at their maximums, while P-45mZ, the greatest Z_eq polymer, dominates EBF and EABF values at their minimums. Because P-0Z is a polymer with low-Z components, it could have the highest EBF. Additionally, according to Fig. ##FIG##13##14##a–d, increasing the thickness of the interacting substance, i.e. increasing the penetration depth of the chosen polymers, causes an increase in the scattering events inside the polymer. Consequently, the EBF and EABF values are incredibly high and display the highest values at the penetration depth of 40 MFP. In light of this, it can be said that P-45mZ has more vital X-ray and γ-ray shielding efficiency than P-0Z.</p>" ]

[ "<title>Conclusion</title>", "<p id=\"Par45\">In the current study, seven PMMA-based polymer samples were prepared and reinforced with ZrO₂ MPs and NPs at concentrations of 15, 30, and 45 wt% to examine their radiation shielding capabilities for diverse purposes. The investigated composites are coded as P-0Z, P-15mZ, P-15nZ, P-30mZ, P-30nZ, P-45mZ, and P-45nZ. TEM was used to measure the average size of and ZrO₂ MPs and NPs. Furthermore, the SEM was used to study the morphology and distribution of ZrO₂ MPs and NPs within the prepared composites. The analysis revealed that ZrO₂ NPs had a uniform distribution inside the composites along with a decline in the porosity of the sample in comparison to the ZrO₂ MPs. The characteristics of ZrO₂/PMMA molecules were also investigated using FT-IR. The MAC was calculated experimentally using the HPGe detector and five standard radioactive point sources. The experimental results significantly agreed with those obtained theoretically from the XCOM database, indicating the precision of the setup used for computing the MAC for the prepared composites. The experimental findings showed that the prepared samples’ ability to attenuate γ-rays at all the examined energies depends on the size and concentration of ZrO₂ particles. The findings of this research also demonstrated that PMMA filled with ZrO₂ NPs has higher μ values than PMMA filled with ZrO₂ MPs and pure PMMA at all selected energies. P-45nZ sample had the highest μ values, which varied between 2.6546 and 0.0991 cm⁻¹ as γ-ray photon energy increased from 0.0595 to 1.408 MeV, respectively. Actually, the MAC for the P-45nZ sample is 1.448 cm²/g at 59.5 keV, which is higher than the values reported in the literature and very close to conventional lead at 661.66 keV. Furthermore, the highest relative increase rate in μ values between nano and micro ZrO₂/PMMA composites was 17.84% reported for the sample P-45nZ at 59.53 keV. The HVL, TVL, and MFP also demonstrated the superiority of ZrO₂ NPs over ZrO₂ MPs. Z_eff and N_eff were increased by increasing the content of ZrO₂ to the PMMA, which improved the γ-ray shielding efficiency. Due to their easy and quick manufacture, simple processing, non-toxic, lightweight, cost-effective, and environmental friendliness, the proposed composites have advantages over lead materials. Therefore, the developed nano ZrO₂/PMMA composites are effective shielding materials that can be used to reduce the gamma dose in radiation facilities. Future research could further examine the capability of the proposed composites in shielding neutrons.</p>" ]

[ "<p id=\"Par1\">This research aimed to examine the radiation shielding properties of unique polymer composites for medical and non-medical applications. For this purpose, polymer composites, based on poly methyl methacrylate (PMMA) as a matrix, were prepared and reinforced with micro- and nanoparticles of ZrO₂ fillers at a loading of 15%, 30%, and 45% by weight. Using the high purity germanium (HPGe) detector, the suggested polymer composites’ shielding characteristics were assessed for various radioactive sources. The experimental values of the mass attenuation coefficients (MAC) of the produced composites agreed closely with those obtained theoretically from the XCOM database. Different shielding parameters were estimated at a broad range of photon energies, including the linear attenuation coefficient (μ), tenth value layer (TVL), half value layer (HVL), mean free path (MFP), effective electron density (N_eff), effective atomic number (Z_eff), and equivalent atomic number (Z_eq), as well as exposure buildup factor (EBF) and energy absorption buildup factor (EABF) to provide more shielding information about the penetration of γ-rays into the chosen composites. The results showed that increasing the content of micro and nano ZrO₂ particles in the PMMA matrix increases μ values and decreases HVL, TVL, and MFP values. P-45nZ sample with 45 wt% of ZrO₂ nanoparticles had the highest μ values, which varied between 2.6546 and 0.0991 cm⁻¹ as γ-ray photon energy increased from 0.0595 to 1.408 MeV, respectively. Furthermore, the highest relative increase rate in μ values between nano and micro composites was 17.84%, achieved for the P-45nZ sample at 59.53 keV. These findings demonstrated that ZrO₂ nanoparticles shield radiation more effectively than micro ZrO₂ even at the same photon energy and filler wt%. Thus, the proposed nano ZrO₂/PMMA composites can be used as effective shielding materials to lessen the transmitted radiation dose in radiation facilities.</p>", "<title>Subject terms</title>", "<p>Open access funding provided by The Science, Technology &amp; Innovation Funding Authority (STDF) in cooperation with The Egyptian Knowledge Bank (EKB).</p>" ]

[]

[ "<title>Author contributions</title>", "<p>A.M.E.-K. proposed the main aim and the methodology of the research. A.Y.E. and M.T.A. prepared the composite materials. Mahmoud I. Abbas depicted all the figures. M.T.A. and A.Y. E. wrote the main manuscript text. All authors reviewed and revised the manuscript.</p>", "<title>Funding</title>", "<p>Open access funding provided by The Science, Technology &amp; Innovation Funding Authority (STDF) in cooperation with The Egyptian Knowledge Bank (EKB).</p>", "<title>Data availability</title>", "<p>All data generated or analyzed during this study are included in this published article.</p>", "<title>Competing interests</title>", "<p id=\"Par46\">The authors declare no competing interests.</p>" ]

[ "<fig id=\"Fig1\"><label>Figure 1</label><caption><p>Silicon rubber mold.</p></caption></fig>", "<fig id=\"Fig2\"><label>Figure 2</label><caption><p>The prepared specimen’s final shape.</p></caption></fig>", "<fig id=\"Fig3\"><label>Figure 3</label><caption><p>Experimental arrangement for the γ-rays measuring system.</p></caption></fig>", "<fig id=\"Fig4\"><label>Figure 4</label><caption><p>The obtained spectra using Co-60 radioactive source: (<bold>a</bold>) without absorber and (<bold>b</bold>) with P-0Z sample.</p></caption></fig>", "<fig id=\"Fig5\"><label>Figure 5</label><caption><p>TEM images of (<bold>a</bold>) ZrO₂ MPs and (<bold>b</bold>) ZrO₂ NPs.</p></caption></fig>", "<fig id=\"Fig6\"><label>Figure 6</label><caption><p>SEM images of (<bold>a</bold>) P-0Z, (<bold>b</bold>) P-15mZ, (<bold>c</bold>) P-15nZ, (<bold>d</bold>) P-45mZ, and (<bold>e</bold>) P-45nZ composites.</p></caption></fig>", "<fig id=\"Fig7\"><label>Figure 7</label><caption><p>FT-IR spectrum of (<bold>a</bold>) P-0Z, (<bold>b</bold>) ZrO₂ MPs, (<bold>c</bold>) P-15mZ, (<bold>d</bold>) P-45mZ, (<bold>e</bold>) ZrO₂ NPs, (<bold>f</bold>) P-15nZ, and (<bold>g</bold>) P-45nZ samples.</p></caption></fig>", "<fig id=\"Fig8\"><label>Figure 8</label><caption><p>Comparison between linear attenuation coefficients of micro- and nano-ZrO₂/PMMA composites at different ZrO₂ concentrations as a function of photon energy.</p></caption></fig>", "<fig id=\"Fig9\"><label>Figure 9</label><caption><p>Relative increase rates (δ%) in relation to photon energy at various ZrO₂weight percentages.</p></caption></fig>", "<fig id=\"Fig10\"><label>Figure 10</label><caption><p>The variance of the (<bold>a</bold>) half value layer, (<bold>b</bold>) tenth value layer, (<bold>c</bold>) as well as mean free path, (<bold>d</bold>) with respect to energy for the studied polymers.</p></caption></fig>", "<fig id=\"Fig11\"><label>Figure 11</label><caption><p>Z_eff of pure PMMA, and micro ZrO₂/PMMA composites at different energies.</p></caption></fig>", "<fig id=\"Fig12\"><label>Figure 12</label><caption><p>N_eff of pure PMMA, and micro ZrO₂/PMMA composites at different energies.</p></caption></fig>", "<fig id=\"Fig13\"><label>Figure 13</label><caption><p>Z_eq of pure PMMA, and micro ZrO₂/PMMA composites at different energies.</p></caption></fig>", "<fig id=\"Fig14\"><label>Figure 14</label><caption><p>EBF and EABF of P-0Z, P-15mZ, P-30mZ, and P-45mZ as a function of energy at penetration depths of 1, 10, 20, 30, and 40 MFP.</p></caption></fig>" ]

[ "<table-wrap id=\"Tab1\"><label>Table 1</label><caption><p>The physical properties of MMA^{##UREF##31##46##}.</p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th align=\"left\">Physical properties</th><th align=\"left\">Boiling point</th><th align=\"left\">Density</th><th align=\"left\">Vapor pressure</th><th align=\"left\">Molecular weight</th></tr></thead><tbody><tr><td align=\"left\">Clear, colorless liquid of strong odor</td><td align=\"left\">100 °C</td><td align=\"left\">0.943 g/cm³</td><td align=\"left\">38 hPa at 20 °C</td><td align=\"left\">100 g/mol</td></tr></tbody></table></table-wrap>", "<table-wrap id=\"Tab2\"><label>Table 2</label><caption><p>The physical properties of PMMA^{##UREF##31##46##}.</p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th align=\"left\">Physical properties</th><th align=\"left\">Bead diameter</th><th align=\"left\">Density</th><th align=\"left\">Molecular weight</th></tr></thead><tbody><tr><td align=\"left\">Fine powder, polymer beads, soluble in liquid polymer</td><td align=\"left\">1–120 µm</td><td align=\"left\">1.18 g/</td><td align=\"left\">0.8 g/mol</td></tr></tbody></table></table-wrap>", "<table-wrap id=\"Tab3\"><label>Table 3</label><caption><p>The physical properties of ZrO₂ MPs and ZrO₂ NPs.</p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th align=\"left\">Zirconium powder</th><th align=\"left\">Purity (%)</th><th align=\"left\">APS</th><th align=\"left\">Color</th><th align=\"left\">Density (g/cm³)</th></tr></thead><tbody><tr><td align=\"left\">ZrO₂ MPs</td><td char=\".\" align=\"char\">99.9</td><td align=\"left\">40–50 µm</td><td align=\"left\">White</td><td char=\".\" align=\"char\">5.68</td></tr><tr><td align=\"left\">ZrO₂ NPs</td><td char=\".\" align=\"char\">99.9</td><td align=\"left\">80 nm</td><td align=\"left\">White</td><td char=\".\" align=\"char\">5.68</td></tr></tbody></table></table-wrap>", "<table-wrap id=\"Tab4\"><label>Table 4</label><caption><p>Sample codes, weight fractions, and densities of ZrO₂/PMMA composites.</p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th align=\"left\" rowspan=\"3\">Sample codes</th><th align=\"left\" colspan=\"3\">Composition (wt%)</th><th align=\"left\" rowspan=\"3\">Density (g/cm³)</th></tr><tr><th align=\"left\" rowspan=\"2\">PMMA</th><th align=\"left\" colspan=\"2\">ZrO₂</th></tr><tr><th align=\"left\">Micro</th><th align=\"left\">Nano</th></tr></thead><tbody><tr><td align=\"left\">P-0Z</td><td align=\"left\">100</td><td align=\"left\">–</td><td align=\"left\">–</td><td char=\"±\" align=\"char\">1.176 ± 0.007</td></tr><tr><td align=\"left\">P-15mZ</td><td align=\"left\">85</td><td align=\"left\">15</td><td align=\"left\">–</td><td char=\"±\" align=\"char\">1.328 ± 0.006</td></tr><tr><td align=\"left\">P-15nZ</td><td align=\"left\">85</td><td align=\"left\">–</td><td align=\"left\">15</td><td char=\"±\" align=\"char\">1.390 ± 0.005</td></tr><tr><td align=\"left\">P-30mZ</td><td align=\"left\">70</td><td align=\"left\">30</td><td align=\"left\">–</td><td char=\"±\" align=\"char\">1.516 ± 0.004</td></tr><tr><td align=\"left\">P-30nZ</td><td align=\"left\">70</td><td align=\"left\">–</td><td align=\"left\">30</td><td char=\"±\" align=\"char\">1.593 ± 0.009</td></tr><tr><td align=\"left\">P-45mZ</td><td align=\"left\">55</td><td align=\"left\">45</td><td align=\"left\">–</td><td char=\"±\" align=\"char\">1.731 ± 0.005</td></tr><tr><td align=\"left\">P-45nZ</td><td align=\"left\">55</td><td align=\"left\">–</td><td align=\"left\">45</td><td char=\"±\" align=\"char\">1.833 ± 0.008</td></tr></tbody></table></table-wrap>", "<table-wrap id=\"Tab5\"><label>Table 5</label><caption><p>Standard radioactive point sources and their emitted photon energies and activities.</p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th align=\"left\">Radioactive sources</th><th align=\"left\">Photon energy (MeV)</th><th align=\"left\">Activity (kBq)</th></tr></thead><tbody><tr><td align=\"left\">Am-241</td><td char=\".\" align=\"char\">0.05953</td><td char=\".\" align=\"char\">253.250</td></tr><tr><td align=\"left\" rowspan=\"2\">Ba-133</td><td char=\".\" align=\"char\">0.08099</td><td char=\".\" align=\"char\" rowspan=\"2\">109.598</td></tr><tr><td char=\".\" align=\"char\">0.35601</td></tr><tr><td align=\"left\">Cs-137</td><td char=\".\" align=\"char\">0.66166</td><td char=\".\" align=\"char\">278.997</td></tr><tr><td align=\"left\" rowspan=\"2\">Co-60</td><td char=\".\" align=\"char\">1.17323</td><td char=\".\" align=\"char\" rowspan=\"2\">33.658</td></tr><tr><td char=\".\" align=\"char\">1.3325</td></tr><tr><td align=\"left\" rowspan=\"6\">Eu-152</td><td char=\".\" align=\"char\">0.12178</td><td char=\".\" align=\"char\" rowspan=\"6\">141.590</td></tr><tr><td char=\".\" align=\"char\">0.24469</td></tr><tr><td char=\".\" align=\"char\">0.34428</td></tr><tr><td char=\".\" align=\"char\">0.7789</td></tr><tr><td char=\".\" align=\"char\">0.96413</td></tr><tr><td char=\".\" align=\"char\">1.4081</td></tr></tbody></table></table-wrap>", "<table-wrap id=\"Tab6\"><label>Table 6</label><caption><p>Measured and theoretical values of MACs and their relative deviation (Δ%) for pure PMMA, micro ZrO₂/PMMA and nano ZrO₂/PMMA composites.</p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th align=\"left\" rowspan=\"2\">Sample</th><th align=\"left\" rowspan=\"2\">Photon energy (keV)</th><th align=\"left\" colspan=\"4\">MAC (cm² g⁻¹)</th></tr><tr><th align=\"left\">Nano</th><th align=\"left\">Micro</th><th align=\"left\">XCOM</th><th align=\"left\">Δ%</th></tr></thead><tbody><tr><td align=\"left\" rowspan=\"11\">Pure PMMA</td><td char=\".\" align=\"char\">0.05953</td><td align=\"left\" rowspan=\"11\"/><td char=\".\" align=\"char\">0.18248</td><td char=\".\" align=\"char\">0.1819</td><td char=\".\" align=\"char\">0.32</td></tr><tr><td char=\".\" align=\"char\">0.08099</td><td char=\".\" align=\"char\">0.16879</td><td char=\".\" align=\"char\">0.1682</td><td char=\".\" align=\"char\">0.35</td></tr><tr><td char=\".\" align=\"char\">0.12178</td><td char=\".\" align=\"char\">0.15298</td><td char=\".\" align=\"char\">0.1522</td><td char=\".\" align=\"char\">0.51</td></tr><tr><td char=\".\" align=\"char\">0.24469</td><td char=\".\" align=\"char\">0.12276</td><td char=\".\" align=\"char\">0.1233</td><td char=\".\" align=\"char\"> − 0.44</td></tr><tr><td char=\".\" align=\"char\">0.35601</td><td char=\".\" align=\"char\">0.10731</td><td char=\".\" align=\"char\">0.1076</td><td char=\".\" align=\"char\"> − 0.27</td></tr><tr><td char=\".\" align=\"char\">0.66166</td><td char=\".\" align=\"char\">0.08401</td><td char=\".\" align=\"char\">0.0832</td><td char=\".\" align=\"char\">0.94</td></tr><tr><td char=\".\" align=\"char\">0.7789</td><td char=\".\" align=\"char\">0.07679</td><td char=\".\" align=\"char\">0.0773</td><td char=\".\" align=\"char\"> − 0.65</td></tr><tr><td char=\".\" align=\"char\">0.96413</td><td char=\".\" align=\"char\">0.06956</td><td char=\".\" align=\"char\">0.0699</td><td char=\".\" align=\"char\"> − 0.45</td></tr><tr><td char=\".\" align=\"char\">1.17323</td><td char=\".\" align=\"char\">0.06395</td><td char=\".\" align=\"char\">0.0634</td><td char=\".\" align=\"char\">0.81</td></tr><tr><td char=\".\" align=\"char\">1.3325</td><td char=\".\" align=\"char\">0.05927</td><td char=\".\" align=\"char\">0.0594</td><td char=\".\" align=\"char\"> − 0.25</td></tr><tr><td char=\".\" align=\"char\">1.4081</td><td char=\".\" align=\"char\">0.05825</td><td char=\".\" align=\"char\">0.0578</td><td char=\".\" align=\"char\">0.85</td></tr><tr><td align=\"left\" rowspan=\"11\">15 wt% ZrO₂/PMMA</td><td char=\".\" align=\"char\">0.05953</td><td align=\"left\">0.61951</td><td char=\".\" align=\"char\">0.56627</td><td char=\".\" align=\"char\">0.5667</td><td char=\".\" align=\"char\">− 0.08  </td></tr><tr><td char=\".\" align=\"char\">0.08099</td><td align=\"left\">0.34900</td><td char=\".\" align=\"char\">0.32229</td><td char=\".\" align=\"char\">0.3227</td><td char=\".\" align=\"char\">− 0.13 </td></tr><tr><td char=\".\" align=\"char\">0.12178</td><td align=\"left\">0.21015</td><td char=\".\" align=\"char\">0.19578</td><td char=\".\" align=\"char\">0.1957</td><td char=\".\" align=\"char\">0.04</td></tr><tr><td char=\".\" align=\"char\">0.24469</td><td align=\"left\">0.13533</td><td char=\".\" align=\"char\">0.12726</td><td char=\".\" align=\"char\">0.1262</td><td char=\".\" align=\"char\">0.84</td></tr><tr><td char=\".\" align=\"char\">0.35601</td><td align=\"left\">0.11382</td><td char=\".\" align=\"char\">0.10768</td><td char=\".\" align=\"char\">0.107</td><td char=\".\" align=\"char\">0.64</td></tr><tr><td char=\".\" align=\"char\">0.66166</td><td align=\"left\">0.08489</td><td char=\".\" align=\"char\">0.08133</td><td char=\".\" align=\"char\">0.08165</td><td char=\".\" align=\"char\">− 0.40 </td></tr><tr><td char=\".\" align=\"char\">0.7789</td><td align=\"left\">0.07950</td><td char=\".\" align=\"char\">0.07605</td><td char=\".\" align=\"char\">0.07574</td><td char=\".\" align=\"char\">0.41</td></tr><tr><td char=\".\" align=\"char\">0.96413</td><td align=\"left\">0.07051</td><td char=\".\" align=\"char\">0.06852</td><td char=\".\" align=\"char\">0.06839</td><td char=\".\" align=\"char\">0.20</td></tr><tr><td char=\".\" align=\"char\">1.17323</td><td align=\"left\">0.06331</td><td char=\".\" align=\"char\">0.06250</td><td char=\".\" align=\"char\">0.06205</td><td char=\".\" align=\"char\">0.73</td></tr><tr><td char=\".\" align=\"char\">1.3325</td><td align=\"left\">0.05828</td><td char=\".\" align=\"char\">0.05798</td><td char=\".\" align=\"char\">0.05813</td><td char=\".\" align=\"char\">−0.25  </td></tr><tr><td char=\".\" align=\"char\">1.4081</td><td align=\"left\">0.05662</td><td char=\".\" align=\"char\">0.05648</td><td char=\".\" align=\"char\">0.05651</td><td char=\".\" align=\"char\"> − 0.06</td></tr><tr><td align=\"left\" rowspan=\"11\">30 wt% ZrO₂/PMMA</td><td char=\".\" align=\"char\">0.05953</td><td align=\"left\">1.03767</td><td char=\".\" align=\"char\">0.94591</td><td char=\".\" align=\"char\">0.9514</td><td char=\".\" align=\"char\">−0.58  </td></tr><tr><td char=\".\" align=\"char\">0.08099</td><td align=\"left\">0.52244</td><td char=\".\" align=\"char\">0.47889</td><td char=\".\" align=\"char\">0.4771</td><td char=\".\" align=\"char\">0.38</td></tr><tr><td char=\".\" align=\"char\">0.12178</td><td align=\"left\">0.26084</td><td char=\".\" align=\"char\">0.24077</td><td char=\".\" align=\"char\">0.2393</td><td char=\".\" align=\"char\">0.61</td></tr><tr><td char=\".\" align=\"char\">0.24469</td><td align=\"left\">0.13943</td><td char=\".\" align=\"char\">0.12995</td><td char=\".\" align=\"char\">0.1292</td><td char=\".\" align=\"char\">0.58</td></tr><tr><td char=\".\" align=\"char\">0.35601</td><td align=\"left\">0.11175</td><td char=\".\" align=\"char\">0.10554</td><td char=\".\" align=\"char\">0.1065</td><td char=\".\" align=\"char\">−0.90  </td></tr><tr><td char=\".\" align=\"char\">0.66166</td><td align=\"left\">0.08412</td><td char=\".\" align=\"char\">0.08047</td><td char=\".\" align=\"char\">0.08008</td><td char=\".\" align=\"char\">0.49</td></tr><tr><td char=\".\" align=\"char\">0.7789</td><td align=\"left\">0.07646</td><td char=\".\" align=\"char\">0.07388</td><td char=\".\" align=\"char\">0.07418</td><td char=\".\" align=\"char\">− 0.41 </td></tr><tr><td char=\".\" align=\"char\">0.96413</td><td align=\"left\">0.06906</td><td char=\".\" align=\"char\">0.06728</td><td char=\".\" align=\"char\">0.06691</td><td char=\".\" align=\"char\">0.56</td></tr><tr><td char=\".\" align=\"char\">1.17323</td><td align=\"left\">0.06165</td><td char=\".\" align=\"char\">0.06069</td><td char=\".\" align=\"char\">0.06067</td><td char=\".\" align=\"char\">0.03</td></tr><tr><td char=\".\" align=\"char\">1.3325</td><td align=\"left\">0.05738</td><td char=\".\" align=\"char\">0.05673</td><td char=\".\" align=\"char\">0.05684</td><td char=\".\" align=\"char\">−0.20  </td></tr><tr><td char=\".\" align=\"char\">1.4081</td><td align=\"left\">0.05481</td><td char=\".\" align=\"char\">0.05475</td><td char=\".\" align=\"char\">0.05527</td><td char=\".\" align=\"char\">−0.94  </td></tr><tr><td align=\"left\" rowspan=\"11\">45 wt% ZrO₂/PMMA</td><td char=\".\" align=\"char\">0.05953</td><td align=\"left\">1.44818</td><td char=\".\" align=\"char\">1.32409</td><td char=\".\" align=\"char\">1.336</td><td char=\".\" align=\"char\">−0.89  </td></tr><tr><td char=\".\" align=\"char\">0.08099</td><td align=\"left\">0.68241</td><td char=\".\" align=\"char\">0.62912</td><td char=\".\" align=\"char\">0.6315</td><td char=\".\" align=\"char\">−0.38  </td></tr><tr><td char=\".\" align=\"char\">0.12178</td><td align=\"left\">0.30566</td><td char=\".\" align=\"char\">0.28365</td><td char=\".\" align=\"char\">0.2829</td><td char=\".\" align=\"char\">0.27</td></tr><tr><td char=\".\" align=\"char\">0.24469</td><td align=\"left\">0.13988</td><td char=\".\" align=\"char\">0.13114</td><td char=\".\" align=\"char\">0.1322</td><td char=\".\" align=\"char\">−0.80  </td></tr><tr><td char=\".\" align=\"char\">0.35601</td><td align=\"left\">0.11304</td><td char=\".\" align=\"char\">0.10630</td><td char=\".\" align=\"char\">0.1059</td><td char=\".\" align=\"char\">0.37</td></tr><tr><td char=\".\" align=\"char\">0.66166</td><td align=\"left\">0.08303</td><td char=\".\" align=\"char\">0.07915</td><td char=\".\" align=\"char\">0.07851</td><td char=\".\" align=\"char\">0.81</td></tr><tr><td char=\".\" align=\"char\">0.7789</td><td align=\"left\">0.07605</td><td char=\".\" align=\"char\">0.07279</td><td char=\".\" align=\"char\">0.07262</td><td char=\".\" align=\"char\">0.23</td></tr><tr><td char=\".\" align=\"char\">0.96413</td><td align=\"left\">0.06754</td><td char=\".\" align=\"char\">0.06528</td><td char=\".\" align=\"char\">0.06543</td><td char=\".\" align=\"char\">−0.23  </td></tr><tr><td char=\".\" align=\"char\">1.17323</td><td align=\"left\">0.06105</td><td char=\".\" align=\"char\">0.05950</td><td char=\".\" align=\"char\">0.05928</td><td char=\".\" align=\"char\">0.38</td></tr><tr><td char=\".\" align=\"char\">1.3325</td><td align=\"left\">0.05624</td><td char=\".\" align=\"char\">0.05546</td><td char=\".\" align=\"char\">0.05556</td><td char=\".\" align=\"char\">−0.18  </td></tr><tr><td char=\".\" align=\"char\">1.4081</td><td align=\"left\">0.05406</td><td char=\".\" align=\"char\">0.05373</td><td char=\".\" align=\"char\">0.05402</td><td char=\".\" align=\"char\">−0.54  </td></tr></tbody></table></table-wrap>" ]

[ "<inline-formula id=\"IEq1\"><alternatives><tex-math id=\"M1\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${{\\text{cm}}}^{3}$$\\end{document}</tex-math><mml:math id=\"M2\"><mml:msup><mml:mrow><mml:mtext>cm</mml:mtext></mml:mrow><mml:mn>3</mml:mn></mml:msup></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ1\"><label>1</label><alternatives><tex-math id=\"M3\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\mu =1/t \\,{\\text{ln}}\\frac{{I}_{0}}{I},$$\\end{document}</tex-math><mml:math id=\"M4\" display=\"block\"><mml:mrow><mml:mi>μ</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">/</mml:mo><mml:mi>t</mml:mi><mml:mspace width=\"0.166667em\"/><mml:mtext>ln</mml:mtext><mml:mfrac><mml:msub><mml:mi>I</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mi>I</mml:mi></mml:mfrac><mml:mo>,</mml:mo></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq2\"><alternatives><tex-math id=\"M5\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$)$$\\end{document}</tex-math><mml:math id=\"M6\"><mml:mo stretchy=\"false\">)</mml:mo></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ2\"><label>2</label><alternatives><tex-math id=\"M7\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$MAC=\\frac{\\mu }{{\\rho }_{s} },$$\\end{document}</tex-math><mml:math id=\"M8\" display=\"block\"><mml:mrow><mml:mi>M</mml:mi><mml:mi>A</mml:mi><mml:mi>C</mml:mi><mml:mo>=</mml:mo><mml:mfrac><mml:mi>μ</mml:mi><mml:msub><mml:mi>ρ</mml:mi><mml:mi>s</mml:mi></mml:msub></mml:mfrac><mml:mo>,</mml:mo></mml:mrow></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ3\"><label>3</label><alternatives><tex-math id=\"M9\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\rho }_{s}=\\frac{({m}_{1})}{\\left({m}_{1}-{m}_{2}\\right)}\\cdot {\\rho }_{1},$$\\end{document}</tex-math><mml:math id=\"M10\" display=\"block\"><mml:mrow><mml:msub><mml:mi>ρ</mml:mi><mml:mi>s</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mfrac><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>m</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mfenced close=\")\" open=\"(\"><mml:msub><mml:mi>m</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>-</mml:mo><mml:msub><mml:mi>m</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:mfenced></mml:mfrac><mml:mo>·</mml:mo><mml:msub><mml:mi>ρ</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>,</mml:mo></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq3\"><alternatives><tex-math id=\"M11\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\rho }_{1}$$\\end{document}</tex-math><mml:math id=\"M12\"><mml:msub><mml:mi>ρ</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq4\"><alternatives><tex-math id=\"M13\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${m}_{1}$$\\end{document}</tex-math><mml:math id=\"M14\"><mml:msub><mml:mi>m</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ4\"><label>4</label><alternatives><tex-math id=\"M15\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$HVL=\\frac{{\\text{ln}}2}{\\mu },$$\\end{document}</tex-math><mml:math id=\"M16\" display=\"block\"><mml:mrow><mml:mi>H</mml:mi><mml:mi>V</mml:mi><mml:mi>L</mml:mi><mml:mo>=</mml:mo><mml:mfrac><mml:mrow><mml:mtext>ln</mml:mtext><mml:mn>2</mml:mn></mml:mrow><mml:mi>μ</mml:mi></mml:mfrac><mml:mo>,</mml:mo></mml:mrow></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ5\"><label>5</label><alternatives><tex-math id=\"M17\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$TVL=\\frac{{\\text{ln}}10}{\\mu }.$$\\end{document}</tex-math><mml:math id=\"M18\" display=\"block\"><mml:mrow><mml:mi>T</mml:mi><mml:mi>V</mml:mi><mml:mi>L</mml:mi><mml:mo>=</mml:mo><mml:mfrac><mml:mrow><mml:mtext>ln</mml:mtext><mml:mn>10</mml:mn></mml:mrow><mml:mi>μ</mml:mi></mml:mfrac><mml:mo>.</mml:mo></mml:mrow></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ6\"><label>6</label><alternatives><tex-math id=\"M19\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$MFP=\\frac{1}{\\mu }.$$\\end{document}</tex-math><mml:math id=\"M20\" display=\"block\"><mml:mrow><mml:mi>M</mml:mi><mml:mi>F</mml:mi><mml:mi>P</mml:mi><mml:mo>=</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:mi>μ</mml:mi></mml:mfrac><mml:mo>.</mml:mo></mml:mrow></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ7\"><label>7</label><alternatives><tex-math id=\"M21\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\sigma }_{a}=\\frac{( \\frac{\\mu }{\\rho } )}{N{\\Sigma }_{i}\\frac{{w}_{i}}{{A}_{i}}},$$\\end{document}</tex-math><mml:math id=\"M22\" display=\"block\"><mml:mrow><mml:msub><mml:mi>σ</mml:mi><mml:mi>a</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mfrac><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mfrac><mml:mi>μ</mml:mi><mml:mi>ρ</mml:mi></mml:mfrac><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mrow><mml:mi>N</mml:mi><mml:msub><mml:mi mathvariant=\"normal\">Σ</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mfrac><mml:msub><mml:mi>w</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:msub><mml:mi>A</mml:mi><mml:mi>i</mml:mi></mml:msub></mml:mfrac></mml:mrow></mml:mfrac><mml:mo>,</mml:mo></mml:mrow></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ8\"><label>8</label><alternatives><tex-math id=\"M23\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\sigma }_{e}= \\frac{1}{N}{\\Sigma }_{i}\\left(\\frac{\\mu }{\\rho }\\right)\\frac{{f}_{i}{A}_{i}}{{z}_{i}},$$\\end{document}</tex-math><mml:math id=\"M24\" display=\"block\"><mml:mrow><mml:msub><mml:mi>σ</mml:mi><mml:mi>e</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:mi>N</mml:mi></mml:mfrac><mml:msub><mml:mi mathvariant=\"normal\">Σ</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mfenced close=\")\" open=\"(\"><mml:mfrac><mml:mi>μ</mml:mi><mml:mi>ρ</mml:mi></mml:mfrac></mml:mfenced><mml:mfrac><mml:mrow><mml:msub><mml:mi>f</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:msub><mml:mi>A</mml:mi><mml:mi>i</mml:mi></mml:msub></mml:mrow><mml:msub><mml:mi>z</mml:mi><mml:mi>i</mml:mi></mml:msub></mml:mfrac><mml:mo>,</mml:mo></mml:mrow></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ9\"><label>9</label><alternatives><tex-math id=\"M25\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${Z}_{eff}=\\frac{{\\sigma }_{a}}{{\\sigma }_{e}}.$$\\end{document}</tex-math><mml:math id=\"M26\" display=\"block\"><mml:mrow><mml:msub><mml:mi>Z</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">eff</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mfrac><mml:msub><mml:mi>σ</mml:mi><mml:mi>a</mml:mi></mml:msub><mml:msub><mml:mi>σ</mml:mi><mml:mi>e</mml:mi></mml:msub></mml:mfrac><mml:mo>.</mml:mo></mml:mrow></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ10\"><label>10</label><alternatives><tex-math id=\"M27\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${{\\text{N}}}_{eff}=N\\frac{{{\\text{Z}}}_{eff}}{{{\\Sigma }_{i}f}_{i}{A}_{i}}.$$\\end{document}</tex-math><mml:math id=\"M28\" display=\"block\"><mml:mrow><mml:msub><mml:mtext>N</mml:mtext><mml:mrow><mml:mi mathvariant=\"italic\">eff</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mi>N</mml:mi><mml:mfrac><mml:msub><mml:mtext>Z</mml:mtext><mml:mrow><mml:mi mathvariant=\"italic\">eff</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:msub><mml:mrow><mml:msub><mml:mi mathvariant=\"normal\">Σ</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mi>f</mml:mi></mml:mrow><mml:mi>i</mml:mi></mml:msub><mml:msub><mml:mi>A</mml:mi><mml:mi>i</mml:mi></mml:msub></mml:mrow></mml:mfrac><mml:mo>.</mml:mo></mml:mrow></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ11\"><label>11</label><alternatives><tex-math id=\"M29\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${Z}_{eq}=\\frac{{Z}_{1}\\left({\\text{log}}{R}_{2}-{\\text{log}}R\\right)+{Z}_{2}({\\text{log}}R-{\\text{log}}{R}_{1})}{({\\text{log}}{R}_{2}-{\\text{log}}{R}_{1})},$$\\end{document}</tex-math><mml:math id=\"M30\" display=\"block\"><mml:mrow><mml:msub><mml:mi>Z</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">eq</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mfrac><mml:mrow><mml:msub><mml:mi>Z</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mfenced close=\")\" open=\"(\"><mml:mtext>log</mml:mtext><mml:msub><mml:mi>R</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mo>-</mml:mo><mml:mtext>log</mml:mtext><mml:mi>R</mml:mi></mml:mfenced><mml:mo>+</mml:mo><mml:msub><mml:mi>Z</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mtext>log</mml:mtext><mml:mi>R</mml:mi><mml:mo>-</mml:mo><mml:mtext>log</mml:mtext><mml:msub><mml:mi>R</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mtext>log</mml:mtext><mml:msub><mml:mi>R</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mo>-</mml:mo><mml:mtext>log</mml:mtext><mml:msub><mml:mi>R</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mfrac><mml:mo>,</mml:mo></mml:mrow></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ12\"><label>12</label><alternatives><tex-math id=\"M31\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$C=\\frac{{c}_{1}\\left({\\text{log}}{Z}_{2}-{\\text{log}}{Z}_{eq}\\right)+{c}_{2}\\left({\\text{log}}{Z}_{eq}-{\\text{log}}{Z}_{1}\\right)}{({\\text{log}}{Z}_{2}-{\\text{log}}{Z}_{1})}.$$\\end{document}</tex-math><mml:math id=\"M32\" display=\"block\"><mml:mrow><mml:mi>C</mml:mi><mml:mo>=</mml:mo><mml:mfrac><mml:mrow><mml:msub><mml:mi>c</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mfenced close=\")\" open=\"(\"><mml:mtext>log</mml:mtext><mml:msub><mml:mi>Z</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mo>-</mml:mo><mml:mtext>log</mml:mtext><mml:msub><mml:mi>Z</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">eq</mml:mi></mml:mrow></mml:msub></mml:mfenced><mml:mo>+</mml:mo><mml:msub><mml:mi>c</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mfenced close=\")\" open=\"(\"><mml:mtext>log</mml:mtext><mml:msub><mml:mi>Z</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">eq</mml:mi></mml:mrow></mml:msub><mml:mo>-</mml:mo><mml:mtext>log</mml:mtext><mml:msub><mml:mi>Z</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:mfenced></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mtext>log</mml:mtext><mml:msub><mml:mi>Z</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mo>-</mml:mo><mml:mtext>log</mml:mtext><mml:msub><mml:mi>Z</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mfrac><mml:mo>.</mml:mo></mml:mrow></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ13\"><label>13</label><alternatives><tex-math id=\"M33\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$B\\left(E,x\\right) =\\left[ 1+\\frac{b-1}{k-1}\\left({k}^{x}-1\\right)\\right], k\\ne 1,$$\\end{document}</tex-math><mml:math id=\"M34\" display=\"block\"><mml:mrow><mml:mi>B</mml:mi><mml:mfenced close=\")\" open=\"(\"><mml:mi>E</mml:mi><mml:mo>,</mml:mo><mml:mi>x</mml:mi></mml:mfenced><mml:mo>=</mml:mo><mml:mfenced close=\"]\" open=\"[\"><mml:mn>1</mml:mn><mml:mo>+</mml:mo><mml:mfrac><mml:mrow><mml:mi>b</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mi>k</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:mfrac><mml:mfenced close=\")\" open=\"(\"><mml:msup><mml:mrow><mml:mi>k</mml:mi></mml:mrow><mml:mi>x</mml:mi></mml:msup><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mfenced></mml:mfenced><mml:mo>,</mml:mo><mml:mi>k</mml:mi><mml:mo>≠</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo></mml:mrow></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ14\"><label>14</label><alternatives><tex-math id=\"M35\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$B\\left(E,x\\right) =\\left[1+\\left( b-1\\right)x\\right] , k=1,$$\\end{document}</tex-math><mml:math id=\"M36\" display=\"block\"><mml:mrow><mml:mi>B</mml:mi><mml:mfenced close=\")\" open=\"(\"><mml:mi>E</mml:mi><mml:mo>,</mml:mo><mml:mi>x</mml:mi></mml:mfenced><mml:mo>=</mml:mo><mml:mfenced close=\"]\" open=\"[\"><mml:mn>1</mml:mn><mml:mo>+</mml:mo><mml:mfenced close=\")\" open=\"(\"><mml:mi>b</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mfenced><mml:mi>x</mml:mi></mml:mfenced><mml:mo>,</mml:mo><mml:mi>k</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo></mml:mrow></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ15\"><label>15</label><alternatives><tex-math id=\"M37\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$k\\left(E,x\\right) =\\left[c{x}^{a}+d\\frac{{\\text{tan}}h\\left(x/ {X}_{k}-2\\right )-{\\text{tan}}h\\left(-2\\right)}{1-{\\text{tan}}h\\left(-2\\right)}\\right]\\, for \\,x\\le 40 \\, {\\text{mfp}},$$\\end{document}</tex-math><mml:math id=\"M38\" display=\"block\"><mml:mrow><mml:mi>k</mml:mi><mml:mfenced close=\")\" open=\"(\"><mml:mi>E</mml:mi><mml:mo>,</mml:mo><mml:mi>x</mml:mi></mml:mfenced><mml:mo>=</mml:mo><mml:mfenced close=\"]\" open=\"[\"><mml:mi>c</mml:mi><mml:msup><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mi>a</mml:mi></mml:msup><mml:mo>+</mml:mo><mml:mi>d</mml:mi><mml:mfrac><mml:mrow><mml:mtext>tan</mml:mtext><mml:mi>h</mml:mi><mml:mfenced close=\")\" open=\"(\"><mml:mi>x</mml:mi><mml:mo stretchy=\"false\">/</mml:mo><mml:msub><mml:mi>X</mml:mi><mml:mi>k</mml:mi></mml:msub><mml:mo>-</mml:mo><mml:mn>2</mml:mn></mml:mfenced><mml:mo>-</mml:mo><mml:mtext>tan</mml:mtext><mml:mi>h</mml:mi><mml:mfenced close=\")\" open=\"(\"><mml:mo>-</mml:mo><mml:mn>2</mml:mn></mml:mfenced></mml:mrow><mml:mrow><mml:mn>1</mml:mn><mml:mo>-</mml:mo><mml:mtext>tan</mml:mtext><mml:mi>h</mml:mi><mml:mfenced close=\")\" open=\"(\"><mml:mo>-</mml:mo><mml:mn>2</mml:mn></mml:mfenced></mml:mrow></mml:mfrac></mml:mfenced><mml:mspace width=\"0.166667em\"/><mml:mi>f</mml:mi><mml:mi>o</mml:mi><mml:mi>r</mml:mi><mml:mspace width=\"0.166667em\"/><mml:mi>x</mml:mi><mml:mo>≤</mml:mo><mml:mn>40</mml:mn><mml:mspace width=\"0.166667em\"/><mml:mtext>mfp</mml:mtext><mml:mo>,</mml:mo></mml:mrow></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ16\"><label>16</label><alternatives><tex-math id=\"M39\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\Delta \\%=\\left[{MAC}_{exp.} - {MAC}_{XCOM}\\right)/{MAC}_{XCOM}]\\times 100.$$\\end{document}</tex-math><mml:math id=\"M40\" display=\"block\"><mml:mrow><mml:mi mathvariant=\"normal\">Δ</mml:mi><mml:mo>%</mml:mo><mml:mo>=</mml:mo><mml:mfenced close=\")\" open=\"[\"><mml:msub><mml:mrow><mml:mi mathvariant=\"italic\">MAC</mml:mi></mml:mrow><mml:mrow><mml:mi>e</mml:mi><mml:mi>x</mml:mi><mml:mi>p</mml:mi><mml:mo>.</mml:mo></mml:mrow></mml:msub><mml:mo>-</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant=\"italic\">MAC</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"italic\">XCOM</mml:mi></mml:mrow></mml:msub></mml:mfenced><mml:mo stretchy=\"false\">/</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant=\"italic\">MAC</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"italic\">XCOM</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo stretchy=\"false\">]</mml:mo><mml:mo>×</mml:mo><mml:mn>100</mml:mn><mml:mo>.</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ17\"><label>17</label><alternatives><tex-math id=\"M41\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\delta \\%=[({\\mu }_{\\mathrm{nano }}- {\\mu }_{{\\text{micro}}}/{\\mu }_{{\\text{micro}}}]\\times 100.$$\\end{document}</tex-math><mml:math id=\"M42\" display=\"block\"><mml:mrow><mml:mi>δ</mml:mi><mml:mo>%</mml:mo><mml:mo>=</mml:mo><mml:mo stretchy=\"false\">[</mml:mo><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>μ</mml:mi><mml:mi mathvariant=\"normal\">nano</mml:mi></mml:msub><mml:mo>-</mml:mo><mml:msub><mml:mi>μ</mml:mi><mml:mtext>micro</mml:mtext></mml:msub><mml:mo stretchy=\"false\">/</mml:mo><mml:msub><mml:mi>μ</mml:mi><mml:mtext>micro</mml:mtext></mml:msub><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow><mml:mo>×</mml:mo><mml:mn>100</mml:mn><mml:mo>.</mml:mo></mml:mrow></mml:math></alternatives></disp-formula>" ]

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{ "acronym": [], "definition": [] }

[ "<title>Introduction</title>", "<p id=\"Par3\">Antimicrobial stewardship aims to optimise drug use to prolong current therapeutic effectiveness and combat antimicrobial resistance (AMR)^{##REF##28882725##1##}. One key aspect of antimicrobial stewardship is the route of administration. It is common for critically ill patients to be given empirical intravenous (IV) antibiotic therapy upon admission due to rapid delivery, high bioavailability, and uncertainty surrounding a potential infection. Then later in the treatment regime once the patient is stabilized and the infection is understood, their antibiotics are often switched to an oral administration route. There is a well described focus to switch from IV-to-oral administration as early as possible and to use more oral drugs when appropriate, given they are often equally effective and can reduce side effects during prolonged exposure^{##REF##17090560##2##–##REF##35187565##5##}. In a range of infectious diseases that were traditionally treated with IV only (e.g., bacteremia, endocarditis, and bone and joint infections), recent studies have demonstrated that oral therapy can be non-inferior to IV in efficacy^{##REF##36394002##6##–##REF##34715060##11##}. Furthermore, reducing the unnecessary use of indwelling IV devices is a well established patient safety and infection prevention priority to minimise the risk of healthcare associated infections (HCAIs)^{##REF##31005606##12##}. Beyond the infection complications of IV catheters, oral administration is more comfortable for the patient, reduces nurses’ workload, and allows for easy discharge from the hospital. Furthermore, oral therapy is cheaper and more cost-effective^{##UREF##4##13##}. The UK Health Security Agency recently published national antimicrobial IV-to-oral switch (IVOS) criteria for early switching^{##UREF##5##14##}. The requirements were developed based on expert consensus and primarily revolve around the patient’s clinical and infection markers improving as well as specific points with regards to absorption, bioavailability, and infection type.</p>", "<p id=\"Par4\">Despite significant evidence, the uptake of early oral therapy remains low^{##REF##33335941##15##,##REF##35869753##16##}, and beyond guidelines^{##UREF##5##14##} there is a lack of research in IV-to-oral decision support systems. Given this, we decided to investigate if a machine learning based clinical decision support system (CDSS) could assist antibiotic switch decision making at the individual patient level. More specifically neural network models were developed to predict, based on routinely collected clinical parameters, whether a patient could be suitable for switching from IV-to-oral antibiotics on any given day. ICU data was utilised given it is widely available, comprehensive, and if a CDSS can be developed for critical patients then it can likely be adapted to less severe settings. Many CDSSs utilising machine learning have been developed to assist with other aspects of antimicrobial use^{##REF##28268133##17##–##UREF##6##19##}; however, limited clinical utilisation and adoption has been seen^{##REF##35603280##20##}. As such, when tackling this problem we wanted to ensure our CDSS solution was simple, fair, interpretable, and generalisable to maximise the ability for clinical translation. By simple we mean the model architecture can be understood by non-experts, while fair infers model performance is not biased to particular sensitive attributes or protected characteristics. Interpretability means predictions can more easily be understood, explained, and trusted. Finally, a model is generalisable if it can be applied to many healthcare settings with consistent performance. We imagine by providing individualised antibiotic switch estimations such a system could support patient-centric decisions and provide assurance on if switching could be appropriate or not in a given clinical context. Figure ##FIG##0##1## shows an overview of this research.</p>" ]

[ "<title>Methods</title>", "<title>Datasets</title>", "<p id=\"Par24\">Two publicly available large de-identified real-world clinical datasets containing routinely collected EHR information were used within this research. MIMIC-IV (4th version of the Medical Information Mart for Intensive Care database) which contains over 40,000 patients admitted to the Beth Israel Deaconess Medical Center (BIDMC) in Boston, Massachusetts between 2008 and 2019^{##UREF##7##21##,##REF##10851218##22##}, was used for feature selection, model optimisation and hold out testing. Meanwhile, the eICU Collaborative Research Database contains data for over 200,000 admissions to ICUs across the United States from 2014 to 2015^{##REF##10851218##22##–##UREF##8##24##}, was used for transfer learning to confirm generlisability. Our study complies with all the data use and ethical regulations required to access the datasets. For both datasets, the patient population was filtered to those who received IV and oral antibiotic treatment within the ICU (IV treatment was limited to less than 8 days). Unfortunately, the datasets used in this research do not contain explicit information on if, when, or why an IV-to-oral switch was considered. However, by utilising the available prescribing data and taking what the clinicians actually did as a label we can approximate the prescribing behaviour and train a machine learning model. We therefore focused on making a route of administration prediction for each day the patient was on antibiotics given clinical decisions regarding antimicrobial treatment are most often made on a daily basis. As such negative switch labels were defined as each day a patient was on IV antibiotics, while positive labels were defined as every other day (i.e., where the patient was on oral but not IV antibiotics). The antibiotic spectrum index (ASI) from^{##REF##28560946##25##} was used to assess the average breadth of activity of IV and oral treatment regimes. By looking at the ASI on the day before switching and the first day of only oral administration we can understand how a change in route of administration is most often associated with the ASI.</p>", "<title>Feature selection</title>", "<p id=\"Par25\">Our aim was to make a model that through utilising routinely available patient vitals could act as a starting point for the decision making process and flag when a switch could be considered for a particular patient. The latest UK Health Security Agency (UKHSA) IVOS criteria^{##UREF##5##14##} were analysed and ten related features were extracted from the datasets. Specifically: heart rate, respiratory rate, temperature, O2 saturation pulseoxymetry, systolic blood pressure, diastolic blood pressure, mean blood pressure, GCS motor response, GCS verbal response, and GCS motor eye opening (Supplementary Table ##SUPPL##0##1)##. White Cell Count and C-Reactive Protein were excluded due to data missingness, requirement for a blood test, and UKHSA guidelines stating that they should be considered but are not necessary for a switch. Other important aspects of the guidelines such as infection type and absorption status, were also not included as input features to the model as much of this data was unavailable or collected in a way that makes it unsuitable for machine learning. Furthermore, evidence surrounding these is constantly changing^{##UREF##3##8##,##REF##34715060##11##,##UREF##18##43##}. We aimed to create a simple, generalisable model that uses only routinely available patient data and has the potential to be used in many different healthcare settings. The Canonical Time-series Characteristics (Catch22) methodology^{##UREF##19##44##} (along with the mean and variance) was utilised through sktime^{##UREF##20##45##,##UREF##21##46##} to transform temporal data into daily tabular values. This was done for each specific day and the whole of the current stay. In addition, the difference between transformed values for a given day and the preceding day was calculated. SHapley Additive exPlanations (SHAP) values^{##UREF##9##26##} and a genetic algorithm^{##UREF##22##47##} were then used for feature selection. Specifically, an excessively large neural network with 851,729 trainable parameters was preliminary trained, SHAP values were calculated and those features with a value of greater than or equal to 0.5 were selected for use in the genetic algorithm. The genetic algorithm optimised for AUROC and was run twice. Once for a simple set of 5 features and the second without a limitation on the number of features. 10 iterations with 50 individuals and 25 iterations with 20 individuals were used respectively.</p>", "<title>Model development</title>", "<p id=\"Par26\">The MIMIC-IV EHR dataset was randomly split (50%, 50%) based on patients ICU ‘stay_id’ into a preprocessing and a hold-out set in order to generalise switching prescribing behavior and get a reliable unbiased estimate of the models performance given the selected hyperparameters and feature set. The preprocessing set was split randomly into training, validation, and testing sets for feature selection as discussed above with Pytorch^{##UREF##23##48##} used to create the neural networks. After feature selection, optuna^{##UREF##24##49##} with the objective of maximising the AUROC was used to select the models hyperparameters, and optimal alternative cutoff thresholds were determined from the preprocessing validation subset. Youden’s Index^{##REF##15405679##27##} was used to optimise the AUROC, while finding the point where precision, recall and the F1 score were equal was used as a stringent cutoff for reducing the FPR. Subsequently, once the features and models were finalised the unseen hold-out set was randomly split 10 times into stratified training, validation, and testing sets for evaluation. Specifically, 10 naive models based on the previously identified features and model hyperparameters were trained and the final performance of such models was evaluated. The synthetic minority oversampling technique^{##UREF##25##50##} was used during training to address label class imbalance. The Adam optimiser^{##UREF##26##51##} was used with binary cross entropy with logits loss. The training utilised 10 epochs, and the model with the greatest AUROC on the validation dataset was selected as the final model to obtain results on the unseen test set.</p>", "<title>Model evaluation</title>", "<p id=\"Par27\">Standard ML metrics were used to evaluate model performance. Specifically for the switch classification task the AUROC, accuracy, precision, TPR, FPR, F1 score, and Area Under the Precision Recall curve (AUPRC) were calculated. The standard deviation was calculated to indicate the variation in results. To provide a baseline for comparison two infection markers that are clearly defined within the latest guidelines^{##UREF##5##14##} were separately also used for predicting when switching could be appropriate in each patient. Specifically, their temperature must have been between 36 °C and 38 °C for the past 24 h and the Early Warning Score must be decreasing, upon which a switch would then be suggested for the rest of that patient’s stay. It was not possible to include every aspect of the guidelines due to many being ambiguous and not recorded within the data. However, it acts as a fair comparison to our models as it utilises similar patient data and actually contains additional information not fed into our models such as the inspired O2 fraction. The best performing final model and its respective hold-out split were used to break the distribution of labels and predictions down by IV treatment duration, to evaluate how predictions compare temporally to the real labels and discern when the model performed well vs poorly (Fig. ##FIG##1##2##). To understand the value of the models switch predictions and how they relate to patient outcomes, the difference in days between our predicted switch events and real switch events was calculated and the mean LOS and mortality outcomes were taken (Fig. ##FIG##2##3##, Supplementary Table ##SUPPL##0##4)##. Furthermore, we analysed whether there was a variation in the remaining ICU length of stay (LOS) for patients who remained on IV vs those who switched on that day (Supplementary Fig. ##SUPPL##0##3)##. This was done for dates with 2 to 7 days of previous IV treatment based on the dissimilarity between model predictions and labels on those days (Fig. ##FIG##1##2##). For statistical analysis the non-parametric Wilcoxon rank-sum (Mann-Whitney U) test with alpha set at 0.05 was used to test if the difference in means was statistical significant given the non-normal data distribution. Effect sizes were calculated using Cohen’s d method with pooled standard deviation. Models were evaluated using functions and metrics from the Scikit-learn and SciPy libraries^{##UREF##27##52##,##REF##32015543##53##}.</p>", "<p id=\"Par28\">To further validate findings, evaluations were performed on specific patient populations and infectious diseases within MIMIC. Antibiotics with incomplete oral absorption (bioavailability &lt; 90%) were determined through consultation with a pharmacist and a literature search on PubMed, the Electronic Medicines Compendium, and UpToDate. The final list of antibiotics with incomplete oral absorption is shown in Supplementary Table ##SUPPL##0##7##. Total parenteral nutrition was used as a proxy for poor oral absorption (malabsorption) while hospital ICD diagnostic codes were used to identify patients with UTI’s, pneumonia, and sepsis. These infections were chosen as they are highly prevalent in the dataset and UTI’s/pneumonia are commonly treated with oral antibiotics but sepsis sees less oral utilisation. Note that infection types in MIMIC are linked to the hospital stay ‘hadm_id’ only and not the specific ICU stay ‘stay_id’ as diagnoses are only coded for billing purposes upon hospital discharge. The best performing short and long models from the MIMIC hold-out set were then evaluated on data extracted from the eICU database via transfer learning to re-train and subsequently test the models. The same data processing pipeline was used for eICU and transfer learning utilised the same procedure as with evaluation on the MIMIC hold-out set except the models parameters were initialised with the best performing final MIMIC trained models.</p>", "<p id=\"Par29\">The best performing final short and long models trained on the MIMIC hold-out set were used for fairness and interpretability research. SimplEx^{##UREF##10##29##} was used as a post-hoc explanation methodology to extract similar patient examples, their importance, and the contribution of each feature for each example. To this extent first, the corpus and test latent representations are computed. SimplEx was then fitted and the integrated jacobian decomposition for a particular patient was calculated and displayed. To simplify visualisations only those examples with an importance greater than 0.1 are shown. To assess model fairness the demanding equalised odds (EO) metric was used given we want to acknowledge and ideally minimise false positives as well as obtain equal performance across sensitive attribute classes. We defined that EO was achieved for a given sensitive attribute group if the TPR was not less than 0.1 from the global average and the FPR was not greater than 0.1 from the global average. EO was assessed utilising the 1st threshold for the sensitive attributes age (grouped into brackets based on the nearest decade), sex, race, insurance, language, and marital status. Threshold optimisation^{##UREF##11##30##} was then employed to see if the models fairness could be improved. Specifically, the postprocessing thresholdoptimizer method from fairlearn^{##UREF##28##54##} was used with the balanced accuracy objective and either the equalised odds, FPR parity or TPR parity constraint.</p>", "<title>Reporting summary</title>", "<p id=\"Par30\">Further information on research design is available in the ##SUPPL##1##Nature Portfolio Reporting Summary## linked to this article.</p>" ]

[ "<title>Results</title>", "<title>Data</title>", "<p id=\"Par5\">8694 unique intensive care unit (ICU) stays, were extracted from the MIMIC dataset^{##UREF##7##21##,##REF##10851218##22##}, with 1,668 from eICU^{##REF##10851218##22##–##UREF##8##24##}. 10 clinical features were selected based on the UK antimicrobial IVOS criteria^{##UREF##5##14##} (Supplementary Table. ##SUPPL##0##1)##. Transformation of those temporally dynamic values into time series features as detailed in Section 4.1 resulted in 960 unique features for each day of each patient’s stay. Details of the MIMIC preprocessing, MIMIC hold-out, and eICU datasets are shown in Table ##TAB##0##1##. All the datasets are relatively equally balanced, however, the specific antibiotic utilisation distribution varies between MIMIC and eICU. Furthermore, eICU represents a more unwell population given the higher proportion of life-threatening infections such as sepsis.</p>", "<p id=\"Par6\">The antibiotic spectrum index (ASI)^{##REF##28560946##25##} demonstrates if an antibiotic treatment regime shows broad or narrow activity. Larger values indicate a broader spectrum, while smaller values correlate with more targeted activity. A statistically significant (<italic>p</italic>-value &lt; 0.01, statistic 1686390, alpha 0.05, effect size 0.87) difference was found between the mean ASI for IV and oral antibiotics upon switching (8.25 and 5.89 respectively) through the Wilcoxon rank-sum test. In addition, the majority of patients (70.03%) see a decrease in their treatments ASI upon switching, with a mean decrease of 23.04% although this was highly variable (Supplementary Fig. ##SUPPL##0##1)##.</p>", "<title>Feature and model optimisation</title>", "<p id=\"Par7\">The first excessively large neural network trained on the preprocessing training subset achieved an Area Under the Receiver Operating Characteristic curve (AUROC) of 0.76 on the preprocessing test subset. SHapley Additive exPlanations (SHAP) values^{##UREF##9##26##} were calculated and the top 98 features were selected for input into a genetic algorithm. The genetic algorithm produced two sets of features, one short set, containing only 5 features, and another longer set of 37. The short and long feature sets achieved an AUROC of 0.80 and 0.82 on the preprocessing test subset respectively. The final features for each set are shown in Supplementary Table. ##SUPPL##0##1## with the respective Catch22 time-series transformations listed in Supplementary Table. ##SUPPL##0##2##. During hyperparameter optimisation our objective was to find the most simple models whilst maintaining performance. For both feature sets, this was achieved with new less complex models being found to achieve the same AUROC. The final hyperparameters of each model are shown in Supplementary Table. ##SUPPL##0##3##. Finally, alternative cutoff thresholds were explored for both models to maximise the AUROC and minimise the FPR (Supplementary Fig. ##SUPPL##0##2)##. This then allows for a traffic light system to be employed at deployment for simplicity and interpretability (Fig. ##FIG##3##4##). Youden’s index^{##REF##15405679##27##} which optimises the AUROC, found the 1st cutoff point of 0.54 and 0.52 for the short and long models respectively. The point where precision, recall and the F1 score were equal acted as a 2nd stringent threshold. This cutoff point was 0.74 for the short models and 0.79 for the long models, resulting in a lower AUROC (0.70 and 0.74 respectively) on the preprocessing test subset but a superior false positive rate (FPR) (0.11 and 0.09 respectively versus 0.26 and 0.22 using Youden’s threshold).</p>", "<title>Model evaluation</title>", "<p id=\"Par8\">The final short models trained and tested on the hold-out set obtained a mean AUROC of 0.78 (SD 0.02), FPR 0.25 (SD 0.02) with the 1st Youden’s threshold, and a mean AUROC of 0.69 (SD 0.03), FPR 0.10 (SD 0.02) with the 2nd threshold. Meanwhile, the final long models achieved a mean AUROC of 0.80 (SD 0.01), FPR 0.25 (SD 0.04) with the 1st cutoff, and a mean AUROC of 0.75 (SD 0.02), FPR 0.10 (SD 0.03) with the 2nd cutoff. Further evaluation metrics for each model and threshold can be found in Table ##TAB##1##2##. For comparison, a baseline that utilised two clear infection markers (temperature and Early Warning Score) from the latest guidelines^{##UREF##5##14##} obtained worse results with an AUROC of 0.66, accuracy of 0.61, TPR of 0.75, and FPR of 0.43. Predictions and labels broken down by IV treatment duration (Fig. ##FIG##1##2##) shows that the majority of incorrect predictions occurred in the middle of IV treatment days when the models predicted to switch but the real label indicated the patient continued with IV. The short model on average predicted 70% and 38% of patients could switch earlier than they did with the 1st and 2nd thresholds respectively. Arguably the long model demonstrated a more balanced profile with 51/28% early, 38/41% agreement, and 11/31% late switch predictions with the 1st and 2nd thresholds. When the difference between the real and predicted switch event was minimal, mean patient LOS outcomes were reduced (Fig. ##FIG##2##3##). Furthermore, a statistically significant difference (Wilcoxon rank-sum test, alpha 0.05) in remaining LOS was observed between those who received oral versus those who had IV treatment, with 2, 3, and 4 prior days of IV treatment (oral mean, IV mean, <italic>p</italic>-value, statistic and effect size of 1.03;1.70; &lt; 0.01;555588;0.39, 0.91;1.89; &lt; 0.01;227473;0.56 and 0.95;2.02; &lt; 0.01;24572;0.57 respectively). No statistically significant differences were observed on the later days 5, 6, and 7 (Supplementary Fig. ##SUPPL##0##3)##. No mortality differences were observed due to imbalanced data (Supplementary Table. ##SUPPL##0##4)##.</p>", "<p id=\"Par9\">eICU is a different dataset from MIMIC covering distinct hospitals with a separate patient population and unique data distribution. These differences can often cause problems for machine learning models but allows us to validate our features and modeling approach on an external dataset. When applied to eICU data via transfer learning a mean AUROC of 0.72 (SD 0.02), 0.65 (SD 0.05), 0.72 (SD 0.02), 0.64 (SD 0.06), and a FPR of 0.24 (SD 0.04), 0.05 (SD 0.02), 0.24 (SD 0.04) and 0.06 (SD 0.03) was obtained for the short and long models 1st and 2nd thresholds respectively (Table ##TAB##1##2##). Both models outperformed the eICU baseline which obtained an AUROC of 0.55, accuracy of 0.67, TPR of 0.38, and FPR 0.28.</p>", "<p id=\"Par10\">Achieving target drug exposure against the pathogenic organism is important during antibiotic treatment and is often a concern when deciding to switch to oral administration^{##REF##33743592##28##}. For those patients who were on oral antibiotics with incomplete absorption a mean AUROC of 0.73 (SD 0.03), 0.67 (SD 0.05), 0.77 (SD 0.02), 0.73 (SD 0.03), and a FPR of 0.33 (SD 0.06), 0.12 (SD 0.04), 0.28 (SD 0.07) and 0.12 (SD 0.07) was achieved for the short and long models 1st and 2nd cutoffs respectively (Table ##TAB##1##2##).</p>", "<p id=\"Par11\">If patients have issues with enteral absorption, oral antibiotic therapy is less likely to be suitable^{##UREF##5##14##}. When tested on patients with poor absorption a mean AUROC of 0.76 (SD 0.10), 0.75 (SD 0.11), 0.75 (SD 0.07), 0.71 (SD 0.16), and a FPR of 0.48 (SD 0.20), 0.28 (SD 0.12), 0.43 (SD 0.14) and 0.12 (SD 0.12) was obtained for the short and long models 1st and 2nd thresholds respectively (Table ##TAB##1##2##).</p>", "<p id=\"Par12\">Results were then examined for patients with specific infections. For urinary tract infection (UTI) patients a mean AUROC of 0.77 (SD 0.03), 0.74 (SD 0.04), 0.78 (SD 0.02), 0.77 (SD 0.04), and an FPR of 0.33 (SD 0.03), 0.15 (SD 0.03), 0.31 (SD 0.04) and 0.13 (SD 0.05) was achieved for the short and long models 1st and 2nd cutoffs respectively (Table ##TAB##1##2##). When tested on patients with pneumonia a mean AUROC of 0.76 (SD 0.03), 0.76 (SD 0.03), 0.77 (SD 0.02), 0.74 (SD 0.04), and a FPR of 0.35 (SD 0.03), 0.16 (SD 0.04), 0.32 (SD 0.04) and 0.14 (SD 0.04) was obtained for the short and long models 1st and 2nd thresholds respectively (Table ##TAB##1##2##). Finally, for sepsis patients, a mean AUROC of 0.82 (SD 0.05), 0.79 (SD 0.12), 0.77 (SD 0.07), 0.76 (SD 0.18), and a FPR of 0.36 (SD 0.10), 0.17 (SD 0.09), 0.35 (SD 0.08) and 0.16 (SD 0.07) was achieved for the short and long models 1st and 2nd cutoffs respectively (Table ##TAB##1##2##).</p>", "<title>Interpretability</title>", "<p id=\"Par13\">Two cutoff thresholds allows for a simple traffic light system to be presented to clinicians with regards to if a switch could be appropriate at a particular time. To further improve interpretability and model understanding the SimplEx^{##UREF##10##29##} methodology was applied. Once fitted the decomposition for a particular patient was computed to get corpus examples, their importance, and feature contribution. This data was combined and infectious disease clinicians consulted to create informative visual representations. Figure ##FIG##3##4## shows an example of these for short model predictions.</p>", "<title>Fairness</title>", "<p id=\"Par14\">Overall the models demonstrated equalised odds (EO) across the majority of sensitive attribute groups. Table ##TAB##2##3## shows the AUROC, TPR, and FPR for both short and long models by sensitive attribute group. The short model did not obtain EO for those in the age bracket of 90, of Native American descendance, or with Medicaid insurance (Table ##TAB##2##3##). On the other hand, the long model only showed a discrepancy for patients in the age bracket of 30. For the short model, threshold optimisation^{##UREF##11##30##} with the true positive rate (TPR) parity constraint enabled EO to be achieved for those in the age bracket of 90, while the EO constraint standardised performance across insurance groups (Supplementary Table. ##SUPPL##0##5##, Supplementary Fig. ##SUPPL##0##4)##. No constraint enabled the model to demonstrate EO for the native group. For the long model, the FPR parity constraint caused EO to be obtained for those in the age bracket of 30 (Supplementary Table. ##SUPPL##0##6##, Supplementary Fig. ##SUPPL##0##4)##.</p>" ]

[ "<title>Discussion</title>", "<p id=\"Par15\">To maximise clinical utility we aimed to minimise complexity during feature selection and model development. Through the genetic algorithm, two feature sets of interest were identified. The short set utilised only 5 features but maintained performance, while the long set enabled slight improvements in the evaluation metrics. The two most important SHAP features utilised the same time series transformation (SB_MotifThree_quantile_hh) for systolic blood pressure over the whole ICU stay and heart rate over the current day respectively. This measure uses equiprobable binning to indicate the predictability of a time series. This is medically relevant to switching the administration route as clinicians look for vitals to stabilise before switching. Interestingly the 3rd and 4th SHAP ranked features represent the same type of feature (IN_AutoMutualInfoStats_40_gaussian_fmmi calculated over the whole of the ICU stay) for two different clinical parameters, respiratory rate, and the mean blood pressure. Furthermore, their feature values (shown in Supplementary Fig. ##SUPPL##0##5)## are very similar, indicating why having both features was likely redundant to the models. Other features in the short set also demonstrate clinical importance. For example, the first minimum of an O2 saturation pulseoximetry autocorrelation function (CO_FirstMin_ac) would indicate variable stability and hence clinical improvement or deterioration. While a value indicating the importance of low frequencies in the Glasgow Coma Scale (GCS) motor response (SP_Summaries_welch_rect_area_5_1), could show if the patient is retaining consciousness or not over a long period, which is often necessary for administering oral medication. Overall these features combine to provide a comprehensive but succinct overview of the general health status of the patient which can be used to determine if switching could be appropriate.</p>", "<p id=\"Par16\">Results on specific infections and antibiotic characteristics demonstrate the models have stable performance across numerous different patient groups. Particularly important is understanding when oral antibiotics with incomplete absorption can be used, given concerns surrounding achieving therapeutic concentrations. Our long model achieved an AUROC of 0.77 (SD 0.02) in this subpopulation. Furthermore, in conditions such as sepsis where patients are critically-ill for prolonged periods and fewer oral therapies are utilised, our short model obtained an AUROC of 0.82 (SD 0.02). Indicating that such a support system could be utilised in severe infections. Transfer learning results on the eICU dataset were stringent with regards to predicting when switching could be appropriate (Supplementary Fig. ##SUPPL##0##3)##, this is to be expected considering the patients in eICU are on average more severely unwell than in MIMIC (Table ##TAB##0##1##). Switching administration route is influenced by many behavioural factors that are not easily modeled. Given eICU contains data from many different hospitals the prescribing behaviour with regards to oral switching is likely much more heterogeneous than in MIMIC whose data is from a single institution. As such, the eICU model is having to approximate many different behaviours, which results in varying performance across institutions (Supplementary Fig. ##SUPPL##0##6)##, and likely causes it to be more stringent with regard to predicting a switch to optimise performance. Similar behavior is observed with the baseline eICU results which confirms predicting the route of administration is a more challenging task in eICU when compared to MIMIC (AUROC of 0.66 and 0.55 respectively). Further research into subpopulations and other datasets could identify unfavourable IV-to-oral switch characteristics, such as individuals with abnormal pharmacokinetics or immunosuppression. Specific thresholding or separate models^{##REF##35603280##31##} could then ensure patients with such attributes require a larger output to be flagged as suitable for switching. Combining this with alternative thresholds to ensure fairness though can very quickly make CDSSs excessively complex, leading to misunderstanding, misuse, and reluctant adoption^{##REF##28268133##17##,##REF##35603280##20##,##UREF##12##32##}. We believe this research strikes a practical balance between performance and usefulness for IV-to-oral switch decision support. Overall the results demonstrate our methods and models are generalisable as similar performance was obtained across all MIMIC tests with different patient populations, and between two distinct ICU datasets indicating the feature sets identified are informative and that the selected hyperparameters can model the underlying data.</p>", "<p id=\"Par17\">Overall the models demonstrated reasonably fair performance across all sensitive attribute groups. When equalised odds were not achieved, threshold optimisation^{##UREF##11##30##} was able to improve the results for a given group in all cases, except for that of the Native American group. This population was the most underrepresented within the data with an average of only 11 patients in the test set, highlighting the need for further good quality real or synthetic data on minority populations. When threshold optimisation was undertaken a trade-off between groups in a sensitive attribute class was sometimes observed. For example, the TPR parity constraint on the short model achieved EO for those in the age bracket of 90. However, it caused the FPR of those in the minority group around 20 years old to increase from 0.29 to 0.61 (Supplementary Table. ##SUPPL##0##5)##. This loss in performance for the 20-year-old group was also partially seen for the FPR parity constraint on the long model (Supplementary Table. ##SUPPL##0##6)##. This shows the importance of balance when considering if a model is defined as fair or not, in particular for drastically different patient populations, such as 90 versus 20-year-olds. Prioritising one group or sensitive attribute can hinder model performance in others. As such honest and decent precautions and analysis are needed to ensure algorithms are equal and reasonable without discrimination. Moreover, for antibiotic decision making further ethical considerations need to be taken into account including the effect on other individuals outside of the patient being treated^{##UREF##13##33##}. We believe this analysis demonstrates such CDSSs can be fair; however, further validation is certainly required.</p>", "<p id=\"Par18\">Two feature sets were used in this research to evaluate the trade-off between simplicity and explainability vs performance which has been widely discussed in the machine learning literature^{##UREF##14##34##}. Overall results show that the long model often demonstrates slightly superior performance to the short model. However, it is inherently more complex and in some scenarios such as in those with sepsis, it performs worse than the short model. Further research including understanding clinicians opinions is required to determine what model is most appropriate in specific circumstances. Alternative cutoff thresholds were also investigated for our binary classification task to maximise the AUROC and minimise the FPR. Results show that this was achieved for both the short and long models by fixing the thresholds from the preprocessing validation subset. With the 1st threshold achieving a reasonable AUROC and the 2nd threshold having a lower FPR, although as expected this comes at the expense of a worse AUROC score. We envisage such thresholds being utilised similar to a traffic light, whereby suggestions can be split into don’t, potentially, or do switch based on the model’s level of confidence (Fig. ##FIG##3##4##). This type of structure is simple, familiar to individuals and should ensure along with interpretability methods that such a model acts as an appropriate CDSS and allows for the end user to understand the output alongside other information in order to make the final decision.</p>", "<p id=\"Par19\">Explainability and interpretability are critical aspects of using machine learning models in the real world^{##UREF##15##35##,##UREF##16##36##}. To ensure our model and its outputs could be understood and interrogated SimplEx^{##UREF##10##29##} was utilised and visual representations created (Fig. ##FIG##3##4##). These visual summaries include a number of aspects that were noted as important for understanding by clinical colleagues. Firstly textual descriptions enable key information to be conveyed quickly and reduce the barrier to adoption through universal understanding. Secondly, related patient examples are shown and scored. Clinicians rely heavily on prior experience when undertaking antibiotic treatment decisions^{##UREF##17##37##}; as such, showing historical examples and how they compare to the current patient of interest is perceived as appropriate. In conjunction, highlighting whether the model was correct on previous examples at each threshold provides some level of reassurance on how well the model performs on this type of patient and therefore if the predictions should be trusted or not. Finally, patient-specific feature contribution can be shown to illustrate how the model arrived at that conclusion. Figure ##FIG##3##4## shows that while in many cases a clear switch decision is apparent, inherently some days (e.g., day 3) and patients present a particularly complex case. This reflects what is often seen in reality with decisions regarding antimicrobial switching not being clear-cut. By incorporating interpretability methods, models such as those developed in this research can become clinically useful CDSSs.</p>", "<p id=\"Par20\">The objective of a CDSS to support IV-to-oral switch decision making is to facilitate antimicrobial stewardship. ASI results are in-line with current literature indicating frequent oral prescribing may use less broad spectrum IV antibiotics overall and therefore could be beneficial from an AMR and HCAIs perspective^{##REF##31005606##12##,##REF##32346210##38##}. As such, this evidence supports the drive to maximise the use of oral therapies and alongside limited adoption^{##REF##33335941##15##,##REF##35869753##16##} highlights why a switch focused CDSS may be useful. It is however notoriously difficult to discern the value of predictions from a CDSS. A retrospective analysis was conducted to understand how such switch models may benefit healthcare institutions and patients. Figure ##FIG##1##2## shows for the first two days upon starting IV treatment our models predict that the majority of patients should not switch which corresponds with the true labels. This is in line with the latest UK guidelines whereby the IV-to-oral switch should be considered daily after 48 hours^{##UREF##5##14##}. For dates with 2 to 7 prior days of IV treatment though there develops a disconnect between the labels and model predictions. This is particularly apparent for the short model and the first more lenient threshold. Model outputs indicate that by day 4 almost all patients could be suitable for switching to oral administration from a clinical parameter, health status perspective. For some patients, there will be risk factors beyond the models input features that the clinician considered meaning they did not switch, but for others, the clinician may have been unaware or neglected the decision meaning switching earlier may have been suitable. Furthermore, results show that LOS is minimised when predictions and the true labels align, and upon switching patients usually see prompt discharge. Our models may therefore be able to provide useful decision support by raising awareness of when switching could be suitable for a particular patient. Given this decision is often neglected and postponed, such a CDSS may be able to promote switching when appropriate which could potentially support efforts to stop AMR, prevent HCAIs, and benefit patients.</p>", "<p id=\"Par21\">To improve the clinical applicability of our solution a number of logic-based rules could be implemented. For example, if a patient has a certain type of infection, malabsorption, immunosuppression, has recently vomited, or could have compliant issues, an overriding rule based on the latest guidelines^{##UREF##5##14##} could suggest not to switch. Furthermore, the number of days of IV treatment should be highlighted alongside conditions, such as sepsis, in which extra care should be taken, as these factors influence switch decision making. If a patient is receiving an IV antibiotic and there is a similar oral version available this could be flagged alongside model outputs as a ‘simple’ switch. Moreover, given the potential comfort, workload, and discharge benefits when patients have no IV catheters, CDSSs should consider the wider patient treatment paradigm, and potentially further encourage switching when IV access is only for antibiotic treatment. Finally, to improve practice it is important for clinicians to document when a switch occurred and why that decision was made. This ensures in the future such individualised antibiotic decision making can be data-driven based on real evidence, rather than decided by habit or general population evidence. By combining machine learning approaches with clinical logic we can ensure patient safety while driving a positive change in antimicrobial utilisation. In the future we will conduct further research on how such solutions could be combined and implemented in real-time to create a complete CDSS for antibiotic optimisation, that is well received by the clinical community and provides novel, useful information.</p>", "<p id=\"Par22\">There are limitations to this research study. Firstly, the use of historical patient data means that all of our models predictions are based on historical prescribing practices. Due to concerns surrounding AMR, there has been a large amount of research into antibiotic prescribing over recent years^{##REF##28268133##17##,##REF##26603922##39##–##REF##36065724##42##} and hence it is plausible our models switch suggestions are ‘out of date’. Secondly, our model only analyses a snapshot of the patient and not all the factors that are clinically used to assess a patient’s suitability for switching^{##UREF##5##14##}. As discussed in the methods, this is due to data challenges, but incorporating additional criteria into the model so that under certain circumstances a switch suggestion cannot be given is an avenue for future work. However, we believe by anayzing and summarising multiple variables regarding the patients clinical and infection status such a system could support switch decision making with the final decision always made by the clinician. Finally, the current work presented only evaluates such models on US based ICU data. How such a system could perform in other medical settings and health-systems such as infectious diseases wards, the UK’s NHS and low and middle income countries remains an outstanding question. But given the results presented and the routine, standardised nature of the raw input data we believe our approach is generalisable and there is potential to translate this research into other non-ICU medical settings where oral therapy may be more commonly utilised.</p>", "<p id=\"Par23\">In summary, we have identified clinically relevant features and developed simple, fair, interpretable, and generalisable models to estimate when a patient could switch from IV-to-oral antibiotic treatment. In the future, this research will require further analysis and prospective evaluation to understand its safety, clinical benefit, and how it can influence antimicrobial decision making. But given AMR, HCAIs, and the interest in promoting oral therapies, such a system holds great promise to provide clinically useful antimicrobial decision support.</p>" ]

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[ "<p id=\"Par1\">Antimicrobial resistance (AMR) and healthcare associated infections pose a significant threat globally. One key prevention strategy is to follow antimicrobial stewardship practices, in particular, to maximise targeted oral therapy and reduce the use of indwelling vascular devices for intravenous (IV) administration. Appreciating when an individual patient can switch from IV to oral antibiotic treatment is often non-trivial and not standardised. To tackle this problem we created a machine learning model to predict when a patient could switch based on routinely collected clinical parameters. 10,362 unique intensive care unit stays were extracted and two informative feature sets identified. Our best model achieved a mean AUROC of 0.80 (SD 0.01) on the hold-out set while not being biased to individuals protected characteristics. Interpretability methodologies were employed to create clinically useful visual explanations. In summary, our model provides individualised, fair, and interpretable predictions for when a patient could switch from IV-to-oral antibiotic treatment. Prospectively evaluation of safety and efficacy is needed before such technology can be applied clinically.</p>", "<p id=\"Par2\">The decision to switch patients from intravenous to oral antibiotic therapy is important for the individual and wider society. Here, authors show a machine learning model using routine clinical data can predict when a patient could switch.</p>", "<title>Subject terms</title>" ]

[ "<title>Supplementary information</title>", "<p>\n\n\n\n</p>", "<title>Source data</title>", "<p>\n\n</p>" ]

[ "<title>Supplementary information</title>", "<p>The online version contains supplementary material available at 10.1038/s41467-024-44740-2.</p>", "<title>Acknowledgements</title>", "<p>William Bolton was supported by the UKRI CDT in AI for Healthcare <ext-link ext-link-type=\"uri\" xlink:href=\"http://ai4health.io\">http://ai4health.io</ext-link> (Grant No. P/S023283/1) and by the National Institute for Health, Health Protection Research Unit in Healthcare Associated Infections and Antimicrobial Resistance at Imperial College London in partnership with the UK Health Security Agency (previously PHE), in collaboration with, Imperial Healthcare Partners, University of Cambridge and University of Warwick. He is also affiliated to the Department of Health and Social Care, Centre for Antimicrobial Optimisation. The authors would like to acknowledge (1) the National Institute for Health Research Health Protection Research Unit (NIHR HPRU) in Healthcare Associated Infection and Antimicrobial Resistance at Imperial College London and (2) The Department for Health and Social Care funded Centre for Antimicrobial Optimisation (CAMO) at Imperial College London. This study is independent research partly funded by the National Institute for Health Research. The views expressed in this publication are those of the authors and not necessarily those of the NHS, the National Institute for Health Research the Department of Health and Social Care or the UK Health Security Agency.</p>", "<title>Author contributions</title>", "<p>W.B., R.W., and T.M.R. contributed to study concept and design. W.B. contributed to data acquisition and analysis. W.B. and T.M.R. contributed to the manuscript drafting, discussion of the results, and review of the data. All authors contributed to data interpretation, as well as final revisions of the manuscript. All authors had full access to all the data in the study and had final responsibility for the decision to submit for publication.</p>", "<title>Peer review</title>", "<title>Peer review information</title>", "<p id=\"Par31\"><italic>Nature Communications</italic> thanks the anonymous reviewers for their contribution to the peer review of this work. A peer review file is available.</p>", "<title>Data availability</title>", "<p>Publicly available datasets were analyzed in this study. The MIMIC-IV dataset can be found at <ext-link ext-link-type=\"uri\" xlink:href=\"https://physionet.org/content/mimiciv/2.0/\">https://physionet.org/content/mimiciv/2.0/</ext-link>and the eICU dataset at <ext-link ext-link-type=\"uri\" xlink:href=\"https://physionet.org/content/eicu-crd/2.0/\">https://physionet.org/content/eicu-crd/2.0/</ext-link>. Both are accessible once you are a credentialed user on physionet, have completed the required training and signed the appropriate data use agreement. Specific additional data can be provided upon request to the authors, provided that it is in line with the datasets data use and ethical regulations. <xref ref-type=\"sec\" rid=\"Sec16\">Source data</xref> are provided with this paper.</p>", "<title>Code availability</title>", "<p>The computer code used in this research is available at <ext-link ext-link-type=\"uri\" xlink:href=\"https://github.com/WilliamBolton/iv_to_oral\">https://github.com/WilliamBolton/iv_to_oral</ext-link>^{##UREF##29##55##}.</p>", "<title>Competing interests</title>", "<p id=\"Par32\">Author T.M.R. was employed by Sandoz (2020), Roche Diagnostics Ltd (2021), and bioMerieux (2021–2022). These commercial entities were not involved in the study design, collection, analysis, interpretation of data, the writing of this article or the decision to submit it for publication. All authors declare no other competing interests.</p>" ]

[ "<fig id=\"Fig1\"><label>Fig. 1</label><caption><title>Overview of the steps taken in this research study to develop fair and interpretable machine learning models for antimicrobial switch decision making.</title><p>MIMIC Medical Information Mart for Intensive Care, ICU Intensive care unit, IV Intravenous, IVOS Intravenous-to-oral switch, SHAP SHapley Additive exPlanations, UTI Urinary tract infection.</p></caption></fig>", "<fig id=\"Fig2\"><label>Fig. 2</label><caption><title>Labels and predictions by IV treatment duration.</title><p>Plots for the short model <bold>A</bold> and the long model <bold>B</bold>. IV Intravenous.</p></caption></fig>", "<fig id=\"Fig3\"><label>Fig. 3</label><caption><title>Mean patient LOS outcomes by days between the real and predicted switch event.</title><p>Plots for the short model <bold>A</bold> and the long model <bold>B</bold>. A negative number on the <italic>x</italic> axis indicates the predicted switch event was before the real switch event. The opposite is true for positive numbers, while 0 means they occurred on the same day. IV Intravenous, LOS Length of stay.</p></caption></fig>", "<fig id=\"Fig4\"><label>Fig. 4</label><caption><title>Example visual representation for a particular patient.</title><p>`Traffic light' suggestions are initially displayed in a temporal manner as the patient progresses. If required clinicians can obtain more information for any given day of the patient’s stay. Simple textual descriptions are provided alongside detailed tabular graphics to maximise clarity. The text quickly gets across key points, while tables show similar example patients and their features, both with their relative importance and contribution respectively. Furthermore, switch labels and predictions across both thresholds are displayed. Note that the patients label would obviously not be available during clinical use, but is shown here to be comprehensive. Finally, the limitations and use cases for the system are clearly labeled. ICU Intensive care unit, IV Intravenous.</p></caption></fig>" ]

[ "<table-wrap id=\"Tab1\"><label>Table 1</label><caption><p>Dataset demographics and statistics</p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th/><th/><th colspan=\"3\">Dataset</th></tr><tr><th colspan=\"2\">Statistic</th><th>MIMIC preprocessing</th><th>MIMIC hold-out</th><th>eICU</th></tr></thead><tbody><tr><td colspan=\"2\">Number of stays</td><td>4347</td><td>4347</td><td>1668</td></tr><tr><td colspan=\"2\">Age (mean)</td><td>65.30 (SD 15.16)</td><td>65.44 (SD 15.23)</td><td>64.74 (SD 15.91)</td></tr><tr><td colspan=\"2\">Length of stay (mean)</td><td>3.14 (SD 2.78)</td><td>3.12 (SD 2.71)</td><td>3.17 (SD 2.83)</td></tr><tr><td>Sex (%)</td><td>Male</td><td>58.82</td><td>58.92</td><td>50.30</td></tr><tr><td/><td>Female</td><td>41.18</td><td>41.08</td><td>49.70</td></tr><tr><td>Race (%)</td><td>White</td><td>67.97</td><td>68.09</td><td>78.77</td></tr><tr><td/><td>Black</td><td>9.55</td><td>9.92</td><td>15.08</td></tr><tr><td/><td>Unknown</td><td>10.28</td><td>9.35</td><td>3.50</td></tr><tr><td/><td>Other</td><td>6.22</td><td>5.48</td><td/></tr><tr><td/><td>Hispanic</td><td>3.27</td><td>3.94</td><td>1.39</td></tr><tr><td/><td>Asian</td><td>2.34</td><td>2.94</td><td>0.84</td></tr><tr><td/><td>Native American</td><td>0.36</td><td>0.27</td><td>0.42</td></tr><tr><td>Antimicrobial treatment length (mean)</td><td>Overall</td><td>3.34 (SD 2.16)</td><td>3.29 (SD 2.01)</td><td>2.97 (SD 1.94)</td></tr><tr><td/><td>IV</td><td>2.79 (SD 1.47)</td><td>2.75 (SD 1.46)</td><td>2.46 (SD 1.55)</td></tr><tr><td/><td>Oral</td><td>2.76 (SD 1.90)</td><td>2.69 (SD 1.81)</td><td>1.14 (SD 1.78)</td></tr><tr><td>Infection type (%, most common shown)</td><td>UTI</td><td>65.10</td><td>64.69</td><td>10.19</td></tr><tr><td/><td>Pneumonia</td><td>26.30</td><td>26.62</td><td>31.71</td></tr><tr><td/><td>Sepsis</td><td>18.20</td><td>19.63</td><td>32.91</td></tr><tr><td>IV antibiotics (%, those with a frequency of greater than 5% shown)</td><td>Vancomycin</td><td>35.96</td><td>35.56</td><td>41.98</td></tr><tr><td/><td>Cefepime</td><td>12.28</td><td>14.03</td><td>2.86</td></tr><tr><td/><td>Cefazolin</td><td>12.47</td><td>12.08</td><td>1.71</td></tr><tr><td/><td>Piperacillin-Tazobactam</td><td>9.35</td><td>8.42</td><td>8.92</td></tr><tr><td/><td>Ceftriaxone</td><td>7.82</td><td>8.04</td><td>6.10</td></tr><tr><td/><td>Levofloxacin</td><td>1.76</td><td>1.99</td><td>14.12</td></tr><tr><td>Oral antibiotics (%, those with a frequency of greater than 5% shown)</td><td>Azithromycin</td><td>22.96</td><td>23.18</td><td>12.24</td></tr><tr><td/><td>Vancomycin</td><td>13.45</td><td>14.23</td><td>12.37</td></tr><tr><td/><td>Ciprofloxacin</td><td>12.77</td><td>12.14</td><td>-</td></tr><tr><td/><td>Levofloxacin</td><td>12.69</td><td>12.37</td><td>51.70</td></tr><tr><td/><td>Sulfameth/Trimethoprim</td><td>11.82</td><td>10.04</td><td>-</td></tr><tr><td/><td>Metronidazole</td><td>7.90</td><td>9.32</td><td>22.72</td></tr><tr><td>Microbiology (%, those with a frequency of greater than 5% shown)</td><td>Positive growth</td><td>32.76</td><td>33.07</td><td>13.01</td></tr><tr><td/><td><italic>Escherichia coli</italic></td><td>6.86</td><td>6.51</td><td>-</td></tr><tr><td/><td><italic>Staphylococcus aureus</italic></td><td>6.69</td><td>7.33</td><td>-</td></tr></tbody></table></table-wrap>", "<table-wrap id=\"Tab2\"><label>Table 2</label><caption><p>Evaluation results</p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th/><th/><th/><th colspan=\"7\">Evaluation Metric</th></tr><tr><th>Dataset</th><th>Model</th><th>Threshold</th><th>AUROC</th><th>Accuracy</th><th>Precision</th><th>F1 Score</th><th>AUPRC</th><th>TPR</th><th>FPR</th></tr></thead><tbody><tr><td>MIMIC</td><td>Short</td><td>1st</td><td>0.78 (SD 0.02)</td><td>0.76 (SD 0.01)</td><td>0.39 (SD 0.02)</td><td>0.53 (SD 0.03)</td><td>0.35 (SD 0.03)</td><td>0.80 (SD 0.05)</td><td>0.25 (SD 0.02)</td></tr><tr><td/><td/><td>2nd</td><td>0.69 (SD 0.03)</td><td>0.83 (SD 0.01)</td><td>0.49 (SD 0.03)</td><td>0.48 (SD 0.04)</td><td>0.32 (SD 0.03)</td><td>0.48 (SD 0.06)</td><td>0.10 (SD 0.02)</td></tr><tr><td/><td>Long</td><td>1st</td><td>0.80 (SD 0.01)</td><td>0.77 (SD 0.03)</td><td>0.41 (SD 0.04)</td><td>0.55 (SD 0.03)</td><td>0.37 (SD 0.03)</td><td>0.85 (SD 0.04)</td><td>0.25 (SD 0.04)</td></tr><tr><td/><td/><td>2nd</td><td>0.75 (SD 0.02)</td><td>0.85 (SD 0.02)</td><td>0.55 (SD 0.05)</td><td>0.57 (SD 0.03)</td><td>0.40 (SD 0.03)</td><td>0.61 (SD 0.06)</td><td>0.10 (SD 0.03)</td></tr><tr><td>MIMIC Incomplete absorption</td><td>Short</td><td>1st</td><td>0.73 (SD 0.03)</td><td>0.72 (SD 0.03)</td><td>0.64 (SD 0.04)</td><td>0.71 (SD 0.03)</td><td>0.59 (SD 0.03)</td><td>0.79 (SD 0.06)</td><td>0.33 (SD 0.06)</td></tr><tr><td/><td/><td>2nd</td><td>0.67 (SD 0.05)</td><td>0.71 (SD 0.04)</td><td>0.74 (SD 0.06)</td><td>0.56 (SD 0.11)</td><td>0.57 (SD 0.05)</td><td>0.47 (SD 0.12)</td><td>0.12 (SD 0.04)</td></tr><tr><td/><td>Long</td><td>1st</td><td>0.77 (SD 0.02)</td><td>0.76 (SD 0.03)</td><td>0.68 (SD 0.05)</td><td>0.74 (SD 0.03)</td><td>0.63 (SD 0.04)</td><td>0.82 (SD 0.06)</td><td>0.28 (SD 0.07)</td></tr><tr><td/><td/><td>2nd</td><td>0.73 (SD 0.03)</td><td>0.75 (SD 0.03)</td><td>0.79 (SD 0.09)</td><td>0.66 (SD 0.06)</td><td>0.63 (SD 0.05)</td><td>0.57 (SD 0.09)</td><td>0.12 (SD 0.07)</td></tr><tr><td>MIMIC Mal- absorption</td><td>Short</td><td>1st</td><td>0.76 (SD 0.10)</td><td>0.64 (SD 0.17)</td><td>0.44 (SD 0.25)</td><td>0.57 (SD 0.23)</td><td>0.44 (SD 0.25)</td><td>1.00 (SD 0.00)</td><td>0.48 (SD 0.21)</td></tr><tr><td/><td/><td>2nd</td><td>0.75 (SD 0.11)</td><td>0.73 (SD 0.06)</td><td>0.41 (SD 0.20)</td><td>0.50 (SD 0.21)</td><td>0.39 (SD 0.15)</td><td>0.78 (SD 0.33)</td><td>0.28 (SD 0.12)</td></tr><tr><td/><td>Long</td><td>1st</td><td>0.75 (SD 0.07)</td><td>0.65 (SD 0.10)</td><td>0.39 (SD 0.16)</td><td>0.53 (SD 0.17)</td><td>0.38 (SD 0.15)</td><td>0.93 (SD 0.13)</td><td>0.43 (SD 0.14)</td></tr><tr><td/><td/><td>2nd</td><td>0.71 (SD 0.16)</td><td>0.79 (SD 0.10)</td><td>0.57 (SD 0.35)</td><td>0.48 (SD 0.26)</td><td>0.45 (SD 0.24)</td><td>0.53 (SD 0.34)</td><td>0.12 (SD 0.12)</td></tr><tr><td>MIMIC UTI</td><td>Short</td><td>1st</td><td>0.77 (SD 0.03)</td><td>0.71 (SD 0.03)</td><td>0.38 (SD 0.07)</td><td>0.52 (SD 0.07)</td><td>0.35 (SD 0.07)</td><td>0.87 (SD 0.05)</td><td>0.33 (SD 0.03)</td></tr><tr><td/><td/><td>2nd</td><td>0.74 (SD 0.04)</td><td>0.81 (SD 0.02)</td><td>0.49 (SD 0.08)</td><td>0.54 (SD 0.08)</td><td>0.38 (SD 0.08)</td><td>0.63 (SD 0.10)</td><td>0.15 (SD 0.03)</td></tr><tr><td/><td>Long</td><td>1st</td><td>0.78 (SD 0.02)</td><td>0.72 (SD 0.03)</td><td>0.39 (SD 0.06)</td><td>0.54 (SD 0.06)</td><td>0.36 (SD 0.06)</td><td>0.87 (SD 0.04)</td><td>0.31 (SD 0.04)</td></tr><tr><td/><td/><td>2nd</td><td>0.77 (SD 0.04)</td><td>0.83 (SD 0.02)</td><td>0.54 (SD 0.09)</td><td>0.58 (SD 0.07)</td><td>0.42 (SD 0.07)</td><td>0.66 (SD 0.12)</td><td>0.13 (SD 0.05)</td></tr><tr><td>MIMIC Pneumonia</td><td>Short</td><td>1st</td><td>0.76 (SD 0.03)</td><td>0.68 (SD 0.03)</td><td>0.28 (SD 0.05)</td><td>0.42 (SD 0.07)</td><td>0.26 (SD 0.05)</td><td>0.86 (SD 0.07)</td><td>0.35 (SD 0.03)</td></tr><tr><td/><td/><td>2nd</td><td>0.76 (SD 0.03)</td><td>0.82 (SD 0.02)</td><td>0.41 (SD 0.05)</td><td>0.51 (SD 0.04)</td><td>0.32 (SD 0.05)</td><td>0.68 (SD 0.07)</td><td>0.16 (SD 0.03)</td></tr><tr><td/><td>Long</td><td>1st</td><td>0.77 (SD 0.02)</td><td>0.70 (SD 0.04)</td><td>0.30 (SD 0.06)</td><td>0.44 (SD 0.07)</td><td>0.28 (SD 0.05)</td><td>0.87 (SD 0.03)</td><td>0.32 (SD 0.04)</td></tr><tr><td/><td/><td>2nd</td><td>0.74 (SD 0.04)</td><td>0.82 (SD 0.03)</td><td>0.41 (SD 0.09)</td><td>0.48 (SD 0.08)</td><td>0.31 (SD 0.07)</td><td>0.62 (SD 0.10)</td><td>0.14 (SD 0.04)</td></tr><tr><td>MIMIC Sepsis</td><td>Short</td><td>1st</td><td>0.82 (SD 0.05)</td><td>0.71 (SD 0.05)</td><td>0.38 (SD 0.15)</td><td>0.53 (SD 0.17)</td><td>0.38 (SD 0.15)</td><td>1.00 (SD 0.00)</td><td>0.36 (SD 0.10)</td></tr><tr><td/><td/><td>2nd</td><td>0.79 (SD 0.12)</td><td>0.82 (SD 0.08)</td><td>0.53 (SD 0.25)</td><td>0.56 (SD 0.20)</td><td>0.43 (SD 0.18)</td><td>0.76 (SD 0.27)</td><td>0.17 (SD 0.09)</td></tr><tr><td/><td>Long</td><td>1st</td><td>0.77 (SD 0.07)</td><td>0.68 (SD 0.08)</td><td>0.35 (SD 0.14)</td><td>0.48 (SD 0.14)</td><td>0.33 (SD 0.13)</td><td>0.90 (SD 0.13)</td><td>0.35 (SD 0.08)</td></tr><tr><td/><td/><td>2nd</td><td>0.76 (SD 0.18)</td><td>0.80 (SD 0.10)</td><td>0.46 (SD 0.21)</td><td>0.52 (SD 0.22)</td><td>0.42 (SD 0.16)</td><td>0.68 (SD 0.33)</td><td>0.16 (SD 0.07)</td></tr><tr><td>eICU</td><td>Short</td><td>1st</td><td>0.72 (SD 0.02)</td><td>0.75 (SD 0.03)</td><td>0.36 (SD 0.05)</td><td>0.47 (SD 0.05)</td><td>0.29 (SD 0.04)</td><td>0.68 (SD 0.06)</td><td>0.24 (SD 0.04)</td></tr><tr><td/><td/><td>2nd</td><td>0.65 (SD 0.05)</td><td>0.85 (SD 0.02)</td><td>0.60 (SD 0.12)</td><td>0.42 (SD 0.10)</td><td>0.31 (SD 0.06)</td><td>0.34 (SD 0.10)</td><td>0.05 (SD 0.02)</td></tr><tr><td/><td>Long</td><td>1st</td><td>0.72 (SD 0.02)</td><td>0.75 (SD 0.02)</td><td>0.36 (SD 0.06)</td><td>0.46 (SD 0.04)</td><td>0.29 (SD 0.04)</td><td>0.67 (SD 0.07)</td><td>0.24 (SD 0.04)</td></tr><tr><td/><td/><td>2nd</td><td>0.64 (SD 0.06)</td><td>0.84 (SD 0.02)</td><td>0.48 (SD 0.17)</td><td>0.38 (SD 0.15)</td><td>0.28 (SD 0.07)</td><td>0.33 (SD 0.15)</td><td>0.06 (SD 0.03)</td></tr></tbody></table></table-wrap>", "<table-wrap id=\"Tab3\"><label>Table 3</label><caption><p>Fairness results</p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th/><th/><th colspan=\"4\">Short model</th><th colspan=\"4\">Long model</th></tr><tr><th>Sensitive attribute</th><th>Group</th><th>AUROC</th><th>TPR</th><th>FPR</th><th>EO</th><th>AUROC</th><th>TPR</th><th>FPR</th><th>EO</th></tr></thead><tbody><tr><td>—</td><td>All patients</td><td>0.78 (SD 0.02)</td><td>0.80 (SD 0.05)</td><td>0.25 (SD 0.02)</td><td>—</td><td>0.80 SD (0.01)</td><td>0.85 SD (0.04)</td><td>0.25 SD (0.04)</td><td>—</td></tr><tr><td>Sex</td><td>Female</td><td>0.74 (SD 0.18)</td><td>0.79 (SD 0.35)</td><td>0.29 (SD 0.13)</td><td><italic>✓</italic></td><td>0.80 SD (0.15)</td><td>0.85 SD (0.34)</td><td>0.25 SD (0.10)</td><td><italic>✓</italic></td></tr><tr><td/><td>Male</td><td>0.80 (SD 0.08)</td><td>0.82 (SD 0.22)</td><td>0.23 (SD 0.06)</td><td><italic>✓</italic></td><td>0.80 SD (0.16)</td><td>0.84 SD (0.33)</td><td>0.23 SD (0.08)</td><td><italic>✓</italic></td></tr><tr><td>Age</td><td>20</td><td>0.73 (SD 0.08)</td><td>0.74 (SD 0.15)</td><td>0.27 (SD 0.06)</td><td><italic>✓</italic></td><td>0.76 SD (0.09)</td><td>0.77 SD (0.16)</td><td>0.24 SD (0.05)</td><td><italic>✓</italic></td></tr><tr><td/><td>30</td><td>0.80 (SD 0.02)</td><td>0.86 (SD 0.06)</td><td>0.26 (SD 0.04)</td><td><italic>✓</italic></td><td>0.72 SD (0.03)</td><td>0.64 SD (0.09)</td><td>0.20 SD (0.05)</td><td>✗</td></tr><tr><td/><td>40</td><td>0.78 (SD 0.04)</td><td>0.81 (SD 0.08)</td><td>0.25 (SD 0.03)</td><td><italic>✓</italic></td><td>0.77 SD (0.02)</td><td>0.80 SD (0.06)</td><td>0.26 SD (0.06)</td><td><italic>✓</italic></td></tr><tr><td/><td>50</td><td>0.76 (SD 0.04)</td><td>0.78 (SD 0.09)</td><td>0.25 (SD 0.04)</td><td><italic>✓</italic></td><td>0.80 SD (0.04)</td><td>0.87 SD (0.08)</td><td>0.26 SD (0.05)</td><td><italic>✓</italic></td></tr><tr><td/><td>60</td><td>0.79 (SD 0.02)</td><td>0.82 (SD 0.04)</td><td>0.23 (SD 0.04)</td><td><italic>✓</italic></td><td>0.80 SD (0.03)</td><td>0.84 SD (0.03)</td><td>0.24 SD (0.05)</td><td><italic>✓</italic></td></tr><tr><td/><td>70</td><td>0.73 (SD 0.08)</td><td>0.69 (SD 0.19)</td><td>0.23 (SD 0.07)</td><td><italic>✓</italic></td><td>0.81 SD (0.06)</td><td>0.86 SD (0.12)</td><td>0.23 SD (0.05)</td><td><italic>✓</italic></td></tr><tr><td/><td>80</td><td>0.77 (SD 0.02)</td><td>0.81 (SD 0.04)</td><td>0.26 (SD 0.03)</td><td><italic>✓</italic></td><td>0.81 SD (0.01)</td><td>0.85 SD (0.06)</td><td>0.23 SD (0.05)</td><td><italic>✓</italic></td></tr><tr><td/><td>90</td><td>0.78 (SD 0.03)</td><td>0.79 (SD 0.07)</td><td>0.23 (SD 0.02)</td><td>✗</td><td>0.78 SD (0.02)</td><td>0.78 SD (0.04)</td><td>0.22 SD (0.04)</td><td><italic>✓</italic></td></tr><tr><td>Race</td><td>Asian</td><td>0.79 (SD 0.08)</td><td>0.83 (SD 0.12)</td><td>0.24 (SD 0.11)</td><td><italic>✓</italic></td><td>0.80 SD (0.11)</td><td>0.84 SD (0.18)</td><td>0.24 SD (0.08)</td><td><italic>✓</italic></td></tr><tr><td/><td>Black</td><td>0.78 (SD 0.04)</td><td>0.83 (SD 0.07)</td><td>0.27 (SD 0.05)</td><td><italic>✓</italic></td><td>0.80 SD (0.04)</td><td>0.85 SD (0.07)</td><td>0.24 SD (0.06)</td><td><italic>✓</italic></td></tr><tr><td/><td>Hispanic</td><td>0.80 (SD 0.07)</td><td>0.85 (SD 0.12)</td><td>0.25 (SD 0.08)</td><td><italic>✓</italic></td><td>0.80 SD (0.08)</td><td>0.84 SD (0.16)</td><td>0.25 SD (0.08)</td><td><italic>✓</italic></td></tr><tr><td/><td>Native</td><td>0.78 (SD 0.17)</td><td>0.97 (SD 0.07)</td><td>0.43 (SD 0.35)</td><td>✗</td><td>0.82 SD (0.13)</td><td>1.00 SD (0.00)</td><td>0.35 SD (0.23)</td><td><italic>✓</italic></td></tr><tr><td/><td>Other</td><td>0.76 (SD 0.06)</td><td>0.72 (SD 0.10)</td><td>0.19 (SD 0.05)</td><td><italic>✓</italic></td><td>0.79 SD (0.07)</td><td>0.77 SD (0.09)</td><td>0.20 SD (0.09)</td><td><italic>✓</italic></td></tr><tr><td/><td>Unknown</td><td>0.79 (SD 0.05)</td><td>0.83 (SD 0.11)</td><td>0.25 (SD 0.03)</td><td><italic>✓</italic></td><td>0.82 SD (0.03)</td><td>0.87 SD (0.06)</td><td>0.23 SD (0.05)</td><td><italic>✓</italic></td></tr><tr><td/><td>White</td><td>0.77 (SD 0.02)</td><td>0.79 (SD 0.06)</td><td>0.24 (SD 0.03)</td><td><italic>✓</italic></td><td>0.80 SD (0.02)</td><td>0.84 SD (0.04)</td><td>0.24 SD (0.05)</td><td><italic>✓</italic></td></tr><tr><td>Insurance</td><td>Medicaid</td><td>0.72 (SD 0.07)</td><td>0.69 (SD 0.17)</td><td>0.26 (SD 0.06)</td><td>✗</td><td>0.76 SD (0.08)</td><td>0.77 SD (0.16)</td><td>0.26 SD (0.05)</td><td><italic>✓</italic></td></tr><tr><td/><td>Medicare</td><td>0.78 (SD 0.03)</td><td>0.81 (SD 0.06)</td><td>0.25 (SD 0.02)</td><td><italic>✓</italic></td><td>0.81 SD (0.02)</td><td>0.85 SD (0.04)</td><td>0.24 SD (0.05)</td><td><italic>✓</italic></td></tr><tr><td/><td>Other</td><td>0.78 (SD 0.02)</td><td>0.80 (SD 0.05)</td><td>0.24 (SD 0.03)</td><td><italic>✓</italic></td><td>0.80 SD (0.02)</td><td>0.84 SD (0.05)</td><td>0.23 SD (0.04)</td><td><italic>✓</italic></td></tr><tr><td>Language</td><td>English</td><td>0.77 (SD 0.04)</td><td>0.79 (SD 0.09)</td><td>0.25 (SD 0.04)</td><td><italic>✓</italic></td><td>0.81 SD (0.06)</td><td>0.85 SD (0.11)</td><td>0.24 SD (0.05)</td><td><italic>✓</italic></td></tr><tr><td/><td>Other</td><td>0.78 (SD 0.02)</td><td>0.80 (SD 0.05)</td><td>0.25 (SD 0.02)</td><td><italic>✓</italic></td><td>0.77 SD (0.01)</td><td>0.78 SD (0.04)</td><td>0.24 SD (0.04)</td><td><italic>✓</italic></td></tr><tr><td>Marital status</td><td>Divorced</td><td>0.78 (SD 0.04)</td><td>0.80 (SD 0.10)</td><td>0.24 (SD 0.03)</td><td><italic>✓</italic></td><td>0.79 SD (0.05)</td><td>0.82 SD (0.09)</td><td>0.24 SD (0.05)</td><td><italic>✓</italic></td></tr><tr><td/><td>Married</td><td>0.77 (SD 0.03)</td><td>0.77 (SD 0.06)</td><td>0.22 (SD 0.02)</td><td><italic>✓</italic></td><td>0.81 SD (0.01)</td><td>0.83 SD (0.05)</td><td>0.22 SD (0.05)</td><td><italic>✓</italic></td></tr><tr><td/><td>Single</td><td>0.78 (SD 0.02)</td><td>0.84 (SD 0.05)</td><td>0.28 (SD 0.03)</td><td><italic>✓</italic></td><td>0.79 SD (0.03)</td><td>0.84 SD (0.06)</td><td>0.27 SD (0.04)</td><td><italic>✓</italic></td></tr><tr><td/><td>Widowed</td><td>0.79 (SD 0.04)</td><td>0.82 (SD 0.08)</td><td>0.24 (SD 0.02)</td><td><italic>✓</italic></td><td>0.81 SD (0.03)</td><td>0.85 SD (0.07)</td><td>0.24 SD (0.06)</td><td><italic>✓</italic></td></tr><tr><td/><td>Unknown</td><td>0.77 (SD 0.05)</td><td>0.83 (SD 0.09)</td><td>0.29 (SD 0.05)</td><td><italic>✓</italic></td><td>0.84 SD (0.04)</td><td>0.93 SD (0.06)</td><td>0.24 SD (0.07)</td><td><italic>✓</italic></td></tr></tbody></table></table-wrap>" ]

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[ "<supplementary-material content-type=\"local-data\" id=\"MOESM1\"></supplementary-material>", "<supplementary-material content-type=\"local-data\" id=\"MOESM2\"></supplementary-material>", "<supplementary-material content-type=\"local-data\" id=\"MOESM3\"></supplementary-material>", "<supplementary-material content-type=\"local-data\" id=\"MOESM4\"></supplementary-material>" ]

[ "<table-wrap-foot><p>Note that infection types in MIMIC are determined through ‘hadm_id’ and are not definitively linked to the antimicrobial switch of interest as diagnoses are only coded for billing purposes upon hospital discharge. This results in totals over 100% as patients have multiple infection episodes. Sex was determined based on the information contained within the dataset. <italic>MIMIC</italic> Medical Information Mart for Intensive Care, <italic>ICU</italic> Intensive care unit, <italic>IV</italic> Intravenous, <italic>SD</italic> Standard deviation, <italic>UTI</italic> Urinary tract infection.</p></table-wrap-foot>", "<table-wrap-foot><p>MIMIC;Medical Information Mart for Intensive Care, ICU;Intensive care unit, UTI;Urinary tract infection, SD;Standard deviation, AUROC;Area under the receiver operating characteristic, AUPRC;Area under precision-recall, TPR;True positive rate, FPR;False positive rate</p></table-wrap-foot>", "<table-wrap-foot><p><italic>SD</italic> Standard deviation, <italic>AUROC</italic> Area under the receiver operating characteristic, <italic>TPR</italic> True positive rate, <italic>FPR</italic> False positive rate, <italic>EO</italic> Equalised odds</p></table-wrap-foot>", "<fn-group><fn><p><bold>Publisher’s note</bold> Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.</p></fn></fn-group>" ]

[ "<graphic xlink:href=\"41467_2024_44740_Fig1_HTML\" id=\"d32e321\"/>", "<graphic xlink:href=\"41467_2024_44740_Fig2_HTML\" id=\"d32e1421\"/>", "<graphic xlink:href=\"41467_2024_44740_Fig3_HTML\" id=\"d32e1439\"/>", "<graphic xlink:href=\"41467_2024_44740_Fig4_HTML\" id=\"d32e1492\"/>" ]

[ "<media xlink:href=\"41467_2024_44740_MOESM1_ESM.pdf\"><caption><p>Supplementary information</p></caption></media>", "<media xlink:href=\"41467_2024_44740_MOESM2_ESM.pdf\"><caption><p>Reporting Summary</p></caption></media>", "<media xlink:href=\"41467_2024_44740_MOESM3_ESM.pdf\"><caption><p>Peer Review File</p></caption></media>", "<media xlink:href=\"41467_2024_44740_MOESM4_ESM.xlsx\"><caption><p>Source Data</p></caption></media>" ]

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[ "<title>Introduction</title>", "<p id=\"Par2\">In recent years, intelligent transportation have become increasingly complex due to rapid urbanization and population growth. There is a growing need for abnormal event detection in transportation networks. The Shanghai Bund trampling incident that occurred on December 31, 2014, in China is a widely known tragedy closely associated with traffic anomaly detection^{##UREF##0##1##}. Furthermore, on January 26, 2017, in Harbin, the largest city in northeastern China, a single traffic incident resulted in a chain of rear-end collisions, leading to eight fatalities and thirty-two injuries^{##UREF##1##2##}. The above events show that early detection and prediction of anomalies before they occur are of significant value in preventing serious incidents. Therefore, an efficient and accurate anomaly detection system holds significant research value as it enables continuous monitoring of specific indicators and effective prevention of potential anomalies.</p>", "<p id=\"Par3\">Anomaly detection is widely used in smart cities, especially in intelligent transportation systems. The intelligent transportation system discussed in this paper is an artificial intelligence-based technology that aims to detect intelligent traffic anomalies. By learning the relationships between sensors, we could detect anomalies from sensors data^{##REF##34982699##3##–##UREF##3##5##}. However, traffic anomalies usually exhibit complex forms due to two aspects: high dimensionality, sparsity, abnormal scarcity (i.e., the need to correlate time and space, including speed or flow), and difficulty in capturing the hidden relationship between nodes (i.e., spatial modeling in the face of different data sources with varying degrees of anomalies in density or distribution and scale)^{##UREF##4##6##,##UREF##5##7##}. Therefore, it is important to explore ways to capture complex inter-sensor relationships and detect anomalies from node relationships. Several methods are based on Generative Adversarial Networks (GANs) based method^{##REF##34982699##3##}. However, the generator of GANs may be ineffective in fully capturing the hidden distribution of the data, which leads to a high false alarm rate and miss alarm rate due to the combination of the Binary CrossEntropyLoss (BCE) loss function. Most previous methods for anomaly detection are variants of Long Short Term Memory (LSTM)^{##UREF##6##8##,##UREF##7##9##}, such as FC-LSTM^{##UREF##8##10##}, which focuses on capturing both the various static factors and dynamic interactions that affect traffic flow. Moreover, there exists a category of networks, such as TCNs (temporal convolutional networks), designed to address temporal dependencies, which can capture global temporal information^{##UREF##9##11##}. However, TCNs may not be as flexible in the context of traffic timing due to variations in the amount of historical information needed for model predictions across different domains. When TCNs (temporal convolutional networks) face a dynamic transportation network, their performance may be poor because their perceptual field is not large enough to describe the dynamics, complexity, and capture the global contextual information^{##UREF##10##12##}.</p>", "<p id=\"Par4\">The most advanced approach employs a graphical convolutional neural network (GNN) for spatial modeling reuse and combines LSTM to deal with anomaly prediction in time series^{##REF##34982699##3##}. There is also a method of passing adversarial training, learning the spatiotemporal features of traffic dynamics and traffic anomalies, respectively^{##UREF##11##13##}. Existing methods of anomaly detection using graph convolutional neural networks (GCNs) do not well-address data sparsity and capture unseen nodes and the spatiotemporal correlation between nodes in the traffic network.</p>", "<p id=\"Par5\">To solve the above problem, we propose a mirror temporal convolutional module (MTCM) to capture the anomalous information related to input data and hidden dynamic nodes in traffic networks. We mainly design two modules in MTGAE: mirror temporal convolutional module (MTCM) and graph convolutional gate recurrent unit cell (GCGRU CELL). Combined with self-adaptive, the MTCM can efficiently input into its modules in the face of sections of varying lengths in the dataset. MTCM explores the potential association between nodes and nodes by learning the complex hidden relationships and dependencies between nodes in traffic networks. GCGRU CELL module fully uses the existing prior knowledge (historical data). It captures road information, hidden node relationships, and dependencies for anomaly information redistribution, thus allowing us to obtain anomaly information more easily. We summarize the contributions of this paper as follows:<list list-type=\"bullet\"><list-item><p id=\"Par6\">We propose an anomalous detection framework called MTGAE, which maximizes the exploration of possible anomalies between nodes in the complex interdependencies, and better captures the hidden features between node-to-node in the traffic network.</p></list-item><list-item><p id=\"Par7\">We construct a mirror temporal convolutional module, which is self-adapt to dataset and captures and cascades the hidden information between nodes by maximizing the breakthrough of the perceptual field of view of TCN.</p></list-item><list-item><p id=\"Par8\">We propose the GCGRU CELL module, which captures long-term and short-term dependent anomalies in the traffic network space-time and maximizes the extraction of spatiotemporal features and possible anomaly information by cooperating with MTCM.</p></list-item></list></p>" ]

[ "<title>Methodology</title>", "<p id=\"Par14\">Although many traffic anomaly detection methods have achieved optimal performance, they often overlook the hidden relationships between nodes during the detection process. For instance, traffic congestion during peak periods upstream can impact downstream traffic. This oversight results in many models lacking the ability to capture long-term temporal correlations, spatial characteristics, and high periodic trends. To address this, we aim to identify abnormal information and potential anomalies in the complex interdependencies among nodes in traffic networks. Consequently, we propose a traffic anomaly detection framework, MTGAE, with node interaction (see Fig. ##FIG##1##2##).</p>", "<p id=\"Par15\">MTGAE consists of two main modules: MTCM and GCGRU CELL. The original input first passed through an adaptive process. This allows our module to better self-adapt to existing datasets by converting graph signals in low-dimensional spaces into potential vectors in high-dimensional spaces. Then we construct MTCM and GCGRU CELL. Specifically, we built MTCM to expand the hidden information in spacetime. MTCM internally expands <italic>x</italic> to the latent variables by mirror flip, and increases dilation factors and generates the hidden states <italic>H</italic> to capture both long-term spatiotemporal complex dependencies combining with TCN. Meanwhile, we built GCGRU CELL module to capture long-term and short-term 84 dependent anomalies in the traffic network. It combines original inputs and the hidden spatiotemporal states <italic>H</italic> as prior information. We first redistribute it through the Gaussian kernel module but without changing the overall structure of the traffic network (see in Fig. ##FIG##1##2##), then combine with our GCN modules to extract more spatiotemporal information. Subsequently, based on the output of the first GCGRU CELL, the spatio-temporal information and MTCM’s hidden information <italic>H</italic>, the second GCGRU CELL module adds more hidden details to correct the defects generated. Finally, we link the reconstructed results with the loss function to determine whether there are anomalies. In this section, we introduce the details of the MTGAE.</p>", "<title>Problem definition</title>", "<p id=\"Par16\">In this paper, traffic anomaly is monitored and detected in discrete time series . We denote the adjacency matrix representation graph as where <italic>V</italic> indicates different nodes, such as two nodes and , <italic>E</italic> denotes the set of edges between two nodes and <italic>W</italic> is the weighted adjacency matrix. A larger weight between two nodes means they are closer in the road networks and vice versa (see Fig. ##FIG##0##1##). Given , we aim to find the abnormal event in the graph <italic>G</italic> that disrupts the regular traffic operation.</p>", "<p id=\"Par17\">We aim to find the event in the graph <italic>G</italic> that disrupts the regular traffic operation. We get the hidden state through a specially designed contextual encoder, embed the information as a coded low-dimensional embedding, and then decode it to derive the average reconstruction error that minimizes the weighted adjacency matrix. It should be noted that our model is specifically trained using data representing normal traffic conditions. Consequently, when an anomaly occurs in the traffic operation, it deviates significantly from this ’normal’ baseline. This deviation is captured as a high reconstruction error by our model, effectively indicating the presence of an anomaly.</p>", "<title>Encoder</title>", "<p id=\"Par18\">Our encoder process comprises three steps: the adaptive process, the mirror temporal convolutional module (MTCM), and the graph convolutional neural network recurrent cell (GCGRU CELL). Initially, the original data, denoted as x, passes through the adaptive process, and MTCM is constructed to capture the evolving states that are not visible in the spacetime continuum among the road network nodes over time. In the GCGRU CELL, based on prior knowledge of the hidden states H from MTCM, our GCN layer, through the Gaussian kernel module, explores potential anomalies in the complex interdependencies between nodes. The encoder is trained to learn up to 24 hours in a day and 7 days in a week, facilitating interaction between the GCGRU CELL and a full connection (refer to Fig. ##FIG##1##2##). Finally, the graph embedding is applied.</p>", "<title>Mirror temporal convolutional module (MTCM)</title>", "<p id=\"Par19\">Inspired by TCN^{##UREF##9##11##} (see Fig. ##FIG##2##3##a), we proposed a superior module named MTCM that wides application in traffic prediction. Although TCNs can use the extended convolution to expand the perceptual field, they are weaker than advanced networks (e.g., Transformer) which can use correlation information of arbitrary length. Moreover, TCNs need strong adaptability to different historical information, which may have uneven predictive power and perceptual field. To overcome the above situations, we adapt the TCN before transmitting the traffic network features to reduce the fluctuation of different historical information on the ability of the TCN. We then perform a mirror flip to further preserve the features and capture the complex hidden relationships and dependencies between nodes in the traffic network. This explores the potential associations between nodes. Furthermore, thanks to the one-dimensional convolution of the TCN, we can keep the output sequence consistent with the original input in length. Finally, this output sequence will be passed as the subsequent hidden state <italic>H</italic>. More formally, for a 1-D sequence input and a filter , k is the kernel size (the kernel size in the Fig. ##FIG##2##3## is 2), and <italic>d</italic> is the causal factor (see Fig. ##FIG##2##3##a). The dilated convolution operation <italic>f</italic> on element <italic>x</italic> of the sequence is defined as:where is the sequence input in mirror flipping, denotes concatenate. This further increases the range of perceptual field and prevents more historical data from being lost in the process of inflated convolution.</p>", "<title>GCGRU CELL</title>", "<p id=\"Par20\">It mainly includes the Gaussian kernel module and GCN layer. We did not adopt GRU model (as shown in Fig. ##FIG##3##4##) but construct the GCN model inspired by GRU after the Gaussian kernel module. In GCGRU CELL, we replace the original gated cyclic unit of GRU to our GCN, which has the following two significant: the reset gate helps to capture short-term dependencies in the sequence, and the update gate helps to capture long-term dependencies in the sequence. This effectively predicts both long-term and short-term traffic network cycles, and combine with Gaussian kernel processing and prior knowledge <italic>H</italic> (hidden information of MTCM), GCN can capture anomaly information and possible anomalies in complex interdependencies among nodes while predicting. Unlike image data, Graph convolution is an essential operation to extract a node’s features. Figure ##FIG##2##3##b gives examples of an origin node (orange node) to take the average value of the node features within its neighbours (white nodes in ellipse).</p>", "<p id=\"Par21\"><bold>(1) Gaussian kernel module</bold> To further enhance the anomaly detection capability of our module, we employ Gaussian kernel function. It could maintain the ability of high-dimensional data distribution characteristics, which is crucial for traffic network anomaly detection. Specifically, Gaussian kernels facilitate the mapping of data from its original space to a higher-dimensional feature space where complex traffic network patterns and potential anomalies are more easily identified and processed. Moreover, Gaussian kernel exhibit the stability: It could manage minor fluctuations by adjusting learned scale parameter (see Eq. ##FORMU##23##2##) or utilizing a minimax strategy^{##UREF##46##49##}, thereby ensuring more stable anomaly detection results. In summary, embedding Gaussian kernels in the GCGRU CELL module aims to enhance the model’s performance and accuracy in detecting anomalies within complex traffic networks. Experimental data demonstrate that using Gaussian kernels to alter the data distribution effectively improves the accuracy of traffic anomaly detection (see Table ##TAB##2##3##). Building on this foundation, we further explored the anomaly detection capabilities of the Gaussian kernel module. As depicted in Fig. ##FIG##4##5##, we performed an intermediate variable exploration of the eight feature points generated by 490 edges entering the GCGRU CELL. This demonstrates the stability and the data mapping capability of our module by conducting visualization operations on intermediate variables before and after integrating the Gaussian kernel module into the GCGRU cell. Throughout the experimentation, the overall structure of the data remains unchanged, ensuring consistency and reliability. Our GCGRU CELL receives two input modes. The first input is from the original input <italic>x</italic> after adaptation and receives the hidden information <italic>H</italic> from the MTCM. Then set the input as . The second input is the output of the first GCGRU CELL , which also receives hidden information <italic>H</italic>. We also set this input as . Then, the formula calculated by the Gaussian kernel module is as follows:where is generated based on the learned scale (we usually set the value between 0.5 and 1) and the <italic>i</italic>-th element corresponds to the <italic>i</italic>-th time point. Specifically, for the <italic>i</italic>-th time point, its association weight to the -th point is calculated by the Gaussian kernel.</p>", "<p id=\"Par22\"><bold>(2) GCN layer</bold> Generally, the traffic network is presented as a weighted digraph. Traditional graph convolution networks only operate on adjacent nodes, which results in better short-term prediction than long-term prediction. Therefore, the spectral graph theory is used in this paper. Let and establish spectral matrix , where <italic>I</italic> is the identity matrix and is the degree matrix, is the adjacent matrix. To explore deeper and more complex traffic networks, we extend the graph convolution network to a higher level and divide the traffic graph <italic>g</italic>(<italic>x</italic>) sent by the Gaussian kernel module into subgraph , and the subgraph considers its neighbour nodes , which achieves more high-order information aggregation.where represents learnable weights, and denotes the computed results of graph convolution as time <italic>t</italic>increases.</p>", "<p id=\"Par23\">In a separate aspect, the use of GRU^{##UREF##46##49##} simplifies the model, reducing complexity and enabling a faster, more effective characterization of sentence semantics. Compared to LSTM, GRU reduces the number of gating parameters, utilizes fewer training parameters, requires less memory, and offers faster execution and training. Owing to these advantages, our model adopts the GRU architecture over the traditional LSTM approach. We have transformed the gating unit into a graph convolution layer, as outlined in Eq. (##FORMU##34##3##). This adaptation allows the GRU architecture to imitate the gating unit effectively. Consequently, the GCN layer can discern more hidden states from data processed by the Gaussian kernel module, capturing the dynamic spatial correlations within the traffic network and identifying previously unseen network connections. Formally,where is the previous memory state, , and , , <italic>U</italic> are the weight parameters, is the current feature input, and is a sigmoid activation function. We combine GCN and GRU to capture the long-term dependencies between nodes in the graph.</p>", "<p id=\"Par24\"><bold>(3) Graph Embedding (GE)</bold> We construct a time embedding (referred to as the GE module in Fig. ##FIG##1##2##) after the second GCGRU CELL to effectively capture the intricate weekly and hourly periodicity inherent in the mobility data. The time embedding consists of two components: represents the time of day embedding, and represents the day of week embedding. For example, at a specific time t (e.g., 13:00 on Saturday, July 30), we use (i.e., 13:00) and (i.e., Saturday) as the time embeddings. These embeddings serve the purpose of incorporating additional temporal information as context for the conditioned encoder and decoder. By incorporating these temporal factors as graph embeddings , the model could accurately capture and represent the patterns and variations in mobility data associated with different times and days.where is the graph embedding at time <italic>t</italic>, from the formula 4 and is weight matrix.</p>", "<title>Decoder</title>", "<p id=\"Par25\">In the decoder, we begin with information extraction about the node embedding from the graph embedding . For each pair of node embeddings , we embed the time information into the information of each pair of nodes and compute the corresponding weight in the weighted adjacency matrix. We then combine these node embeddings and time embeddings to form a graph embedding information that varies over time <italic>t</italic>. It contains both the information of the nodes and the time information (that is, the embedding includes the collective features of all nodes in the graph at that moment <italic>t</italic>). Subsequently, a fully connected layer is used to process this graph embedding, to recover useful vector representations from it. After processing by the fully connected layer, the vectors <italic>i</italic> and <italic>j</italic>, corresponding to and , are unstacked to recover the embedding of each individual node at time <italic>t</italic>. Consequently, the outcome of this process is the embedding representation of a particular node <italic>n</italic> under specific time <italic>t</italic> conditions. Finally, to obtain the reconstructed edge weights, we first used the <italic>ReLU</italic> activation function to process the graph embeddings, resulting in a feature vector that has undergone a nonlinear transformation. Then, the reconstructed edge weights are obtained from the feature vector and the <italic>Sigmoid</italic> function.</p>", "<p id=\"Par26\">The presence of a bilinear module in the decoder is significant. The bilinear module applies a transformation to the incoming data, serving two main benefits: 1) The bilinear module ensures that edge weight predictions consider directionality. In the directed graph, the edge weight from node <italic>i</italic> to node <italic>j</italic> could differ from the weight from node j to node i. 2) The bilinear module employs the formula to calculate the edge weights, where <italic>A</italic> is a learned parameter. This approach enables the model to distinguish edge weights based on direction, more accurately depicting directed graph relationships.where is weight matrix, is the weight matrix of feature vector and is the weight matrix of . The Sigmoid ensure the output .</p>", "<title>Loss function</title>", "<p id=\"Par27\">We use the mean squared error (MSE) as the loss function, a measure of the difference between the actual value <italic>y</italic> and the predict , to evaluate our model. Formally:where <italic>i</italic> is the value of each point in the sequence. And the reconstructed weights are and the actual weights are . During testing, the loss function Eq. (##FORMU##74##7##) for each testing instance is used as its anomaly score.</p>" ]

[ "<title>Result and analysis</title>", "<p id=\"Par43\"><bold>(1) Comparison with state-of-the-art work</bold> Initially, we compared our MTGAE model with some baseline models using the AUC as evaluation metric. The calculation of AUC considers both the classification ability of the classifier for positive and negative cases, which can still make a reasonable evaluation of the classifier in the case of sample imbalance. We fixed and in the pollution magnitude and used the anomaly rate to compare the model’s ability to detect anomalies. It can be seen from Table ##TAB##0##1## that our MTGAE is significantly better than the other models. Our model outperforms others by about 0.1–0.4 at different anomaly rates.</p>", "<p id=\"Par44\">After that, we fixed the time slice of pollution to study the abnormal magnitude to change the pollution magnitude differently. As shown in Table ##TAB##1##2##, we controlled as 25% and 50% respectively, and was controlled as the same pollution magnitude under . We can see that our models are higher than the baseline but in the higher . For example, the AUC of most models with and is above 0.9, and most of the baseline models with are not performing well. Instead, they all gradually increase in AUC capacity after increases, while our model has an excellent performance in all aspects, so it is a more competitive model.</p>", "<p id=\"Par45\"><bold>(2) Ablation study</bold> In the ablation study, we evaluated several MTGAE variants to evaluate the effects of different parts of our MTGAE (see Table ##TAB##2##3##). The variants include: (i) MTGAE-gan: We used the framework of GAN instead of an autoencoder. (ii) MTGAE-ot: We adopted the approach of using an autoencoder and only employed the original TCN instead of our proposed MTCM. (iii) MTGAE-mt: We removed TCN and incorporated our proposed MTCM module. (iv) MTGAE-lstm: We removed the GRU and replaced it with the LSTM. (v) MTGAE-grumt: We removed the LSTM and replaced it with the GCGRU CELL. (vi) MTGAE-Transformer: We removed the GCGRU CELL and replaced it with the Transformer. (vii) MTGAE-gb: We incorporated the Gaussian kernel module into MTCM and GCGRU CELL, placing it in the final data processing stage of the GCGRU CELL. (viii) MTGAE: Our complete model framework. The study indicates that the basic TCN variant underperforms unless combined with the Gaussian kernel function for processing, which illustrates the importance of MTCM and GCGRU CELL for anomaly detection. Notably, incorporating a mirror into TCN significantly improves its efficacy in enhancing GCGRU CELL performance, this demonstrates superior ability in capturing both long and short-term memory and temporal information in time series.</p>", "<p id=\"Par46\"><bold>(3) Real world reflects abnormal traffic</bold> We used the NYC dataset from January 1, 2019, to January 7, 2019, to test the real-world traffic situation to prove the effectiveness of our model. We used the reconstruction loss to represent the possibility of anomalies, as shown in Fig. ##FIG##5##6##. January 4 is Friday in the real world, and we can see that the possibility of anomalies in the afternoon distribution of this day is very intensive, from which we can infer that Black Friday Shopping is prone to traffic anomalies due to traffic jams.</p>", "<p id=\"Par47\"><bold>(4) Sensitivity analysis</bold> To study how MTGAE varies for weekly, hourly, and node embedding, we put and \n. We explored the model’s affectivity on spacetime, and we changed the dimension of node embedding to 25 to 200 (the dimensionality is acceptable for the first GCN and the second GCN) and the week and hour dimension of temporal embedding to 10 to 200 for training. As shown in Fig. ##FIG##5##6##b, our model does not change much, and the AUCs all remain between 0.9 and 1, indicating that our model works well in most environments. Moreover, we can further see that the AUC of our model is lower when the time node embedding is large than when the embedding is small.</p>", "<p id=\"Par48\"><bold>(5) Generalization ability</bold> To explore the generalization ability of MTGAE, we performed experiments on a large-scale dynamic graph dataset DGraphFin in the financial domain^{##UREF##56##60##}. It contains over 3.7 million nodes and 4.3 million dynamic edges. Nodes represent financial loan users, and directed edges represent emergency contact relationships. Each dimension represents 17 different elements of personal profiles, such as age and gender. Among the nodes in the dataset, 15,509 are categorized as fraudsters, 1,210,092 as normal users, and the remaining 66.8% of nodes (2,474,949 nodes) are registered users who have not borrowed from the platform. Based on the officially published baseline and code, we input the DGraphFin data into our MTGAE, then carry out feature learning through the 17 features in the structure of MTGAE, and finally divide into two categories (other baselines also divide into two categories) for anomaly detection, with results shown in the Table ##TAB##3##4##. Compered with the network specifically designed for DGraphFin dataset, experiments results illustrates that our MTGAE possesses certain generalization capabilities.</p>" ]

[]

[ "<title>Conclusions</title>", "<p id=\"Par49\">This paper proposes MTGAN, a spatio-temporal anomaly detection framework for traffic. In the encoder, we propose two modules: the Mirror TCN (MTCM) and a variant of GCGRU, namely the GCGRU CELL that captures correlations in spatial and temporal dimensions, and a practical approach: adaptive TCN. We then performed anomaly injection on the dataset by three contamination metrics and tested it on the NCY dataset. Experiment results show that our framework outperforms the baseline in traffic anomaly detection, particularly in aspects of sparsity and high dimensionality, thereby contributing to further research. In future work, we will explore additional extensions of MTGAE in more datasets and further explore methods for learning dynamic spatial correlations.</p>" ]

[ "<p id=\"Par1\">Traffic time series anomaly detection has been intensively studied for years because of its potential applications in intelligent transportation. However, classical traffic anomaly detection methods often overlook the evolving dynamic associations between road network nodes, which leads to challenges in capturing the long-term temporal correlations, spatial characteristics, and abnormal node behaviors in datasets with high periodicity and trends, such as morning peak travel periods. In this paper, we propose a mirror temporal graph autoencoder (MTGAE) framework to explore anomalies and capture unseen nodes and the spatiotemporal correlation between nodes in the traffic network. Specifically, we propose the mirror temporal convolutional module to enhance feature extraction capabilities and capture hidden node-to-node features in the traffic network. Morever, we propose the graph convolutional gate recurrent unit cell (GCGRU CELL) module. This module uses Gaussian kernel functions to map data into a high-dimensional space, and enables the identification of anomalous information and potential anomalies within the complex interdependencies of the traffic network, based on prior knowledge and input data. We compared our work with several other advanced deep-learning anomaly detection models. Experimental results on the NYC dataset illustrate that our model works best compared to other models for traffic anomaly detection.</p>", "<title>Subject terms</title>" ]

[ "<title>Related work</title>", "<p id=\"Par9\">In this section, we introduce the graph convolution networks, temporal convolutional networks, and autoencoder-based anomaly detection.</p>", "<title>Graph convolution networks</title>", "<p id=\"Par10\">Recently, Graph Neural Network (GNN) variants, such as Graph Convolutional Networks (GCN), have demonstrated ground-breaking performances on many deep-learning tasks. In addition, it is modular, scalable, stronger in generalization ability, and explores insights that direct further research^{##UREF##12##14##}. GCN captures the complex dependencies of node embeddings through information across vertices^{##UREF##13##15##}. Due to these powerful features, in variants of GCN, the sensors on the road of the traffic network are considered nodes in intelligent transportation, and each node’s traffic speed or flow rate is regarded as a dynamic input feature. Among them, the Graph attention network (GAT) updates the node features through a pairwise function between the nodes with learnable weights^{##UREF##14##16##}. However, it only computes one restricted form of static attention. To address this limitation, GATv2^{##UREF##15##17##} introduces dynamic attention alongside static attention, allowing for more dynamic and adaptive computation of graph attention. In the subsequent development of the GCN, CorrSTN^{##UREF##16##18##} effectively incorporates correlation information into the spatial structure. PDFormer^{##UREF##17##19##} captures both short-range and long-range spatial dependencies by utilizing various graph masking, which enables the learning of dynamic urban traffic patterns and overcomes the restriction of modeling spatial dependencies statically. Moreover, STAEformer^{##UREF##18##20##} takes into account the intrinsic spatial-temporal relationships and temporal ordering information in traffic time series. These methods are widely used in traffic forecasting, while graph embedding for traffic anomaly detection is less studied. For example, ST-Decompn solves the legal problem caused by changes in location and time in traffic cities through decomposition, as well as anomalies that may show up differently in the face of different datasets^{##UREF##19##21##}, ConGAE detects traffic anomalies using semi-supervised frameworks such as autoencoders only for OD (origin-destination pairs) datasets on data washing and high dimensionality^{##UREF##11##13##}. Besides, Graph Convolutional Adversarial Network (STGAN) uses adversarial training, which is divided into three modules to capture different features respectively: the recent module for local, the trend module for Long-term, and the external module for other traffic dynamics and anomalies, but the unsupervised learning, like an adversarial neural network, brings instability for anomaly detection^{##REF##34982699##3##}. Influenced state of the art, we borrowed the graph convolutional gated recurrent unit (GCGRU)^{##UREF##20##22##} simultaneously to solve the problem of Spatiotemporal characteristics of traffic anomalies. Our work is focused on the traffic anomaly prediction capabilities of GCN.</p>", "<p id=\"Par11\">Graph autoencoders (GAEs) are a kind of unsupervised learning method, which means they map nodes to a potential vector space through an encoding process, reconstructing graph information from the vector to generate a graph similar to the original one (decoding)^{##UREF##13##15##,##UREF##21##23##}. For example, ADN^{##UREF##22##24##} is a graph autoencoder structure and achieves information diffusion through alternating spatial and temporal self-attention. Due to the power of GAE^{##UREF##23##25##}, it is widely used in different research directions, such as link prediction^{##UREF##24##26##–##UREF##28##30##}, graph clustering^{##UREF##29##31##,##UREF##30##32##}, hyperspectral anomaly detection^{##UREF##31##33##}. While the traditional GCN takes node features and adjacency matrix as input and node embedding as output, GAEs compresses the node embeddings of all nodes in a graph to a single graph embedding to obtain information about the context.</p>", "<title>Temporal convolutional networks</title>", "<p id=\"Par12\">Earlier research methods focus on traffic-related problems but have shown significant inaccuracies in anomaly prediction. Deep learning has gradually dominated time series prediction tasks with sophisticated data modeling capabilities and autonomous learning abilities in recent years. Most studies in the field of transportation rely on gated linear Unit (GLU)^{##UREF##32##34##}, or gated recursive units (GRU)^{##UREF##33##35##} to capture the dynamic temporal correlation of time series data. Moreover, based on the transformer architecture, STGM^{##UREF##34##36##} introduces a novel attention mechanism to capture both long-term and short-term temporal dependencies. Temporal convolutional networks (TCNs) also have significant advantages in addressing temporal dependencies, especially in time series prediction tasks. However, most traffic flow anomaly prediction frameworks use the original Temporal Convolutional Network (TCN)^{##UREF##35##37##,##UREF##36##38##} structure without modification, and traffic anomaly detection is still under-explored. In this study, we have enhanced the TCN to better detect anomalies within this domain, allowing for a more comprehensive analysis of time series data.</p>", "<title>Autoencoder-based anomaly detection</title>", "<p id=\"Par13\">The autoencoder, an unsupervised neural network, has seen significant success across various fields. This success is largely due to its superior ability to discriminate between abnormal and regular inputs, making it widely used in anomaly detection^{##UREF##37##39##–##UREF##41##44##}. In the field of graph convolutional networks (GCN), GCN-based autoencoders are also employed for anomaly detection^{##UREF##42##45##–##UREF##45##48##}. They are mainly studied in graph embedding, which is consistent with the direction of our work, thanks to the network structure of the graph, which can connect various points in the intricate world for anomaly detection.</p>", "<title>Experiments</title>", "<title>Datasets and implementation</title>", "<p id=\"Par28\">To ensure the model’s credibility, we focused on general datasets that target traffic anomaly detection in our experiments. We verify our MTGAE method on two public traffic network datasets.<list list-type=\"bullet\"><list-item><p id=\"Par29\"><bold>PEMS-BAY dataset:</bold> It is collected in real-time from nearly 40,000 individual detectors spanning the freeway system across all major metropolitan areas of California^{##UREF##47##50##}. The dataset comprises 365 sensors located in the bay area, and it contains traffic data recorded from April to May 2014. For our analysis, we selected a subgraph of six sensors, each with recorded speed and traffic flow information pertaining to our network. Furthermore, we extended the duration of each traffic incident from CHP (CHP Traffic Incident Information <ext-link ext-link-type=\"uri\" xlink:href=\"https://www.chp.ca.gov/traffic\">https://www.chp.ca.gov/traffic</ext-link>), by one hour to account for the impact of traffic accidents.</p></list-item><list-item><p id=\"Par30\"><bold>New York City (NYC) taxi dataset:</bold> The New York City (NYC) taxi trips dataset is publicly released by the Taxi and Limousine Commission (TLC). We use it to record the time and location of each taxi pick-up and drop-off and pool the records formed for each hour of that taxi into a matrix. This dataset includes six months of data, from January 2019 to March 2019. Since the NYC dataset lacks exception tagging points, we utilized exception injection to add exceptions into the timing of the dataset^{##UREF##48##51##,##UREF##49##52##}.</p></list-item></list></p>", "<title>Baselines</title>", "<p id=\"Par31\">To validate our method’s effectiveness in anomaly detection within the NYC dataset. We obtained these methods from their official public code repositories and employed their optimal experimental setups, running all models on the NYC dataset to guarantee fairness:<list list-type=\"bullet\"><list-item><p id=\"Par32\"><bold>Con-GAE</bold>^{##UREF##11##13##}: The method was developed to tackle the challenges posed by extreme data sparsity and high dimensionality, specifically to address anomalies in traffic conditions. Moreover, It utilizes context-enhanced graph autoencoders to enhance the effectiveness of anomaly detection.</p></list-item><list-item><p id=\"Par33\"><bold>SuperGAT</bold>^{##UREF##50##53##}: A self-supervised graph attention network, uses edge information to guide attention learning. SuperGAT analyzes two common attention forms, revealing their limitations in capturing label agreement and edge presence, and proposes enhanced attention mechanisms tailored to graph characteristics.</p></list-item><list-item><p id=\"Par34\"><bold>EG</bold>^{##UREF##51##54##}: The Efficient Graph Convolution (EGC) method is an isotropic Graph Neural Network (GNN) architecture.EGC outperforms comparable anisotropic models like GAT and PNA in terms of accuracy and efficiency. This finding challenges the prevalent belief that anisotropic GNNs are inherently superior.</p></list-item><list-item><p id=\"Par35\"><bold>GraphGPS</bold>^{##UREF##52##55##}: A modular and scalable framework designed to build graph transformers, integrating message passing with global attention. This framework also categorizes positional and structural encodings, thereby injecting useful inductive biases. GraphGPS demonstrates state-of-the-art performance in various graph learning tasks and scales effortlessly to thousands of nodes.</p></list-item><list-item><p id=\"Par36\"><bold>GATv2</bold>^{##UREF##15##17##}: Graph Attention Networks (GATs) are limited by their computation of restricted “static” attention, inhibiting their ability to dynamically prioritize neighbors. To overcome this limitation, GATv2 alters the order of operations in the scoring function, enabling more expressive dynamic attention.</p></list-item><list-item><p id=\"Par37\"><bold>Dir-GNN</bold>^{##UREF##53##56##}: The method enhances message passing neural networks (MPNNs) by incorporating edge directionality and conducting distinct aggregations for incoming and outgoing edges. Moreover, It significantly betters learning on heterophilic graphs, where neighboring nodes often have different labels, and maintains performance on homophilic graphs, characterized by label-sharing neighbors.</p></list-item><list-item><p id=\"Par38\"><bold>PMLP</bold>^{##UREF##54##57##}: The method introduces propagational MLPs, which employ MLP architecture for training and add message passing layers before inference. This approach bridges the gap between MLPs and GNNs, achieving performance that is comparable to or surpasses that of GNNs. It demonstrates the effectiveness of GNN architectures for generalization, even without training in a graph context. Additionally, PMLPs offer faster and more robust training than GNNs.</p></list-item></list></p>", "<title>Experimental setups</title>", "<p id=\"Par39\">Our experiments were conducted using a GPU 2080TI and an Intel(R) Core(TM) i7-7700K CPU @ 4.20GHz. Considering the anomaly problem, we experimentally used anomaly injection, randomly selecting time slices in each sequence to inject anomalies. Then extract a portion of the time series corresponding uniformly distributed time slices for perturbation factors for anomaly perturbation (e.g., 10 am for 10 pm). In this experiment, we set three pollution ratios and magnitudes , , and on the data set NYC. Anomalies in traffic networks are mainly divided into two types^{##UREF##55##58##,##REF##36927733##59##}: (1) Spatial anomalies: where the current traffic conditions are inconsistent with normal traffic conditions (for example, the flow of traffic vehicles is inconsistent with the normal flow of travel in the past). (2) Temporal anomalies: where the current traffic conditions conform to the normal spatial pattern, but not to the current time. In this paper, we perform some anomaly handling on the dataset: Let represent the proportion of time slices randomly selected for contamination, which is applicable to the injection of both spatial and temporal anomalies; Let denote the proportion of origin-destination pairs selected for contamination; Let defines the range of the uniform distribution used to perturb the travel time. In fact, defines the magnitude of spatial anomalies, i.e., the maximum possible value of travel time perturbation.</p>", "<p id=\"Par40\">In the experiments, we adjust the levels of pollution ratios and magnitudes (, , and ), to evaluate the effectiveness of anomaly detection under different scenarios. The specific steps are as follows: For spatial anomalies, we first randomly select a certain proportion () of time slices and randomly choose a certain proportion () of origin-destination pairs in each contaminated time slice, and then perturb the travel time of these pairs by factors drawn from the uniform distribution <italic>U</italic>(, ). Temporal anomalies are created by randomly selecting a certain proportion () of time slices and shifting the time in the data by 12 hours (e.g., changing 8 PM to 8 AM, and vice versa). We set , , and .</p>", "<p id=\"Par41\">For the training process, we initially set the epoch number at 150 and the batch size at 10 per epoch. In the previously mentioned day of the week and hour of the day metrics, set both and to 100, and the dimension of the graph embedding we set to 150 and 50, respectively, the discard rate was set to 0.2, the learning rate is 0.001 by default. Then, we set the learning rate decay in the process, each time, the growth is 0.1 times the last learning rate so that the model can learn the parameters better. Finally, we selected the NYC datasets from January 8 to March 31, 2019, as the training set and extracted 10% from it for validation. We used the NYC datasets from January 1 to January 7, 2019, and a portion of the Uber Movement as the test set. Note that the sampling process was based on uniform distribution random sampling, and both training set and test set were mutually exclusive (i.e., the same data point would not appear in both the training set and test set).</p>", "<p id=\"Par42\">In addition, ablation experiments were performed on the PEMS dataset to verify the effectiveness of the proposed module, which was evaluated using MAE metrics. The loss functions that MAE and RMSE are more credible test methods in some anomaly detection, especially in the traffic area^{##REF##34982699##3##}. Six epochs were set for training. Each period was divided into 128 batches, the generator loss function was 500, the learning rate was 0.001, and the decayed by a factor of 0.1 per epoch. In this dataset, we set the number of layers of TCN to 9 and transformed the head nodes in GCN to GAT to improve the model’s parallelism. In learning, we set the hidden layer to 64.</p>" ]

[ "<title>Acknowledgements</title>", "<p>This work was supported by the National Key Research and Development Program of China (2022ZD0115604) and National Natural Science Foundation of China (Grant Nos. 42130608, 42075142) and the Sichuan Science and Technology program (Grant Nos. 2023ZHCG0018, 2023NSFSC0470, 2021YFQ0053, 2022YFG0152, 23NSFSC2224, 2020JDTD0020, 2022YFG0026, 2021YFG0018, 2020YJ0241).</p>", "<title>Author contributions</title>", "<p>Conceptualization, X.L., Q.T., C. S. and X.W.; Data curation, Z.R.; Formal analysis, X.L. and J.P.; Investigation, Z.R., X.L. and C.S.; Methodology, Z.R.; Project administration, Z.R.; Resources, X.L. and X.W.; Software, Z.R.; Supervision, X.L. and K.C.; Validation, Z.R. and X.L.; Visualization, Z.R.; Writing—original draft, Z.R.; Writing—review &amp; editing, X.L. All authors reviewed the manuscript.</p>", "<title>Data availability</title>", "<p>The datasets analyzed during the current study are available in the GitHub repository, including the NYC dataset and the PEMS dataset, which can be found at <ext-link ext-link-type=\"uri\" xlink:href=\"https://github.com/yuehu9/Con-GAE\">https://github.com/yuehu9/Con-GAE</ext-link> and <ext-link ext-link-type=\"uri\" xlink:href=\"https://github.com/dleyan/STGAN\">https://github.com/dleyan/STGAN</ext-link>, respectively. And the datasets used for testing the generalization capabilities: <ext-link ext-link-type=\"uri\" xlink:href=\"https://dgraph.xinye.com/dataset\">https://dgraph.xinye.com/dataset</ext-link>. Additionally, we collected and analyzed some of the data used in our experiments. The experimental data collected during the current study available from the corresponding author on reasonable request.</p>", "<title>Competing interests</title>", "<p id=\"Par50\">The authors declare no competing interests.</p>" ]

[ "<fig id=\"Fig1\"><label>Figure 1</label><caption><p>Illustration of two parts of NYC dataset graph (first column) and corresponding weighted adjacency matrix (second column) corresponding to scaled inverse travel times between points on the graph. Note that the mobility data is modeled as a series of time-dependent directed weighted graphs. (<bold>a</bold>) A portion of the visualizable NYC mobility data. (<bold>b</bold>) the adjacency matrix corresponding to subfigure (<bold>a</bold>). (<bold>c</bold>) Another portion of the visualizable NYC mobility data, but it is not part of subgraph (<bold>a</bold>). (<bold>d</bold>) the adjacency matrix corresponding to subfigure (<bold>c</bold>).</p></caption></fig>", "<fig id=\"Fig2\"><label>Figure 2</label><caption><p>The architecture of the MTGAE. The architecture consists of two main components: an Encoder and a Decoder. The Encoder includes an adaptive process, the MTCM, and the GCGRU CELL. The MTCM is designed to effectively capture relevant information from data of variable length. It incorporates the TCN for processing data that has undergone a ’mirroring’ transformation, adjusting the length of mirrored data back to its original state prior to input. The GCGRU CELL, comprising a Gaussian Kernel and Graph Convolutional Networks (GCNs), is instrumental in mapping finite-dimensional data to a higher dimensional space. This mapping aids in anomaly detection while preserving data distribution. The GCNs within the GCGRU CELL leverage the GRU architecture to extract spatial-temporal dependencies. Lastly, the Decoder’s primary function, facilitated by the Bilinear module, is to resample features derived from the Encoder’s output, enhancing the overall data interpretation process.</p></caption></fig>", "<fig id=\"Fig3\"><label>Figure 3</label><caption><p>(<bold>a</bold>) The illustration explains: MTCM uses a hole convolution kernel with a size of 2. The left is the mirror image feature of , uses the expansion factor K, selects the input of each k step, and then uses 1D convolution. (<bold>b</bold>) The figure explains how an embedded node and surrounding embedded nodes are connected through GCN, where the orange node is the original node, and its neighbour nodes are white and enclosed in the ellipse. (<bold>c</bold>) The figure explains ur GCN layer is different from the original GCN, our GCN layer can associate more sub-nodes. (<bold>d</bold>) The diagram shows how our GCN layer is associated with its child nodes (blue nodes) through the example orange node, and then the child nodes (blue nodes) spread to its child nodes (yellow nodes).</p></caption></fig>", "<fig id=\"Fig4\"><label>Figure 4</label><caption><p>The image shows the GRU architecture, on which we were inspired to change the gating unit of GRU to GCN, giving it the same ability as GRU to capture short-term and long-term dependencies in a sequence. The update gating and reset gating are reflected in the derivation equations of GCGRU in this paper.</p></caption></fig>", "<fig id=\"Fig5\"><label>Figure 5</label><caption><p>We extracted the intermediate variables before and after using the Gaussian kernel module to visually demonstrate this module’s importance in our model. (<bold>a</bold>) Before the Gaussian kernel module. (<bold>b</bold>) After the Gaussian kernel module.</p></caption></fig>", "<fig id=\"Fig6\"><label>Figure 6</label><caption><p>(<bold>a</bold>) Our model’s ability to detect traffic anomalies. The horizontal axis denotes an hour, the vertical axis denotes date, and the color depth indicates the possibility of traffic anomalies (reconstruction loss). (<bold>b</bold>) The sensitivity experiment of the model. It is guaranteed to be between 0.9 and 1.0 under different node embedding and time embedding.</p></caption></fig>" ]

[ "<table-wrap id=\"Tab1\"><label>Table 1</label><caption><p>Given fixed and , the AUC scores of different models with the fraction of the time slices chosen to be polluted.</p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th align=\"left\">anomaly rate </th><th align=\"left\">5%</th><th align=\"left\">10%</th><th align=\"left\"> 20%</th></tr></thead><tbody><tr><td align=\"left\">HA (1981)</td><td align=\"left\">0.728</td><td align=\"left\">0.687</td><td align=\"left\">0.711</td></tr><tr><td align=\"left\">RTC (2020)</td><td align=\"left\">0.736</td><td align=\"left\">0.765</td><td align=\"left\">0.790</td></tr><tr><td align=\"left\">AE (2002)</td><td align=\"left\">0.813</td><td align=\"left\">0.812</td><td align=\"left\">0.806</td></tr><tr><td align=\"left\">EncDec-AD (2016)</td><td align=\"left\">0.584</td><td align=\"left\">0.582</td><td align=\"left\">0.582</td></tr><tr><td align=\"left\">REBM (2018)</td><td align=\"left\">0.844</td><td align=\"left\">0.859</td><td align=\"left\">0.833</td></tr><tr><td align=\"left\">DAGMM (2016)</td><td align=\"left\">0.550</td><td align=\"left\">0.546</td><td align=\"left\">0.507</td></tr><tr><td align=\"left\">GraphSAGE (2017)</td><td align=\"left\">0.842</td><td align=\"left\">0.840</td><td align=\"left\">0.860</td></tr><tr><td align=\"left\">GCN (2016)</td><td align=\"left\">0.708</td><td align=\"left\">0.717</td><td align=\"left\">0.744</td></tr><tr><td align=\"left\">Con-GAE (2021)</td><td align=\"left\">0.903</td><td align=\"left\">0.908</td><td align=\"left\">0.913</td></tr><tr><td align=\"left\">SuperGAT (2021)</td><td align=\"left\">0.901</td><td align=\"left\">0.908</td><td align=\"left\">0.910</td></tr><tr><td align=\"left\">EG (2021)</td><td align=\"left\">0.883</td><td align=\"left\">0.902</td><td align=\"left\">0.912</td></tr><tr><td align=\"left\">GraphGPS (2022)</td><td align=\"left\">0.894</td><td align=\"left\">0.906</td><td align=\"left\">0.911</td></tr><tr><td align=\"left\">GATv2 (2022)</td><td align=\"left\">0.887</td><td align=\"left\">0.899</td><td align=\"left\">0.902</td></tr><tr><td align=\"left\">Dir-GNN (2023)</td><td align=\"left\">0.882</td><td align=\"left\">0.899</td><td align=\"left\">0.911</td></tr><tr><td align=\"left\">PMLP-layer128 (2023)</td><td align=\"left\">0.960</td><td align=\"left\">0.965</td><td align=\"left\">0.968</td></tr><tr><td align=\"left\">MTGAE</td><td align=\"left\">1.000</td><td align=\"left\">1.000</td><td align=\"left\">1.000</td></tr></tbody></table></table-wrap>", "<table-wrap id=\"Tab2\"><label>Table 2</label><caption><p>Given fixed fixed , the AUC scores for anomaly detection. Results are shown under different and .</p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th align=\"left\">Spatial anomaly rate </th><th align=\"left\" colspan=\"3\">25%</th><th align=\"left\" colspan=\"3\">50%</th></tr><tr><th align=\"left\">Anomaly magnitude </th><th align=\"left\">5%</th><th align=\"left\">10%</th><th align=\"left\">20%</th><th align=\"left\">5%</th><th align=\"left\">10%</th><th align=\"left\">20%</th></tr></thead><tbody><tr><td align=\"left\">HA (1981)</td><td align=\"left\">0.405</td><td align=\"left\">0.533</td><td align=\"left\">0.804</td><td align=\"left\">0.455</td><td align=\"left\">0.687</td><td align=\"left\">0.934</td></tr><tr><td align=\"left\">AE (2002)</td><td align=\"left\">0.294</td><td align=\"left\">0.572</td><td align=\"left\">0.936</td><td align=\"left\">0.405</td><td align=\"left\">0.812</td><td align=\"left\">0.994</td></tr><tr><td align=\"left\">EncDec-AD (2016)</td><td align=\"left\">0.410</td><td align=\"left\">0.483</td><td align=\"left\">0.727</td><td align=\"left\">0.452</td><td align=\"left\">0.582</td><td align=\"left\">0.896</td></tr><tr><td align=\"left\">GCN (2016)</td><td align=\"left\">0.443</td><td align=\"left\">0.564</td><td align=\"left\">0.844</td><td align=\"left\">0.498</td><td align=\"left\">0.717</td><td align=\"left\">0.966</td></tr><tr><td align=\"left\">DAGMM (2016)</td><td align=\"left\">0.511</td><td align=\"left\">0.527</td><td align=\"left\">0.567</td><td align=\"left\">0.525</td><td align=\"left\">0.546</td><td align=\"left\">0.639</td></tr><tr><td align=\"left\">GraphSAGE (2017)</td><td align=\"left\">0.381</td><td align=\"left\">0.627</td><td align=\"left\">0.963</td><td align=\"left\">0.491</td><td align=\"left\">0.840</td><td align=\"left\">1.000</td></tr><tr><td align=\"left\">REBM (2018)</td><td align=\"left\">0.389</td><td align=\"left\">0.633</td><td align=\"left\">0.958</td><td align=\"left\">0.491</td><td align=\"left\">0.859</td><td align=\"left\">0.997</td></tr><tr><td align=\"left\">RTC (2020)</td><td align=\"left\">0.626</td><td align=\"left\">0.699</td><td align=\"left\">0.863</td><td align=\"left\">0.648</td><td align=\"left\">0.765</td><td align=\"left\">0.942</td></tr><tr><td align=\"left\">Con-GAE (2021)</td><td align=\"left\">0.946</td><td align=\"left\">0.755</td><td align=\"left\">0.985</td><td align=\"left\">0.610</td><td align=\"left\">0.908</td><td align=\"left\">1.000</td></tr><tr><td align=\"left\">SuperGAT (2021)</td><td align=\"left\">0.904</td><td align=\"left\">0.910</td><td align=\"left\">0.911</td><td align=\"left\">0.893</td><td align=\"left\">0.908</td><td align=\"left\">0.911</td></tr><tr><td align=\"left\">EG (2021)</td><td align=\"left\">0.897</td><td align=\"left\">0.904</td><td align=\"left\">0.906</td><td align=\"left\">0.888</td><td align=\"left\">0.902</td><td align=\"left\">0.906</td></tr><tr><td align=\"left\">GraphGPS (2022)</td><td align=\"left\">0.900</td><td align=\"left\">0.909</td><td align=\"left\">0.911</td><td align=\"left\">0.890</td><td align=\"left\">0.906</td><td align=\"left\">0.911</td></tr><tr><td align=\"left\">GATv2 (2022)</td><td align=\"left\">0.894</td><td align=\"left\">0.901</td><td align=\"left\">0.902</td><td align=\"left\">0.885</td><td align=\"left\">0.899</td><td align=\"left\">0.902</td></tr><tr><td align=\"left\">Dir-GNN (2023)</td><td align=\"left\">0.893</td><td align=\"left\">0.902</td><td align=\"left\">0.904</td><td align=\"left\">0.883</td><td align=\"left\">0.899</td><td align=\"left\">0.904</td></tr><tr><td align=\"left\">PMLP-layer1 (2023)</td><td align=\"left\">0.895</td><td align=\"left\">0.902</td><td align=\"left\">0.905</td><td align=\"left\">0.884</td><td align=\"left\">0.900</td><td align=\"left\">0.905</td></tr><tr><td align=\"left\">PMLP-layer2 (2023)</td><td align=\"left\">0.880</td><td align=\"left\">0.887</td><td align=\"left\">0.888</td><td align=\"left\">0.871</td><td align=\"left\">0.885</td><td align=\"left\">0.889</td></tr><tr><td align=\"left\">PMLP-layer64 (2023)</td><td align=\"left\">0.916</td><td align=\"left\">0.924</td><td align=\"left\">0.926</td><td align=\"left\">0.906</td><td align=\"left\">0.922</td><td align=\"left\">0.926</td></tr><tr><td align=\"left\">PMLP-layer128 (2023)</td><td align=\"left\">0.960</td><td align=\"left\">0.967</td><td align=\"left\">0.968</td><td align=\"left\">0.952</td><td align=\"left\">0.965</td><td align=\"left\">0.968</td></tr><tr><td align=\"left\">MTGAE</td><td align=\"left\">1.000</td><td align=\"left\">1.000</td><td align=\"left\">1.000</td><td align=\"left\">1.000</td><td align=\"left\">1.000</td><td align=\"left\">1.000</td></tr></tbody></table></table-wrap>", "<table-wrap id=\"Tab3\"><label>Table 3</label><caption><p>Ablation study.</p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th align=\"left\">Model</th><th align=\"left\">MAE</th><th align=\"left\">RMSE</th><th align=\"left\">Note</th></tr></thead><tbody><tr><td align=\"left\">MTGAE-gan</td><td align=\"left\">3.076</td><td align=\"left\">6.770</td><td align=\"left\">Replace our model architecture with GAN</td></tr><tr><td align=\"left\">MTGAE-ot</td><td align=\"left\">3.150</td><td align=\"left\">6.880</td><td align=\"left\">The temporal convolution module without mirror</td></tr><tr><td align=\"left\">MTGAE-mt</td><td align=\"left\">3.019</td><td align=\"left\">6.713</td><td align=\"left\">The temporal convolution module with mirror</td></tr><tr><td align=\"left\">MTGAE-lstm</td><td align=\"left\">3.224</td><td align=\"left\">6.940</td><td align=\"left\">Replace GRU with LSTM</td></tr><tr><td align=\"left\">MTGAE-grumt</td><td align=\"left\">3.052</td><td align=\"left\">6.751</td><td align=\"left\">Add the GCGRU-Cell module to the architecture</td></tr><tr><td align=\"left\">MTGAE-Transformer</td><td align=\"left\">17.020</td><td align=\"left\">32.507</td><td align=\"left\">Replace our model architecture with Transformer</td></tr><tr><td align=\"left\">MTGAE-gb</td><td align=\"left\">3.088</td><td align=\"left\">6.780</td><td align=\"left\">The Gaussian kernel module as a post-processing step</td></tr><tr><td align=\"left\">MTGAE</td><td align=\"left\">2.990</td><td align=\"left\">6.679</td><td align=\"left\">Our method</td></tr></tbody></table></table-wrap>", "<table-wrap id=\"Tab4\"><label>Table 4</label><caption><p>The AUC scores of our model and other baselines on the DGraphFin dataset.</p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th align=\"left\">Model</th><th align=\"left\">AUC</th></tr></thead><tbody><tr><td align=\"left\">GCN</td><td align=\"left\">0.707</td></tr><tr><td align=\"left\">MLP</td><td align=\"left\">0.719</td></tr><tr><td align=\"left\">GAT</td><td align=\"left\">0.733</td></tr><tr><td align=\"left\">GATv2</td><td align=\"left\">0.762</td></tr><tr><td align=\"left\">TGN</td><td align=\"left\">0.774</td></tr><tr><td align=\"left\">SAGE</td><td align=\"left\">0.776</td></tr><tr><td align=\"left\">MTGAE</td><td align=\"left\">0.768</td></tr></tbody></table></table-wrap>" ]

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\n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$v_j$$\\end{document}</tex-math><mml:math id=\"M12\"><mml:msub><mml:mi>v</mml:mi><mml:mi>j</mml:mi></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq7\"><alternatives><tex-math id=\"M13\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$G(T) = (V, E, W)$$\\end{document}</tex-math><mml:math id=\"M14\"><mml:mrow><mml:mi>G</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>T</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>=</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>V</mml:mi><mml:mo>,</mml:mo><mml:mi>E</mml:mi><mml:mo>,</mml:mo><mml:mi>W</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq8\"><alternatives><tex-math id=\"M15\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$t_a \\in T$$\\end{document}</tex-math><mml:math id=\"M16\"><mml:mrow><mml:msub><mml:mi>t</mml:mi><mml:mi>a</mml:mi></mml:msub><mml:mo>∈</mml:mo><mml:mi>T</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq9\"><alternatives><tex-math id=\"M17\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$t_a \\in T$$\\end{document}</tex-math><mml:math id=\"M18\"><mml:mrow><mml:msub><mml:mi>t</mml:mi><mml:mi>a</mml:mi></mml:msub><mml:mo>∈</mml:mo><mml:mi>T</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq10\"><alternatives><tex-math id=\"M19\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$h_2^{(t)}$$\\end{document}</tex-math><mml:math id=\"M20\"><mml:msubsup><mml:mi>h</mml:mi><mml:mn>2</mml:mn><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:msubsup></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq11\"><alternatives><tex-math id=\"M21\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\varvec{x} =\\{x_1, x_2, x_3,\\ldots x_i\\} \\in \\mathbb {R}^i$$\\end{document}</tex-math><mml:math id=\"M22\"><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">x</mml:mi></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mo stretchy=\"false\">{</mml:mo><mml:msub><mml:mi>x</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>x</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>x</mml:mi><mml:mn>3</mml:mn></mml:msub><mml:mo>,</mml:mo><mml:mo>…</mml:mo><mml:msub><mml:mi>x</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo stretchy=\"false\">}</mml:mo></mml:mrow><mml:mo>∈</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant=\"double-struck\">R</mml:mi></mml:mrow><mml:mi>i</mml:mi></mml:msup></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq12\"><alternatives><tex-math id=\"M23\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\varvec{f}: \\{0, \\ldots , j - 1\\} \\rightarrow \\mathbb {R}$$\\end{document}</tex-math><mml:math id=\"M24\"><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">f</mml:mi></mml:mrow><mml:mo>:</mml:mo><mml:mo stretchy=\"false\">{</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mo>…</mml:mo><mml:mo>,</mml:mo><mml:mi>j</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">}</mml:mo><mml:mo stretchy=\"false\">→</mml:mo><mml:mi mathvariant=\"double-struck\">R</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ1\"><label>1</label><alternatives><tex-math id=\"M25\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} (x_{m}\\oplus x)*f(j)=\\sum _{i=0}^{k-1}f(i)\\cdot j-d \\cdot i \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M26\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>x</mml:mi><mml:mi>m</mml:mi></mml:msub><mml:mo>⊕</mml:mo><mml:mi>x</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mrow/><mml:mo>∗</mml:mo><mml:mi>f</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>j</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:munderover><mml:mo>∑</mml:mo><mml:mrow><mml:mi>i</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow><mml:mrow><mml:mi>k</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:munderover><mml:mi>f</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>i</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>·</mml:mo><mml:mi>j</mml:mi><mml:mo>-</mml:mo><mml:mi>d</mml:mi><mml:mo>·</mml:mo><mml:mi>i</mml:mi></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq13\"><alternatives><tex-math id=\"M27\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$x_m$$\\end{document}</tex-math><mml:math id=\"M28\"><mml:msub><mml:mi>x</mml:mi><mml:mi>m</mml:mi></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq14\"><alternatives><tex-math id=\"M29\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\oplus$$\\end{document}</tex-math><mml:math id=\"M30\"><mml:mo>⊕</mml:mo></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq15\"><alternatives><tex-math id=\"M31\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$x_m$$\\end{document}</tex-math><mml:math id=\"M32\"><mml:msub><mml:mi>x</mml:mi><mml:mi>m</mml:mi></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq16\"><alternatives><tex-math id=\"M33\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$x$$\\end{document}</tex-math><mml:math id=\"M34\"><mml:mi>x</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq17\"><alternatives><tex-math id=\"M35\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$s^{(t)}$$\\end{document}</tex-math><mml:math id=\"M36\"><mml:msup><mml:mi>s</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:msup></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq18\"><alternatives><tex-math id=\"M37\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$z^{(t)}$$\\end{document}</tex-math><mml:math id=\"M38\"><mml:msup><mml:mi>z</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:msup></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq19\"><alternatives><tex-math id=\"M39\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\sigma$$\\end{document}</tex-math><mml:math id=\"M40\"><mml:mi>σ</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq20\"><alternatives><tex-math id=\"M41\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\varvec{H_{apt}}=x+H$$\\end{document}</tex-math><mml:math id=\"M42\"><mml:mrow><mml:mrow><mml:msub><mml:mi mathvariant=\"bold-italic\">H</mml:mi><mml:mrow><mml:mi mathvariant=\"bold-italic\">apt</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo>=</mml:mo><mml:mi>x</mml:mi><mml:mo>+</mml:mo><mml:mi>H</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq21\"><alternatives><tex-math id=\"M43\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$h_ 1 ^ {(t)}$$\\end{document}</tex-math><mml:math id=\"M44\"><mml:msubsup><mml:mi>h</mml:mi><mml:mn>1</mml:mn><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:msubsup></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq22\"><alternatives><tex-math id=\"M45\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\varvec{H_{apt}}=h_ 1^{(t)} + H$$\\end{document}</tex-math><mml:math id=\"M46\"><mml:mrow><mml:mrow><mml:msub><mml:mi mathvariant=\"bold-italic\">H</mml:mi><mml:mrow><mml:mi mathvariant=\"bold-italic\">apt</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo>=</mml:mo><mml:msubsup><mml:mi>h</mml:mi><mml:mn>1</mml:mn><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:msubsup><mml:mo>+</mml:mo><mml:mi>H</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ2\"><label>2</label><alternatives><tex-math id=\"M47\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} {\\textrm{g}}(H_{apt}) = \\frac{1}{{\\sigma _i \\sqrt{2\\pi } }}{e^{ - \\frac{1}{2}{{\\left( {\\frac{{H_{apt} - i }}{\\sigma _i }} \\right) }^2}}}, \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M48\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:mtext>g</mml:mtext><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>H</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">apt</mml:mi></mml:mrow></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:mrow><mml:msub><mml:mi>σ</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:msqrt><mml:mrow><mml:mn>2</mml:mn><mml:mi>π</mml:mi></mml:mrow></mml:msqrt></mml:mrow></mml:mfrac><mml:msup><mml:mi>e</mml:mi><mml:mrow><mml:mo>-</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:mn>2</mml:mn></mml:mfrac><mml:msup><mml:mrow><mml:mfenced close=\")\" open=\"(\"><mml:mfrac><mml:mrow><mml:msub><mml:mi>H</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">apt</mml:mi></mml:mrow></mml:msub><mml:mo>-</mml:mo><mml:mi>i</mml:mi></mml:mrow><mml:msub><mml:mi>σ</mml:mi><mml:mi>i</mml:mi></mml:msub></mml:mfrac></mml:mfenced></mml:mrow><mml:mn>2</mml:mn></mml:msup></mml:mrow></mml:msup><mml:mo>,</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq23\"><alternatives><tex-math id=\"M49\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$g(H_{apt})$$\\end{document}</tex-math><mml:math id=\"M50\"><mml:mrow><mml:mi>g</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>H</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">apt</mml:mi></mml:mrow></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq24\"><alternatives><tex-math id=\"M51\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\sigma$$\\end{document}</tex-math><mml:math id=\"M52\"><mml:mi>σ</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq25\"><alternatives><tex-math id=\"M53\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\sigma _i$$\\end{document}</tex-math><mml:math id=\"M54\"><mml:msub><mml:mi>σ</mml:mi><mml:mi>i</mml:mi></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq26\"><alternatives><tex-math id=\"M55\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$H_{apt}$$\\end{document}</tex-math><mml:math id=\"M56\"><mml:msub><mml:mi>H</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">apt</mml:mi></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq27\"><alternatives><tex-math id=\"M57\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\varvec{G}=(V, E, W)$$\\end{document}</tex-math><mml:math id=\"M58\"><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">G</mml:mi></mml:mrow><mml:mo>=</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>V</mml:mi><mml:mo>,</mml:mo><mml:mi>E</mml:mi><mml:mo>,</mml:mo><mml:mi>W</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq28\"><alternatives><tex-math id=\"M59\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\varvec{L}=I - \\hat{D} ^ {- 1/2} \\hat{A} \\hat{D} ^ {- 1/2}$$\\end{document}</tex-math><mml:math id=\"M60\"><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">L</mml:mi></mml:mrow><mml:mo>=</mml:mo><mml:mi>I</mml:mi><mml:mo>-</mml:mo><mml:msup><mml:mover accent=\"true\"><mml:mi>D</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover><mml:mrow><mml:mo>-</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">/</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mover accent=\"true\"><mml:mi>A</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover><mml:msup><mml:mover accent=\"true\"><mml:mi>D</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover><mml:mrow><mml:mo>-</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">/</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq29\"><alternatives><tex-math id=\"M61\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\hat{D}$$\\end{document}</tex-math><mml:math id=\"M62\"><mml:mover accent=\"true\"><mml:mi>D</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq30\"><alternatives><tex-math id=\"M63\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\hat{A}$$\\end{document}</tex-math><mml:math id=\"M64\"><mml:mover accent=\"true\"><mml:mi>A</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq31\"><alternatives><tex-math id=\"M65\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$GA_ {sub}= \\{g (H_{apt1}), g (H_{apt2})\\ldots g (H_{aptn}) \\}$$\\end{document}</tex-math><mml:math id=\"M66\"><mml:mrow><mml:mi>G</mml:mi><mml:msub><mml:mi>A</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">sub</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mrow><mml:mo stretchy=\"false\">{</mml:mo><mml:mi>g</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>H</mml:mi><mml:mrow><mml:mi>a</mml:mi><mml:mi>p</mml:mi><mml:mi>t</mml:mi><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>,</mml:mo><mml:mi>g</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>H</mml:mi><mml:mrow><mml:mi>a</mml:mi><mml:mi>p</mml:mi><mml:mi>t</mml:mi><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>…</mml:mo><mml:mi>g</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>H</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">aptn</mml:mi></mml:mrow></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo stretchy=\"false\">}</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq32\"><alternatives><tex-math id=\"M67\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$GA_{sub-neighbour}$$\\end{document}</tex-math><mml:math id=\"M68\"><mml:mrow><mml:mi>G</mml:mi><mml:msub><mml:mi>A</mml:mi><mml:mrow><mml:mi>s</mml:mi><mml:mi>u</mml:mi><mml:mi>b</mml:mi><mml:mo>-</mml:mo><mml:mi>n</mml:mi><mml:mi>e</mml:mi><mml:mi>i</mml:mi><mml:mi>g</mml:mi><mml:mi>h</mml:mi><mml:mi>b</mml:mi><mml:mi>o</mml:mi><mml:mi>u</mml:mi><mml:mi>r</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ3\"><label>3</label><alternatives><tex-math id=\"M69\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} r^{(t)} = ReLU(GA_{sub}^{(t-1)} \\cdot {W_1^{(t)}} + GA_{sub-neighbour}^{(t-1)} \\cdot {W_2^{(t)}}), \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M70\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:msup><mml:mi>r</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:msup><mml:mo>=</mml:mo><mml:mi>R</mml:mi><mml:mi>e</mml:mi><mml:mi>L</mml:mi><mml:mi>U</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>G</mml:mi><mml:msubsup><mml:mi>A</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">sub</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:msubsup><mml:mo>·</mml:mo><mml:msubsup><mml:mi>W</mml:mi><mml:mn>1</mml:mn><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:msubsup><mml:mo>+</mml:mo><mml:mi>G</mml:mi><mml:msubsup><mml:mi>A</mml:mi><mml:mrow><mml:mi>s</mml:mi><mml:mi>u</mml:mi><mml:mi>b</mml:mi><mml:mo>-</mml:mo><mml:mi>n</mml:mi><mml:mi>e</mml:mi><mml:mi>i</mml:mi><mml:mi>g</mml:mi><mml:mi>h</mml:mi><mml:mi>b</mml:mi><mml:mi>o</mml:mi><mml:mi>u</mml:mi><mml:mi>r</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:msubsup><mml:mo>·</mml:mo><mml:msubsup><mml:mi>W</mml:mi><mml:mn>2</mml:mn><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:msubsup><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>,</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq33\"><alternatives><tex-math id=\"M71\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$W_i^{(t)}$$\\end{document}</tex-math><mml:math id=\"M72\"><mml:msubsup><mml:mi>W</mml:mi><mml:mi>i</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:msubsup></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq34\"><alternatives><tex-math id=\"M73\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$r^{(t)}$$\\end{document}</tex-math><mml:math id=\"M74\"><mml:msup><mml:mi>r</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:msup></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ4\"><label>4</label><alternatives><tex-math id=\"M75\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} \\begin{aligned} s^{(t)}&amp;= \\sigma ({W_z}{r^{(t)}} + {U_z}{g(H_{apt})}), \\\\ z^{(t)}&amp;= \\sigma ({W_s}{r^{(t)}} + {U_s}{g(H_{apt})}), \\\\ {\\widehat{h}^{(t)}}&amp;= \\tanh {({W_{{r^{(t)}}}} + U({s^{(t)}} \\odot {g(H_{apt})}))}, \\\\ {h^{(t)}}&amp;= {z^{(t)}} \\odot {h^{(t - 1)}} + (1 - {z^{(t)}}) \\odot {\\widehat{h}^{(t)}}, \\end{aligned} \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M76\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:msup><mml:mi>s</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:msup></mml:mtd><mml:mtd columnalign=\"left\"><mml:mrow><mml:mo>=</mml:mo><mml:mi>σ</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>W</mml:mi><mml:mi>z</mml:mi></mml:msub><mml:msup><mml:mi>r</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:msup><mml:mo>+</mml:mo><mml:msub><mml:mi>U</mml:mi><mml:mi>z</mml:mi></mml:msub><mml:mrow><mml:mi>g</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>H</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">apt</mml:mi></mml:mrow></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>,</mml:mo></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:mrow/><mml:msup><mml:mi>z</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:msup></mml:mrow></mml:mtd><mml:mtd columnalign=\"left\"><mml:mrow><mml:mo>=</mml:mo><mml:mi>σ</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>W</mml:mi><mml:mi>s</mml:mi></mml:msub><mml:msup><mml:mi>r</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:msup><mml:mo>+</mml:mo><mml:msub><mml:mi>U</mml:mi><mml:mi>s</mml:mi></mml:msub><mml:mrow><mml:mi>g</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>H</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">apt</mml:mi></mml:mrow></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>,</mml:mo></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:mrow/><mml:msup><mml:mover accent=\"true\"><mml:mi>h</mml:mi><mml:mo stretchy=\"true\">^</mml:mo></mml:mover><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:msup></mml:mrow></mml:mtd><mml:mtd columnalign=\"left\"><mml:mrow><mml:mo>=</mml:mo><mml:mo>tanh</mml:mo><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>W</mml:mi><mml:msup><mml:mi>r</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:msup></mml:msub><mml:mo>+</mml:mo><mml:mi>U</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msup><mml:mi>s</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:msup><mml:mo>⊙</mml:mo><mml:mrow><mml:mi>g</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>H</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">apt</mml:mi></mml:mrow></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>,</mml:mo></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:mrow/><mml:msup><mml:mi>h</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:msup></mml:mrow></mml:mtd><mml:mtd columnalign=\"left\"><mml:mrow><mml:mo>=</mml:mo><mml:msup><mml:mi>z</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:msup><mml:mo>⊙</mml:mo><mml:msup><mml:mi>h</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:msup><mml:mo>+</mml:mo><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>1</mml:mn><mml:mo>-</mml:mo><mml:msup><mml:mi>z</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:msup><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>⊙</mml:mo><mml:msup><mml:mover accent=\"true\"><mml:mi>h</mml:mi><mml:mo stretchy=\"true\">^</mml:mo></mml:mover><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:msup><mml:mo>,</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq35\"><alternatives><tex-math id=\"M77\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$h^{t-1}$$\\end{document}</tex-math><mml:math id=\"M78\"><mml:msup><mml:mi>h</mml:mi><mml:mrow><mml:mi>t</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq36\"><alternatives><tex-math id=\"M79\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$W_z$$\\end{document}</tex-math><mml:math id=\"M80\"><mml:msub><mml:mi>W</mml:mi><mml:mi>z</mml:mi></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq37\"><alternatives><tex-math id=\"M81\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$W_s$$\\end{document}</tex-math><mml:math id=\"M82\"><mml:msub><mml:mi>W</mml:mi><mml:mi>s</mml:mi></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq38\"><alternatives><tex-math id=\"M83\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$U_z$$\\end{document}</tex-math><mml:math id=\"M84\"><mml:msub><mml:mi>U</mml:mi><mml:mi>z</mml:mi></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq39\"><alternatives><tex-math id=\"M85\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$U_s$$\\end{document}</tex-math><mml:math id=\"M86\"><mml:msub><mml:mi>U</mml:mi><mml:mi>s</mml:mi></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq40\"><alternatives><tex-math id=\"M87\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$r^{(t)}$$\\end{document}</tex-math><mml:math id=\"M88\"><mml:msup><mml:mi>r</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:msup></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq41\"><alternatives><tex-math id=\"M89\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\sigma$$\\end{document}</tex-math><mml:math id=\"M90\"><mml:mi>σ</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq42\"><alternatives><tex-math id=\"M91\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$h_{hour} \\in R^{day}$$\\end{document}</tex-math><mml:math id=\"M92\"><mml:mrow><mml:msub><mml:mi>h</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">hour</mml:mi></mml:mrow></mml:msub><mml:mo>∈</mml:mo><mml:msup><mml:mi>R</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">day</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq43\"><alternatives><tex-math id=\"M93\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$h_{day} \\in R^{week}$$\\end{document}</tex-math><mml:math id=\"M94\"><mml:mrow><mml:msub><mml:mi>h</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">day</mml:mi></mml:mrow></mml:msub><mml:mo>∈</mml:mo><mml:msup><mml:mi>R</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">week</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq44\"><alternatives><tex-math id=\"M95\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$h_{hour}$$\\end{document}</tex-math><mml:math id=\"M96\"><mml:msub><mml:mi>h</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">hour</mml:mi></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq45\"><alternatives><tex-math id=\"M97\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$h_{day}$$\\end{document}</tex-math><mml:math id=\"M98\"><mml:msub><mml:mi>h</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">day</mml:mi></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq46\"><alternatives><tex-math id=\"M99\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${h_G}^{(t)}$$\\end{document}</tex-math><mml:math id=\"M100\"><mml:msup><mml:mrow><mml:msub><mml:mi>h</mml:mi><mml:mi>G</mml:mi></mml:msub></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:msup></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ5\"><label>5</label><alternatives><tex-math id=\"M101\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} \\begin{aligned} {\\widetilde{h}_G}^{(t)}&amp;= \\oplus (h_1^{(t)},h_2^{(t)},\\ldots ,h_n^{(t)},{h_{hour}}(t),{h_{day}}(t)),\\\\ {\\widetilde{u}_G}^{(t)}&amp;= {{\\hat{U}}^{({\\textrm{t}})}}({{\\hat{U}}^{({\\textrm{t}})}}({\\widetilde{h}_G}^{(t)})), \\\\ {h_G}^{(t)}&amp;= {\\mathop {\\textrm{Re}}\\nolimits } LU({U_G}{\\widetilde{u}_G}^{(t)}), \\end{aligned} \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M102\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:msup><mml:mrow><mml:msub><mml:mover accent=\"true\"><mml:mi>h</mml:mi><mml:mo stretchy=\"true\">~</mml:mo></mml:mover><mml:mi>G</mml:mi></mml:msub></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:msup></mml:mtd><mml:mtd columnalign=\"left\"><mml:mrow><mml:mo>=</mml:mo><mml:mo>⊕</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:msubsup><mml:mi>h</mml:mi><mml:mn>1</mml:mn><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:msubsup><mml:mo>,</mml:mo><mml:msubsup><mml:mi>h</mml:mi><mml:mn>2</mml:mn><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:msubsup><mml:mo>,</mml:mo><mml:mo>…</mml:mo><mml:mo>,</mml:mo><mml:msubsup><mml:mi>h</mml:mi><mml:mi>n</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:msubsup><mml:mo>,</mml:mo><mml:msub><mml:mi>h</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">hour</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>,</mml:mo><mml:msub><mml:mi>h</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">day</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>,</mml:mo></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:mrow/><mml:msup><mml:mrow><mml:msub><mml:mover accent=\"true\"><mml:mi>u</mml:mi><mml:mo stretchy=\"true\">~</mml:mo></mml:mover><mml:mi>G</mml:mi></mml:msub></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:msup></mml:mrow></mml:mtd><mml:mtd columnalign=\"left\"><mml:mrow><mml:mo>=</mml:mo><mml:msup><mml:mrow><mml:mover accent=\"true\"><mml:mi>U</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mtext>t</mml:mtext><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msup><mml:mrow><mml:mover accent=\"true\"><mml:mi>U</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mtext>t</mml:mtext><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msup><mml:mrow><mml:msub><mml:mover accent=\"true\"><mml:mi>h</mml:mi><mml:mo stretchy=\"true\">~</mml:mo></mml:mover><mml:mi>G</mml:mi></mml:msub></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:msup><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>,</mml:mo></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:mrow/><mml:msup><mml:mrow><mml:msub><mml:mi>h</mml:mi><mml:mi>G</mml:mi></mml:msub></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:msup></mml:mrow></mml:mtd><mml:mtd columnalign=\"left\"><mml:mrow><mml:mo>=</mml:mo><mml:mtext>Re</mml:mtext><mml:mi>L</mml:mi><mml:mi>U</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>U</mml:mi><mml:mi>G</mml:mi></mml:msub><mml:msup><mml:mrow><mml:msub><mml:mover accent=\"true\"><mml:mi>u</mml:mi><mml:mo stretchy=\"true\">~</mml:mo></mml:mover><mml:mi>G</mml:mi></mml:msub></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:msup><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>,</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq47\"><alternatives><tex-math id=\"M103\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${h_G}^{(t)}$$\\end{document}</tex-math><mml:math id=\"M104\"><mml:msup><mml:mrow><mml:msub><mml:mi>h</mml:mi><mml:mi>G</mml:mi></mml:msub></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:msup></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq48\"><alternatives><tex-math id=\"M105\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\hat{U}$$\\end{document}</tex-math><mml:math id=\"M106\"><mml:mover accent=\"true\"><mml:mi>U</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq49\"><alternatives><tex-math id=\"M107\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$U_G$$\\end{document}</tex-math><mml:math id=\"M108\"><mml:msub><mml:mi>U</mml:mi><mml:mi>G</mml:mi></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq50\"><alternatives><tex-math id=\"M109\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${h_G}^{(t)}$$\\end{document}</tex-math><mml:math id=\"M110\"><mml:msup><mml:mrow><mml:msub><mml:mi>h</mml:mi><mml:mi>G</mml:mi></mml:msub></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:msup></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq51\"><alternatives><tex-math id=\"M111\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$(v_i, v_j)$$\\end{document}</tex-math><mml:math id=\"M112\"><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>v</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>v</mml:mi><mml:mi>j</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq52\"><alternatives><tex-math id=\"M113\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${h_{hour}}^{(t)},{h_{day}}^{(t)}$$\\end{document}</tex-math><mml:math id=\"M114\"><mml:mrow><mml:msup><mml:mrow><mml:msub><mml:mi>h</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">hour</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:msup><mml:mo>,</mml:mo><mml:msup><mml:mrow><mml:msub><mml:mi>h</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">day</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:msup></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq53\"><alternatives><tex-math id=\"M115\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$w_{ij}$$\\end{document}</tex-math><mml:math id=\"M116\"><mml:msub><mml:mi>w</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">ij</mml:mi></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq54\"><alternatives><tex-math id=\"M117\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\widetilde{h}_G^{\\prime }}{(t)}$$\\end{document}</tex-math><mml:math id=\"M118\"><mml:mrow><mml:msubsup><mml:mover accent=\"true\"><mml:mi>h</mml:mi><mml:mo stretchy=\"true\">~</mml:mo></mml:mover><mml:mi>G</mml:mi><mml:mo>′</mml:mo></mml:msubsup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq55\"><alternatives><tex-math id=\"M119\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${h_i}^{(t)}$$\\end{document}</tex-math><mml:math id=\"M120\"><mml:msup><mml:mrow><mml:msub><mml:mi>h</mml:mi><mml:mi>i</mml:mi></mml:msub></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:msup></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq56\"><alternatives><tex-math id=\"M121\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${h_j}^{(t)}$$\\end{document}</tex-math><mml:math id=\"M122\"><mml:msup><mml:mrow><mml:msub><mml:mi>h</mml:mi><mml:mi>j</mml:mi></mml:msub></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:msup></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq57\"><alternatives><tex-math id=\"M123\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${h_n}^{(t)}$$\\end{document}</tex-math><mml:math id=\"M124\"><mml:msup><mml:mrow><mml:msub><mml:mi>h</mml:mi><mml:mi>n</mml:mi></mml:msub></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:msup></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq58\"><alternatives><tex-math id=\"M125\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\hat{W_{ij}}(t)}$$\\end{document}</tex-math><mml:math id=\"M126\"><mml:mrow><mml:mover accent=\"true\"><mml:msub><mml:mi>W</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">ij</mml:mi></mml:mrow></mml:msub><mml:mo stretchy=\"false\">^</mml:mo></mml:mover><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq59\"><alternatives><tex-math id=\"M127\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${W_{ij}^{R}(t)}$$\\end{document}</tex-math><mml:math id=\"M128\"><mml:mrow><mml:msubsup><mml:mi>W</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">ij</mml:mi></mml:mrow><mml:mi>R</mml:mi></mml:msubsup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq60\"><alternatives><tex-math id=\"M129\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\hat{W_{ij}}(t)}$$\\end{document}</tex-math><mml:math id=\"M130\"><mml:mrow><mml:mover accent=\"true\"><mml:msub><mml:mi>W</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">ij</mml:mi></mml:mrow></mml:msub><mml:mo stretchy=\"false\">^</mml:mo></mml:mover><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq61\"><alternatives><tex-math id=\"M131\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$w_ij={h_i}^{(t)}A{h_j}^{(t)}$$\\end{document}</tex-math><mml:math id=\"M132\"><mml:mrow><mml:msub><mml:mi>w</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mi>j</mml:mi><mml:mo>=</mml:mo><mml:msup><mml:mrow><mml:msub><mml:mi>h</mml:mi><mml:mi>i</mml:mi></mml:msub></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:msup><mml:mi>A</mml:mi><mml:msup><mml:mrow><mml:msub><mml:mi>h</mml:mi><mml:mi>j</mml:mi></mml:msub></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:msup></mml:mrow></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ6\"><label>6</label><alternatives><tex-math id=\"M133\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} \\begin{aligned} {\\widetilde{h}_G^{\\prime }}{(t)}&amp;= ReLU({{dec}_G}Concat({h_G}^{(t)},{h_{hour}}^{(t)},{h_{day}}^{(t)})),\\\\ \\{ {h_1}^{(t)},h_2^{(t)},\\ldots ,h_n^{(t)}\\}&amp;= unstack({\\widetilde{h}_G^{\\prime }}{(t)}),\\\\ {\\hat{W_{ij}}(t)}&amp;= ReLU({dec_1} Concat({h_i}^{(t)}, {h_j}^{(t)})), \\\\ {W_{ij}^{R}(t)}&amp;= Sigmoid({dec_2} {\\hat{W_{ij}}(t)}) \\end{aligned} \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M134\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:msubsup><mml:mover accent=\"true\"><mml:mi>h</mml:mi><mml:mo stretchy=\"true\">~</mml:mo></mml:mover><mml:mi>G</mml:mi><mml:mo>′</mml:mo></mml:msubsup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:mtd><mml:mtd columnalign=\"left\"><mml:mrow><mml:mo>=</mml:mo><mml:mi>R</mml:mi><mml:mi>e</mml:mi><mml:mi>L</mml:mi><mml:mi>U</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant=\"italic\">dec</mml:mi></mml:mrow><mml:mi>G</mml:mi></mml:msub><mml:mi>C</mml:mi><mml:mi>o</mml:mi><mml:mi>n</mml:mi><mml:mi>c</mml:mi><mml:mi>a</mml:mi><mml:mi>t</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msup><mml:mrow><mml:msub><mml:mi>h</mml:mi><mml:mi>G</mml:mi></mml:msub></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:msup><mml:mo>,</mml:mo><mml:msup><mml:mrow><mml:msub><mml:mi>h</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">hour</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:msup><mml:mo>,</mml:mo><mml:msup><mml:mrow><mml:msub><mml:mi>h</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">day</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:msup><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>,</mml:mo></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:mrow/><mml:mo stretchy=\"false\">{</mml:mo><mml:msup><mml:mrow><mml:msub><mml:mi>h</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:msup><mml:mo>,</mml:mo><mml:msubsup><mml:mi>h</mml:mi><mml:mn>2</mml:mn><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:msubsup><mml:mo>,</mml:mo><mml:mo>…</mml:mo><mml:mo>,</mml:mo><mml:msubsup><mml:mi>h</mml:mi><mml:mi>n</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:msubsup><mml:mo stretchy=\"false\">}</mml:mo></mml:mrow></mml:mtd><mml:mtd columnalign=\"left\"><mml:mrow><mml:mo>=</mml:mo><mml:mi>u</mml:mi><mml:mi>n</mml:mi><mml:mi>s</mml:mi><mml:mi>t</mml:mi><mml:mi>a</mml:mi><mml:mi>c</mml:mi><mml:mi>k</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:msubsup><mml:mover accent=\"true\"><mml:mi>h</mml:mi><mml:mo stretchy=\"true\">~</mml:mo></mml:mover><mml:mi>G</mml:mi><mml:mo>′</mml:mo></mml:msubsup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>,</mml:mo></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:mrow/><mml:mrow><mml:mover accent=\"true\"><mml:msub><mml:mi>W</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">ij</mml:mi></mml:mrow></mml:msub><mml:mo stretchy=\"false\">^</mml:mo></mml:mover><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:mrow></mml:mtd><mml:mtd columnalign=\"left\"><mml:mrow><mml:mo>=</mml:mo><mml:mi>R</mml:mi><mml:mi>e</mml:mi><mml:mi>L</mml:mi><mml:mi>U</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mi>d</mml:mi><mml:mi>e</mml:mi><mml:msub><mml:mi>c</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:mrow><mml:mi>C</mml:mi><mml:mi>o</mml:mi><mml:mi>n</mml:mi><mml:mi>c</mml:mi><mml:mi>a</mml:mi><mml:mi>t</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msup><mml:mrow><mml:msub><mml:mi>h</mml:mi><mml:mi>i</mml:mi></mml:msub></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:msup><mml:mo>,</mml:mo><mml:msup><mml:mrow><mml:msub><mml:mi>h</mml:mi><mml:mi>j</mml:mi></mml:msub></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:msup><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>,</mml:mo></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:mrow/><mml:mrow><mml:msubsup><mml:mi>W</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">ij</mml:mi></mml:mrow><mml:mi>R</mml:mi></mml:msubsup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:mrow></mml:mtd><mml:mtd columnalign=\"left\"><mml:mrow><mml:mo>=</mml:mo><mml:mi>S</mml:mi><mml:mi>i</mml:mi><mml:mi>g</mml:mi><mml:mi>m</mml:mi><mml:mi>o</mml:mi><mml:mi>i</mml:mi><mml:mi>d</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mi>d</mml:mi><mml:mi>e</mml:mi><mml:msub><mml:mi>c</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:mrow><mml:mrow><mml:mover accent=\"true\"><mml:msub><mml:mi>W</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">ij</mml:mi></mml:mrow></mml:msub><mml:mo stretchy=\"false\">^</mml:mo></mml:mover><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq62\"><alternatives><tex-math id=\"M135\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$dec_G$$\\end{document}</tex-math><mml:math id=\"M136\"><mml:mrow><mml:mi>d</mml:mi><mml:mi>e</mml:mi><mml:msub><mml:mi>c</mml:mi><mml:mi>G</mml:mi></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq63\"><alternatives><tex-math id=\"M137\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$dec_1$$\\end{document}</tex-math><mml:math id=\"M138\"><mml:mrow><mml:mi>d</mml:mi><mml:mi>e</mml:mi><mml:msub><mml:mi>c</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq64\"><alternatives><tex-math id=\"M139\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\hat{W_{ij}}(t)}$$\\end{document}</tex-math><mml:math id=\"M140\"><mml:mrow><mml:mover accent=\"true\"><mml:msub><mml:mi>W</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">ij</mml:mi></mml:mrow></mml:msub><mml:mo stretchy=\"false\">^</mml:mo></mml:mover><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq65\"><alternatives><tex-math id=\"M141\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$dec_2$$\\end{document}</tex-math><mml:math id=\"M142\"><mml:mrow><mml:mi>d</mml:mi><mml:mi>e</mml:mi><mml:msub><mml:mi>c</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq66\"><alternatives><tex-math id=\"M143\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${W_{ij}^{R}(t)}$$\\end{document}</tex-math><mml:math id=\"M144\"><mml:mrow><mml:msubsup><mml:mi>W</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">ij</mml:mi></mml:mrow><mml:mi>R</mml:mi></mml:msubsup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq67\"><alternatives><tex-math id=\"M145\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${W_{ij}^{R}(t)} \\in [0, 1]$$\\end{document}</tex-math><mml:math id=\"M146\"><mml:mrow><mml:mrow><mml:msubsup><mml:mi>W</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">ij</mml:mi></mml:mrow><mml:mi>R</mml:mi></mml:msubsup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow><mml:mo>∈</mml:mo><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq68\"><alternatives><tex-math id=\"M147\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\hat{y}$$\\end{document}</tex-math><mml:math id=\"M148\"><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ7\"><label>7</label><alternatives><tex-math id=\"M149\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} \\begin{aligned} {Loss({W_{ij}^{R}(t)}, W_{ij}) = \\frac{1}{n} \\sum ^{n}_{i=1} ({W_{ij}^{R}(t)}-W_{ij})^2}, \\end{aligned} \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M150\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:mrow><mml:mi>L</mml:mi><mml:mi>o</mml:mi><mml:mi>s</mml:mi><mml:mi>s</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:msubsup><mml:mi>W</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">ij</mml:mi></mml:mrow><mml:mi>R</mml:mi></mml:msubsup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow><mml:mo>,</mml:mo><mml:msub><mml:mi>W</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">ij</mml:mi></mml:mrow></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:mi>n</mml:mi></mml:mfrac><mml:munderover><mml:mo>∑</mml:mo><mml:mrow><mml:mi>i</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mi>n</mml:mi></mml:munderover><mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:msubsup><mml:mi>W</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">ij</mml:mi></mml:mrow><mml:mi>R</mml:mi></mml:msubsup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow><mml:mo>-</mml:mo><mml:msub><mml:mi>W</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">ij</mml:mi></mml:mrow></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:msup></mml:mrow><mml:mo>,</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq69\"><alternatives><tex-math id=\"M151\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$W_{ij}^{R}(t)$$\\end{document}</tex-math><mml:math id=\"M152\"><mml:mrow><mml:msubsup><mml:mi>W</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">ij</mml:mi></mml:mrow><mml:mi>R</mml:mi></mml:msubsup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq70\"><alternatives><tex-math id=\"M153\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$W_{ij}$$\\end{document}</tex-math><mml:math id=\"M154\"><mml:msub><mml:mi>W</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">ij</mml:mi></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq71\"><alternatives><tex-math id=\"M155\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\theta$$\\end{document}</tex-math><mml:math id=\"M156\"><mml:mi>θ</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq72\"><alternatives><tex-math id=\"M157\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\theta$$\\end{document}</tex-math><mml:math id=\"M158\"><mml:mi>θ</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq73\"><alternatives><tex-math id=\"M159\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\alpha$$\\end{document}</tex-math><mml:math id=\"M160\"><mml:mi>α</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq74\"><alternatives><tex-math id=\"M161\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\beta$$\\end{document}</tex-math><mml:math id=\"M162\"><mml:mi>β</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq75\"><alternatives><tex-math id=\"M163\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\gamma$$\\end{document}</tex-math><mml:math id=\"M164\"><mml:mi>γ</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq76\"><alternatives><tex-math id=\"M165\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\gamma$$\\end{document}</tex-math><mml:math id=\"M166\"><mml:mi>γ</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq77\"><alternatives><tex-math id=\"M167\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\alpha$$\\end{document}</tex-math><mml:math id=\"M168\"><mml:mi>α</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq78\"><alternatives><tex-math id=\"M169\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\beta$$\\end{document}</tex-math><mml:math id=\"M170\"><mml:mi>β</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq79\"><alternatives><tex-math id=\"M171\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\beta$$\\end{document}</tex-math><mml:math id=\"M172\"><mml:mi>β</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq80\"><alternatives><tex-math id=\"M173\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\alpha$$\\end{document}</tex-math><mml:math id=\"M174\"><mml:mi>α</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq81\"><alternatives><tex-math id=\"M175\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\beta$$\\end{document}</tex-math><mml:math id=\"M176\"><mml:mi>β</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq82\"><alternatives><tex-math id=\"M177\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\gamma$$\\end{document}</tex-math><mml:math id=\"M178\"><mml:mi>γ</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq83\"><alternatives><tex-math id=\"M179\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\gamma$$\\end{document}</tex-math><mml:math id=\"M180\"><mml:mi>γ</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq84\"><alternatives><tex-math id=\"M181\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\alpha$$\\end{document}</tex-math><mml:math id=\"M182\"><mml:mi>α</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq85\"><alternatives><tex-math id=\"M183\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$-\\beta$$\\end{document}</tex-math><mml:math id=\"M184\"><mml:mrow><mml:mo>-</mml:mo><mml:mi>β</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq86\"><alternatives><tex-math id=\"M185\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\beta$$\\end{document}</tex-math><mml:math id=\"M186\"><mml:mi>β</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq87\"><alternatives><tex-math id=\"M187\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\gamma$$\\end{document}</tex-math><mml:math id=\"M188\"><mml:mi>γ</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq88\"><alternatives><tex-math id=\"M189\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\alpha \\in \\{25\\%, 50\\%\\}$$\\end{document}</tex-math><mml:math id=\"M190\"><mml:mrow><mml:mi>α</mml:mi><mml:mo>∈</mml:mo><mml:mo stretchy=\"false\">{</mml:mo><mml:mn>25</mml:mn><mml:mo>%</mml:mo><mml:mo>,</mml:mo><mml:mn>50</mml:mn><mml:mo>%</mml:mo><mml:mo stretchy=\"false\">}</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq89\"><alternatives><tex-math id=\"M191\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\beta \\in \\{5\\%, 10\\%, 20\\%\\}$$\\end{document}</tex-math><mml:math id=\"M192\"><mml:mrow><mml:mi>β</mml:mi><mml:mo>∈</mml:mo><mml:mo stretchy=\"false\">{</mml:mo><mml:mn>5</mml:mn><mml:mo>%</mml:mo><mml:mo>,</mml:mo><mml:mn>10</mml:mn><mml:mo>%</mml:mo><mml:mo>,</mml:mo><mml:mn>20</mml:mn><mml:mo>%</mml:mo><mml:mo stretchy=\"false\">}</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq90\"><alternatives><tex-math id=\"M193\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\gamma \\in \\{5\\%, 10\\%, 20\\%\\}$$\\end{document}</tex-math><mml:math id=\"M194\"><mml:mrow><mml:mi>γ</mml:mi><mml:mo>∈</mml:mo><mml:mo stretchy=\"false\">{</mml:mo><mml:mn>5</mml:mn><mml:mo>%</mml:mo><mml:mo>,</mml:mo><mml:mn>10</mml:mn><mml:mo>%</mml:mo><mml:mo>,</mml:mo><mml:mn>20</mml:mn><mml:mo>%</mml:mo><mml:mo stretchy=\"false\">}</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq91\"><alternatives><tex-math id=\"M195\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$h_{day}$$\\end{document}</tex-math><mml:math id=\"M196\"><mml:msub><mml:mi>h</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">day</mml:mi></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq92\"><alternatives><tex-math id=\"M197\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$h_{hour}$$\\end{document}</tex-math><mml:math id=\"M198\"><mml:msub><mml:mi>h</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">hour</mml:mi></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq93\"><alternatives><tex-math id=\"M199\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\alpha =50\\%$$\\end{document}</tex-math><mml:math id=\"M200\"><mml:mrow><mml:mi>α</mml:mi><mml:mo>=</mml:mo><mml:mn>50</mml:mn><mml:mo>%</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq94\"><alternatives><tex-math id=\"M201\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\beta =10\\%$$\\end{document}</tex-math><mml:math id=\"M202\"><mml:mrow><mml:mi>β</mml:mi><mml:mo>=</mml:mo><mml:mn>10</mml:mn><mml:mo>%</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq95\"><alternatives><tex-math id=\"M203\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\alpha$$\\end{document}</tex-math><mml:math id=\"M204\"><mml:mi>α</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq96\"><alternatives><tex-math id=\"M205\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\beta$$\\end{document}</tex-math><mml:math id=\"M206\"><mml:mi>β</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq97\"><alternatives><tex-math id=\"M207\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\gamma$$\\end{document}</tex-math><mml:math id=\"M208\"><mml:mi>γ</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq98\"><alternatives><tex-math id=\"M209\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\gamma$$\\end{document}</tex-math><mml:math id=\"M210\"><mml:mi>γ</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq99\"><alternatives><tex-math id=\"M211\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\gamma$$\\end{document}</tex-math><mml:math id=\"M212\"><mml:mi>γ</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq100\"><alternatives><tex-math id=\"M213\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\alpha$$\\end{document}</tex-math><mml:math id=\"M214\"><mml:mi>α</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq101\"><alternatives><tex-math id=\"M215\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\beta$$\\end{document}</tex-math><mml:math id=\"M216\"><mml:mi>β</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq102\"><alternatives><tex-math id=\"M217\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\alpha$$\\end{document}</tex-math><mml:math id=\"M218\"><mml:mi>α</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq103\"><alternatives><tex-math id=\"M219\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\beta$$\\end{document}</tex-math><mml:math id=\"M220\"><mml:mi>β</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq104\"><alternatives><tex-math id=\"M221\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\alpha =50\\%$$\\end{document}</tex-math><mml:math id=\"M222\"><mml:mrow><mml:mi>α</mml:mi><mml:mo>=</mml:mo><mml:mn>50</mml:mn><mml:mo>%</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq105\"><alternatives><tex-math id=\"M223\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\beta =20\\%$$\\end{document}</tex-math><mml:math id=\"M224\"><mml:mrow><mml:mi>β</mml:mi><mml:mo>=</mml:mo><mml:mn>20</mml:mn><mml:mo>%</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq106\"><alternatives><tex-math id=\"M225\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\beta =5\\%$$\\end{document}</tex-math><mml:math id=\"M226\"><mml:mrow><mml:mi>β</mml:mi><mml:mo>=</mml:mo><mml:mn>5</mml:mn><mml:mo>%</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq107\"><alternatives><tex-math id=\"M227\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\beta$$\\end{document}</tex-math><mml:math id=\"M228\"><mml:mi>β</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq108\"><alternatives><tex-math id=\"M229\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\gamma$$\\end{document}</tex-math><mml:math id=\"M230\"><mml:mi>γ</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq109\"><alternatives><tex-math id=\"M231\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\alpha$$\\end{document}</tex-math><mml:math id=\"M232\"><mml:mi>α</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq110\"><alternatives><tex-math id=\"M233\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\beta$$\\end{document}</tex-math><mml:math id=\"M234\"><mml:mi>β</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq111\"><alternatives><tex-math id=\"M235\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\alpha$$\\end{document}</tex-math><mml:math id=\"M236\"><mml:mi>α</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq112\"><alternatives><tex-math id=\"M237\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\beta$$\\end{document}</tex-math><mml:math id=\"M238\"><mml:mi>β</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq113\"><alternatives><tex-math id=\"M239\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\alpha =50\\%$$\\end{document}</tex-math><mml:math id=\"M240\"><mml:mrow><mml:mi>α</mml:mi><mml:mo>=</mml:mo><mml:mn>50</mml:mn><mml:mo>%</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq114\"><alternatives><tex-math id=\"M241\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\beta =10\\%$$\\end{document}</tex-math><mml:math id=\"M242\"><mml:mrow><mml:mi>β</mml:mi><mml:mo>=</mml:mo><mml:mn>10</mml:mn><mml:mo>%</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq115\"><alternatives><tex-math id=\"M243\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\gamma =10\\%$$\\end{document}</tex-math><mml:math id=\"M244\"><mml:mrow><mml:mi>γ</mml:mi><mml:mo>=</mml:mo><mml:mn>10</mml:mn><mml:mo>%</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>" ]

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[ "<fn-group><fn><p><bold>Publisher's note</bold></p><p>Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.</p></fn><fn><p>These authors contributed equally: Zhiyu Ren and Xiaojie Li.</p></fn></fn-group>" ]

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[ "<title>Introduction</title>", "<p id=\"Par3\">High-throughput spatial omics technologies are at the forefront of modern molecular biology, and promise to provide topographic context to the wealth of available transcriptomic data. Recent breakthroughs in profiling technology have revolutionised our understanding of multicellular biological systems, and the collection of Subcellular Spatial Transcriptomics (SST) technologies (e.g. 10x Genomics Xenium^{##REF##38114474##1##}; NanoString CosMx^{##REF##36203011##2##}; BGI Stereo-seq^{##REF##35512705##3##}; and Vizgen MERSCOPE) now offer the promise to tackle biological problems that were previously inaccessible and better understand intercellular communication by preserving tissue architecture. Depending on the commercial platforms, these ultra-high resolution, spatially resolved single-cell data contain mixtures of nuclear, cytoplasmic, and/or cell membrane signals, and create new data challenges in information extraction. More specifically, the aim is to ensure all available data can be capitalised to automatically and accurately distinguish the boundaries of individual cells, as the fundamental goal of SST technologies is to understand how single-cell transcriptomes behave in situ within a given tissue^{##UREF##0##4##}.</p>", "<p id=\"Par4\">Limited attempts have been made to address these data challenges and to date, three conceptual categories have emerged. The first employs morphological operations originally designed for lower-resolution imaging technologies such as microscopy. Within this category, initial nuclei segmentation is accomplished with a nuclear marker, using thresholding or pretrained models such as Cellpose^{##REF##33318659##5##} and Mesmer^{##REF##34795433##6##}. Cell boundaries are then identified using either morphological expansion by a prespecified distance^{##REF##38114474##1##} or using a watershed algorithm on a mask of the cell bodies^{##REF##35512705##3##}. Chen et al. applied a global threshold to the density of all molecules in SST data to estimate the cell body mask. The limitation of Cellpose^{##REF##33318659##5##} and similar approaches is that they were primarily designed for microscopy modalities and fluorescent markers, so they may not always be suitable for SST due to dissimilar visual characteristics.</p>", "<p id=\"Par5\">Secondly, an alternative approach to cell segmentation does not identify cell boundaries directly, but classifies or clusters individual transcripts into distinct measurement categories that pertain to cells. These include segmentation-free and transcript-based methods, as exemplified by Baysor^{##REF##34650268##7##}, StereoCell^{##UREF##1##8##}, pciSeq^{##REF##31740815##9##}, Sparcle^{##REF##36699413##10##}, and ClusterMap^{##REF##34625546##11##}. However, a key limitation of these approaches is their assumption that expression of all RNAs within a cell body are homogeneous, and in the case of Baysor, that cell shapes (morphologies) can be well approximated with a multivariate normal prior. This can result in visually unrealistic segmentations that do not correspond well to imaging data.</p>", "<p id=\"Par6\">Thirdly, more recent approaches have begun to leverage deep learning (DL) methods. DL models such as U-Net^{##UREF##2##12##} have provided solutions for many image analysis challenges. However, they require ground truth to be generated for training. DL-based methods for SST cell segmentation include GeneSegNet^{##REF##37858204##13##} and SCS^{##REF##37429992##14##}, though supervision is still required in the form of initial cell labels or based on hard-coded rules. Further limitations of existing methods encountered during our benchmarking, such as lengthy code runtimes, are included in Supplementary Table ##SUPPL##0##1##. The self-supervised learning (SSL) paradigm can provide a solution to overcome the requirement of annotations. While SSL-based methods have shown promise for other imaging modalities^{##REF##36323790##15##,##REF##36323790##16##}, direct application to SST images remains challenging. SST data are considerably different from other cellular imaging modalities and natural images (e.g., regular RGB images), as they typically contain hundreds of channels, and there is a lack of clear visual cues that indicate cell boundaries. This creates new challenges such as (i) accurately delineating cohesive masks for cells in densely-packed regions, (ii) handling high sparsity within gene channels, and (iii) addressing the lack of contrast for cell instances.</p>", "<p id=\"Par7\">While these morphological and DL-based approaches have shown promise, they have not fully exploited the high-dimensional expression information contained within SST data. It has become increasingly clear that relying solely on imaging information may not be sufficient to accurately segment cells. There is growing interest in leveraging large, well-annotated scRNA-seq datasets^{##REF##36321662##17##}, as exemplified by JSTA^{##REF##34057817##18##}, which proposed a joint cell segmentation and cell type annotation strategy. While much of the literature has emphasised the importance of accounting for biological information such as transcriptional composition, cell type, and cell morphology, the impact of incorporating such information into segmentation approaches remains to be fully understood.</p>", "<p id=\"Par8\">Here, we present a biologically-informed deep learning-based cell segmentation (BIDCell) framework (Fig. ##FIG##0##1##a), that addresses the challenges of cell body segmentation in SST images through key innovations in the framework and learning strategies. We introduce (a) biologically-informed loss functions with multiple synergistic components; and (b) explicitly incorporate prior knowledge from single-cell sequencing data to enable the estimation of different cell shapes. The combination of our losses and use of existing scRNA-seq data in supplement to subcellular imaging data improves performance, and BIDCell is generalisable across different SST platforms. Along with the development of our segmentation method, we created a comprehensive evaluation framework for cell segmentation, CellSPA, that assesses five complementary categories of criteria for identifying the optimal segmentation strategies. This framework aims to promote the adoption of new segmentation methods for novel biotechnological data.</p>" ]

[ "<title>Methods</title>", "<title>Datasets and preprocessing</title>", "<p id=\"Par31\">We used publicly available data resources from three different SST commercial platforms (10 × Genomics Xenium, NanoString CosMx, and Vizgen MERSCOPE), and sequencing data from Human Cell Atlas.</p>", "<title>Subcellular spatial transcriptomics data</title>", "<p id=\"Par32\">For all datasets and for each gene, detected transcripts were converted into a 2D image where the value of each pixel represents the number of detected transcripts at its location. The images were combined channel-wise, resulting in an image volume , where <italic>H</italic> is the height of the sample, <italic>W</italic> is the width of the sample, and <italic>n</italic>_{<italic>g</italic><italic>e</italic><italic>n</italic><italic>e</italic><italic>s</italic>} is the number of genes in the panel.</p>", "<title>(i) Xenium-BreastCancer1 and Xenium-BreastCancer2</title>", "<p id=\"Par33\">The Breast Cancer datasets included in this study were downloaded from <ext-link ext-link-type=\"uri\" xlink:href=\"https://www.10xgenomics.com/products/xenium-in-situ/preview-dataset-human-breast\">https://www.10xgenomics.com/products/xenium-in-situ/preview-dataset-human-breast</ext-link>(accessed 9 Feb 2023), and included two replicates. Low-quality transcripts for 10 × Genomics Xenium data with a phred-scaled quality value score below 20 were removed, as suggested by the vendor^{##REF##38114474##1##}. Negative control transcripts, blanks, and antisense transcripts were also filtered out. This resulted in 313 unique genes with the overall pixel dimension of the images being 5475 × 7524 × 313 for Xenium breast cancer replicate 1 (Xenium-BreastCancer1) and 5474 × 7524 × 313 for Xenium breast cancer replicate 2 (Xenium-BreastCancer2).</p>", "<title>(ii) Xenium-MouseBrain</title>", "<p id=\"Par34\">The Mouse Brain data included in this study was downloaded from <ext-link ext-link-type=\"uri\" xlink:href=\"https://www.10xgenomics.com/resources/datasets/fresh-frozen-mouse-brain-replicates-1-standard\">https://www.10xgenomics.com/resources/datasets/fresh-frozen-mouse-brain-replicates-1-standard</ext-link> (accessed 14 Feb 2023) and were processed following the steps in (i). There were 248 unique genes, and the resulting size of the image was 7038 × 10,277 × 248 pixels.</p>", "<title>(iii) CosMx-Lung</title>", "<p id=\"Par35\">The CosMx NSCLC Lung dataset included in this study was downloaded from <ext-link ext-link-type=\"uri\" xlink:href=\"https://nanostring.com/products/cosmx-spatial-molecular-imager/nsclc-ffpe-dataset/\">https://nanostring.com/products/cosmx-spatial-molecular-imager/nsclc-ffpe-dataset/</ext-link> (accessed 24 Mar 2023). We used data for Lung5-1, which comprised 30 fields of view. Transcripts containing “NegPrb” were removed, resulting in 960 unique genes and an overall image dimension of 7878 × 9850 × 960 pixels.</p>", "<title>(iv) MERSCOPE-Melanoma</title>", "<p id=\"Par36\">The MERSCOPE melanoma data included in this study were downloaded from <ext-link ext-link-type=\"uri\" xlink:href=\"https://info.vizgen.com/merscope-ffpe-solution\">https://info.vizgen.com/merscope-ffpe-solution</ext-link> (for patient 2, accessed 26 Mar 2023). Transcripts with “Blank-” were filtered out, resulting in 500 unique genes and an image with 6841 × 7849 × 500 pixels.</p>", "<title>(v) Stereo-seq-MouseEmbryo</title>", "<p id=\"Par37\">The Stereo-seq data used in this study, including the DAPI image and detected gene expressions (bin 1), were downloaded from <ext-link ext-link-type=\"uri\" xlink:href=\"https://db.cngb.org/stomics/mosta/download/\">https://db.cngb.org/stomics/mosta/download/</ext-link> for sample E12.5_E1S3. Stereo-seq data contains a far greater number of genes compared to Xenium, CosMx, and MERSCOPE. For efficiency, we selected a panel of 275 highly variable genes (HVGs) as the input to BIDCell. The HVGs are the common genes of the top 1000 HVGs from both Stereo-seq data and the single-cell reference data.</p>", "<title>Nuclei segmentation</title>", "<p id=\"Par38\">DAPI images were directly downloaded from the websites of their respective datasets. In cases where the maximum intensity projection (MIP) DAPI image was not provided, we computed the MIP DAPI by finding the maximum intensity value for each <italic>(x,y)</italic> location for each stack of DAPI. DAPI images were resized to align with the lateral resolutions of spatial transciptomic maps using bilinear interpolation. Nuclei segmentation was performed on the MIP DAPI using the pretrained Cellpose model with automatic estimation of nuclei diameter^{##REF##33318659##5##}. We used the “cyto\" model as we found the “nuclei\" model to undersegment or omit a considerable number (e.g., 21k for Xenium-BreastCancer1) of nuclei given the same MIP DAPI image, which is consistent with another study^{##UREF##4##29##}. Other nuclei segmentation methods may be used with BIDCell as our framework is not limited to Cellpose.</p>", "<title>Transcriptomics sequencing data</title>", "<p id=\"Par39\">We used five publicly available single-cell RNA-seq data collections as references to guide the cell segmentation in BIDCell and evaluation with CellSPA. For the reference data with multiple datasets, we constructed cell-type specific profiles by aggregating the gene expression by cell type per dataset.</p>", "<title>(i) TISCH-BRCA</title>", "<p id=\"Par40\">The reference for Xenium-BreastCancer used in BIDCell was based on 10 single-cell breast cancer datasets downloaded from The Tumor Immune Single Cell Hub 2 (TISCH2)^{##REF##36321662##17##} from <ext-link ext-link-type=\"uri\" xlink:href=\"http://tisch.comp-genomics.org/gallery/?cancer=BRCA&amp;species=Human\">http://tisch.comp-genomics.org/gallery/?cancer=BRCA&amp;species=Human</ext-link>, which contains the gene by cell expressions and cell annotations of the data. We used the “celltype major lineage\" as the cell type labels. We combined the “CD4Tconv\" and “Treg\" as “CD4Tconv/Treg\" and “CD8T\" and “CD8Tex\" as “CD8T/CD8Tex\", which results in 17 cell types in total.</p>", "<title>(ii) Chromium-BreastCancer</title>", "<p id=\"Par41\">To evaluate the performance of Xenium-BreastCancer, we downloaded the Chromium scFFPE-seq data from the same experiment from <ext-link ext-link-type=\"uri\" xlink:href=\"https://www.10xgenomics.com/products/xenium-in-situ/preview-dataset-human-breast\">https://www.10xgenomics.com/products/xenium-in-situ/preview-dataset-human-breast</ext-link> (accessed 22 March 2023), which contains 30,365 cells and 18,082 expressed genes. We then performed Louvain clustering on the k-nearest neighbour graph with <italic>k</italic> = 20, based on the top 50 principal components (PCs) to obtain 22 clusters. We then annotated each cluster based on the markers and annotation provided in the original publication^{##REF##38114474##1##}.</p>", "<title>(iii) Allen Brain Map</title>", "<p id=\"Par42\">The reference for Xenium-MouseBrain data was based on Mouse Whole Cortex and Hippocampus SMART-seq data downloaded from <ext-link ext-link-type=\"uri\" xlink:href=\"https://portal.brain-map.org/atlases-and-data/rnaseq/mouse-whole-cortex-and-hippocampus-smart-seq\">https://portal.brain-map.org/atlases-and-data/rnaseq/mouse-whole-cortex-and-hippocampus-smart-seq</ext-link>, which contains both gene by cell expressions and cell annotations of the data. We used the cluster annotation from “cell_type_alias_label\" as the cell type labels and combined some of the labels with a small number of cells. For example, we combined all “Sst\" subtypes as “Sst\" and all “Vip\" subtypes as “Vip\", which results in 59 cell types in total.</p>", "<title>(iv) HLCA and TISCH-NSCLC</title>", "<p id=\"Par43\">The reference for CosMx-Lung for both BIDCell and CellSPA was based on Human Lung Cell Atlas (HLCA)^{##REF##37291214##30##}, provided in the “HLCA_v1.h5ad\" file from <ext-link ext-link-type=\"uri\" xlink:href=\"https://beta.fastgenomics.org/p/hlca\">https://beta.fastgenomics.org/p/hlca</ext-link>, including both gene expressions and cell type annotations of the data. We used “ann_finest_level\" as cell type labels, which contained 50 cell types in total.</p>", "<p id=\"Par44\">As HLCA only contains single-cell datasets from non-cancer lung tissue, we complemented the reference data with malignant cells provided in TISCH2, where we downloaded 6 single-cell NSCLC datasets with tumour samples from <ext-link ext-link-type=\"uri\" xlink:href=\"http://tisch.comp-genomics.org/gallery/?cancer=NSCLC&amp;species=Human\">http://tisch.comp-genomics.org/gallery/?cancer=NSCLC&amp;species=Human</ext-link>. We only included the cells labelled as malignant cells in the reference.</p>", "<title>(v) TISCH-SKCM</title>", "<p id=\"Par45\">The reference for MERSCOPE-Melanoma for both BIDCell and CellSPA was based on 10 single-cell melanoma datasets downloaded from TISCH2 from <ext-link ext-link-type=\"uri\" xlink:href=\"http://tisch.comp-genomics.org/gallery/?cancer=SKCM&amp;species=Human\">http://tisch.comp-genomics.org/gallery/?cancer=SKCM&amp;species=Human</ext-link>, which contains the gene by cell expressions and cell annotations of the data. We used the “celltype major lineage” as the cell type labels. We combined the “CD4Tconv” and “Treg” as “CD4Tconv/Treg” and “CD8T” and “CD8Tex” as “CD8T/CD8Tex”, which resulted in 15 cell types in total.</p>", "<title>(vi) Mouse Embryo reference</title>", "<p id=\"Par46\">The reference for Stereo-seq-MouseEmbryo was downloaded from GEO database under accession code: <ext-link ext-link-type=\"uri\" xlink:href=\"https://www.ncbi.nlm.nih.gov/geo/query/acc.cgi?acc=GSE119945\">GSE119945</ext-link>^{##REF##30787437##31##}, which contain both counts and cell type annotation data. The E12.5 data was then used as reference.</p>", "<title>Biologically-informed deep learning-based cell segmentation (BIDCell) overview</title>", "<p id=\"Par47\">BIDCell is a self-supervised deep learning framework that computes biologically-informed loss functions to optimise learnable parameters for the prediction of cell segmentation masks for spatial transcriptomic data. BIDCell uses three types of data: (i) spatial transcriptomic maps of genes, (ii) corresponding DAPI image, and (iii) average gene expression profiles of cell types from a reference dataset, such as the Human Cell Atlas. A major innovation in developing BIDCell is the use of biologically-informed prior knowledge via the SSL paradigm to enable DL models to learn complex structures in SST data, to derive cell segmentations that are visually more realistic and capture better expression profiles.</p>", "<p id=\"Par48\">The BIDCell framework has the following four key characteristics:<list list-type=\"bullet\"><list-item><p id=\"Par49\">BIDCell predicts diverse cell shapes for datasets containing various cell types to better capture cell expressions (see section Elongated and non-elongated shapes).</p></list-item><list-item><p id=\"Par50\">BIDCell uses positive and negative markers from sequencing data to enhance the guidance for learning relationships between spatial gene expressions and cell morphology in the form of cell segmentations (see section Positive and negative cell-type markers).</p></list-item><list-item><p id=\"Par51\">BIDCell is parameterised by a deep learning architecture that learns to segment cells from spatial transcriptomic images (see section Deep learning-based segmentation).</p></list-item><list-item><p id=\"Par52\">BIDCell uses biologically-informed, self-supervised loss functions to train the deep learning architecture without the need for manual annotations and better capture cell expressions (see section BIDCell training and loss functions).</p></list-item></list></p>", "<title>Elongated and non-elongated shapes</title>", "<p id=\"Par53\">BIDCell is capable of generating cell segmentations that exhibit different morphologies for different cell types, rather than assume a generally circular profile for all cell types. In particular, BIDCell can distinguish between cell types that typically appear more elongated, such as fibroblasts and smooth muscle cells, and those that are typically more rounded or circular, such as B cells. Elongated cell types can be directly specified for each tissue sample as desired, based on existing biological knowledge.</p>", "<p id=\"Par54\">We used the expression within the nuclei (see section Nuclei segmentation) of cells to perform an initial classification of elongated and non-elongated cell types. Transcripts were mapped to nuclei using nuclei segmentations, and the Spearman correlation was computed between nuclei expression profiles and reference cell types of the Human Cell Atlas. Nuclei were classified as the cell type with which it was most highly correlated to. This initial classification coupled with the eccentricity of the nuclei were used to inform the cell-calling loss function (described in section Cell-calling loss) to produce segmentation morphologies with more variation that are more appropriate for different cell types. We considered epithelial cells, fibroblasts, myofibroblasts, and smooth muscle cells to be elongated for samples of breast cancer and melanoma. Endothelial cells, fibroblasts, myofibroblasts, fibromyocytes, and pericytes were deemed elongated for NSCLC. We considered all cell types in the mouse brain sample to be elongated.</p>", "<title>Positive and negative cell-type markers</title>", "<p id=\"Par55\">BIDCell learns relationships between the spatial distribution of gene expressions and cell morphology in the form of cell segmentations. This relationship can be enhanced by incorporating biological knowledge in the form of cell-type markers, specially, the genes that are typically more expressed (positive markers) and less expressed (negative markers) in different cell types, which allows BIDCell to predict segmentations that lead to more accurate cell expression profiles. Cell-type marker knowledge is drawn from the Human Cell Atlas, which allows BIDCell to be applied without requiring a matched single-cell reference for the same sample of interest. Markers were incorporated into BIDCell through our positive and negative marker losses (described in section Positive and negative marker losses).</p>", "<title>Deep learning-based segmentation</title>", "<p id=\"Par56\">BIDCell is parameterised by a set of learnable parameters <italic>θ</italic> of a deep learning segmentation model. We used the popular UNet 3+^{##UREF##3##19##} as the backbone of our framework to perform cell segmentation by predicting the probability of cell instances at each pixel. This architecture may be swapped out for other segmentation architectures. UNet 3+ was originally proposed for organ segmentation in computed tomography (CT) images. It was built on the original U-Net^{##UREF##2##12##} and incorporated full-scale skip connections that combined low-level details with high-level features across different scales (resolutions). UNet 3+ comprised an encoding branch and decoding branch with five levels of feature scales. We did not adopt the deep supervision component proposed by UNet 3+, and instead only computed training losses at the lateral resolution of the original input.</p>", "<p id=\"Par57\">\n<underline>Input</underline>\n</p>", "<p id=\"Par58\">The input to the UNet 3+ model was a cropped multichannel spatial transcriptomic image , where <italic>n</italic>_{<italic>g</italic><italic>e</italic><italic>n</italic><italic>e</italic><italic>s</italic>} represents the channel axis corresponding to the total number of genes in the dataset, <italic>h</italic> is the height of the input patch, and <italic>w</italic> is the width of the input patch. Prior to being fed into the first convolutional layer, the input was reshaped to [<italic>n</italic>_{<italic>c</italic><italic>e</italic><italic>l</italic><italic>l</italic><italic>s</italic>}, <italic>n</italic>_{<italic>g</italic><italic>e</italic><italic>n</italic><italic>e</italic><italic>s</italic>}, <italic>h</italic>, <italic>w</italic>], effectively placing <italic>n</italic>_{<italic>c</italic><italic>e</italic><italic>l</italic><italic>l</italic><italic>s</italic>} in the <italic>batch size</italic> dimension. In this way, all the cells in a patch were processed simultaneously, and the model could flexibly support an arbitrary number of cells without requiring extra padding or preprocessing. <italic>n</italic>_{<italic>c</italic><italic>e</italic><italic>l</italic><italic>l</italic><italic>s</italic>} was determined by the corresponding patch of nuclei to ensure consistency with predicted cell instances. Input volumes that were empty of nuclei were disregarded during training and yielded no cells during prediction.</p>", "<p id=\"Par59\">\n<underline>Output and segmentation prediction</underline>\n</p>", "<p id=\"Par60\">The softmax function was applied to the output of UNet 3+ to yield probabilities of foreground and background pixels for each cell instance. This produced multiple probabilities for background pixels (i.e., <italic>n</italic>_{<italic>c</italic><italic>e</italic><italic>l</italic><italic>l</italic><italic>s</italic>} probabilities per pixel for a patch containing <italic>n</italic>_{<italic>c</italic><italic>e</italic><italic>l</italic><italic>l</italic><italic>s</italic>}), due to the placement of cell instances in the <italic>batch size</italic> dimension. These probabilities were aggregated by averaging across all the background predictions per pixel. The <italic>argmax</italic> function was applied pixel-wise to the foreground probabilities for all cells and averaged background probabilities. This produced a segmentation map corresponding to the object (cell instance or background) with the highest probability at each pixel.</p>", "<p id=\"Par61\">\n<underline>Morphological processing</underline>\n</p>", "<p id=\"Par62\">The initial segmentation output by the deep learning model was further refined to ensure pixel connectivity within each cell (i.e., all the sections of the cell were connected). The process involved standard morphological image processing techniques to each cell, including dilation, erosion, hole-filling, and removal of isolated islands, while ensuring that the nucleus was captured. First, dilation followed by erosion were applied using a 5 × 5 circular kernel with two iterations each. Hole-filling was then carried out on the cell section with the largest overlap with the nucleus. Any remaining pixels initially predicted for the cell that were still not connected to the main cell section were discarded. After morphological processing, the number of transcripts captured within each cell is slightly higher, while purity metrics and correlation with Chromium are the same or slightly higher (Supplementary Fig. ##SUPPL##0##24##).</p>", "<p id=\"Par63\">\n<underline>Mapping transcripts to predicted cells</underline>\n</p>", "<p id=\"Par64\">The detected transcripts were mapped to cells using the final predicted segmentations. The segmentation map was resized back to the original pixel resolution using nearest neighbour interpolation. Transcripts located in the mask of a cell were added to the expression profile of the cell. This produced a gene-cell matrix <italic>n</italic>_{<italic>c</italic><italic>e</italic><italic>l</italic><italic>l</italic><italic>s</italic>} × <italic>n</italic>_{<italic>g</italic><italic>e</italic><italic>n</italic><italic>e</italic><italic>s</italic>}, which was used for performance evaluation and downstream analysis.</p>", "<title>BIDCell training and loss functions</title>", "<p id=\"Par65\">The BIDCell framework combines several loss functions that automatically derive supervisory signals from the input data and/or predicted segmentations at each step of the training process. This approach to learning is a core aspect of SSL^{##REF##35476575##32##}. Furthermore, the modular and additive design of the loss functions allows each loss to be swapped out with alternative approaches to compute training signals. The SSL label describes the ability of the framework to automatically learn relationships between gene expressions and cell morphology from its inputs.</p>", "<p id=\"Par66\">Our approach for learning the parameters <italic>θ</italic> of the segmentation model relies on minimising a total of 6 loss functions that we propose with our framework. Some of the losses effectively increase the number of pixels predicted for a cell, while others reduce the size of its segmentation. The nuclei encapsulation, cell-calling, over-segmentation, and overlap losses guide the basic morphology of cells. The positive and negative marker losses refine the cell morphologies learned through the other loss functions, by further guiding the model to learn biologically-informed relationships between gene expressions and cell morphology. This is reminiscent of the pretext and downstream (fine-tuning) stages commonly encountered in SSL, where the pretext task aids the model to learn better representations or intermediate weights, while the fine-tuning task refines the weights and further improves performance for a particular prediction task. Taken together, the losses ensure that the segmentation model learns relationships between spatially-localised, high-dimensional gene expression information and the morphology of individual cells.</p>", "<p id=\"Par67\">(A) <bold>Nuclei encapsulation loss</bold></p>", "<p id=\"Par68\">The segmentation of a cell must contain all the pixels of the cell’s nucleus. Additionally, the expressed genes in nuclei can guide the model to learn which genes should be predicted within cells. Hence, we included a loss function <italic>L</italic>_{<italic>n</italic><italic>e</italic>} that incentivises the model to learn to correctly predict nuclei pixels:where <bold>x</bold>_{<bold>nuc</bold>} is the binary nucleus segmentation mask, and is the predicted segmentation for all cells of the corresponding training patch.</p>", "<p id=\"Par69\">(B) <bold>Cell-calling loss</bold></p>", "<p id=\"Par70\">The aim of the cell-calling loss was to increase the number of transcripts assigned to cells. We also designed the cell-calling loss to allow BIDCell to capture cell-type specific morphologies. Unique expansion masks <bold>e</bold>_{<bold>c</bold>} ∈ {0, 1}^{<italic>h</italic>×<italic>w</italic>} were computed for each cell based on the shape of its nucleus and whether its nucleus expression profile was indicative of an elongated cell type. The expansion mask of a non-elongated cell was computed by applying a single iteration of the morphological dilation operator with a circular kernel of 20 × 20 pixels to its binary nucleus mask.</p>", "<p id=\"Par71\">The expansion mask of an elongated cell was computed based on the elongation of its nucleus, defined as the eccentricity of an ellipse fitted to its nucleus mask:where <italic>a</italic> represents the length of the major axis, and <italic>b</italic> is the length of the minor axis.</p>", "<p id=\"Par72\">We found that elongated cell types tended to have nuclei with higher eccentricity (Supplementary Fig. ##SUPPL##0##1##). Hence, the eccentricity of a nucleus could serve as a proxy for the shape of its cell via an elongated expansion mask. We computed each cell-specific elongated expansion mask using an elliptical dilation kernel applied to the nucleus. The horizontal and vertical lengths of the elliptical kernel were computed by:where <italic>α</italic> is a scaling factor set to 0.9, <italic>e</italic><italic>c</italic><italic>c</italic>_{<italic>n</italic><italic>u</italic><italic>c</italic>} is the eccentricity of the nucleus, <italic>l</italic>_{<italic>t</italic>} is the sum of <italic>l</italic>_{<italic>h</italic>} and <italic>l</italic>_{<italic>v</italic>}, which was set to 60 pixels, and <italic>l</italic>_{<italic>v</italic><italic>m</italic>} is the minimum vertical length, which was set to 3 pixels. These values were selected based on visual inspection (e.g., the cells appear reasonably sized), and were kept consistent across the different elongated cell types and datasets used in this study. The elliptical dilation kernel was rotated to align with the nucleus and applied to the nucleus mask to produce the elongated expansion mask of the cell.</p>", "<p id=\"Par73\">The expansion masks were used in our cell-calling loss function that was minimised during training:where <bold>e</bold>_{<bold>c</bold>} is the expansion mask and is the predicted segmentation of cell <italic>c</italic> of <italic>M</italic> cells in an input patch.</p>", "<p id=\"Par74\">(C) <bold>Over-segmentation loss</bold></p>", "<p id=\"Par75\">We introduced the over-segmentation loss to counter the cell size-increasing effects of the cell-calling loss to prevent the segmentations becoming too large and splitting into separate segments. This loss function elicited a penalty whenever the sum of cytoplasmic predictions exceeded the sum of nuclei predictions for a cell in a given patch:where for cell <italic>c</italic> at pixel (<italic>i</italic>, <italic>j</italic>), is the predicted foreground probability for cell <italic>c</italic>, <italic>x</italic>_{<italic>n</italic><italic>u</italic><italic>c</italic>,<italic>i</italic><italic>j</italic>} ∈ {0, 1} is the binary nucleus mask, and <italic>σ</italic> is the sigmoid function. <italic>L</italic>_{<italic>o</italic><italic>s</italic>} was normalised by number of cells <italic>M</italic> to aid smooth training.</p>", "<p id=\"Par76\">(D) <bold>Overlap loss</bold></p>", "<p id=\"Par77\">Cells are often densely-packed together in samples of various human tissues. This poses a challenge to segmentation models in predicting clear boundaries and coherent segmentations for neighbouring cells without overlap. We introduced the overlap loss to penalise the prediction of multiple cells occurring at each pixel:<italic>L</italic>_{<italic>o</italic><italic>v</italic>} was normalised by number of cells <italic>M</italic>, and the lateral dimensions <italic>h</italic> and <italic>w</italic> of the input to aid smooth training.</p>", "<p id=\"Par78\">(E) <bold>Positive and negative marker losses</bold></p>", "<p id=\"Par79\">The purposes of our positive and negative marker losses were to encourage the model to capture pixels that contained positive cell-type markers, and penalise the model when segmentations captured pixels that contained negative cell-type markers for each cell. The marker losses refine the initial morphology learned through the other loss functions, by further guiding the model to learn biologically-informed relationships between gene expressions and cell morphology.</p>", "<p id=\"Par80\">The positive and negative markers for the training loss were those with expressions in the highest and lowest 10 percentile for each cell type of a tissue sample. In our experiments, we found that a higher number of positive markers tended to increase the size of predicted cells as the model learns to capture more markers, and vice versa. We found that removing positive markers that were common to at least a third of cell types in each tissue type was appropriate across the different datasets for training.</p>", "<p id=\"Par81\">The one-hot encoded lists of positive and negative markers of the cell type for cell <italic>c</italic> were converted into sparse maps <bold>m</bold>_{<bold>pos,</bold><bold>c</bold>} ∈ {0, 1}^{<italic>h</italic>×<italic>w</italic>} and <bold>m</bold>_{<bold>neg,c</bold>} ∈ {0, 1}^{<italic>h</italic>×<italic>w</italic>}. At each pixel, 0 indicated the absence of all markers, while 1 indicated the presence of any positive or negative marker for its respective map. <bold>m</bold>_{<bold>pos,</bold><bold>c</bold>} and <bold>m</bold>_{<bold>neg,</bold><bold>c</bold>} were then multiplied element-wise by the expansion mask <bold>e</bold>_{<bold>c</bold>} to remove markers far away from the current cell. Each marker map was dilated by a 3 × 3 kernel, which was based on the assumption that pixels in a 3 × 3 region around each marker were most likely from the same cell. We found this dilation to improve training guidance and segmentation quality, as the maps tended to be quite sparse.</p>", "<p id=\"Par82\">The marker maps were then used to compute the positive and negative marker losses:</p>", "<title>Total loss</title>", "<p id=\"Par83\">The model was trained by minimising the sum of all the loss functions over <italic>N</italic> training patches:where each <italic>λ</italic> represents a hyperparameter that scaled its respective <italic>L</italic>. The value of <italic>λ</italic> for all loss functions was set to 1.0 (except for the ablation and lambdas studies); this ensured our losses were not fine-tuned to any particular datasets.</p>", "<title>Practical implementation</title>", "<title>Details</title>", "<p id=\"Par84\">To address computational efficiency concerns related to memory usage, we partitioned the spatial transcriptomic maps into patches of 48 × 48 × <italic>n</italic>_{<italic>g</italic><italic>e</italic><italic>n</italic><italic>e</italic><italic>s</italic>} for input into UNet 3+. BIDCell has been verified for datasets containing up to 960 genes on a 12 GB GPU. It is also important to note that the number of genes primarily affects the weights of the first convolutional layer, thus having a minor impact on memory usage.</p>", "<p id=\"Par85\">The patch-based predictions could result in effects along the patch boundaries such as sharp or cut-off cells. When dividing the transcriptomic maps into patches, we create two sets of patches of the same lateral dimensions with an overlap equal to half the lateral size of the patches. The predictions for the patches were combined (see Supplementary Fig. ##SUPPL##0##25##), without additional operations to resolve potential disagreement between predictions of the two sets. Only patches from the first set (no overlaps) were selected during training, while all patches were used during inference.</p>", "<p id=\"Par86\">One image patch was input into the model at one time, though batch size was effectively <italic>n</italic>_{<italic>c</italic><italic>e</italic><italic>l</italic><italic>l</italic><italic>s</italic>} due to reshaping (see section Deep learning-based segmentation-Input). Neither normalisation nor standardisation were applied to the input image patches, such that the pixels depicted raw detections of transcripts.</p>", "<p id=\"Par87\">The model was trained end-to-end from scratch for 4000 iterations (i.e., using 4000 training patches). This amounted to a maximum of 22% of the entire image, thereby leaving the rest of the image unseen by the model during inference. Weights of the convolutional layers were initialised using He et al.’s method^{##UREF##5##33##}. We employed standard on-the-fly image data augmentation by randomly applying a flip (horizontal or vertical), rotation (of 90, 180, or 270 degrees) in the <italic>(x,y)</italic> plane. The order of training samples was randomised prior to training. We employed the Adam optimiser^{##UREF##6##34##} to minimise the sum of all losses at a fixed learning rate of 0.00001, with a first moment estimate of 0.9, second moment estimate of 0.999, and weight decay of 0.0001.</p>", "<title>Time and system considerations</title>", "<p id=\"Par88\">We ran BIDCell on a Linux system with a 12GB NVIDIA GTX Titan V GPU, Intel(R) Core(TM) i9-9900K CPU @ 3.60GHz with 16 threads, and 64GB RAM. BIDCell was implemented in Python using PyTorch. For Xenium-BreastCancer1, which contained 109k detected nuclei, 41M pixels <italic>(x,y)</italic>, and 313 genes, training was completed after approximately 10 minutes for 4000 steps. Inference time was about 50 minutes for the complete image. Morphological processing required approximately 30 min to generate the final segmentation. A comparison of the runtimes between different methods is included in Supplementary Fig. ##SUPPL##0##26##.</p>", "<title>Ablation study</title>", "<p id=\"Par89\">We performed an ablation study to determine the contributions from each loss function and effects of different hyperparameter values (Supplementary Figs. ##SUPPL##0##4##, ##SUPPL##0##5##). We used Xenium-BreastCancer1 for these experiments. We evaluated BIDCell without each of the different loss functions by individually setting their corresponding weights <italic>λ</italic> to zero. Furthermore, we evaluated different parameterisations of the cell-calling loss. We experimented with different diameters for the dilation kernel for non-elongated cells, including 10, 20, and 30 pixels, and different total lengths of the minor and major axes <italic>l</italic>_{<italic>t</italic>} of the dilation kernel for elongated cells, including 50, 60, and 70 pixels. We also ran BIDCell without shape-specific expansions, thereby assuming a non-elongated shape for all cells.</p>", "<title>Performance evaluation</title>", "<p id=\"Par90\">We compared our BIDCell framework to vendor-provided cell segmentations, and methods designed to identify cell bodies via cell segmentation. Table ##TAB##1##2## provides a summary of all methods compared from adapting classical approaches including Voronoi expansion, nuclei dilation, and the watershed algorithm, to recently proposed approaches for SST images including Baysor, JSTA, and Cellpose. Methods that were excluded from the evaluations include those that focus on the assignment of transcripts to cells and do not consider the cell boundaries, underperformance on the public datasets, lack of code and instructions to prepare data into the required formats, and failure of the method to detect any cells (Supplementary Table ##SUPPL##0##1##).</p>", "<title>Settings used for other methods</title>", "<p id=\"Par91\">We used publicly available code for Baysor, JSTA, and Cellpose with default parameters unless stated otherwise. All comparison methods that required nuclei information used identical nuclei as BIDCell, which were detected using Cellpose (v2.1.1) (see Nuclei segmentation).<list list-type=\"bullet\"><list-item><p id=\"Par92\">Baysor - Version 0.5.2 was applied either without a prior, or with a prior nuclei segmentation with default prior segmentation confidence of 0.2. For both instances, we followed recommended settings^{##UREF##7##35##}, including 15 for the minimum number of transcripts expected per cell, and not setting a scale value, since the sample contained cells of varying sizes. We found the scale parameter to have a considerable effect on segmentation predictions, and often resulted in cells with unrealistically uniform appearances if explicitly set.</p></list-item><list-item><p id=\"Par93\">JSTA - default parameters were used. We encountered high CPU loading and issues with two regions of Xenium-BreastCancer1, which yielded empty predictions for those regions despite multiple attempts and efforts to reduce input size.</p></list-item><list-item><p id=\"Par94\">Cellpose - Version 2.1.1 was applied to the channel-wise concatenated image comprising DAPI as the “nuclei” channel, and sum of spatial transcriptomic maps across all genes as the “cells” channel, using the pre-trained “cyto” model with automatic estimation of cell diameter.</p></list-item><list-item><p id=\"Par95\">Voronoi - Classical Voronoi expansion was seeded on nuclei centroids and applied using the SciPy library (v1.9.3).</p></list-item><list-item><p id=\"Par96\">Watershed - The watershed algorithm was performed on the sum of transcriptomic maps across all genes. Seeded watershed used nuclei centroids and was applied using OpenCV (v4.6.0).</p></list-item><list-item><p id=\"Par97\">Cellpose nuclei dilation - we applied dilation to nuclei masks as a comparison segmentation method. Each nucleus was enlarged by about 1 micron in radius by applying morphological dilation using a 3 × 3 circular kernel for one iteration. Overlaps between adjacent cell expansions were permitted.</p></list-item></list></p>", "<title>Evaluation metrics and settings</title>", "<p id=\"Par98\">We introduce the CellSPA framework, that captures evaluation metrics across five complementary categories. A summary of this information is provided in Supplementary Table ##SUPPL##0##2##.</p>", "<p id=\"Par99\">\n<bold>[A] Baseline metrics</bold>\n</p>", "<p id=\"Par100\">\n<bold>Overall characteristics</bold>\n<list list-type=\"bullet\"><list-item><p id=\"Par101\">Number of cells</p></list-item><list-item><p id=\"Par102\">Proportion of transcripts assigned</p></list-item></list>\n</p>", "<p id=\"Par103\"><bold>Cell-level QC metrics</bold><list list-type=\"bullet\"><list-item><p id=\"Par104\">Proportion of cells expressing each gene</p></list-item><list-item><p id=\"Par105\">Number of transcripts per cell</p></list-item><list-item><p id=\"Par106\">Number of genes expressed per cell</p></list-item><list-item><p id=\"Par107\">Cell area</p></list-item></list>where ∑_{<italic>i</italic>∈<italic>I</italic>}<italic>n</italic>_{<italic>i</italic>} represents the sum of all total transcripts over a set <italic>I</italic>, and <italic>A</italic> represents the cell area.</p>", "<p id=\"Par108\">\n<bold>Cell morphology metrics</bold>\n</p>", "<p id=\"Par109\">We evaluated multiple morphology-based metrics and provide diagrammatic illustrations in Supplementary Fig. ##SUPPL##0##27##.</p>", "<p id=\"Par110\">• Elongation =where <italic>W</italic>_bb represents the width of the bounding box, and <italic>H</italic>_bb represents the height of the bounding box.</p>", "<p id=\"Par111\">Elongation measures the ratio of height versus the width of the bounding box (Supplementary Fig. ##SUPPL##0##27##f). Elongation is insensitive to concave irregularities and holes present in the shape of the cell. The value of this metric will be 1 for a perfect square bounding box. As the cell becomes more elongated the value will either increase far above 1 or decrease far below 1, depending on whether the elongation occurs along the height or width of the bounding box.</p>", "<p id=\"Par112\">• Circularity =where <italic>A</italic> represents the area, and <italic>P</italic>_convex represents the convex perimeter.</p>", "<p id=\"Par113\">Circularity measures the area to perimeter ratio while excluding local irregularities of the cell. We used the convex perimeter of the object as opposed to its true perimeter to avoid concave irregularities. The value will be 1 for a circle and decreases as a cell becomes less circular.</p>", "<p id=\"Par114\">• Sphericity =where <italic>R</italic>_I represents the radius of the inscribing circle, and <italic>R</italic>_C represents the radius of the circumscribing circle.</p>", "<p id=\"Par115\">Sphericity measures the rate at which an object approaches the shape of a sphere while accounting for the largest local irregularity of the cell by comparing the ratio of the radius largest circle that fits inside the cell (inscribing circle) to the radius of the smallest circle that contains the whole cell (circumscribing circle). The value is 1 for a sphere and decreases as the cell becomes less spherical.</p>", "<p id=\"Par116\">• Compactness =where <italic>A</italic> represents the area, and <italic>P</italic>_cell represents the cell perimeter.</p>", "<p id=\"Par117\">Compactness measures the ratio of the area of an object to the area of a circle with the same perimeter. Compactness uses the perimeter of the cell thus it considers local irregularities in the cell perimeter. A circle will have a value of 1, and the less smooth or more irregular the perimeter of a cell, the smaller the value will be. For most cells the numerical values for compactness and circularity are expected to be similar. Identifying which cells have large differences between these metrics can identify cells with highly irregular perimeters which may be of interest for downstream analysis and quality control for segmentation.</p>", "<p id=\"Par118\">• Convexity =where <italic>P</italic>_convex represents the convex perimeter and <italic>P</italic>_cell represents the cell perimeter.</p>", "<p id=\"Par119\">Convexity measures the ratio of the convex perimeter of a cell to its perimeter. The value will be 1 for a circle and decrease the more irregular the perimeter of a cell becomes, similar to compactness.</p>", "<p id=\"Par120\">• Eccentricity =where <italic>L</italic>_minor represents the length of the minor axis and <italic>L</italic>_major represents the length of the major axis.</p>", "<p id=\"Par121\">Eccentricity (or ellipticity) measures the ratio of the major axis to the minor axis of a cell. The major axis is the longest possible line that can be drawn between the inner boundary of a cell without intersecting its boundary. The minor axis is the longest possible line can be drawn within the inner boundary of a cell while while also being perpendicular to the major axis. This gives a value of 1 for a circle and decreases the more flat the cell becomes.</p>", "<p id=\"Par122\">• Solidity =where <italic>A</italic> represents the area, and <italic>A</italic>_convex represents the convex area.</p>", "<p id=\"Par123\">Solidity measures the ratio of the area of a cell to the convex area of a cell. This measures the density of a cell by detecting holes and irregular boundaries in the cell shape. The maximum value will be 1 for a cell with a perfectly convex and smooth boundary and will decrease as the cell shape becomes more concave and/or irregular.</p>", "<p id=\"Par124\">\n<bold>Gene-level QC characteristics</bold>\n</p>", "<p id=\"Par125\">• Proportion of cells expressing each gene</p>", "<p id=\"Par126\"><bold>[B] Segmented cell expression purity</bold>. We implemented two broad classes of statistics to capture (i) the concordance of expression profile with scRNA-seq data and (ii) the expression purity or homogeneity of cell type markers. The scRNA-seq data used are described in Section Datasets and preprocessing and listed in Table ##TAB##0##1##.</p>", "<p id=\"Par127\">• Concordance with scRNA-seq data - We calculated the similarity of the expression pattern between the segmented cells and publicly available single-cell datasets. Here the similarity was measured by Pearson correlation of the average log-normalised gene expression for each cell type. We also calculated the concordance of the proportion of non-zero expression for each cell type between the segmented cells and scRNA-seq data. For data with paired Chromium data from the same experiment, i.e., Xenium-Brain, we also compared the cell type proportion and quantify the concordance using the Pearson correlation. We annotated the cell type annotation for segmented cells using scClassify^{##UREF##8##36##} with scRNA-seq data as reference.</p>", "<p id=\"Par128\">• Purity of expression - We first curated a list of positive markers and negative markers from the scRNA-seq reference data. For each cell type, we selected the highest and lowest 10 percentile of the genes with difference of expression compared to other cell types. We also removed the positive markers that were common to more than 25% of cell types for a more pure positive marker list. For each segmented cell, we then consider the genes with the highest 10 percentile of expression as positive genes and lowest 10 percentile as negative markers. We then calculated the Precision, Recall and F1 score for both positive and negative markers. We further summarised the average positive marker F1 scores and negative marker F1 scores into one Purity F1 score for each method, where we first scaled the average positive and negative marker F1 scores into the range of [0, 1] and then calculated the F1 score of transformed metrics as the following:</p>", "<p id=\"Par129\">\n<bold>[C] Spatial characteristics</bold>\n</p>", "<p id=\"Par130\">In this category, we measured the association between cell type diversity in local spatial regions and all the cell-level baseline characteristics provided in [A]. We first divided each image into multiple small regions. Then, for each local spatial region, we calculated the cell type diversity using Shannon entropy with the R package ’entropy’, where a higher entropy indicates a more diverse cell type composition. Next, we assessed the variability of cell-level baseline characteristics within each local region using the coefficient of variation. Subsequently, for each of the cell-level baseline characteristics mentioned in [A], we calculated the Pearson correlation between the cell type diversity (measured using Shannon entropy) and the coefficient of variation of these characteristics across all local regions. Here, we anticipate that regions with more diverse cell type compositions will exhibit higher variability in cell-level characteristics, leading to a stronger correlation between these two metrics.</p>", "<p id=\"Par131\">\n<bold>[D] Neighbouring contamination</bold>\n</p>", "<p id=\"Par132\">This metric is designed for cell segmentation to ensure that the expression signals between neighboring cells are not contaminated. For a pair of cell types (e.g., cell type A and B), we computed the Euclidean distance from each cell in cell type A to its nearest neighbor belonging to cell type B. We then grouped the cells of cell type A based on a range of distances. Within each group, we calculated the proportion of cells expressing a selected negative marker, which is a cell type marker for cell type B. We anticipate that the method with less contamination will result in segmented cells expressing lower levels of the negative marker, even when the distance to a different cell type is minimal.</p>", "<p id=\"Par133\">\n<bold>[E] Replicability</bold>\n</p>", "<p id=\"Par134\">Our analysis involved assessing the agreement between the Xenium-BreastCancer1 and Xenium-BreastCancer2 datasets, which are closely related in terms of all the cell-level baseline characteristics provided in [A]. As these datasets are considered to be sister regions, we anticipated that the distribution of all the baseline characteristics, as well as the cell type composition, would be similar. We use Pearson correlation to quantify the degree of concordance.</p>", "<title>Statistics and reproducibility</title>", "<p id=\"Par135\">All analysis was done in R version version (4.3.0). No statistical method was used to predetermine sample size. No data were excluded from the analyses. All cells that passed quality control were included in the analyses. The experiments were not randomized. The Investigators were not blinded to allocation during experiments and outcome assessment.</p>", "<title>Reporting summary</title>", "<p id=\"Par136\">Further information on research design is available in the ##SUPPL##2##Nature Portfolio Reporting Summary## linked to this article.</p>" ]

[ "<title>Results</title>", "<title>BIDCell: Incorporating biological insights using deep learning to improve cell shape representation</title>", "<p id=\"Par9\">BIDCell is a DL-based cell segmentation method that identifies each individual cell and all its pixels as a cohesive mask. BIDCell uses subcellular spatial transcriptomic maps, corresponding DAPI images, and relevant average expression profiles of cell types from single-cell sequencing datasets; the latter is obtained from public repositories such as the Human Cell Atlas. Given the lack of ground truth and visual features that indicate cell boundaries in the SST images, BIDCell instead focuses on the relationships between the high-dimensional spatial gene expressions and cell morphology. The BIDCell framework automatically derives supervisory signals from the input data and/or predicted segmentations, which is an approach to learning that we borrow from SSL.</p>", "<p id=\"Par10\">To achieve this, we designed multiple loss functions that represent various criteria based on biological knowledge, that work synergistically to produce accurate segmentations (Fig. ##FIG##0##1##a; see Methods and Supplementary Materials for a detailed description). BIDCell learns to use the locations of highly- and lowly-expressed marker genes to calibrate the segmentation to capture higher “cell expression purity”, thereby ensuring transcripts within each cell share the same profile. Furthermore, BIDCell captures local expression patterns using a data-driven, cell-type-informed morphology. We found that the eccentricity measure of nuclei could reveal diverse cell morphologies that correspond to established knowledge, such as elongated morphologies for fibroblasts (Supplementary Fig. ##SUPPL##0##1##). By capturing a diverse set of cell shapes and leveraging marker information from previous single-cell experiments (Table ##TAB##0##1##), BIDCell generates superior segmentations (Fig. ##FIG##0##1b## and Supplementary Figs. ##SUPPL##0##2## and ##SUPPL##0##3##), and overcomes the limitations of many existing methods (Table ##TAB##1##2##) that rely primarily on SST image intensity values for cell segmentation.</p>", "<p id=\"Par11\">We further ensure the integrity of cell segmentations by proposing three other cooperative loss functions. Appropriate cell sizes are supported by capturing expression patterns local to nuclei using guidance from cell-type informed morphologies (cell-calling), while ensuring the cohesiveness of cell instances (over-segmentation) and enhancing segmentation in densely-populated regions (overlap loss). BIDCell also leverages expression patterns within nuclei to guide the identification of cell body pixels.</p>", "<p id=\"Par12\">We investigated removing individual losses in an ablation study with Xenium-BreastCancer1 data (Supplementary Figs. ##SUPPL##0##4##, ##SUPPL##0##5##). Our investigation shows that the losses work synergistically; e.g., there was a marked increase in purity F1 relative to the amount of captured transcripts when the losses were combined. With the inclusion of single-cell data (which informs the positive and negative losses, and contributes to the ability to predict elongated cell shapes), performance improved considerably, particularly in purity metrics and correlation to Chromium data. The use of single-cell data helped the model to better capture transcripts that are more biologically meaningful within cells. By default, the weights of the losses are all 1.0 and do not need to be tuned for BIDCell to perform well, though further fine-tuning is possible (Supplementary Fig. ##SUPPL##0##6##). The popular UNet 3+^{##UREF##3##19##} serves as the segmentation backbone architecture in BIDCell, though this is not a requirement and it may be replaced with alternative architectures (Supplementary Fig. ##SUPPL##0##7##).</p>", "<title>CellSPA comprehensive evaluation framework captures diverse sets of metrics of segmentation aspects across five complementary categories</title>", "<p id=\"Par13\">To ensure an unbiased comparison, we introduce a Cell Segmentation Performance Assessment (CellSPA) framework (Fig. ##FIG##1##2##a) that captures cell segmentation metrics across five complementary categories. These categories, detailed in Fig. ##FIG##1##2##a and Supplementary Table ##SUPPL##0##2##, include (i) <underline>baseline characteristics</underline> at both the cell and gene levels; (ii) measures of segmented cell expression, where we assess the “<underline>expression purity”</underline> of our assigned segmented cells based on how well transcripts within the segmented cell share a similar expression profile; (iii) measures of baseline cell characteristics in its <underline>spatial environment</underline>, including spatial region diversity and corresponding diversity in morphology; (iv) a measure of contamination between <underline>nearest neighbours</underline> (Supplementary Fig. ##SUPPL##0##8##); and (v) measures of <underline>replicability</underline>.</p>", "<p id=\"Par14\">Using CellSPA, we compared the performance of BIDCell with several recently developed methods for the segmentation of SST data. These methods included classical segmentation-based approaches such as simple dilation, watershed, and Voroni; and transcript-based approaches including Baysor. Additionally, we evaluated JSTA^{##REF##34057817##18##}, which attempts to jointly determine cell (sub) types and cell segmentation based on an extension from the traditional watershed approach. In all comparisons, we limited the computational time to within 72 h, which we deemed a practical requirement for the solutions provided by each approach (see Discussion).</p>", "<p id=\"Par15\">To ensure the minimal appropriateness of segmented cells, we examine a series of quality control (QC) statistics. As an illustrative example using Xenium-BreastCancer1 data, we segmented cells using BIDCell, generating 100,000 number of cells, with 53.4% of transcripts assigned (Fig. ##FIG##1##2##b). We first confirm that the total number of transcripts per cell and the number of genes per cell were greater in the whole cell (cell body + nuclei) compared to just the nuclei (Fig. ##FIG##1##2##c and Supplementary Fig. ##SUPPL##0##9##).</p>", "<p id=\"Par16\">Similarly, using the percentage of cells expressing each gene between the nuclei vs. the cell body, we further evaluate the level of information presented in the nuclei and the cell body from the gene level (Fig. ##FIG##1##2##d). We find that the segmented cells of some of the methods (e.g. Baysor) did not yield any additional transcript information beyond that of the nuclei, where we see a tight concordance (lying on a 45-degree line) between the segmented cell body and the cell nuclei. However, BIDCell, 10x, Cellpose, and JSTA are all able to capture additional transcript information. Moving forward, we will focus on methods that provide “additional\" information to the nuclei, with an emphasis on the ability to better capture cell boundaries.</p>", "<p id=\"Par17\">Lastly, we examine the cell morphology of the segmented cells against the segmented nuclei, including cell area, elongation, compactness, sphericity, convexity, eccentricity, solidity and circularity (See Methods and Supplementary Fig. ##SUPPL##0##10##). Through these metrics, we are able to identify the outliers of the segmented cells, such as cells with extremely large areas in JSTA, Voronoi and Watershed in the sparse areas (Supplementary Fig. ##SUPPL##0##11##). We illustrate that as intended from our cell-mask, BIDCell has cell morphology that is highly correlated with the nuclei morphology (Fig. ##FIG##1##2##e). Furthermore, we find that segmented cells from BIDCell exhibit more diverse cell morphology characteristics compared to other methods (Supplementary Fig. ##SUPPL##0##12##).</p>", "<title>BIDCell captures improved purity of cell expression, leading to less contamination from neighbouring cells</title>", "<p id=\"Par18\">To determine whether various cell segmentation methods can improve spatial resolution without sacrificing detection efficiency, we first compare the correlation between cell type signatures in the Xenium and Chromium V2 platforms for Xenium-BreastCancer1 data (Fig. ##FIG##2##3##a). We observed that the performance of correlation for average expression between the spatial and sequencing profile ranges between 0.72 and 0.8 across all methods. Interestingly, we observe a trade-off between the size of the cell (average total transcript per cell) and the level of correlation. Fig. ##FIG##2##3##a demonstrates the importance of employing two metrics to quantify segmentation performance. While Cellpose achieved the highest Pearson correlation overall, BIDCell achieved the highest Pearson correlation among methods that detect a similar number of transcripts as Chromium data (i.e., cell sizes that are more similar to segmentation of the cell body as opposed to the nuclei). Similar results are shown in the average percentage of expressed genes (Fig. ##FIG##2##3##b). Furthermore, Fig. ##FIG##2##3##c highlights a high level of consistency in cell type proportion between the segmented cells generated by BIDCell and Chromium (cor = 0.95). BIDCell also has a higher presence of positive markers and a lower presence of negative markers in large cells (Fig. ##FIG##2##3##d and Supplementary Fig. ##SUPPL##0##13##), demonstrating an improvement in the expression purity of segmentation.</p>", "<p id=\"Par19\">In category III of CellSPA, we investigate the potential contamination between neighbouring cells by comparing the percentage of B cells that expressed negative markers, such as CD3D and CD3E, which are positive T cell markers but are considered negative markers in B cells. The presence of T cell marker genes in B cells suggests potential contamination during the cell segmentation process. Fig. ##FIG##2##3##e and Supplementary Fig. ##SUPPL##0##14## indicate that BIDCell showed the smallest percentage of contamination cells, indicating its ability to reduce contamination in a densely populated region.</p>", "<p id=\"Par20\">Lastly, we investigate the spatial diversity by examining the association between the cell type composition and the various cell level characteristics of spatial local regions. Here, we expect the region with a diverse composition of cell types would have a high variety of cell sizes and morphologies. We first divide the image into several local regions and then quantify the diversity of the cell type composition of a region using entropy (Fig. ##FIG##2##3##f). As shown in Fig. ##FIG##2##3##g and Supplementary Fig. ##SUPPL##0##15##, we find that BIDCell achieves a higher correlation of the coefficient of variation of the cell-level characteristics (the total transcripts, the total genes expressed and cell area) with the cell type entropy compared to the other methods. Similarly, we observe that the variety of cell elongation in BIDCell is highly correlated with the proportion of fibroblasts, one of the dominant cell types in the data (Fig. ##FIG##2##3##h).</p>", "<p id=\"Par21\">Together, with a comprehensive benchmarking using CellSPA, we demonstrate that the BIDCell segmentation achieves a better balance between high cell expression purity and a large cell body compared to the other state-of-the-art methods, which capture a more diverse range of cell morphologies and provide the potential for a more accurate representation of the topographic context of neighbouring cellular interactions.</p>", "<title>BIDCell is replicable and generalisable to multiple SST platforms</title>", "<p id=\"Par22\">As an additional sensitivity analysis to the ablation study, we evaluated the replicability of BIDCell. We compared the results between the two replicated studies (Xenium-BreastCancer1 and Xenium-BreastCancer2). Figure ##FIG##2##3##i displays images of the two replicates, with corresponding cell types highlighted in Fig. ##FIG##2##3##j (left panel). The results are very similar, demonstrating that BIDCell is replicable. The tSNE plot in Fig. ##FIG##2##3##j (right panel) shows a well-mixed population of cells between the two replicated studies. The high correlation of the cell morphology metrics of segmented cells from BIDCell between the two replicates further confirm the replicability of our method (Supplementary Fig. ##SUPPL##0##16##).</p>", "<p id=\"Par23\">We demonstrate the generalisability of BIDCell to other SST platforms and tissue types by applying BIDCell to data generated by CosMx from NanoString (Fig. ##FIG##3##4##a–c, Supplementary Figs. ##SUPPL##0##17##, ##SUPPL##0##18##) and MERSCOPE data from Vizgen (Fig. ##FIG##3##4##d–f, a–c, Supplementary Figs. ##SUPPL##0##19##, ##SUPPL##0##20##). In particular, we observed that BIDCell had a lower percentage of B cells expressing negative markers (markers indicating contamination) for the CosMx-Lung data (Fig. ##FIG##3##4##c), suggesting more accurate cell segmentation. Additionally, in MERSCOPE-Melanoma data, regions with more diverse cell types corresponded to more diverse cell type characteristics (Fig. ##FIG##3##4##f). Furthermore, we also applied BIDCell to Stereo-seq from BGI (Supplementary Fig. ##SUPPL##0##21##). We have now demonstrated the applicability of BIDCell on data from four major platforms, and from five different tissue types. We believe that our method has the flexibility and generalisability to other data from other SST platforms and tissues.</p>", "<title>Accurate cell segmentation can reveal region-specific subtypes among neuronal cells</title>", "<p id=\"Par24\">To further assess the performance of BIDCell in accurately segmenting closely packed cells, we performed an evaluation on another case study from Xenium-MouseBrain data. The hippocampus is critical for learning and memory^{##REF##18270514##20##}, and the tripartite synapses formed between the dentate gyrus and cornu ammonis (CA) have been well studied^{##REF##31632251##21##}. Because of the density of pyramidal neurons within the CA region, we asked whether or not BIDCell could accurately distinguish CA1, 2, and 3 from one another. Figure ##FIG##4##5##a, b show the spatial image and highlight the neuronal cell type and neuronal regions using scClassify trained existing sequencing data (Table ##TAB##0##1)##. Fig. ##FIG##4##5##c compares the segmentation pattern obtained using 10x vs. BIDCell. Note that BIDCell generates a more finely textured and tighter pattern of cells than 10x, and the output more closely resembles the pattern seen in Fig. ##FIG##4##5##a. The superior performance of BIDCell is further confirmed by the evaluation metrics. With similar size of the segmented cells with 10x (Supplementary Fig. ##SUPPL##0##22##), BIDCell achieves a higher similarity with scRNA-seq and expression purity score (Fig. ##FIG##4##5##d–e, Supplementary Fig. ##SUPPL##0##23##). BIDCell can identify neuronal subtype markers that distinguish granule neurons in the dentate gyrus (<italic>Prox1</italic>) from pyramidal neurons in CA1-3 (<italic>Neurod6</italic>) (^{##REF##28549853##22##}; Fig. ##FIG##4##5##f). Furthermore, it is able to spatially subdivide pyramidal neurons in the CA region despite their close proximity to one another. Fig. ##FIG##4##5##f shows the expression patterns of <italic>Wfs1</italic> in CA1^{##REF##19561297##23##}, <italic>Necab2</italic> in CA2^{##REF##23850650##24##} and <italic>Slit2</italic> in CA3^{##REF##34686348##25##}, consistent with prior studies. Interestingly, we found a new gene (<italic>Cpne8</italic>) that is enriched in CA1, consistent with in situ data from the Allen Brain Atlas and illustrates BIDCell’s capacity for biological discovery.</p>" ]

[ "<title>Discussion</title>", "<p id=\"Par25\">Here we presented BIDCell, a method for cell segmentation in subcellular spatially resolved transcriptomics data. BIDCell leverages DL with its biologically-informed loss functions that allow the model to self-learn and capture both cell type and cell shape information, while optimising for cell expression purity. Its default components (such as the backbone architecture and use of cell type profiles) may be exchanged for other architectures and Atlas datasets. We have demonstrated the effectiveness of BIDCell by comparing it to state-of-the-art methods and have shown that BIDCell provides better cell body delineation. Moreover, our flexible approach can be applied to different technology platforms, and different gene panels. Our study highlights the potential of BIDCell for accurate cell segmentation and its potential impact on the field of subcellular spatially resolved transcriptomics.</p>", "<p id=\"Par26\">The typical approach to leverage advancements in DL relies on ground truth to guide models to learn relationships between inputs and outputs. However, manual annotation of individual pixels is unattainable for SST that contain hundreds of molecular units per pixel, given the time and effort of manual labour. Further, we have shown (e.g., with Cellpose) that models pretrained on other imaging modalities do not transfer well to SST images. BIDCell innovates through its integrated loss functions that inject biological knowledge of cell morphology and expressions, to allow the model to self-learn from the given spatial transcriptomic and DAPI images, and produce superior visual and quantitative performance compared to previous methods. Our loss functions also allow BIDCell to be broadly applicable across diverse tissue types and various SST platforms. Therefore, BIDCell can facilitate faster research outputs and new discoveries.</p>", "<p id=\"Par27\">Establishing an easy-to-use evaluation system is crucial for promoting reproducible science and transparency, as well as facilitating further methods development. In CellSPA, we have extended beyond a single accuracy metric and introduced metrics that represent important downstream properties or biological characteristics recognised by scientists. This concept of evaluation by human-recognised criteria is also discussed by the computer vision community as “empirical evaluation”^{##REF##26263899##26##}. Another aspect that is often overlooked is related to the practical establishment of benchmarking studies. As benchmarking studies gain recognition, they can be time-consuming due to challenges with software versioning and different operating systems, and different methods may require varying degrees of ease of use and time to adjust the code for comparison. The CellSPA tool is available as a R package with all necessary dependencies, simplifying its installation and usage on local systems, and promoting reproducible science and transparency. Rather than generating a comprehensive comparison of existing methods, which can quickly become outdated, evaluation metrics are generated to allow new methods to be compared to a database of existing methods, without the need to re-implement a large collection of methods. This approach reduces redundancy, allows for direct comparison with state-of-the-art methods, and saves time and effort. Examples of this approach include those for cell deconvolution^{##REF##35577954##27##} and simulation methods^{##REF##34824223##28##}.</p>", "<p id=\"Par28\">A comprehensive evaluation framework is vital when comparing diverse segmentation approaches in the absence of a ground truth. It is important to recognise that different segmentation approaches may purposefully have different priorities and outcomes. For example, a segmentation approach such as a seeded Voronoi tessellation will identify larger cells than a fixed expansion around the nuclei, such as Cellpose cell. The former will typically assign more transcripts and produce a denser map of which cells are touching. In contrast, the latter may produce more homogenous profiles of the cells with fewer assigned molecules and tighter cell boundaries, limiting its capability to estimate physical cell interactions. While achieving more homogenous cell bodies is desired, it can also result from the arbitrary over-segmentation of nuclei. This emphasises that the use of employing a variety of metrics to quantify segmentation performance enables a systematic assessment and reveals the desirable properties of each approach.</p>", "<p id=\"Par29\">Cells have a three-dimensional structure, thus analyses in a two-dimensional perspective may achieve limited representation. BIDCell can be further adapted (e.g., via its cell-calling loss) to incorporate cell membrane markers to enhance segmentation. In MERSCOPE data that display cell membrane markers, there is a percentage (25%) of cells that lack nuclei in their segmentation, likely due to being elongated melanocytes or fibroblasts in a section without a nucleus. While platforms like MERSCOPE can utilise cell membrane markers as cell masks to perform cell segmentation, it is necessary to conduct further research to understand whether a cell’s slicing affects the measurement of expression in tissues. Similarly, in the nervous system, a future challenge will be to accurately identify and segment dendritic and axon morphologies. Like melanocytes and fibroblasts, the varied and elongated nature of these cell morphologies will make it challenging to accurately identify cell boundaries in the absence of nearby nuclei. Because of these difficulties, most approaches may instead generate similar results between the segmentation of the whole cell and the corresponding segmentation of the cell nuclei.</p>", "<p id=\"Par30\">In conclusion, the development of subcellular spatial transcriptomics technologies is revolutionising molecular biology. We have introduced a deep learning approach that does not require ground truth supervision and incorporates prior biological knowledge by leveraging the myriad of single-cell datasets in Atlas databases. We illustrate that our BIDCell method outperforms the current state-of-the-art cell segmentation methods, and we are able to uncover region-specific subtypes in the brain with explicit highlighting of cell bodies and boundaries. Furthermore, recognising the importance of evaluation, we developed CellSPA, a Cell Segmentation Performance Assessment framework, that covers a wide variety of metrics across five complementary categories of cell segmentation characteristics.</p>" ]

[]

[ "<p id=\"Par1\">Recent advances in subcellular imaging transcriptomics platforms have enabled high-resolution spatial mapping of gene expression, while also introducing significant analytical challenges in accurately identifying cells and assigning transcripts. Existing methods grapple with cell segmentation, frequently leading to fragmented cells or oversized cells that capture contaminated expression. To this end, we present BIDCell, a self-supervised deep learning-based framework with biologically-informed loss functions that learn relationships between spatially resolved gene expression and cell morphology. BIDCell incorporates cell-type data, including single-cell transcriptomics data from public repositories, with cell morphology information. Using a comprehensive evaluation framework consisting of metrics in five complementary categories for cell segmentation performance, we demonstrate that BIDCell outperforms other state-of-the-art methods according to many metrics across a variety of tissue types and technology platforms. Our findings underscore the potential of BIDCell to significantly enhance single-cell spatial expression analyses, enabling great potential in biological discovery.</p>", "<p id=\"Par2\">Subcellular in situ spatial transcriptomics offers the promise to address biological problems that were previously inaccessible but requires accurate cell segmentation to uncover insights. Here, authors present BIDCell, a biologically informed, deep learning-based cell segmentation framework.</p>", "<title>Subject terms</title>" ]

[ "<title>Supplementary information</title>", "<p>\n\n\n\n</p>", "<title>Source data</title>", "<p>\n\n</p>" ]

[ "<title>Supplementary information</title>", "<p>The online version contains supplementary material available at 10.1038/s41467-023-44560-w.</p>", "<title>Acknowledgements</title>", "<p>The authors thank all their colleagues, particularly at the Sydney Precision Data Science Centre and Charles Perkins Centre for their support and intellectual engagement. Special thanks to Yue Cao, Lijia Yu, Andy Tran, and Bárbara Zita Peters Couto for their contributions in weekly discussions, and to Nick Robertson for his contribution to the BIDCell package. Thanks also go to Brett Kennedy and Daniel Dlugolenski from the 10x Genomics team in Australia for providing the initial motivation in discussions.</p>", "<p>This work is supported by the AIR@innoHK programme of the Innovation and Technology Commission of Hong Kong to J.Y.H.Y., J.K., E.P., X.F., Y.L. The work is also supported by Judith and David Coffey funding to J.Y.H.Y. and Y.L.; NHMRC Investigator APP2017023 to J.Y.H.Y. and D.M. Australian Research Council Discovery project (DP200103748) to J.K.; Discovery Early Career Researcher Awards (DE220100964) to S.G. and (DE200100944) to E.P. Research Training Program Tuition Fee Offset and Stipend Scholarship to F.A.; Chan Zuckerberg Initiative Single Cell Biology Data Insights grant (2022-249319) to S.G.; and USyd-Cornell Partnership Collaboration Awards to S.G. and D.L. The funding source had no role in the study design, in the collection, analysis, and interpretation of data, in the writing of the manuscript, or in the decision to submit the manuscript for publication.</p>", "<title>Author contributions</title>", "<p>J.Y.H.Y. conceived and led the study with design input from E.P. and S.G. X.F. led the development of the method with input and guidance from J.K., J.Y.H.Y., E.P., and Y.L. Y.L. led the development and interpretation of the evaluation framework with input from E.P., S.G., J.Y.H.Y., D.M., and X.F. D.L. performed the data analysis and interpretation of the mouse brain data with input from J.Y.H.Y. and Y.L. Y.L. and X.F. performed all data curation and processing. D.M., C.W., and F.A. contributed to the refinement of the code and evaluation framework with guidance from Y.L. and X.F. All authors contributed to the writing, editing, and approval of the manuscript.</p>", "<title>Peer review</title>", "<title>Peer review information</title>", "<p id=\"Par137\"><italic>Nature Communications</italic> thanks Jordao Bragantini, Qinghua Jiang and the other, anonymous, reviewer(s) for their contribution to the peer review of this work. A peer review file is available.</p>", "<title>Data availability</title>", "<p>All datasets used in this study are publicly available and were downloaded from the following links (more details including accession codes are provided in Table ##TAB##0##1##). 10x Genomics Xenium breast cancer replicates 1 and 2: <ext-link ext-link-type=\"uri\" xlink:href=\"https://www.10xgenomics.com/products/xenium-in-situ/preview-dataset-human-breast\">https://www.10xgenomics.com/products/xenium-in-situ/preview-dataset-human-breast</ext-link>. 10x Genomics Xenium mouse brain: <ext-link ext-link-type=\"uri\" xlink:href=\"https://www.10xgenomics.com/resources/datasets/fresh-frozen-mouse-brain-replicates-1-standard\">https://www.10xgenomics.com/resources/datasets/fresh-frozen-mouse-brain-replicates-1-standard</ext-link>. NanoString CosMx NSCLC: <ext-link ext-link-type=\"uri\" xlink:href=\"https://nanostring.com/products/cosmx-spatial-molecular-imager/nsclc-ffpe-dataset/\">https://nanostring.com/products/cosmx-spatial-molecular-imager/nsclc-ffpe-dataset/</ext-link>. Vizgen MERSCOPE melanoma2: <ext-link ext-link-type=\"uri\" xlink:href=\"https://info.vizgen.com/merscope-ffpe-solution\">https://info.vizgen.com/merscope-ffpe-solution</ext-link> (requires filling in the form to access). The Stereo-seq E12.5_E1S3 data were downloaded from <ext-link ext-link-type=\"uri\" xlink:href=\"https://db.cngb.org/stomics/mosta/download/\">https://db.cngb.org/stomics/mosta/download/</ext-link>. Tumor Immune Single Cell Hub 2 (TISCH2) BRCA: <ext-link ext-link-type=\"uri\" xlink:href=\"http://tisch.comp-genomics.org/gallery/?cancer=BRCA&amp;species=Human\">http://tisch.comp-genomics.org/gallery/?cancer=BRCA&amp;species=Human</ext-link>. 10x Chromium breast cancer: <ext-link ext-link-type=\"uri\" xlink:href=\"https://www.10xgenomics.com/products/xenium-in-situ/preview-dataset-human-breast\">https://www.10xgenomics.com/products/xenium-in-situ/preview-dataset-human-breast</ext-link>. Allen Brain Map Mouse Whole Cortex and Hippocampus SMART-seq: <ext-link ext-link-type=\"uri\" xlink:href=\"https://portal.brain-map.org/atlases-and-data/rnaseq/mouse-whole-cortex-and-hippocampus-smart-seq\">https://portal.brain-map.org/atlases-and-data/rnaseq/mouse-whole-cortex-and-hippocampus-smart-seq</ext-link>. Human Lung Cell Atlas: <ext-link ext-link-type=\"uri\" xlink:href=\"https://beta.fastgenomics.org/p/hlca\">https://beta.fastgenomics.org/p/hlca</ext-link>. TISCH-NSCLC: <ext-link ext-link-type=\"uri\" xlink:href=\"http://tisch.comp-genomics.org/gallery/?cancer=NSCLC&amp;species=Human\">http://tisch.comp-genomics.org/gallery/?cancer=NSCLC&amp;species=Human</ext-link>. TISCH-SKCM: <ext-link ext-link-type=\"uri\" xlink:href=\"http://tisch.comp-genomics.org/gallery/?cancer=SKCM&amp;species=Human\">http://tisch.comp-genomics.org/gallery/?cancer=SKCM&amp;species=Human</ext-link>. The mouse embryo reference was downloaded from GEO database under accession code [<ext-link ext-link-type=\"uri\" xlink:href=\"https://www.ncbi.nlm.nih.gov/geo/query/acc.cgi?acc=GSE119945\">GSE119945</ext-link>]. The TISCH-BRCA datasets were downloaded from GEO database under accession codes [<ext-link ext-link-type=\"uri\" xlink:href=\"https://www.ncbi.nlm.nih.gov/geo/query/acc.cgi?acc=GSE110686\">GSE110686</ext-link>], [<ext-link ext-link-type=\"uri\" xlink:href=\"https://www.ncbi.nlm.nih.gov/geo/query/acc.cgi?acc=GSE114727\">GSE114727</ext-link>], [<ext-link ext-link-type=\"uri\" xlink:href=\"https://www.ncbi.nlm.nih.gov/geo/query/acc.cgi?acc=GSE138536\">GSE138536</ext-link>], [<ext-link ext-link-type=\"uri\" xlink:href=\"https://www.ncbi.nlm.nih.gov/geo/query/acc.cgi?acc=GSE143423\">GSE143423</ext-link>], [<ext-link ext-link-type=\"uri\" xlink:href=\"https://www.ncbi.nlm.nih.gov/geo/query/acc.cgi?acc=GSE176078\">GSE176078</ext-link>], [<ext-link ext-link-type=\"uri\" xlink:href=\"https://www.ncbi.nlm.nih.gov/geo/query/acc.cgi?acc=GSE148673\">GSE148673</ext-link>], [<ext-link ext-link-type=\"uri\" xlink:href=\"https://www.ncbi.nlm.nih.gov/geo/query/acc.cgi?acc=GSE150660\">GSE150660</ext-link>]; from EBI database under accession code [<ext-link ext-link-type=\"uri\" xlink:href=\"https://www.ebi.ac.uk/biostudies/arrayexpress/studies/E-MTAB-8107\">E-MTAB-8107</ext-link>]; and from SRA under accession code [<ext-link ext-link-type=\"uri\" xlink:href=\"https://trace.ncbi.nlm.nih.gov/Traces/?view=study&amp;acc=SRP114962\">SRP114962</ext-link>]. The original published datasets of HLCA can be accessed under GEO accession number [<ext-link ext-link-type=\"uri\" xlink:href=\"https://www.ncbi.nlm.nih.gov/geo/query/acc.cgi?acc=GSE135893\">GSE135893</ext-link>] for Banovich_Kropski_2020; URL [<ext-link ext-link-type=\"uri\" xlink:href=\"https://www.synapse.org/#!Synapse:syn21041850\">https://www.synapse.org/#!Synapse:syn21041850</ext-link>] for Krasnow_2020; [<ext-link ext-link-type=\"uri\" xlink:href=\"https://www.ncbi.nlm.nih.gov/geo/query/acc.cgi?acc=GSE128033\">GSE128033</ext-link>] for Lafyatis_Rojas_2019; URL [<ext-link ext-link-type=\"uri\" xlink:href=\"https://explore.data.humancellatlas.org/projects/c4077b3c-5c98-4d26-a614-246d12c2e5d7\">https://explore.data.humancellatlas.org/projects/c4077b3c-5c98-4d26-a614-246d12c2e5d7</ext-link>] for Meyer_2019; [<ext-link ext-link-type=\"uri\" xlink:href=\"https://www.ncbi.nlm.nih.gov/geo/query/acc.cgi?acc=GSE158127\">GSE158127</ext-link>] for Misharin_2021; [<ext-link ext-link-type=\"uri\" xlink:href=\"https://www.ncbi.nlm.nih.gov/geo/query/acc.cgi?acc=GSE122960\">GSE122960</ext-link>] and [<ext-link ext-link-type=\"uri\" xlink:href=\"https://www.ncbi.nlm.nih.gov/geo/query/acc.cgi?acc=GSE121611\">GSE121611</ext-link>] for Misharin_Budinger_2018; European Genome-phenome Archive study ID [<ext-link ext-link-type=\"uri\" xlink:href=\"https://ega-archive.org/datasets/EGAD00001005065\">EGAD00001005065</ext-link>] for Teichmann_Meyer_2019. The TISCH-NSCLC datasets were downloaded from GEO database under accession codes [<ext-link ext-link-type=\"uri\" xlink:href=\"https://www.ncbi.nlm.nih.gov/geo/query/acc.cgi?acc=GSE117570\">GSE117570</ext-link>], [<ext-link ext-link-type=\"uri\" xlink:href=\"https://www.ncbi.nlm.nih.gov/geo/query/acc.cgi?acc=GSE127465\">GSE127465</ext-link>], [<ext-link ext-link-type=\"uri\" xlink:href=\"https://www.ncbi.nlm.nih.gov/geo/query/acc.cgi?acc=GSE143423\">GSE143423</ext-link>], [<ext-link ext-link-type=\"uri\" xlink:href=\"https://www.ncbi.nlm.nih.gov/geo/query/acc.cgi?acc=GSE148071\">GSE148071</ext-link>], [<ext-link ext-link-type=\"uri\" xlink:href=\"https://www.ncbi.nlm.nih.gov/geo/query/acc.cgi?acc=GSE150660\">GSE150660</ext-link>]; and from EBI database under accession code [<ext-link ext-link-type=\"uri\" xlink:href=\"https://www.ebi.ac.uk/biostudies/arrayexpress/studies/E-MTAB-6149\">E-MTAB-6149</ext-link>]. The SKCM datasets were downloaded from GEO database under accession codes [<ext-link ext-link-type=\"uri\" xlink:href=\"https://www.ncbi.nlm.nih.gov/geo/query/acc.cgi?acc=GSE115978\">GSE115978</ext-link>], [<ext-link ext-link-type=\"uri\" xlink:href=\"https://www.ncbi.nlm.nih.gov/geo/query/acc.cgi?acc=GSE120575\">GSE120575</ext-link>], [<ext-link ext-link-type=\"uri\" xlink:href=\"https://www.ncbi.nlm.nih.gov/geo/query/acc.cgi?acc=GSE123139\">GSE123139</ext-link>], [<ext-link ext-link-type=\"uri\" xlink:href=\"https://www.ncbi.nlm.nih.gov/geo/query/acc.cgi?acc=GSE139249\">GSE139249</ext-link>], [<ext-link ext-link-type=\"uri\" xlink:href=\"https://www.ncbi.nlm.nih.gov/geo/query/acc.cgi?acc=GSE148190\">GSE148190</ext-link>], [<ext-link ext-link-type=\"uri\" xlink:href=\"https://www.ncbi.nlm.nih.gov/geo/query/acc.cgi?acc=GSE72056\">GSE72056</ext-link>], [<ext-link ext-link-type=\"uri\" xlink:href=\"https://www.ncbi.nlm.nih.gov/geo/query/acc.cgi?acc=GSE134388\">GSE134388</ext-link>], [<ext-link ext-link-type=\"uri\" xlink:href=\"https://www.ncbi.nlm.nih.gov/geo/query/acc.cgi?acc=GSE159251\">GSE159251</ext-link>], [<ext-link ext-link-type=\"uri\" xlink:href=\"https://www.ncbi.nlm.nih.gov/geo/query/acc.cgi?acc=GSE166181\">GSE166181</ext-link>], and [<ext-link ext-link-type=\"uri\" xlink:href=\"https://www.ncbi.nlm.nih.gov/geo/query/acc.cgi?acc=GSE179373\">GSE179373</ext-link>]. <xref ref-type=\"sec\" rid=\"Sec41\">Source data</xref> are provided with this paper.</p>", "<title>Code availability</title>", "<p>We provide our code for data pre-processing, BIDCell training and inference in <ext-link ext-link-type=\"uri\" xlink:href=\"https://github.com/SydneyBioX/BIDCell\">https://github.com/SydneyBioX/BIDCell</ext-link>, 10.5281/zenodo.10070794^{##UREF##9##37##}. We provide our CellSPA framework in <ext-link ext-link-type=\"uri\" xlink:href=\"https://github.com/SydneyBioX/CellSPA\">https://github.com/SydneyBioX/CellSPA</ext-link>, 10.5281/zenodo.10295991^{##UREF##10##38##}.</p>", "<title>Competing interests</title>", "<p id=\"Par138\">The authors declare no competing interests.</p>" ]

[ "<fig id=\"Fig1\"><label>Fig. 1</label><caption><title>BIDCell framework.</title><p><bold>a</bold> Schematic illustration of the BIDCell framework and the loss functions used for training. In the deep learning model, E1 to E5 and D1 to D4 are respectively the encoding and decoding layers, while the connectivity between layers to each decoding layer is indicated by arrows of a unique colour (e.g., green for D3). <bold>b</bold> Comparative illustration of the predictions from BIDCell and other cell segmentation methods on the public Xenium-BreastCancer1 dataset. BIDCell captures cell morphologies with better correspondence to the input images, with a more diverse set of cell shapes that include elongated types. The H&amp;E images are provided for illustration purposes only and were not used as an input for any of the methods shown.</p></caption></fig>", "<fig id=\"Fig2\"><label>Fig. 2</label><caption><title>CellSPA performance evaluation framework.</title><p><bold>a</bold> Schematic showing the cell segmentation evaluation framework with five complementary categories. <bold>b</bold> Bar plots showing overall characteristics, including the number of cells [left], and the number of transcripts [right] for each of the 11 methods. <bold>c</bold> Boxplots of cell-level quality metrics with total number of transcripts [left] and total number of genes [right]. The number points for each box includes the number of cells detected by each method (N = Chromium: 22,294; Cellpose (nuclei): 99,693; BIDCell: 103,209; 10x (nuclei): 126,515; 10x: 160,254; JSTA: 107,131; Cellpose nuclei dilated: 104,307; Cellpose cell: 87,046; Voronoi: 106,227; Watershed: 105,527; Baysor: 177,437; Baysor (no prior): 191,698), and ranges from the first to third quartile with the median as the horizontal line. The boxplot’s lower whisker extends 1.5 times the interquartile range below the first quartile, while the upper whisker extends 1.5 times the interquartile range above the third quartile. <bold>d</bold> Gene-level quality metric represented by a scatter plot of the percentage of cells expressed for each gene in the segmented cells (<italic>y</italic>-axis) vs. the nuclei (<italic>x</italic>-axis). <bold>e</bold> Cell morphology metrics represented by the elongation values between the segmented cells (<italic>y</italic>-axis) and nuclei (<italic>x</italic>-axis), where each dot represents the average elongation for each cell type and the Pearson correlation between the elongation values of nuclei and segmented cells is noted in the top left corner. <bold>f</bold> Scatter plot between correlation the elongation values of nuclei and segmented cells (<italic>y</italic>-axis) and average total number of transcripts per cell (<italic>x</italic>-axis) based on average expression. Source data are provided as a Source Data file.</p></caption></fig>", "<fig id=\"Fig3\"><label>Fig. 3</label><caption><title>CellSPA graphical representation of comparison study using Xenium-BreastCancer.</title><p><bold>a</bold> Correlation heatmap of average expression between segmented cells from BIDCell (<italic>y</italic>-axis) and expression from Chromium data (<italic>x</italic>-axis) [left]. Scatter plot between correlation with Chromium expression (<italic>y</italic>-axis) and average total number of transcripts per cell (<italic>x</italic>-axis) based on average expression [right]. Each dot represents a different method. <bold>b</bold> Scatter plot between correlation with Chromium expression (<italic>y</italic>-axis) and average total number of transcripts per cell (<italic>x</italic>-axis), where each dot represents a different method. <bold>c</bold> Scatter plot between BIDCell (<italic>y</italic>-axis) and expression from Chromium data (<italic>x</italic>-axis) based on the cell type proportion extracted from each of the methods. <bold>d</bold> Scatter plot showing the expression between the F1 score for positive markers in BIDCell (<italic>y</italic>-axis) and in 10x segmentation (<italic>x</italic>-axis) [left], and scatter plot showing the purity F1 score against the average total transcripts per cell [right]. Each dot represents a method. <bold>e</bold> Line plots showing the percentage of B cells expressing the unwanted T cell marker CD4, CD8A, and CD8B against its distance from the nearest T cell, where the B cells are grouped by distance ranges. A lower percentage is better, and each line represents a different method. <bold>f</bold>–<bold>h</bold> Spatial characteristics diversity. <bold>f</bold> indicates the local spatial regions being divided in the images where the left panel indicates the cell type proportions of each local region and the right panel indicates the cell type entropy of the local region. <bold>g</bold> Scatter plots showing the association between the cell type entropy and the coefficient of variation of the total transcripts of three methods: 10x, BIDCell, and Watershed, where each dot represents each local region shown in (<bold>f</bold>). <bold>h</bold> Scatter plots showing the association between the coefficient of variation of elongation and proportion of fibroblasts in the data. <bold>i</bold> Spatial imaging of two replicates in Xenium-BreastCancer, where each dot represents the segmented cells coloured by the annotated cell type. <bold>j</bold> UMAP plots of the two replicates, coloured by cell type [left] and replicate [right]. Source data are provided as a Source Data file.</p></caption></fig>", "<fig id=\"Fig4\"><label>Fig. 4</label><caption><title>Generalisability of BIDCell.</title><p><bold>a</bold> CosMx-Lung image with UMAP plot highlighting different cell types. <bold>b</bold> Comparative illustration of the predictions from BIDCell, NanoString and Cellpose nuclei for CosMx-Lung. <bold>c</bold> Line plots showing the percentage of B cells expressing the unwanted T cell marker CD4, CD8A, and CD8B against its distance from the nearest T cell, where the B cells are grouped by the distance ranges. A lower percentage is better, and each line represents a different method with BIDCell (red), NanoString (orange), and Cellpose nuclei (grey). <bold>d</bold> MERSCOPE-Melanoma image with UMAP highlighting different cell types. <bold>e</bold> Comparative illustration of the predictions from BIDCell, Vizgen and Cellpose nuclei for MERSCOPE-Melanoma. <bold>f</bold> Scatter plot showing the coefficient of variation of the total number of genes against cell type entropy in a given region for cells segmented from BIDCell [left], nuclei cells [middle], and cells segmented from Vizgen [right]. Source data are provided as a Source Data file.</p></caption></fig>", "<fig id=\"Fig5\"><label>Fig. 5</label><caption><title>Assessment using Xenium-MouseBrain data.</title><p><bold>a</bold> Spatial image highlighting the cell type and neuronal regions using scClassify trained on SMART-seq2 data. <bold>b</bold> Comparative illustration of the predictions from BIDCell and other methods. <bold>c</bold> Hippocampus cell segmentation region by 10x [top] and BIDCell [bottom]. <bold>d</bold> Scatter plot showing the Pearson correlation with SMART-seq2 data between 10x and BIDCell for each cell type, where each dot is coloured by the cell type with the same colours as the legend in (<bold>c</bold>). <bold>e</bold> Scatter plot showing the positive purity score between 10x and BIDCell for each cell type, where each dot is coloured by the cell type. <bold>f</bold> The top panel indicates the neurons in the hippocampus region (CA1-CA3, DG) and the bottom panels are 6 x 2 panels showing the five distinct spatial regions with different neuronal markers in the hippocampal regions. From top to bottom, <italic>Prox1</italic> was expressed only in DG, <italic>Neurod6</italic> was expressed in all CA regions, <italic>Slit2</italic> was expressed in CA3, <italic>Necab2</italic> was expressed in CA2, and <italic>Wfs1</italic> and <italic>Cpne8</italic> were expressed in CA1. Source data are provided as a Source Data file.</p></caption></fig>" ]

[ "<table-wrap id=\"Tab1\"><label>Table 1</label><caption><p>Single-cell RNA-seq references used in this study</p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th>Data collection</th><th>Data</th><th># of cell types</th><th>Source</th></tr></thead><tbody><tr><td rowspan=\"10\">TISCH-BRCA</td><td><ext-link ext-link-type=\"uri\" xlink:href=\"https://www.ncbi.nlm.nih.gov/geo/query/acc.cgi?acc=GSE110686\">GSE110686</ext-link></td><td>17</td><td><ext-link ext-link-type=\"uri\" xlink:href=\"http://tisch.comp-genomics.org/gallery/?cancer=BRCA&amp;species=Human\">http://tisch.comp-genomics.org/gallery/?cancer=BRCA&amp;species=Human</ext-link></td></tr><tr><td><ext-link ext-link-type=\"uri\" xlink:href=\"https://www.ncbi.nlm.nih.gov/geo/query/acc.cgi?acc=GSE114727\">GSE114727_10X</ext-link></td><td/><td/></tr><tr><td><ext-link ext-link-type=\"uri\" xlink:href=\"https://www.ncbi.nlm.nih.gov/geo/query/acc.cgi?acc=GSE114727\">GSE114727_inDrop</ext-link></td><td/><td/></tr><tr><td><ext-link ext-link-type=\"uri\" xlink:href=\"https://www.ncbi.nlm.nih.gov/geo/query/acc.cgi?acc=GSE138536\">GSE138536</ext-link></td><td/><td/></tr><tr><td><ext-link ext-link-type=\"uri\" xlink:href=\"https://www.ncbi.nlm.nih.gov/geo/query/acc.cgi?acc=GSE143423\">GSE143423</ext-link></td><td/><td/></tr><tr><td><ext-link ext-link-type=\"uri\" xlink:href=\"https://www.ncbi.nlm.nih.gov/geo/query/acc.cgi?acc=GSE176078\">GSE176078</ext-link></td><td/><td/></tr><tr><td><ext-link ext-link-type=\"uri\" xlink:href=\"https://www.ncbi.nlm.nih.gov/bioproject/PRJNA396019\">SRP114962</ext-link></td><td/><td/></tr><tr><td><ext-link ext-link-type=\"uri\" xlink:href=\"https://www.ebi.ac.uk/biostudies/arrayexpress/studies/EMTAB8107\">EMTAB8107</ext-link></td><td/><td/></tr><tr><td><ext-link ext-link-type=\"uri\" xlink:href=\"https://www.ncbi.nlm.nih.gov/geo/query/acc.cgi?acc=GSE148673\">GSE148673</ext-link></td><td/><td/></tr><tr><td><ext-link ext-link-type=\"uri\" xlink:href=\"https://www.ncbi.nlm.nih.gov/geo/query/acc.cgi?acc=GSE150660\">GSE150660</ext-link></td><td/><td/></tr><tr><td>Chromium-BreastCancer</td><td>Single Cell Gene Expression Flex (FRP)</td><td>22</td><td><ext-link ext-link-type=\"uri\" xlink:href=\"https://www.10xgenomics.com/products/xenium-in-situ/preview-dataset-human-breast\">https://www.10xgenomics.com/products/xenium-in-situ/preview-dataset-human-breast</ext-link></td></tr><tr><td>Mouse brain</td><td>Allen brain map</td><td>59</td><td><ext-link ext-link-type=\"uri\" xlink:href=\"https://portal.brain-map.org/atlases-and-data/rnaseq/mouse-whole-cortex-and-hippocampus-smart-seq\">https://portal.brain-map.org/atlases-and-data/rnaseq/mouse-whole-cortex-and-hippocampus-smart-seq</ext-link></td></tr><tr><td rowspan=\"7\">HLCA</td><td><ext-link ext-link-type=\"uri\" xlink:href=\"https://www.ncbi.nlm.nih.gov/geo/query/acc.cgi?acc=GSE135893\">Banovich_Kropski_2020</ext-link></td><td>50</td><td><ext-link ext-link-type=\"uri\" xlink:href=\"https://beta.fastgenomics.org/p/hlca\">https://beta.fastgenomics.org/p/hlca</ext-link></td></tr><tr><td><ext-link ext-link-type=\"uri\" xlink:href=\"https://www.synapse.org/#!Synapse:syn21041850\">Krasnow_2020</ext-link></td><td/><td/></tr><tr><td><ext-link ext-link-type=\"uri\" xlink:href=\"https://www.ncbi.nlm.nih.gov/geo/query/acc.cgi?acc=GSE128033\">Lafyatis_Rojas_2019</ext-link></td><td/><td/></tr><tr><td><ext-link ext-link-type=\"uri\" xlink:href=\"https://explore.data.humancellatlas.org/projects/c4077b3c-5c98-4d26-a614-246d12c2e5d7\">Meyer_2019</ext-link></td><td/><td/></tr><tr><td><ext-link ext-link-type=\"uri\" xlink:href=\"https://www.ncbi.nlm.nih.gov/geo/query/acc.cgi?acc=GSE158127\">Misharin_2021</ext-link></td><td/><td/></tr><tr><td><ext-link ext-link-type=\"uri\" xlink:href=\"https://www.ncbi.nlm.nih.gov/geo/query/acc.cgi?acc=GSE121611\">Misharin_Budinger_2018</ext-link></td><td/><td/></tr><tr><td><ext-link ext-link-type=\"uri\" xlink:href=\"https://ega-archive.org/datasets/EGAD00001005065\">Teichmann_Meyer_2019</ext-link></td><td/><td/></tr><tr><td rowspan=\"6\">TISCH-NSCLC</td><td><ext-link ext-link-type=\"uri\" xlink:href=\"https://www.ebi.ac.uk/biostudies/arrayexpress/studies/E-MTAB-6149\">EMTAB6149</ext-link></td><td>1</td><td><ext-link ext-link-type=\"uri\" xlink:href=\"http://tisch.comp-genomics.org/gallery/?cancer=SCLC&amp;species=Human\">http://tisch.comp-genomics.org/gallery/?cancer=SCLC&amp;species=Human</ext-link></td></tr><tr><td><ext-link ext-link-type=\"uri\" xlink:href=\"https://www.ncbi.nlm.nih.gov/geo/query/acc.cgi?acc=GSE117570\">GSE117570</ext-link></td><td/><td/></tr><tr><td><ext-link ext-link-type=\"uri\" xlink:href=\"https://www.ncbi.nlm.nih.gov/geo/query/acc.cgi?acc=GSE127465\">GSE127465</ext-link></td><td/><td/></tr><tr><td><ext-link ext-link-type=\"uri\" xlink:href=\"https://www.ncbi.nlm.nih.gov/geo/query/acc.cgi?acc=GSE143423\">GSE143423</ext-link></td><td/><td/></tr><tr><td><ext-link ext-link-type=\"uri\" xlink:href=\"https://www.ncbi.nlm.nih.gov/geo/query/acc.cgi?acc=GSE148071\">GSE148071</ext-link></td><td/><td/></tr><tr><td><ext-link ext-link-type=\"uri\" xlink:href=\"https://www.ncbi.nlm.nih.gov/geo/query/acc.cgi?acc=GSE150660\">GSE150660</ext-link></td><td/><td/></tr><tr><td rowspan=\"10\">SKCM atlas</td><td><ext-link ext-link-type=\"uri\" xlink:href=\"https://www.ncbi.nlm.nih.gov/geo/query/acc.cgi?acc=GSE115978\">GSE115978</ext-link></td><td>15</td><td><ext-link ext-link-type=\"uri\" xlink:href=\"http://tisch.comp-genomics.org/gallery/?cancer=SKCM&amp;species=Human\">http://tisch.comp-genomics.org/gallery/?cancer=SKCM&amp;species=Human</ext-link></td></tr><tr><td><ext-link ext-link-type=\"uri\" xlink:href=\"https://www.ncbi.nlm.nih.gov/geo/query/acc.cgi?acc=GSE120575\">GSE120575</ext-link></td><td/><td/></tr><tr><td><ext-link ext-link-type=\"uri\" xlink:href=\"https://www.ncbi.nlm.nih.gov/geo/query/acc.cgi?acc=GSE123139\">GSE123139</ext-link></td><td/><td/></tr><tr><td><ext-link ext-link-type=\"uri\" xlink:href=\"https://www.ncbi.nlm.nih.gov/geo/query/acc.cgi?acc=GSE139249\">GSE139249</ext-link></td><td/><td/></tr><tr><td><ext-link ext-link-type=\"uri\" xlink:href=\"https://www.ncbi.nlm.nih.gov/geo/query/acc.cgi?acc=GSE148190\">GSE148190</ext-link></td><td/><td/></tr><tr><td><ext-link ext-link-type=\"uri\" xlink:href=\"https://www.ncbi.nlm.nih.gov/geo/query/acc.cgi?acc=GSE72056\">GSE72056</ext-link></td><td/><td/></tr><tr><td><ext-link ext-link-type=\"uri\" xlink:href=\"https://www.ncbi.nlm.nih.gov/geo/query/acc.cgi?acc=GSE134388\">GSE134388</ext-link></td><td/><td/></tr><tr><td><ext-link ext-link-type=\"uri\" xlink:href=\"https://www.ncbi.nlm.nih.gov/geo/query/acc.cgi?acc=GSE159251\">GSE159251</ext-link></td><td/><td/></tr><tr><td><ext-link ext-link-type=\"uri\" xlink:href=\"https://www.ncbi.nlm.nih.gov/geo/query/acc.cgi?acc=GSE166181\">GSE166181</ext-link></td><td/><td/></tr><tr><td><ext-link ext-link-type=\"uri\" xlink:href=\"https://www.ncbi.nlm.nih.gov/geo/query/acc.cgi?acc=GSE179373\">GSE179373</ext-link></td><td/><td/></tr></tbody></table></table-wrap>", "<table-wrap id=\"Tab2\"><label>Table 2</label><caption><p>Summary of existing methods used for comparison</p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th>Types</th><th>Method</th><th>Nuclei segmentation</th><th>Cell body segmetation</th><th>Public code</th><th>Reference</th></tr></thead><tbody><tr><td rowspan=\"2\">Nuclei</td><td>10x (Nuclei)</td><td>10x</td><td>NA</td><td>N/A</td><td/></tr><tr><td>Cellpose (Nuclei)</td><td>Cellpose</td><td>NA</td><td>Version 2.1.1</td><td>^{##REF##33318659##5##}</td></tr><tr><td rowspan=\"2\">Adapted from classical approach</td><td>Cellpose nuclei dilated</td><td>Cellpose</td><td>Dilation</td><td>OpenCV (v4.6.0)</td><td/></tr><tr><td>Voronoi</td><td>Cellpose</td><td>Voronoi expansion</td><td>SciPy library (v1.9.3)</td><td/></tr><tr><td/><td>Watershed</td><td>Cellpose</td><td>Watershed algorithm</td><td>OpenCV (v4.6.0)</td><td/></tr><tr><td rowspan=\"4\">Deep learning-based</td><td>10x</td><td>10x</td><td>10x</td><td>N/A</td><td/></tr><tr><td>BIDCell</td><td>Cellpose</td><td>BIDCell</td><td>Version 4494e02</td><td/></tr><tr><td>Cellpose cell</td><td>Cellpose</td><td>Cellpose</td><td>Version 2.1.1</td><td>^{##REF##33318659##5##}</td></tr><tr><td>JSTA</td><td>Cellpose</td><td>JSTA</td><td>Version ccce064</td><td>^{##REF##34057817##18##}</td></tr><tr><td>Transcript-based</td><td>Baysor</td><td>N/A or Cellpose</td><td>Baysor</td><td>Version 0.5.2</td><td>^{##REF##34650268##7##}</td></tr></tbody></table></table-wrap>" ]

[ "<inline-formula id=\"IEq1\"><alternatives><tex-math id=\"M1\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${{{{{{{\\boldsymbol{X}}}}}}}}\\in {{\\mathbb{R}}}^{H\\times W\\times {n}_{genes}}$$\\end{document}</tex-math><mml:math id=\"M2\"><mml:mi mathvariant=\"bold-italic\">X</mml:mi><mml:mo>∈</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant=\"double-struck\">R</mml:mi></mml:mrow><mml:mrow><mml:mi>H</mml:mi><mml:mo>×</mml:mo><mml:mi>W</mml:mi><mml:mo>×</mml:mo><mml:msub><mml:mrow><mml:mi>n</mml:mi></mml:mrow><mml:mrow><mml:mi>g</mml:mi><mml:mi>e</mml:mi><mml:mi>n</mml:mi><mml:mi>e</mml:mi><mml:mi>s</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msup></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq2\"><alternatives><tex-math id=\"M3\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${{{{{{{\\bf{x}}}}}}}}\\in {{\\mathbb{R}}}^{h\\times w\\times {n}_{genes}}$$\\end{document}</tex-math><mml:math id=\"M4\"><mml:mi mathvariant=\"bold\">x</mml:mi><mml:mo>∈</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant=\"double-struck\">R</mml:mi></mml:mrow><mml:mrow><mml:mi>h</mml:mi><mml:mo>×</mml:mo><mml:mi>w</mml:mi><mml:mo>×</mml:mo><mml:msub><mml:mrow><mml:mi>n</mml:mi></mml:mrow><mml:mrow><mml:mi>g</mml:mi><mml:mi>e</mml:mi><mml:mi>n</mml:mi><mml:mi>e</mml:mi><mml:mi>s</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msup></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ1\"><label>1</label><alternatives><tex-math id=\"M5\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${L}_{ne}({{{{{{{{\\bf{x}}}}}}}}}_{{{{{{{{\\bf{nuc}}}}}}}}},\\hat{{{{{{{{\\bf{y}}}}}}}}})=-{{{{{{{{\\bf{x}}}}}}}}}_{{{{{{{{\\bf{nuc}}}}}}}}}\\log (\\hat{{{{{{{{\\bf{y}}}}}}}}})-(1-{{{{{{{{\\bf{x}}}}}}}}}_{{{{{{{{\\bf{nuc}}}}}}}}})\\log ({{{{{{{\\bf{1}}}}}}}}-\\hat{{{{{{{{\\bf{y}}}}}}}}}),$$\\end{document}</tex-math><mml:math id=\"M6\"><mml:msub><mml:mrow><mml:mi>L</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mi>e</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mrow><mml:mi mathvariant=\"bold\">x</mml:mi></mml:mrow><mml:mrow><mml:mi 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id=\"M10\"><mml:mi>e</mml:mi><mml:mi>c</mml:mi><mml:mi>c</mml:mi><mml:mo>=</mml:mo><mml:msqrt><mml:mrow><mml:mn>1</mml:mn><mml:mo>−</mml:mo><mml:mfrac><mml:mrow><mml:msup><mml:mrow><mml:mi>b</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:mfrac></mml:mrow></mml:msqrt><mml:mo>,</mml:mo></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ3\"><label>3</label><alternatives><tex-math id=\"M11\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${l}_{h}=\\alpha \\times ec{c}_{nuc}\\times {l}_{t},$$\\end{document}</tex-math><mml:math id=\"M12\"><mml:msub><mml:mrow><mml:mi>l</mml:mi></mml:mrow><mml:mrow><mml:mi>h</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mi>α</mml:mi><mml:mo>×</mml:mo><mml:mi>e</mml:mi><mml:mi>c</mml:mi><mml:msub><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mi>u</mml:mi><mml:mi>c</mml:mi></mml:mrow></mml:msub><mml:mo>×</mml:mo><mml:msub><mml:mrow><mml:mi>l</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ4\"><label>4</label><alternatives><tex-math id=\"M13\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${l}_{v}=\\left\\{\\begin{array}{ll}{l}_{t}-{l}_{h},\\quad 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width=\"0.25em\"/><mml:msub><mml:mrow><mml:mi>l</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo>−</mml:mo><mml:msub><mml:mrow><mml:mi>l</mml:mi></mml:mrow><mml:mrow><mml:mi>h</mml:mi></mml:mrow></mml:msub><mml:mo>&gt;</mml:mo><mml:msub><mml:mrow><mml:mi>l</mml:mi></mml:mrow><mml:mrow><mml:mi>v</mml:mi><mml:mi>m</mml:mi></mml:mrow></mml:msub></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign=\"left\"><mml:msub><mml:mrow><mml:mi>l</mml:mi></mml:mrow><mml:mrow><mml:mi>v</mml:mi><mml:mi>m</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo></mml:mtd><mml:mtd columnalign=\"left\"><mml:mspace width=\"0.25em\"/><mml:mstyle><mml:mtext>otherwise</mml:mtext></mml:mstyle></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ5\"><label>5</label><alternatives><tex-math id=\"M15\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} 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mathvariant=\"bold\">y</mml:mi></mml:mrow><mml:mrow><mml:mo>^</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mi>M</mml:mi></mml:mrow></mml:mfrac><mml:munderover accent=\"false\" accentunder=\"false\"><mml:mrow><mml:mo>∑</mml:mo></mml:mrow><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi>M</mml:mi></mml:mrow></mml:munderover><mml:mo>−</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant=\"bold\">e</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold\">c</mml:mi></mml:mrow></mml:msub><mml:mi>log</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mrow><mml:mover accent=\"true\"><mml:mrow><mml:mi mathvariant=\"bold\">y</mml:mi></mml:mrow><mml:mrow><mml:mo>^</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mi>c</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>−</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mn>1</mml:mn><mml:mo>−</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant=\"bold\">e</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold\">c</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mi>log</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi mathvariant=\"bold\">1</mml:mi><mml:mo>−</mml:mo><mml:msub><mml:mrow><mml:mover accent=\"true\"><mml:mrow><mml:mi mathvariant=\"bold\">y</mml:mi></mml:mrow><mml:mrow><mml:mo>^</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mi>c</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>,</mml:mo></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq4\"><alternatives><tex-math id=\"M17\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\hat{{{{{{{{\\bf{y}}}}}}}}}}_{c}$$\\end{document}</tex-math><mml:math id=\"M18\"><mml:msub><mml:mrow><mml:mover accent=\"true\"><mml:mrow><mml:mi mathvariant=\"bold\">y</mml:mi></mml:mrow><mml:mrow><mml:mo>^</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mi>c</mml:mi></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ6\"><label>6</label><alternatives><tex-math id=\"M19\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${p}_{nuc,c}=\\mathop{\\sum}\\limits_{i}\\mathop{\\sum}\\limits_{j}\\sigma ({\\hat{q}}_{ijc}{x}_{nuc,ij}-0.5),$$\\end{document}</tex-math><mml:math id=\"M20\"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mi>u</mml:mi><mml:mi>c</mml:mi><mml:mo>,</mml:mo><mml:mi>c</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:munder><mml:mrow><mml:mo>∑</mml:mo></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:munder><mml:munder><mml:mrow><mml:mo>∑</mml:mo></mml:mrow><mml:mrow><mml:mi>j</mml:mi></mml:mrow></mml:munder><mml:mi>σ</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mrow><mml:mover accent=\"true\"><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mo>^</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi><mml:mi>c</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mi>u</mml:mi><mml:mi>c</mml:mi><mml:mo>,</mml:mo><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo>−</mml:mo><mml:mn>0.5</mml:mn></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>,</mml:mo></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ7\"><label>7</label><alternatives><tex-math id=\"M21\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${p}_{cyto,c}=\\mathop{\\sum}\\limits_{i}\\mathop{\\sum}\\limits_{j}\\sigma ({\\hat{q}}_{ijc}(1-{x}_{nuc,ij})-0.5),$$\\end{document}</tex-math><mml:math id=\"M22\"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi>c</mml:mi><mml:mi>y</mml:mi><mml:mi>t</mml:mi><mml:mi>o</mml:mi><mml:mo>,</mml:mo><mml:mi>c</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:munder><mml:mrow><mml:mo>∑</mml:mo></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:munder><mml:munder><mml:mrow><mml:mo>∑</mml:mo></mml:mrow><mml:mrow><mml:mi>j</mml:mi></mml:mrow></mml:munder><mml:mi>σ</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mrow><mml:mover accent=\"true\"><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mo>^</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi><mml:mi>c</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mn>1</mml:mn><mml:mo>−</mml:mo><mml:msub><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mi>u</mml:mi><mml:mi>c</mml:mi><mml:mo>,</mml:mo><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>−</mml:mo><mml:mn>0.5</mml:mn></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>,</mml:mo></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ8\"><label>8</label><alternatives><tex-math id=\"M23\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${L}_{os}=\\left\\{\\begin{array}{ll}\\frac{1}{M}\\mathop{\\sum }\\limits_{c}^{M}({p}_{cyto,c}-{p}_{nuc,c}),\\quad &amp;\\,{{\\mbox{if}}}\\,\\,\\mathop{\\sum }\\limits_{c}^{M}({p}_{cyto,c}-{p}_{nuc,c})\\, &gt; \\,0\\\\ 0,\\hfill\\quad &amp;\\,{{\\mbox{otherwise}}}\\,\\end{array}\\right.$$\\end{document}</tex-math><mml:math id=\"M24\"><mml:msub><mml:mrow><mml:mi>L</mml:mi></mml:mrow><mml:mrow><mml:mi>o</mml:mi><mml:mi>s</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mfenced open=\"{\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"left\"><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mi>M</mml:mi></mml:mrow></mml:mfrac><mml:munderover accent=\"false\" accentunder=\"false\"><mml:mrow><mml:mo>∑</mml:mo></mml:mrow><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi>M</mml:mi></mml:mrow></mml:munderover><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi>c</mml:mi><mml:mi>y</mml:mi><mml:mi>t</mml:mi><mml:mi>o</mml:mi><mml:mo>,</mml:mo><mml:mi>c</mml:mi></mml:mrow></mml:msub><mml:mo>−</mml:mo><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mi>u</mml:mi><mml:mi>c</mml:mi><mml:mo>,</mml:mo><mml:mi>c</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>,</mml:mo></mml:mtd><mml:mtd columnalign=\"left\"><mml:mspace width=\"0.25em\"/><mml:mstyle><mml:mtext>if</mml:mtext></mml:mstyle><mml:mspace width=\"0.25em\"/><mml:mspace width=\"0.25em\"/><mml:munderover accent=\"false\" accentunder=\"false\"><mml:mrow><mml:mo>∑</mml:mo></mml:mrow><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi>M</mml:mi></mml:mrow></mml:munderover><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi>c</mml:mi><mml:mi>y</mml:mi><mml:mi>t</mml:mi><mml:mi>o</mml:mi><mml:mo>,</mml:mo><mml:mi>c</mml:mi></mml:mrow></mml:msub><mml:mo>−</mml:mo><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mi>u</mml:mi><mml:mi>c</mml:mi><mml:mo>,</mml:mo><mml:mi>c</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mspace width=\"0.25em\"/><mml:mo>&gt;</mml:mo><mml:mspace width=\"0.25em\"/><mml:mn>0</mml:mn></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign=\"left\"><mml:mn>0</mml:mn><mml:mo>,</mml:mo></mml:mtd><mml:mtd columnalign=\"left\"><mml:mspace width=\"0.25em\"/><mml:mstyle><mml:mtext>otherwise</mml:mtext></mml:mstyle></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq5\"><alternatives><tex-math id=\"M25\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\hat{q}}_{ijc}$$\\end{document}</tex-math><mml:math id=\"M26\"><mml:msub><mml:mrow><mml:mover accent=\"true\"><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mo>^</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi><mml:mi>c</mml:mi></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ9\"><label>9</label><alternatives><tex-math id=\"M27\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${s}_{ov,ij}=-(1-{x}_{nuc,ij})+\\mathop{\\sum }\\limits_{c}^{M}\\sigma ({\\hat{q}}_{ijc}(1-{x}_{nuc,ij})-0.5),$$\\end{document}</tex-math><mml:math id=\"M28\"><mml:msub><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mi>o</mml:mi><mml:mi>v</mml:mi><mml:mo>,</mml:mo><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mo>−</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mn>1</mml:mn><mml:mo>−</mml:mo><mml:msub><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mi>u</mml:mi><mml:mi>c</mml:mi><mml:mo>,</mml:mo><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>+</mml:mo><mml:munderover accent=\"false\" accentunder=\"false\"><mml:mrow><mml:mo>∑</mml:mo></mml:mrow><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi>M</mml:mi></mml:mrow></mml:munderover><mml:mi>σ</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mrow><mml:mover accent=\"true\"><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mo>^</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi><mml:mi>c</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mn>1</mml:mn><mml:mo>−</mml:mo><mml:msub><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mi>u</mml:mi><mml:mi>c</mml:mi><mml:mo>,</mml:mo><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>−</mml:mo><mml:mn>0.5</mml:mn></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>,</mml:mo></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ10\"><label>10</label><alternatives><tex-math id=\"M29\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${L}_{ov}=\\left\\{\\begin{array}{ll}\\frac{{\\sum }_{i}{\\sum }_{j}({s}_{ov,ij})}{Mhw},\\quad &amp;\\,{{\\mbox{if}}}\\,\\,{s}_{ov}\\, &gt; \\,0\\\\ 0,\\quad &amp;\\,{{\\mbox{otherwise}}}\\,\\end{array}\\right.$$\\end{document}</tex-math><mml:math id=\"M30\"><mml:msub><mml:mrow><mml:mi>L</mml:mi></mml:mrow><mml:mrow><mml:mi>o</mml:mi><mml:mi>v</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mfenced open=\"{\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"left\"><mml:mfrac><mml:mrow><mml:msub><mml:mrow><mml:mo mathsize=\"big\">∑</mml:mo></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mo mathsize=\"big\">∑</mml:mo></mml:mrow><mml:mrow><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mi>o</mml:mi><mml:mi>v</mml:mi><mml:mo>,</mml:mo><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mrow><mml:mi>M</mml:mi><mml:mi>h</mml:mi><mml:mi>w</mml:mi></mml:mrow></mml:mfrac><mml:mo>,</mml:mo></mml:mtd><mml:mtd columnalign=\"left\"><mml:mspace width=\"0.25em\"/><mml:mstyle><mml:mtext>if</mml:mtext></mml:mstyle><mml:mspace width=\"0.25em\"/><mml:mspace width=\"0.25em\"/><mml:msub><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mi>o</mml:mi><mml:mi>v</mml:mi></mml:mrow></mml:msub><mml:mspace width=\"0.25em\"/><mml:mo>&gt;</mml:mo><mml:mspace width=\"0.25em\"/><mml:mn>0</mml:mn></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign=\"left\"><mml:mn>0</mml:mn><mml:mo>,</mml:mo></mml:mtd><mml:mtd columnalign=\"left\"><mml:mspace width=\"0.25em\"/><mml:mstyle><mml:mtext>otherwise</mml:mtext></mml:mstyle></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ11\"><label>11</label><alternatives><tex-math id=\"M31\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${L}_{pos}({{{{{{{{\\bf{m}}}}}}}}}_{{{{{{{{\\bf{pos}}}}}}}}},\\hat{{{{{{{{\\bf{y}}}}}}}}})=\\frac{1}{M}\\mathop{\\sum }\\limits_{c}^{M}-{{{{{{{{\\bf{m}}}}}}}}}_{{{{{{{{\\bf{pos}}}}}}}},{{{{{{{\\bf{c}}}}}}}}}\\log ({\\hat{{{{{{{{\\bf{y}}}}}}}}}}_{c})-(1-{{{{{{{{\\bf{m}}}}}}}}}_{{{{{{{{\\bf{pos}}}}}}}},{{{{{{{\\bf{c}}}}}}}}})\\log ({{{{{{{\\bf{1}}}}}}}}-{\\hat{{{{{{{{\\bf{y}}}}}}}}}}_{c}),$$\\end{document}</tex-math><mml:math id=\"M32\"><mml:msub><mml:mrow><mml:mi>L</mml:mi></mml:mrow><mml:mrow><mml:mi>p</mml:mi><mml:mi>o</mml:mi><mml:mi>s</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mrow><mml:mi mathvariant=\"bold\">m</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold\">pos</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:mover accent=\"true\"><mml:mrow><mml:mi mathvariant=\"bold\">y</mml:mi></mml:mrow><mml:mrow><mml:mo>^</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mi>M</mml:mi></mml:mrow></mml:mfrac><mml:munderover accent=\"false\" accentunder=\"false\"><mml:mrow><mml:mo>∑</mml:mo></mml:mrow><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi>M</mml:mi></mml:mrow></mml:munderover><mml:mo>−</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant=\"bold\">m</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold\">pos</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant=\"bold\">c</mml:mi></mml:mrow></mml:msub><mml:mi>log</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mrow><mml:mover accent=\"true\"><mml:mrow><mml:mi mathvariant=\"bold\">y</mml:mi></mml:mrow><mml:mrow><mml:mo>^</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mi>c</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>−</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mn>1</mml:mn><mml:mo>−</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant=\"bold\">m</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold\">pos</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant=\"bold\">c</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mi>log</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi mathvariant=\"bold\">1</mml:mi><mml:mo>−</mml:mo><mml:msub><mml:mrow><mml:mover accent=\"true\"><mml:mrow><mml:mi mathvariant=\"bold\">y</mml:mi></mml:mrow><mml:mrow><mml:mo>^</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mi>c</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>,</mml:mo></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ12\"><label>12</label><alternatives><tex-math id=\"M33\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${L}_{neg}({{{{{{{{\\bf{m}}}}}}}}}_{{{{{{{{\\bf{neg}}}}}}}}},\\hat{{{{{{{{\\bf{q}}}}}}}}})=\\frac{1}{M}\\mathop{\\sum }\\limits_{c}^{M}\\sigma ({\\hat{{{{{{{{\\bf{q}}}}}}}}}}_{c}{{{{{{{{\\bf{m}}}}}}}}}_{{{{{{{{\\bf{neg}}}}}}}},{{{{{{{\\bf{c}}}}}}}}}-0.5)$$\\end{document}</tex-math><mml:math id=\"M34\"><mml:msub><mml:mrow><mml:mi>L</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mi>e</mml:mi><mml:mi>g</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mrow><mml:mi mathvariant=\"bold\">m</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold\">neg</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:mover accent=\"true\"><mml:mrow><mml:mi mathvariant=\"bold\">q</mml:mi></mml:mrow><mml:mrow><mml:mo>^</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mi>M</mml:mi></mml:mrow></mml:mfrac><mml:munderover accent=\"false\" accentunder=\"false\"><mml:mrow><mml:mo>∑</mml:mo></mml:mrow><mml:mrow><mml:mi>c</mml:mi></mml:mrow><mml:mrow><mml:mi>M</mml:mi></mml:mrow></mml:munderover><mml:mi>σ</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mrow><mml:mover accent=\"true\"><mml:mrow><mml:mi mathvariant=\"bold\">q</mml:mi></mml:mrow><mml:mrow><mml:mo>^</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mi>c</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi mathvariant=\"bold\">m</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold\">neg</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant=\"bold\">c</mml:mi></mml:mrow></mml:msub><mml:mo>−</mml:mo><mml:mn>0.5</mml:mn></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ13\"><label>13</label><alternatives><tex-math id=\"M35\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\mathop{\\min }\\limits_{\\theta }\\mathop{\\sum }\\limits_{n}^{N}[{\\lambda }_{ne}{L}_{ne}+{\\lambda }_{cc}{L}_{cc}+{\\lambda }_{os}{L}_{os}+{\\lambda }_{ov}{L}_{ov}+{\\lambda }_{pos}{L}_{pos}+{\\lambda }_{neg}{L}_{neg}],$$\\end{document}</tex-math><mml:math id=\"M36\"><mml:munder><mml:mrow><mml:mi>min</mml:mi></mml:mrow><mml:mrow><mml:mi>θ</mml:mi></mml:mrow></mml:munder><mml:munderover accent=\"false\" accentunder=\"false\"><mml:mrow><mml:mo>∑</mml:mo></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow><mml:mrow><mml:mi>N</mml:mi></mml:mrow></mml:munderover><mml:mrow><mml:mo>[</mml:mo><mml:mrow><mml:msub><mml:mrow><mml:mi>λ</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mi>e</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>L</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mi>e</mml:mi></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mi>λ</mml:mi></mml:mrow><mml:mrow><mml:mi>c</mml:mi><mml:mi>c</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>L</mml:mi></mml:mrow><mml:mrow><mml:mi>c</mml:mi><mml:mi>c</mml:mi></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mi>λ</mml:mi></mml:mrow><mml:mrow><mml:mi>o</mml:mi><mml:mi>s</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>L</mml:mi></mml:mrow><mml:mrow><mml:mi>o</mml:mi><mml:mi>s</mml:mi></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mi>λ</mml:mi></mml:mrow><mml:mrow><mml:mi>o</mml:mi><mml:mi>v</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>L</mml:mi></mml:mrow><mml:mrow><mml:mi>o</mml:mi><mml:mi>v</mml:mi></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mi>λ</mml:mi></mml:mrow><mml:mrow><mml:mi>p</mml:mi><mml:mi>o</mml:mi><mml:mi>s</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>L</mml:mi></mml:mrow><mml:mrow><mml:mi>p</mml:mi><mml:mi>o</mml:mi><mml:mi>s</mml:mi></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mi>λ</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mi>e</mml:mi><mml:mi>g</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>L</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mi>e</mml:mi><mml:mi>g</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo>]</mml:mo></mml:mrow><mml:mo>,</mml:mo></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ14\"><label>14</label><alternatives><tex-math id=\"M37\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${{{{{{{\\rm{Density}}}}}}}}=\\frac{{\\sum }_{i\\in I}{n}_{i}}{A},$$\\end{document}</tex-math><mml:math id=\"M38\"><mml:mi mathvariant=\"normal\">Density</mml:mi><mml:mo>=</mml:mo><mml:mfrac><mml:mrow><mml:msub><mml:mrow><mml:mo mathsize=\"big\">∑</mml:mo></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mo>∈</mml:mo><mml:mi>I</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>n</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:mi>A</mml:mi></mml:mrow></mml:mfrac><mml:mo>,</mml:mo></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ15\"><label>15</label><alternatives><tex-math id=\"M39\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\frac{{W}_{{{{{{{{\\rm{bb}}}}}}}}}}{{H}_{{{{{{{{\\rm{bb}}}}}}}}}},$$\\end{document}</tex-math><mml:math id=\"M40\"><mml:mfrac><mml:mrow><mml:msub><mml:mrow><mml:mi>W</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">bb</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">bb</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:mfrac><mml:mo>,</mml:mo></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ16\"><label>16</label><alternatives><tex-math id=\"M41\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\frac{4\\pi \\times A}{{P}_{{{{{{{{\\rm{convex}}}}}}}}}^{2}},$$\\end{document}</tex-math><mml:math id=\"M42\"><mml:mfrac><mml:mrow><mml:mn>4</mml:mn><mml:mi>π</mml:mi><mml:mo>×</mml:mo><mml:mi>A</mml:mi></mml:mrow><mml:mrow><mml:msubsup><mml:mrow><mml:mi>P</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">convex</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup></mml:mrow></mml:mfrac><mml:mo>,</mml:mo></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ17\"><label>17</label><alternatives><tex-math id=\"M43\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\frac{{R}_{{{{{{{{\\rm{I}}}}}}}}}}{{R}_{{{{{{{{\\rm{C}}}}}}}}}},$$\\end{document}</tex-math><mml:math id=\"M44\"><mml:mfrac><mml:mrow><mml:msub><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">I</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">C</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:mfrac><mml:mo>,</mml:mo></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ18\"><label>18</label><alternatives><tex-math id=\"M45\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\frac{4\\pi \\times A}{{P}_{{{{{{{{\\rm{cell}}}}}}}}}^{2}},$$\\end{document}</tex-math><mml:math id=\"M46\"><mml:mfrac><mml:mrow><mml:mn>4</mml:mn><mml:mi>π</mml:mi><mml:mo>×</mml:mo><mml:mi>A</mml:mi></mml:mrow><mml:mrow><mml:msubsup><mml:mrow><mml:mi>P</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">cell</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup></mml:mrow></mml:mfrac><mml:mo>,</mml:mo></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ19\"><label>19</label><alternatives><tex-math id=\"M47\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\frac{{P}_{{{{{{{{\\rm{convex}}}}}}}}}}{{P}_{{{{{{{{\\rm{cell}}}}}}}}}},$$\\end{document}</tex-math><mml:math id=\"M48\"><mml:mfrac><mml:mrow><mml:msub><mml:mrow><mml:mi>P</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">convex</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>P</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">cell</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:mfrac><mml:mo>,</mml:mo></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ20\"><label>20</label><alternatives><tex-math id=\"M49\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\frac{{L}_{{{{{{{{\\rm{minor}}}}}}}}}}{{L}_{{{{{{{{\\rm{major}}}}}}}}}},$$\\end{document}</tex-math><mml:math id=\"M50\"><mml:mfrac><mml:mrow><mml:msub><mml:mrow><mml:mi>L</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">minor</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>L</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">major</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:mfrac><mml:mo>,</mml:mo></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ21\"><label>21</label><alternatives><tex-math id=\"M51\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\frac{A}{{A}_{{{{{{{{\\rm{convex}}}}}}}}}},$$\\end{document}</tex-math><mml:math id=\"M52\"><mml:mfrac><mml:mrow><mml:mi>A</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>A</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">convex</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:mfrac><mml:mo>,</mml:mo></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ22\"><label>22</label><alternatives><tex-math id=\"M53\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$F{1}_{{{{{{{{\\rm{purity}}}}}}}}}=2\\cdot \\frac{(1-F{1}_{{{{{{{{\\rm{negative}}}}}}}}})\\cdot F{1}_{{{{{{{{\\rm{positive}}}}}}}}}}{1-F{1}_{{{{{{{{\\rm{negative}}}}}}}}}+F{1}_{{{{{{{{\\rm{positive}}}}}}}}}}.$$\\end{document}</tex-math><mml:math id=\"M54\"><mml:mi>F</mml:mi><mml:msub><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">purity</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mn>2</mml:mn><mml:mo>⋅</mml:mo><mml:mfrac><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mn>1</mml:mn><mml:mo>−</mml:mo><mml:mi>F</mml:mi><mml:msub><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">negative</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>⋅</mml:mo><mml:mi>F</mml:mi><mml:msub><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mi 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[ "<supplementary-material content-type=\"local-data\" id=\"MOESM1\"></supplementary-material>", "<supplementary-material content-type=\"local-data\" id=\"MOESM2\"></supplementary-material>", "<supplementary-material content-type=\"local-data\" id=\"MOESM3\"></supplementary-material>", "<supplementary-material content-type=\"local-data\" id=\"MOESM4\"></supplementary-material>" ]

[ "<fn-group><fn><p><bold>Publisher’s note</bold> Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.</p></fn><fn><p>These authors contributed equally: Xiaohang Fu, Yingxin Lin.</p></fn></fn-group>" ]

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[ "<media xlink:href=\"41467_2023_44560_MOESM1_ESM.pdf\"><caption><p>Supplementary Information</p></caption></media>", "<media xlink:href=\"41467_2023_44560_MOESM2_ESM.pdf\"><caption><p>Peer Review File</p></caption></media>", "<media xlink:href=\"41467_2023_44560_MOESM3_ESM.pdf\"><caption><p>Reporting Summary</p></caption></media>", "<media xlink:href=\"41467_2023_44560_MOESM4_ESM.xlsx\"><caption><p>Source Data</p></caption></media>" ]

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[ "<title>Introduction</title>", "<p id=\"Par2\">The surface faulting rupture site will rupture under the appropriate bedrock dislocation and damage the tunnel structure on it. With the building and operation of the cross-tunnel structure in an active fault-intensive area, tunnel designs that avoid or only conceptually cross the active fault have been unable to match the rapid development of tunnel engineering transportation. As a result, it is critical to investigate the site rupture mechanism of a cross-fault tunnel, as well as the failure characteristics of the tunnel construction and its anti-rupture measures. The location of the upper breakpoint of the overburden soil and how to destroy the tunnel structure at the location of the earthquake rupture dislocation have become important practical issues of concern to geological engineering, geotechnical engineering, and tunnel engineering researchers.</p>", "<p id=\"Par3\">Currently, the research approaches for studying the failure of cross-fault tunnel structures and their failure mechanisms on site include analyzing earthquake damage cases^{##UREF##0##1##–##UREF##3##4##}, conducting model tests^{##UREF##4##5##–##UREF##11##12##}, and doing numerical simulations^{##UREF##12##13##,##UREF##14##15##}. He et al. conducted an analysis of the data on damage to surrounding rock tunnels in fault rupture zones during the Wenchuan earthquake \"5.12\" and other significant earthquakes^{##UREF##0##1##}. Their findings indicate that tunnels passing through fault rupture zones are prone to damage during earthquakes. The failure of tunnels in fault zones is primarily caused by the disparity in displacement between the surrounding rock in the fault zone and the more stable portion of the surrounding rock. Zhu et al.^{##UREF##1##2##} conducted a comprehensive review of the research conducted by the Tunnel Fault Disaster Mechanism Research Institute. They examined various models, including the engineering geological model, the mechanical model, and the dislocation displacement model. They also analyzed different methods, such as analytical methods, numerical simulation, and physical simulation. Additionally, the existing fault dislocation prevention and control measures, as well as the design and defense principles underlying them, are summarized. Through induction and classification, Li et al.^{##UREF##4##5##} compared and analyzed the key technologies in the current tunnel engineering physical simulation test system and proposed a test system classification method based on the size of the model. Guo et al.^{##UREF##5##6##} performed a 1:80 scale test using a dislocation model test instrument they designed themselves, together with digital image correlation technology. The objective was to investigate the process of shear rupture in the overburden resulting from the dislocation of a 60° dip-reverse fault. Shen^{##UREF##6##7##} used the design method of flexible joint segmented lining to conduct a shaking table test of a cross-fault tunnel in a hilly environment with a geological similarity ratio of 1:30. It is confirmed that it is applied to the structural design of cross-fault tunnels by improving the deformation adaptability of the tunnel structure. Sabagh et al.^{##UREF##7##8##} conducted a tunnel model test with a similarity ratio of 1:60 on a centrifuge for a reverse fault sand overburden site with a bedrock dislocation angle of 60°. The damage state of the tunnel was evaluated by the gradual increase in permanent ground displacement (PGD). Large-scale box test based on cross-fault tunnel seismic damage. Researchers can simulate tunnel stress under seismic fault dislocation and vibration conditions, investigate tunnel failure characteristics and causes, and devise effective fault prevention and control strategies.</p>", "<p id=\"Par4\">Model test analysis and equipment for simulating cross-fault tunnels have garnered attention because of limited on-site seismic damage data, restrictions in research techniques, and funding. However, the test methods are mostly based on the shaking table model test, which simulates the input of near-fault ground motion. However, the shaking table model test cannot simulate the site rock and soil mass rupture caused by the sudden bedrock dislocation of the straight-down fault and the large deformation of the soil near the tunnel foundation or the dislocation of the surrounding rock, and the large deformation of the rock and soil around the tunnel is the main cause of the earthquake damage to the cross-fault tunnel. It is difficult to reasonably simulate the actual cross-fault tunnel structure and its overburden site by using the constant gravity dislocation loading device due to load reasons or the centrifuge model due to the small scale of the test due to the size of the hanging basket box.</p>", "<p id=\"Par5\">As a result, we urgently need It is critical to develop a large-scale constant gravity model test device to simulate the dislocation loading mode of seism genic fault bedrock, analyze the response of tunnels across dip-slip faults under different site soil types, investigate the rupture distribution of overburden soil under dip-slip faults and the failure characteristics of internal tunnel structures, and provide basic data and a foundation for reasonably determining the damage and failure mechanisms of tunnels^{##UREF##15##16##}.</p>", "<p id=\"Par6\">So, this paper combines the large-scale test device platform of the cross-fault tunnel model developed by the research group to conduct research on the failure mechanism of the cross-fault segmented tunnel structure, as well as its site failure characteristics and anti-rupture measures, providing a certain reference value for the design and operation of the cross-fault tunnel structure.</p>" ]

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[ "<title>Conclusion</title>", "<p id=\"Par42\">This article adopts the design concept of \"fuse\", which is to use segmented flexible joint tunnel segments during fault displacement, based on the principle of \"sacrificing small for large\", to reduce the overall damage of soil and surrounding rock to the length of the tunnel body. It’s analyzed that the failure mechanism of segmented tunnel structures in sandy and cohesive soil sites during low-dip reverse fault displacement.<list list-type=\"order\"><list-item><p id=\"Par43\">The tunnel structure's failure characteristics in the soil field across the fault cover layer are determined not only by the degree of bedrock dislocation but also by the failure range of soil with different properties under the action of dislocation. As a result, the position and shape of the overburden soil rupture range induced by bedrock dislocation at different angles should be obvious in the design, particularly the longitudinal soil rupture range in the buried depth where the tunnel is placed.</p></list-item><list-item><p id=\"Par44\">Compared with the free field of cohesive soil overburden, the uneven deformation of the surface of the cohesive soil site with tunnel structure becomes slower as a whole. With the increase in bedrock dislocation, the downward deformation at the tunnel structure at the reverse position of the bedrock dislocation dip angle is significant. The variation of internal earth pressure in the hanging wall area is about 2–4 times that of its free field. In the clay, the damage to the arch bottom of the tunnel structure is more serious, and the strain change is about twice that of the vault. From the longitudinal length of the tunnel, the strain range at the bottom of the arch is about 2 times the length of the top of the arch.</p></list-item><list-item><p id=\"Par45\">Compared with the free field of sand overburden, the surface uneven deformation of the sand site with tunnel structure does not slow down as a whole. With the increase in bedrock dislocation, the downward deformation of the tunnel structure in the V-shaped extrusion shear rupture zone composed of bedrock dislocation dip angle and its reverse position is significant. The length of the influence range is about 5 times that of the free field, and the maximum inclination angle is about 1.8 times that of the free field. The internal earth pressure change in the hanging wall area is about 3–5 times that of the free field. For the ground main rupture zone region across the tunnel length direction, the peak acceleration of the tunnel structure is significantly changed to about 3–5 times of the free field, and the long-period effect of the response spectrum in this region is obvious, and the characteristic period can reach 1 s faster. In sandy soil, the damage to the arch bottom of the tunnel structure is more serious, and the strain change is about three times that of the vault. However, from the longitudinal length of the tunnel, the strain variation range at the top of the arch is about 2 times the length of the bottom of the arch.</p></list-item><list-item><p id=\"Par46\">Compared with the tunnel structure site of cohesive soil, the strain change of the tunnel structure in the sandy soil site is more significant, and the arch top is about 2 times and the arch bottom is about 6 times. In addition, the uplift range of the sand cover layer increases, the rupture zone migrates to the right, the lifting height of the soil is higher, and the uneven deformation is obvious. Therefore, the sandy soil site plays an \" adding seismic \" role in the cross-fault tunnel structure.</p></list-item><list-item><p id=\"Par47\">In the area spanning the ground fault rupture zone, the tunnel structure is affected by soil compression and shear failure. The main structure of the tunnel and the pipeline interface are vulnerable to serious threats. Therefore, it is necessary to take anti-rupture measures, such as over-excavation design, segmented flexible connection, damping joint, damping layer, etc^{##UREF##16##17##–##UREF##19##20##}.</p></list-item><list-item><p id=\"Par48\">This paper utilizes a substantial tunnel model test apparatus to carry out field testing on various soil cover layers over low-dip reverse faults. The experimental data offer valuable insights for the structural design and construction of tunnels across fault rupture.</p></list-item></list></p>" ]

[ "<p id=\"Par1\">The fortification system of the tunnel structure spanning the active fault, such as the failure mechanism and fault-resistant design (measures), has not been thoroughly established. In this study, the self-developed cross-fault large-scale bedrock dislocation loading device platform is utilized to carry out the model test of the tunnel structure and soil site of sand and cohesive soil when the low-angle reverse fault dislocation occurs, based on the earthquake damage. The results demonstrate that: (1) When the fault is staggered, the segmented flexible joint tunnel segment is more favorable in the cohesive soil site. (2) When compared to the cohesive soil tunnel structure site, the strain change of the tunnel structure in the sandy soil site is greater, with the vault increasing by roughly two times and the arch bottom increasing by nearly six times. After the tunnel is buried, the uplift range of the sand cover layer grows, revealing uneven deformation, and the rupture zone migrates to the footwall; hence, the sand site plays a “add seismic” role in the cross-fault tunnel structure. (3) Knowing the location and shape of the rupture range of the overburden soil caused by bedrock dislocation under different inclination angles and soil properties is required in the design in order to place the buried depth and segment length of the tunnel reasonably and take fault-resistant measures.</p>", "<title>Subject terms</title>" ]

[ "<title>Design of a large-scale box test apparatus and scheme for cross-fault rupture tunnel models</title>", "<title>Large-scale model box test apparatus and instrumentation arrangement</title>", "<p id=\"Par7\">The device adopts a self-developed large-scale experimental box platform, including a fault dislocation model platform (soil box, connecting device, base, actuator, and angle support); a bedrock fault dislocation input is simulated using an oil pressure loading device.</p>", "<p id=\"Par8\">The test chamber measures 4.96 m long, 1.85 m wide, 1.85 m high, and 1.4 m high. On the front and back of the chamber, a high-strength organic transparent glass with a thickness of 0.025 m is fitted, and three vertical rigid ribs are positioned on the exterior side. The chamber's side is made of 0.015-m-high-strength steel plate. The box's bottom is made up of two totally rigid steel plates, an L-shaped movable steel plate and a fixed steel plate, which are used to model the dislocation of the top and lower plates of bedrock beneath the overburden dirt.</p>", "<p id=\"Par9\">All the steel plates and the box are connected by a soft canvas around them, and the loading device is placed on the L-shaped movable steel plate. The right side of the soil box is provided with a strip-through hole for inserting the transverse partition board. The partition board is used to bear the gravity caused by the soil compaction in the upper part of the model to avoid damage to the tunnel model. After the soil is rammed, the partition board is drawn out to better simulate the surrounding rock condition. The loading device adjusts the loading direction through the diagonal brace support on the base to simulate the tunnel structure failure under the fault dislocation at different tilt angles (Figs. ##FIG##0##1##, ##FIG##1##2##).</p>", "<p id=\"Par10\">The advantage of this test device is that different test schemes can be formulated based on the tunnel's buried depth. By controlling the loading direction of the loading device, the failure of the tunnel structure under different dip angles and different fault dislocations of the normal and reverse faults is simulated in order to study the tunnel structure's failure mechanism under various working conditions.</p>", "<title>Pilot program design</title>", "<p id=\"Par11\">The investigation primarily conducted four iterations of low-dip reverse fault dislocation simulation tests (Table ##TAB##0##1##), including two in a free field and two in a field with a tunnel structure. In this experiment, the tunnel model in the field test of the tunnel structure was constructed by assembling four sections of models with the same standard. The construction process followed the conventional procedures typically used for building tunnels.</p>", "<p id=\"Par12\">The steps of each trial can be summarized as follows:</p>", "<p id=\"Par13\">Complete the overall installation of the model box, adjust the actuator angle to 45°, and fix it. After the inspection is completed, lay a thin film at the joint between the base plate and the box to prevent soil leakage during the backfilling process. Once everything is in place, start the backfilling process.</p>", "<p id=\"Par14\">Layer the soil and compact it. Fill 15 cm of soil each time and compact it to 10 cm. While compacting, bury sensors such as soil pressure gauges and accelerometers in the soil according to the layout of the testing instruments.</p>", "<p id=\"Par15\">After compacting to a depth of 100 cm (measured from the surface of the soil), install a top bar displacement meter on the surface of the soil, and place cameras and other equipment above and in front of the soil box.</p>", "<p id=\"Par16\">During the tunnel site testing, when compacting to a depth of 70 cm, according to the testing plan for the tunnel site, dig a hole in the soil with a depth of 40 cm and dimensions of 330 × 30 cm. Install strain gauges, displacement meters, and other sensors in the soil to set up the tunnel model. Lift the tunnel model and place it in the dug hole, connect the sections together, and secure them. Install partition boards and backfill the soil, compacting it to a depth of 100 cm.</p>", "<p id=\"Par17\">Control the actuator to jack 10 mm for each sub-stage to record the data once. Each test is conducted under 11 different working conditions. In each working condition, ensure that the data acquisition instrument collects data according to the requirements of the sensor, and make sure to film the test using a camera.</p>", "<title>Model-making</title>", "<p id=\"Par18\">Carry out a 1:30 similarity ratio modeling based on the developed multifunctional fault dislocation loading test device platform (Table ##TAB##1##2##).</p>", "<p id=\"Par19\">Two types of soil, clay and sand, were used for the tests. Determining the similarity of the soils in the scaled-down modeling tests was challenging. Therefore, typical natural cohesive and sandy soils obtained from the field were used as soil samples.</p>", "<p id=\"Par20\">The tunnel structural modeling materials include gypsum, cement, coarse sand, and fine sand. After conducting several proportioning tests based on a 1:30 similarity ratio, the appropriate ratio of these materials for the tunnel structure can be selected (Fig. ##FIG##2##3##). The coefficient of inhomogeneity, Cu, of the sandy soil samples was determined to be 2 through sieve testing and the particle grading curve (Fig. ##FIG##3##4##).</p>", "<p id=\"Par21\">The final selection of the cement: gypsum: fine sand: water ratio is 5:5:30:11. The similarity ratio between the reinforcing mesh used in the test and the actual reinforcement in the project is 1:30, and the reinforcing mesh used is a 25 × 25 mm galvanized iron wire mesh with a diameter of 1 mm. The length of each section of the model is 800 mm. Pour the tunnel model according to the specified ratios and ensure that it is adequately cured until it reaches its required strength (Fig. ##FIG##4##5##). Twist together the protruding main bars and fill them with silicone weather-resistant sealant until the desired strength is achieved.</p>", "<title>Sensor deployment programme</title>", "<p id=\"Par22\">The test data acquisition system is based on the system provided by Jiangsu Dong Hua Company. The primary sensors used in the system include the top-bar displacement gauge, earth pressure gauge, and accelerometer. Accelerometers and earth pressure gauges were installed at 30 cm, 55 cm, 90 cm, 100 cm, and the bottom plate of the soil body. Additionally, 11 top bar displacement gauges were positioned on the soil surface. The model strain gauges are primarily arranged in sections C1 to C13. Among them, 8 strain gauges are arranged in sections C9 to C11, 4 strain gauges are placed in sections C7 and C8, and the remaining sections have 2 strain gauges each, positioned at the model's arch top and arch waist. Based on the prediction that soil rupture is concentrated in the V-shaped rupture zone, four top-bar displacement gauges were arranged inside the segmental model interface on both sides. The specific arrangement of the sensor locations can be determined through a combination of pre-tests and analysis conducted by the subject group (Fig. ##FIG##5##6##, ##FIG##6##7##, ##FIG##7##8##).</p>", "<title>Comparative Analysis of Cross-Fault Tunnel Site Tests</title>", "<title>Comparative analysis of sites with a 45° inclination clay overburden on reverse faults and sites containing tunnels on them</title>", "<p id=\"Par23\">The clay site and its corresponding tunnel site tests, i.e., test numbers ① and ③.These included high-precision top-mounted displacement measurements for surface deformation, internal soil pressure monitoring using miniature pressure gauges, testing of active and passive discs for rock foundations, large dynamic acceleration measurement of the box-soil mass, and the tunnel model's strain test employing a small grid large range sensor. To this end, a systematic analysis of monitoring data was carried out. The specific analysis is as follows (Figs. ##FIG##8##9## and ##FIG##9##10##):</p>", "<p id=\"Par24\">The typical loading levels of 30 mm and 70 mm are chosen for analysis by comparing the front views and top views of the free clay site test ① and the containing tunnel site test ③. Based on the rupture traces, it is observed that the soil cracks in Test ① are more numerous and wider. When the loading amount reaches 30 mm, the soil on the east side of the front view bulges, and cracks appear on the surface of the soil on the west side. Additionally, as the loading amount increases, a crack emerges on the right side of the original crack at approximately 100 mm in the top view (Fig. ##FIG##10##11##a). During Test ③, when loaded to 30 mm, significant cracks appeared above the soil body on the west side in the front view. Additionally, two noticeable cracks appeared on the west side of the box in the top view, and there was substantial soil deformation (Fig. ##FIG##10##11##b).</p>", "<p id=\"Par25\">When loaded to 70 mm, two cracks emerged in the soil above the east side of both Test ① and Test ③. The cracks in Test ① were wider, while those in Test ③ were longer and developed at a larger angle. On closer observation from the top view, it was noticed that both Test ① and Test ③ exhibited two rupture zones. The widths of the rupture zones were similar, but in Test ①, there were more, wider, and more prominent tiny cracks on both sides of the main rupture zone. Furthermore, the uplifted area of the rupture zone in Test ③ was higher. In Test ①, cracks started to appear on the ground surface at a loading amount of 10 mm. However, in Test ③, cracks appeared on the ground surface only when the loading amount reached 20 mm. As a result, when compared to tests ① and ③, test ① damage earlier (Fig. ##FIG##10##11##c,d).</p>", "<p id=\"Par26\">In order to conduct a comparative analysis of the soil pressure on the east and west sides of each layer in Test ① and Test ③ (Fig. ##FIG##8##9##), it is observed that in Test ①, the soil pressure increases with the loading amount. Notably, at a depth of 300 mm, there is a significant change in the soil pressure. In Test ③, the soil pressure undergoes changes at depths of 300 mm and 550 mm, with the largest change occurring at 550 mm. In Test ①, significant changes in earth pressure are observed in the range of -300 ~ 300 mm from the center of the rupture zone at the position of 300 mm from the bedrock. However, the earth pressure undergoes little change at other positions. In Test ③, at a distance of 550 mm and 300 mm from the bedrock on the west side, the earth pressure exhibits significant changes with the increase in loading. On the east side, the earth pressure changes are not as pronounced with increasing loading. However, at a loading amount of 30 mm and a distance of 300 mm from the bedrock, both the 300 mm and 550 mm positions show higher values of earth pressure. Based on the analysis, it can be concluded that in Experiment ③, the tunnel structure is indeed under a significant threat at depths of 550 mm and 300 mm from the bedrock. Specifically, within the range of -550 mm to 550 mm and -300 mm to 300 mm from the center of the rupture zone, there is a serious risk to the integrity and stability of the tunnel structure. Based on the comparison between Test ① and Test ③, it is observed that the change in soil pressure at the center in Test ③ is greater than that in Test ①. This suggests that the area surrounding the tunnel structure, especially at the bedrock dislocation, needs to be strengthened in order to enhance the stability of the tunnel. Reinforcing the soil (surrounding rock) around the tunnel structure would be necessary in this case to ensure the safety and integrity of the tunnel under the increased soil pressure.</p>", "<p id=\"Par27\">Based on the comparative analysis of high-precision top bar displacement and surface deformation, the displacement difference between Test ① and Test ③ at a loading amount of 30 mm does not show significant changes. However, as the loading amount increases to 70 mm, the soil body in Test ③ exhibits a larger lifting height compared to Test ①. This shows that the overall uneven deformation in test ③ is relatively large (Fig. ##FIG##9##10##, Table ##TAB##2##3##).</p>", "<p id=\"Par28\">According to the test ③ comparison diagram of the overall structural damage of the tunnel, the tunnel structure 2/3 and 3/4 pipe interface damage is serious. The western tunnel has minimal damage, while the eastern tunnel is severely damaged. The range of inhomogeneous deformation in test ③ is larger than that in test ①, and the rupture zone region is wider, so the range of soils for the rupture-resistant design of shallow tunnel structures is larger than their free fields. At the same time, this can be analyzed in the structural damage map of the tunnel in Test ③ (Fig. ##FIG##11##12##).</p>", "<title>Comparative analysis of the reverse fault 45° dip sand overburden site and its tunnel-bearing site</title>", "<p id=\"Par29\">The sandy soil site and its corresponding tunnel site tests, i.e., test numbers ② and ④. These included high-precision top-mounted displacement measurements for surface deformation, internal soil pressure monitoring using miniature pressure gauges, testing of active and passive discs for rock foundations, large dynamic acceleration measurement of the box-soil mass, and the tunnel model's strain test employing a small grid large range sensor. To this end, a systematic analysis of monitoring data was carried out. The specific analysis is as follows:</p>", "<p id=\"Par30\">Compared with test ②, the rupture zone area of test ④ obviously moved to the east. The scope of the soil uplift containing the tunnel site increased, the rupture zone migrated to the east side, the height of the soil uplift was higher, and the inhomogeneous deformation was obvious, which had the effect of \"adding seismic\" to the tunnel structure (Fig. ##FIG##17##18##).</p>", "<p id=\"Par31\">Test ② When the loading amount was 30 mm, the soil body was lifted about 30 mm, but there was no soil rupture in the top view (Fig. ##FIG##17##18##a); when the loading amount of 70 mm, the sand soil was on the east side of the soil above a transverse crack. The top view found that the rupture was mainly concentrated on the east side, with the emergence of horizontal and vertical cracks and the soil bulging obvious (Fig. ##FIG##17##18##c). Test ④ When the loading amount was 30 mm the obvious soil bulged, top view found from the east side of the box 800 mm from the development of fine cracks to the middle position but did not penetrate the soil; near the edge of the box body, there is a crack that develops obliquely to the east side (Fig. ##FIG##17##18##b); loading to 70 mm did not appear obvious cracks; top view found that there are two cracks, the east side of the top of the crack is close to the edge of the box, and the east side of the bottom of the cracks away from the edge of the box of about 800 mm (Fig. ##FIG##17##18##d). Cracks appeared on the surface of test ② at a loading amount of 40 mm, and cracks appeared on the surface of test ④ at a loading amount of 30 mm. The surface of test ② was damaged before that of test ④.</p>", "<p id=\"Par32\">Comparative analysis of soil pressure in tests ② and ④(Fig. ##FIG##12##13##) shows that the soil pressure in test ② mainly changes in the middle of the soil body, with the most obvious change in the position of 300 mm and not much change in the other positions; test ④, after being put into the tunnel, shows that the change of soil pressure in the middle of the soil body is more significant; the variation of soil pressure on the west side of test ④ is greater than that on the east side; and the change in the position of 550 mm is more than that in the position of 300 mm, and the change in the position of the top of 900 mm basically does not have any change.</p>", "<p id=\"Par33\">In Test ②, the change in earth pressure is significant within the range of − 300 mm–300 mm from the center of the rupture zone at the 300 mm position (Fig. ##FIG##12##13##).</p>", "<p id=\"Par34\">In Test ④, the change in earth pressure is again evident within the range of − 300 mm–300 mm from the center of the rupture zone at the 300 mm position. Additionally, there is a noticeable change in earth pressure within the range of − 550 mm–550 mm from the center of the rupture zone at the 550 mm position. Thus, both of these ranges suggest potential serious damage to the tunnel structure (Fig. ##FIG##12##13##). Comparing the two tests, Test ④ demonstrates a greater change in earth pressure in the middle of the soil body compared to Test ②.</p>", "<p id=\"Par35\">Based on the analysis of the valid data measured by accelerometers, which were placed along the east and west sides of the boundary of the \"V\" range of influence of 300 mm, 550 mm, and 900 mm, it was found that there was no significant change in acceleration on the east and west sides of tests ② and ④. Consequently, the peak acceleration in the middle of the soil layer remained relatively unchanged in both tests ② and ④ (Fig. ##FIG##13##14##).</p>", "<p id=\"Par36\">The reaction spectrum is further analyzed. The results show that the platform value of the west side of the soil surface in both Test ② and Test ④ is larger, with a value of 2.0. The platform value of the bedrock active disk is 1.5 (Fig. ##FIG##14##15##).</p>", "<p id=\"Par37\">With an increase in the loading amount, the platform value of the west side of the bedrock active disk increases to 2.5 when the loading reaches 70 mm. However, there is no significant change in the platform value for the east side of the soil surface and the bedrock active disk as the loading amount increases (Figs. ##FIG##15##16##, ##FIG##16##17##).</p>", "<p id=\"Par38\">Therefore, it can be concluded that in the influence zone of the rupture, the tunnel outside this zone can be designed using the reaction spectrum theory in the conventional seismic design method, as there are no significant changes observed in the platform values for the east side of the soil surface and the bedrock active disk with the increase in the loading amount (Fig. ##FIG##17##18##).</p>", "<p id=\"Par39\">In Test ② and Test ④, the top bar displacement is indeed approximately the same when the loading amount is 30 mm. However, as the loading amount increases, Test ② shows an increased deformation difference in the rupture zone from the center position, ranging from -900 mm to 600 mm. Furthermore, in the region of 600 mm to 1200 mm, both tests exhibit significant and uneven deformation (Fig. ##FIG##14##15##, Table ##TAB##3##4##).</p>", "<p id=\"Par40\">Based on the comparative diagram of the overall structural damage of the tunnel in Test 4, it is evident that severe damage occurred at the interface of tubes 2/3 and 3/4 of the tunnel structure. Damage to the western tunnels is less severe, while damage to the eastern tunnels remains severe. The range of inhomogeneous deformation in Test ④ is larger than that in Test ②, and the rupture zone region is wider, so the range of soils designed for rupture resistance of shallow tunnel structures is larger than their free fields. Also, this can be analyzed in the damage map of the tunnel structure in test ④ (Fig. ##FIG##18##19##).</p>", "<p id=\"Par41\">The comparison of tests ③ and ④ (Figs. ##FIG##11##12## and ##FIG##18##19##), analyzed under the same inclination angle at the sandy soil site and the clay site, reveals a larger rupture range in the sandy soil site tunnel structure. At the tunnel interface, they are all severely damaged. Therefore, when the reverse fault is the same as the fault dip, the sandy soil site is more dangerous to the underground tunnel structure, and both sites need to strengthen the seismic design of the main tunnel structure and joints.</p>" ]

[ "<title>Acknowledgements</title>", "<p>The authors would like to express their profound gratitude to my supervisor, Jian Yi Zhang, for his invaluable direction and counsel during the study process. My supervisor gave me tolerance and support when I ran into problems, as well as insightful inquiries regarding the direction of my study. My academic and professional progress has benefited immensely from my supervisor's knowledge and instruction, which has been a major factor in this.</p>", "<p>The authors are appreciative of the financial assistance provided by Science for Earthquake Resilience (XH22021A) and National Natural Science Foundation of China (51608118), which allowed me to successfully finish my study. It assisted in acquiring crucial resources, including equipment, internships, and experimental materials, which greatly enhanced the development and outcomes of my study.</p>", "<title>Author contributions</title>", "<p>Z.J.Y Funding acquisition, Supervision, Writing—Review &amp; Editing, Conceptualization. S.Y.J Visualization, Writing—Original Draft, Data Curation, Formal analysis. L.Z.H Project administration, Resources, Data Curation, Methodology. Q.S.H and W.S Resources, Visualization.</p>", "<p>All authors reviewed the manuscript.</p>", "<title>Funding</title>", "<p>National Natural Science Foundation of China,51608118,Earthquake Science and Technology Spark Plan Project, XH22021A.</p>", "<title>Data availability</title>", "<p>The datasets used and/or analyzed during the current study available from the corresponding author on reasonable request.</p>", "<title>Competing interests</title>", "<p id=\"Par49\">The authors declare no competing interests.</p>" ]

[ "<fig id=\"Fig1\"><label>Figure 1</label><caption><p>The actual platform of a model test apparatus for cross-fault tunnels.</p></caption></fig>", "<fig id=\"Fig2\"><label>Figure 2</label><caption><p>Three-dimensional diagram of the platform for the cross-fault tunnel model test apparatus.</p></caption></fig>", "<fig id=\"Fig3\"><label>Figure 3</label><caption><p>Preparation model similar material diagram.</p></caption></fig>", "<fig id=\"Fig4\"><label>Figure 4</label><caption><p>Particle grading curves for sandy soils.</p></caption></fig>", "<fig id=\"Fig5\"><label>Figure 5</label><caption><p>Tunnel model making diagram.</p></caption></fig>", "<fig id=\"Fig6\"><label>Figure 6</label><caption><p>Schematic diagram for sensor placement in a sandy (clayey) soil site.</p></caption></fig>", "<fig id=\"Fig7\"><label>Figure 7</label><caption><p>Schematic diagram for sensor placement in a site with a tunnel.</p></caption></fig>", "<fig id=\"Fig8\"><label>Figure 8</label><caption><p>Schematic diagram illustrating the on-site placement of sensors.</p></caption></fig>", "<fig id=\"Fig9\"><label>Figure 9</label><caption><p>Comparison of earth pressure on the west side with that on the east side.</p></caption></fig>", "<fig id=\"Fig10\"><label>Figure 10</label><caption><p>Comparison of the uneven deformation of the soil and the corresponding position of the top bar displacement gauge on the soil surface.</p></caption></fig>", "<fig id=\"Fig11\"><label>Figure 11</label><caption><p>On-site test three-dimensional comparison diagram.</p></caption></fig>", "<fig id=\"Fig12\"><label>Figure 12</label><caption><p>Comparison of overall structural damage to the tunnel.</p></caption></fig>", "<fig id=\"Fig13\"><label>Figure 13</label><caption><p>Comparison of earth pressure on the west side with that on the east side.</p></caption></fig>", "<fig id=\"Fig14\"><label>Figure 14</label><caption><p>Comparison of peak acceleration on the east and west sides of the plot.</p></caption></fig>", "<fig id=\"Fig15\"><label>Figure 15</label><caption><p>Comparison of the uneven deformation of the soil and the corresponding position of the top bar displacement gauge on the soil surface.</p></caption></fig>", "<fig id=\"Fig16\"><label>Figure 16</label><caption><p>Comparison of the reaction spectra of the east side and west side with a loading volume of 30 mm.</p></caption></fig>", "<fig id=\"Fig17\"><label>Figure 17</label><caption><p>Comparison of the reaction spectra of the east side and west side with a loading volume of 70 mm.</p></caption></fig>", "<fig id=\"Fig18\"><label>Figure 18</label><caption><p>On-site test three-dimensional comparison diagram.</p></caption></fig>", "<fig id=\"Fig19\"><label>Figure 19</label><caption><p>Comparison of overall structural damage to the tunnel.</p></caption></fig>" ]

[ "<table-wrap id=\"Tab1\"><label>Table 1</label><caption><p>Model test working condition parameter table.</p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th align=\"left\"/><th align=\"left\">Test number</th><th align=\"left\">Soil cover thickness (mm)</th><th align=\"left\">Soil type</th><th align=\"left\">Individual load (mm)</th><th align=\"left\">Total loading (mm)</th></tr></thead><tbody><tr><td align=\"left\" rowspan=\"2\">Free-field</td><td align=\"left\">①</td><td align=\"left\">1000</td><td align=\"left\">Clays</td><td align=\"left\">10</td><td align=\"left\">110</td></tr><tr><td align=\"left\">②</td><td align=\"left\">1000</td><td align=\"left\">Sandy soil</td><td align=\"left\">10</td><td align=\"left\">110</td></tr><tr><td align=\"left\" rowspan=\"2\">Sites with tunnels</td><td align=\"left\">③</td><td align=\"left\">1000</td><td align=\"left\">Clays</td><td align=\"left\">10</td><td align=\"left\">110</td></tr><tr><td align=\"left\">④</td><td align=\"left\">1000</td><td align=\"left\">Sandy soil</td><td align=\"left\">10</td><td align=\"left\">110</td></tr></tbody></table></table-wrap>", "<table-wrap id=\"Tab2\"><label>Table 2</label><caption><p>Soil parameters and their similarity constants in the model.</p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th align=\"left\">Physical quantity</th><th align=\"left\"><italic>L:</italic> Length</th><th align=\"left\"><italic>ρ:</italic> Density</th><th align=\"left\"><italic>g:</italic> Gravity acceleration</th><th align=\"left\"><italic>τ:</italic> Soil pressure</th></tr></thead><tbody><tr><td align=\"left\">The constant of similarity of soil</td><td align=\"left\"></td><td align=\"left\"></td><td align=\"left\"></td><td align=\"left\"></td></tr><tr><td align=\"left\">The similarity constant of a tunnel</td><td align=\"left\"></td><td align=\"left\"></td><td align=\"left\"></td><td align=\"left\">–</td></tr></tbody></table><table frame=\"hsides\" rules=\"groups\"><thead><tr><th align=\"left\"><italic>DL:</italic> Rod displacement</th><th align=\"left\"><italic>σ:</italic> Stress</th><th align=\"left\"><italic>ε:</italic> Strain</th><th align=\"left\"><italic>a:</italic> Tunnel acceleration</th><th align=\"left\"><italic>t:</italic> Acceleration time</th></tr></thead><tbody><tr><td align=\"left\" rowspan=\"2\">-</td><td align=\"left\">–</td><td align=\"left\">–</td><td align=\"left\">–</td><td align=\"left\">–</td></tr><tr><td align=\"left\"></td><td align=\"left\"></td><td align=\"left\"></td><td align=\"left\"></td></tr></tbody></table></table-wrap>", "<table-wrap id=\"Tab3\"><label>Table 3</label><caption><p>Comparison and analysis table of uneven deformation in experiments.</p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th align=\"left\"/><th align=\"left\">Range (mm)</th><th align=\"left\">Slope</th><th align=\"left\">Inclination angle (°)</th></tr></thead><tbody><tr><td align=\"left\" rowspan=\"3\">Experiment ① Slope and inclination angle of inhomogeneous deformation</td><td align=\"left\">− 300 ~ 300</td><td char=\".\" align=\"char\">0.006</td><td char=\".\" align=\"char\">0.337</td></tr><tr><td align=\"left\">300 ~ 900</td><td char=\".\" align=\"char\">0.034</td><td char=\".\" align=\"char\">1.922</td></tr><tr><td align=\"left\">900 ~ 1200</td><td char=\".\" align=\"char\">0.022</td><td char=\".\" align=\"char\">1.241</td></tr><tr><td align=\"left\" rowspan=\"3\">Experiment ③ Slope and inclination angle of inhomogeneous deformation</td><td align=\"left\">− 600 ~ -300</td><td char=\".\" align=\"char\">0.024</td><td char=\".\" align=\"char\">1.38</td></tr><tr><td align=\"left\">− 300 ~ 0</td><td char=\".\" align=\"char\">0.018</td><td char=\".\" align=\"char\">1.015</td></tr><tr><td align=\"left\">600 ~ 900</td><td char=\".\" align=\"char\">0.033</td><td char=\".\" align=\"char\">1.88</td></tr></tbody></table></table-wrap>", "<table-wrap id=\"Tab4\"><label>Table 4</label><caption><p>Comparison and analysis table of uneven deformation in experiments.</p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th align=\"left\"/><th align=\"left\">Range (mm)</th><th align=\"left\">Slope</th><th align=\"left\">Inclination angle (°)</th></tr></thead><tbody><tr><td align=\"left\" rowspan=\"3\">Experiment ② Slope and inclination angle of inhomogeneous deformation</td><td align=\"left\">− 900 ~ -300</td><td char=\".\" align=\"char\">0.027</td><td char=\".\" align=\"char\">1.535</td></tr><tr><td align=\"left\">− 300 ~ 600</td><td char=\".\" align=\"char\">0.008</td><td char=\".\" align=\"char\">0.460</td></tr><tr><td align=\"left\">600 ~ 1200</td><td char=\".\" align=\"char\">0.035</td><td char=\".\" align=\"char\">1.981</td></tr><tr><td align=\"left\" rowspan=\"3\">Experiment ④ Slope and inclination angle of inhomogeneous deformation</td><td align=\"left\">0 ~ 300</td><td char=\".\" align=\"char\">0.016</td><td char=\".\" align=\"char\">0.898</td></tr><tr><td align=\"left\">300 ~ 900</td><td char=\".\" align=\"char\">0.023</td><td char=\".\" align=\"char\">1.311</td></tr><tr><td align=\"left\">900 ~ 1200</td><td char=\".\" align=\"char\">0.065</td><td char=\".\" align=\"char\">3.730</td></tr></tbody></table></table-wrap>" ]

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\n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${C}_{\\rho }=1$$\\end{document}</tex-math><mml:math id=\"M4\"><mml:mrow><mml:msub><mml:mi>C</mml:mi><mml:mi>ρ</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq3\"><alternatives><tex-math id=\"M5\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${C}_{g}=1$$\\end{document}</tex-math><mml:math id=\"M6\"><mml:mrow><mml:msub><mml:mi>C</mml:mi><mml:mi>g</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq4\"><alternatives><tex-math id=\"M7\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${C}_{\\uptau }={C}_{L}\\cdot {C}_{\\rho }\\cdot {C}_{g}=30$$\\end{document}</tex-math><mml:math id=\"M8\"><mml:mrow><mml:msub><mml:mi>C</mml:mi><mml:mi mathvariant=\"normal\">τ</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:msub><mml:mi>C</mml:mi><mml:mi>L</mml:mi></mml:msub><mml:mo>·</mml:mo><mml:msub><mml:mi>C</mml:mi><mml:mi>ρ</mml:mi></mml:msub><mml:mo>·</mml:mo><mml:msub><mml:mi>C</mml:mi><mml:mi>g</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mn>30</mml:mn></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq5\"><alternatives><tex-math id=\"M9\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${C}_{L}=30$$\\end{document}</tex-math><mml:math id=\"M10\"><mml:mrow><mml:msub><mml:mi>C</mml:mi><mml:mi>L</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mn>30</mml:mn></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq6\"><alternatives><tex-math id=\"M11\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${C}_{\\rho }=1.25$$\\end{document}</tex-math><mml:math 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\n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${D}_{L}=30$$\\end{document}</tex-math><mml:math id=\"M16\"><mml:mrow><mml:msub><mml:mi>D</mml:mi><mml:mi>L</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mn>30</mml:mn></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq9\"><alternatives><tex-math id=\"M17\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${C}_{\\sigma }={C}_{L}\\cdot {C}_{\\rho }\\cdot {C}_{g}=37.5$$\\end{document}</tex-math><mml:math id=\"M18\"><mml:mrow><mml:msub><mml:mi>C</mml:mi><mml:mi>σ</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:msub><mml:mi>C</mml:mi><mml:mi>L</mml:mi></mml:msub><mml:mo>·</mml:mo><mml:msub><mml:mi>C</mml:mi><mml:mi>ρ</mml:mi></mml:msub><mml:mo>·</mml:mo><mml:msub><mml:mi>C</mml:mi><mml:mi>g</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mn>37.5</mml:mn></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq10\"><alternatives><tex-math id=\"M19\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${C}_{\\varepsilon }=1$$\\end{document}</tex-math><mml:math id=\"M20\"><mml:mrow><mml:msub><mml:mi>C</mml:mi><mml:mi>ε</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq11\"><alternatives><tex-math id=\"M21\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${C}_{a}=1$$\\end{document}</tex-math><mml:math id=\"M22\"><mml:mrow><mml:msub><mml:mi>C</mml:mi><mml:mi>a</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq12\"><alternatives><tex-math id=\"M23\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${C}_{t}=\\sqrt{{C}_{L}/{C}_{g}}=5.48$$\\end{document}</tex-math><mml:math id=\"M24\"><mml:mrow><mml:msub><mml:mi>C</mml:mi><mml:mi>t</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:msqrt><mml:mrow><mml:msub><mml:mi>C</mml:mi><mml:mi>L</mml:mi></mml:msub><mml:mo stretchy=\"false\">/</mml:mo><mml:msub><mml:mi>C</mml:mi><mml:mi>g</mml:mi></mml:msub></mml:mrow></mml:msqrt><mml:mo>=</mml:mo><mml:mn>5.48</mml:mn></mml:mrow></mml:math></alternatives></inline-formula>" ]

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[ "<title>Introduction</title>", "<p id=\"Par2\">Neurological illnesses are characterized by the progressive and permanent loss of neurons in particular brain areas^{##REF##34716723##1##}. Many neurodegenerative disorders are the result of several causes, most likely involving a variety of mechanistic pathways^{##REF##35682615##2##}. Monoamine oxidase-B (MAO-B) represents such a pathway and may play a significant role in neurological disorders such as Alzheimer's disease (AD) and Parkinson's disease (PD)^{##REF##34780749##3##}. The mitochondrial outer membranes of neurons, glia, and other mammalian cells are closely related to the C-terminal transmembrane polypeptide components of MAOs, also known as flavin adenine dinucleotide (FAD)-carrying enzymes^{##REF##36037788##4##,##REF##29335210##5##}. For synaptic connections to function appropriately, xenobiotic and biogenic amine oxidation must be stimulated^{##REF##34915314##6##}. The three-dimensional shapes of the MAO isoforms MAO-A and MAO-B share 70% identical amino acid residues. With only a change of six amino acids between the 16 active-site residues of the two MAOs, their active-site geometries are also similar^{##REF##34806105##7##–##REF##17573034##10##}. The exact location of MAO isoforms in the brain is not yet fully elucidated. In contrast to studies that employed cell cultures and suggested MAO-A localization in glial cells, tests in both primate and non-primate species demonstrated that the glial enzyme is primarily present as type B in the intact brain^{##REF##27803666##11##,##REF##36321487##12##}. MAO-B catalyzes phenyl ethylamine and phenyl methylamine disintegration, whereas MAO-A catalyzes noradrenaline, adrenaline, and serotonin deamination. MAO-A and MAO-B also metabolize dopamine, tryptamine, and tyramine. While MAO-B inhibition increases dopamine levels in the Parkinsonian brain, partially depletes dopaminergic neurons in the substantia nigra pars compacta, and has anti-Parkinsonian effects, selective MAO-A inhibition increases neurotransmitter levels in central nervous system (CNS) noradrenergic and 5-hydroxytryptaminergic neurons^{##REF##34957948##13##–##REF##26164425##15##}.</p>", "<p id=\"Par3\">The mechanism-based inhibitors, selegiline and rasagiline (both MAO-B inhibitors) and clorgyline (a MAO-A inhibitor), are among the isoform-specific inhibitors described. Both MAO isoforms are inhibited by pargyline, a different propargylamine molecule. Other notable irreversible MAO inhibitors include the nonspecific inhibitors phenelzine and tranylcypromine^{##REF##35447344##16##,##UREF##0##17##}. The MAO-A inhibitors, toloxatone and moclobemide, and the MAO-B inhibitor, safinamide, are well-known examples of isoform-specific reversible inhibitors^{##REF##28456030##18##–##REF##29324067##20##}. Dry mouth, nausea, diarrhea, constipation, drowsiness, sleeplessness, dizziness, and light-headedness are the most frequently reported side effects of the current medications used for treatment. When using a patch, skin irritation may also develop at the patch site. The search for novel MAO-A and MAO-B inhibitors has extensively used a variety of heterocycle families as scaffolds, including pyrazolines, chromones, chalcones, xanthines, benzyloxy, thiazoles, coumarins, and their precursors, isatin congeners, thiazolidiniones, and betacarboline^{##UREF##1##21##–##REF##35601305##26##}. As a result, isatin was identified as an effective MAO inhibitor.</p>", "<p id=\"Par4\">Isatin (Fig. ##FIG##0##1##) is an endogenous small molecule with an indole-containing moiety and exhibits a broad range of biological and pharmacological activities. It comprises a nitrogen atom at position 1 and two carbonyl groups at positions 2 and 3. It further contains two rings: a six-membered aromatic ring and a five-membered antiaromatic ring^{##UREF##3##27##,##REF##30654239##28##}. It is widely distributed in the body fluids and different tissues of mammals and occurs naturally in plants^{##REF##33302835##29##}. In addition to clinical studies on the anticancer medications Toceranib, Semaxinib, and Orantinib, the Food and Drug Administration (FDA) has approved isatin-based therapies, such as Sunitinib (anti-tumor) and Nintedanib (anti-tumor)^{##REF##35631362##30##–##REF##35337070##32##} (Fig. ##FIG##0##1##).</p>", "<p id=\"Par5\">Isatin reversibly inhibits human MAO-A and MAO-B, with K_i values of 15 and 3 µM, respectively^{##REF##29336068##33##}. According to previous studies, isatin is located close to the FAD cofactor in the MAO-B substrate cavity. The entrance cavity of the enzyme is free as isatin binds to its substrate cavity^{##REF##30792039##34##,##REF##21134756##35##}. We hypothesized that the C-3 position could be exploited with hydrophobic moieties to improve MAO efficacy^{##REF##35601305##26##}. Therefore, we selected the C-3 position and replaced it with an acyl hydrazone linker and a halogenated phenyl (hydrophobic) moiety. The structural cores of acyl hydrazones, which include two distinctly connected nitrogen atoms, are generally responsible for the physical and chemical properties of these compounds. Therefore, acyl hydrazones are frequently used to develop novel molecules with various functions. Hydrazone derivatives have also been linked to MAO inhibition. Recently, Vishnu et al. synthesized piperonylic hydrazone-based isatin derivatives, however, insignificant interactions were observed in 3,4-methylenedioxy groups with MAO-B binding pocket^{##REF##31177620##36##}. Therefore, we replaced piperonylic with phenyl moiety and designed new approach toward acylhydrazone-based isatin derivatives (Fig. ##FIG##1##2##) to get a new family of effective MAO inhibitors in this study.</p>" ]

[ "<title>Materials and methods</title>", "<title>Chemicals</title>", "<p id=\"Par6\">For synthesis, isatin derivatives, hydrazine hydrate, and benzoic acid were purchased from Sigma-Aldrich (St. Louis, MO, USA) and TCI Chemical (Toshima, Tokyo, Japan). Substrates (benzylamine and kynuramine) and reference inhibitors (clorgyline, lazabemide, pargyline, and toloxatone) as well as recombinant human MAO-A and MAO-B were purchased from Sigma-Aldrich. Reversibility test was performed by using Dialyzer DiaEasy™ (6–8 kDa, BioVision, St. Grove, MA, USA).</p>", "<title>Synthesis</title>", "<p id=\"Par7\">Benzoic acid (1 eq.) and hydrazine hydrate (2.5 eq.) were combined, and the reaction was conducted by using a microwave synthesizer (Monowave 50 Synthesizer, Anton-Paar, Graz, Austria) at 200 °C for 10–20 min. Upon completion of the reaction, the product benzohydrazide was recrystallized from methanol. Then, the mixture of isatin or substituted isatin (0.001 mol) and benzohydrazide (0.001 mol)^{##UREF##5##37##,##REF##21561768##38##} in methanol, by adding a catalytic amount of acetic acid, was placed in a reaction vial and subjected to the microwave synthesizer at 100–120 °C for 5–10 min. The reaction progress was monitored by using thin-layer chromatography (TLC) with an eluent of ethyl acetate and hexane (50:50). Cold ethanol was used to wash the reaction mixture upon completion, and the resulting product was dried to obtain acylhydrazone-based isatin derivatives (76–96% yield). The synthetic scheme of the isatin derivatives is illustrated in Scheme ##FIG##2##1##.</p>", "<title>MAO-A and MAO-B inhibition studies</title>", "<p id=\"Par8\">MAO-A and MAO-B activities were determined using 0.06 mM kynuramine and 0.3 mM benzylamine, respectively^{##REF##27575476##39##}. In preliminary kinetic study, the K_m values of MAO-A and MAO-B were about 0.039 and 0.20 mM, respectively. Concentrations of the substrates used were around 2.0 times of their K_m values for the enzyme assay. The absorbance was measured using the continuous assay method described previously^{##REF##32087226##40##}. Compound inhibitions were compared to those of the reference inhibitors of MAO-A (toloxatone and clorgyline) and MAO-B (lazabemide and pargyline).</p>", "<title>Enzyme kinetics</title>", "<p id=\"Par9\">The compounds IC₅₀ values were calculated using the GraphPad Prism software 5^{##REF##35908673##41##}. The selectivity index (SI) values of the compounds were calculated as follows: (IC₅₀ of MAO-A)/(IC₅₀ of MAO-B)^{##REF##30396116##42##}. The type of enzyme inhibition was determined at five different substrate concentrations (0.0075–0.12 μM of MAO-A and 0.0375–0.6 μM of MAO-B). The inhibitor was used at three concentrations (approximately 0.5, 1.0, and 1.5) times its IC₅₀ value^{##REF##35908673##41##}. Enzyme inhibition patterns and K_i values were determined by comparing Lineweaver–Burk (LB) plots and their secondary plots, respectively^{##REF##32087226##40##}.</p>", "<title>Reversibility studies</title>", "<p id=\"Par10\">The lead compounds reversibility for MAO-A and MAO-B inhibition was evaluated by comparing the undialyzed and dialyzed values at a concentration of 1.5 × the IC₅₀ value after incubation for 30 min prior to measurement, as previously described^{##REF##27575476##39##,##REF##28109809##43##}. The restored activities of the compounds were compared to those of the reference compounds toloxatone and clorgyline (reversible and irreversible inhibitors, respectively) of MAO-A, and lazabemide and pargyline (reversible and irreversible inhibitors, respectively) of MAO-B. Reversibility patterns were determined by comparing the activities of undialyzed (A_U) and dialyzed (A_D) compounds^{##REF##35908673##41##,##REF##28109809##43##}.</p>", "<title>Parallel artificial membrane permeability assay (PAMPA)</title>", "<p id=\"Par11\">The blood–brain barrier (BBB) permeation abilities of the four lead molecules were analyzed using the PAMPA method^{##REF##36575270##44##,##REF##12667689##45##}. The detailed procedure is explained in the Supplementary Data.</p>", "<title>Cytotoxicity, reactive oxygen species (ROS), anti-oxidant, and anti-inflammatory characteristics of the cell line-based assay</title>", "<title>Cell culture and treatments</title>", "<p id=\"Par12\">The National Center for Cell Science (NCCS), Pune, India, supplied the SH-SY5Y-human bone marrow neuroblastoma cell line, which was maintained in accordance with the recommended protocol in DMEM-high-glucose medium (Cat No. AL111, Himedia) supplemented with 10% fetal bovine serum (FBS) and a 1% antibiotic–antimycotic solution at 37 °C in a CO₂ incubator. Subcultures were performed every two days. Briefly, 5 × 10⁵ cells/mL were cultured on a plate and incubated for 24 h to promote cell attachment and reach the required cell density. Neuroinflammation was induced in the SH-SY5Y cells with 1 ug/mL for 2 h followed by cell treatment with different concentrations of the test molecules (<bold>IS6</bold>, <bold>IS7</bold>, <bold>IS13</bold>, and <bold>IS15</bold>) and incubation for 24 h. LPS-treated cells served as positive controls, and untreated cells served as controls.</p>", "<title>3-(4,5-Dimethylthiazol-2-yl)-2,5-diphenyltetrazolium bromide (MTT) assay</title>", "<p id=\"Par13\">The cytotoxicity of synthetic compounds <bold>IS6, IS7,</bold> and <bold>IS13</bold> was assessed using the common MTT test using SHSY-5Y cells. The cells were grown in 96-well plates with 20,000 cells per well. Different compound concentrations were applied to the cells, and the treated cells were then left to incubate for 24 h. After applying 50 µL of MTT (0.5 mg/mL) to each well, the plates were incubated for 3 h, then 100 µL of DMSO was added to dissolve the purple formazan crystal, and an absorbance was measured at 570 nm using a microplate reader (ELX-800, BioTek, CA, USA). Compounds' growth inhibitory concentration (IC₅₀) values were computed.</p>", "<title>Superoxide dismutase (SOD) activity</title>", "<p id=\"Par14\">Using a superoxide dismutase assay kit (KrishGen Biosystems, India), which measures the concentration of formazan crystals using a colorimetric assay, SOD activity was determined. In this test, a tetrazolium salt was used to detect the superoxide radicals produced by xanthine oxidase and hypoxanthine. After treatment, the medium was removed, centrifuged at 2000 rpm for 5 min at room temperature, and placed on ice. A diluted radical detector and lysed cell supernatant or a standard were applied to each well of a 96-well plate to evaluate SOD activity using an ELISA kit, according to the manufacturer’s instructions. The absorbance of the wells was then determined after 5 min at a wavelength of 450 nm using a microplate reader (Safire2, Tecan Group Ltd., Maennedorf, Switzerland). The results were displayed as ng/mL^{##REF##11499385##46##–##UREF##7##49##}.</p>", "<title>Glutathione (GSH) activity</title>", "<p id=\"Par15\">A glutathione assay kit (KrishGen Biosystems) was used to determine the GSH levels. The assay kit was based on the disulfide dimer-oxidized GSH reductase recycling method for 5,5'-dithiobis-2-(nitrobenzoic acid) (DTNB). After treatment, adherent cells were removed by scraping the media from the wells. Following suspension in 50 mM phosphate solution (0.5 mL) with a pH of 6.5 and 1 mM ethylenediaminetetraacetic acid, the cells were chilled. The lysed cell supernatant was used to test GSH levels using an ELISA kit. The absorbance of the yellow product was determined at a wavelength of 450 nm. The total GSH activity was estimated using a GSH standard curve. The results were obtained as ng/mL^{##REF##11499385##46##–##UREF##7##49##}.</p>", "<title>Glutathione peroxidase (GPx) activity</title>", "<p id=\"Par16\">A GPx assay kit (KrishGen Biosystems) was used to evaluate the GPx activity. The kit uses a colorimetric assay to determine the quantity of GPx. Glutathione reductase (GR) mediates GPx activity Oxidized glutathione (GSSG) is produced via hydroperoxide reduction by GPx. This glutathione is recycled back to its reduced state by GR and NADPH. The NADPH to NADP + oxidation was accompanied by a decrease in absorbance at 450 nm (A450). When GPx activity was rate-limiting, the rate of decline in A450 was directly correlated with GPx activity. Following treatment, adherent cells were removed from the wells, suspended in cold PBS, sonicated, and frozen. In accordance with the ELISA kit’s instructions, the lysed cell supernatant, or standard, was applied to all 96 wells of a plate together with a diluted radical detector to assay the activity of GPx. A microplate reader was used to measure the absorbance of the wells after 5 min^{##REF##11499385##46##–##UREF##7##49##}. Results were obtained in ng/mL.</p>", "<title>Catalase (CAT) activity</title>", "<p id=\"Par17\">CAT activity was determined according to Aebi^{##REF##6727660##50##}. A human CAT ELISA kit was purchased commercially (KrishGen Biosystems). Following treatment, the adherent cells were scraped off, suspended in cold PBS, sonicated, and placed on ice. The medium was then removed from each well. The 3 mL CAT assay combination comprised of extract (0.05 mL), phosphate buffer (1.5 mL, 100 mM buffer, pH 7.0), H₂O₂ (0.5 mL), and distilled water (0.95 mL). The absorbance decreased at 450 nm. CAT activity was reported in terms of ng/mL of H₂O₂ oxidized per min per gram^{##REF##11499385##46##–##UREF##7##49##}.</p>", "<title>ROS assays</title>", "<p id=\"Par18\">The OxiSelect Intracellular ROS Assay Kit (Cell Biolabs Inc., San Diego, CA, USA) was used to quantify the levels of the fluorescent probe 20,70-dichlorodihydrofluorescin diacetate (DCFH-DA). In a microplate reader, fluorescence was measured using excitation and emission filters at wavelengths of 488 and 535 nm, respectively^{##UREF##8##51##}.</p>", "<title>IL-6, TNF-α, and NF-kB expression</title>", "<p id=\"Par19\">TNF-α, IL-6, and NF-kB expression in cell lysates was assessed using the respective antibodies (PerCP-Cy5.5, PE, and p65–FITC) according to the manufacturer’s protocol. Briefly, the spent medium was aspirated, and the cells were treated with LPS (1 µg/mL) for 2 h. Then, the required concentrations of experimental compounds and controls were added and incubated for 24 h. The cells were harvested into polystyrene tubes and centrifuged at 25 °C, washed with PBS, and 70% cold ethanol was added drop wise to create a cell pellet while vortexing. The mixture was then incubated at − 20 °C. The cells were pelleted at a high speed, washed twice with PBS, antibodies added (10 µL), mixed thoroughly, and incubated for 30 min in the dark at 20–25 °C. PBS (500 µL) was added and mixed thoroughly, and the reaction was analyzed using BD FACS—Cell Quest pro software^{##REF##11499385##46##–##UREF##7##49##,##REF##31782098##52##}.</p>", "<title>Statistical analysis</title>", "<p id=\"Par20\">Statistical significance was determined by one-way ANOVA followed by Dunnett-t test using Graph Pad Prism Version 8.0.2.</p>", "<title>Computational studies</title>", "<title>Molecular docking</title>", "<p id=\"Par21\">The Schrödinger suite^{##UREF##9##53##} was used to perform the molecular docking investigation of <bold>IS3</bold>, <bold>IS6</bold>, <bold>IS7</bold> and <bold>IS13</bold>, and <bold>IS15</bold>. The human MAO-A (hMAO-A, 2Z5Z) and MAO-B (hMAO-B, 2V5Z) X-ray solved structure was obtained from the Protein Data Bank^{##REF##12913124##9##,##REF##17915852##54##}. The both crystal structures were improved and optimized using the protein preparation wizard included in the Schrödinger suite, which performed energy minimization, hydrogen atom addition, protonation-state correction, and protonation-state addition. The LigPrep tool was used to construct the ligand structures. The co-crystallized ligands served as the automated center of the grid box. For docking simulations, the force Field OPLS4 default settings and extra precision (XP) docking protocol default settings were used^{##REF##27159243##55##,##REF##31606547##56##}.</p>", "<title>Molecular dynamic simulation</title>", "<p id=\"Par22\">Schrödinger LLC’s Desmond simulation program was used to run the molecular dynamics (MD) simulations^{##REF##31782098##52##}. The protein–ligand combination was initially created for the Desmond system builder panel using compound <bold>IS7</bold> against MAO-B in an aqueous solvent system. For complete protein–ligand simulations and stability trajectory analysis (RMSD, RMSF, and protein–ligand contact), the simulation parameters were 100 ns at 300 K, 1.01325 bar pressure, and 1000 frames^{##REF##27159243##55##,##REF##31606547##56##}.</p>", "<title>MM-GBSA</title>", "<p id=\"Par23\">The Generalized Born and Surface Area (MM-GBSA) solvation technique in Molecular Mechanics was utilized to compute the free binding energies of the ligands to the proteins. In this case, we used several postures from MD simulations of the docked complex to evaluate macromolecular stability and protein–ligand binding affinity. The free energy was calculated using the following formula at the post-processing stage that comes after the MD studies.</p>", "<p id=\"Par24\">The contributions of the internal, electrostatic, and van der Waals energies to molecular mechanics are denoted by the symbols E_int, E_ele, and E_vdw, respectively. Within the equation, the free energy contributions of the polar and non-polar solvation systems are denoted by G_pol and G_np, respectively. S is an estimate of the entropy, and T is the absolute temperature. The following formula was used to estimate the binding free energy, or ΔG Bind, between the ligand and the protein.</p>", "<p id=\"Par25\">The protein, ligand, and protein–ligand complex are denoted by the letters P, L, and PL, respectively. The equation above expressed the free energy for each of these entities. We used the solvated systems that we obtained prior to performing MD calculations to determine free binding energies. In this case, solvent molecules more than 5 Å away from the bound ligand were replaced with an implicit model via the GB approach, in their post-processing stages^{##REF##23988151##57##,##UREF##10##58##}.</p>" ]

[ "<title>Results and discussion</title>", "<title>Synthesis</title>", "<p id=\"Par26\">The target molecules were synthesized in two steps. In the first step, an intermediate <italic>acylhydrazide</italic> molecule was synthesized by reacting benzoic acid with hydrazine hydrate. This intermediate was then reacted with isatin and halogenated substituted isatins to obtain the final molecules (substituted acylhydrazone-based isatin derivatives: (<bold>IS1</bold>–<bold>IS16</bold>) via an acid-catalyzed nucleophilic addition reaction. All the procedures were performed using the microwave reactor. The structures of all synthesized compounds were confirmed by ¹H and ¹³C nuclear magnetic resonance ((Bruker Advance Neo 400 MHz NMR spectrometer). The de-shielded protons in all compounds were NH atoms from isatin, and the hydrazone linker exhibited ranges of 11.5–11.0 δ and 12.50–14.0 δ, respectively. Sharp de-shielded Sp² carbonyl carbons of the isatin and hydrazone linkers were observed at 163.60 δ and 141.10 δ, respectively (Supporting Information Figs. ##SUPPL##0##S1##–##SUPPL##0##S48##).</p>", "<title>MAO-A and MAO-B inhibition studies</title>", "<p id=\"Par27\">Of the 16 compounds, <bold>IS7</bold> most potently inhibited MAO-B with an IC₅₀ value of 0.082 μM, followed by <bold>IS13</bold> (IC₅₀ = 0.104 μM) (Table ##TAB##0##1##, Fig. ##SUPPL##0##S52##). Compounds <bold>IS7</bold> and <bold>IS6</bold> (<italic>para</italic>-Br and –Cl in the B-ring, respectively) showed higher MAO-B inhibition than the basic compound <bold>IS5</bold> (–H in B-ring, IC₅₀ = 4.136 μM), i.e., –Br &gt; –Cl &gt; –H &gt; –F in order. In contrast, MAO-B inhibition decreased in the order of <italic>meta</italic>-position substitution in the A-ring, that is, <bold>IS5</bold> (–H) &gt; <bold>IS13</bold> (–Cl) &gt; <bold>IS1</bold> (–Br) &gt; <bold>IS9</bold> (–F), suggesting that the <italic>meta</italic>-F substituent of the A-ring contributed to a decrease in MAO-B inhibition.</p>", "<p id=\"Par28\">These IC₅₀ values were lower than those of the aldoxime- and hydroxy-functionalized chalcones <bold>ACE7</bold> and <bold>HC6</bold> (IC₅₀ = 0.012 and 0.0046 μM, respectively)^{##UREF##11##59##}, but higher than those of the dimethoxy-halogenated chalcone <bold>DM2</bold> (IC₅₀ = 0.067 μM)^{##REF##36145373##60##}. In contrast, compound <bold>IS15</bold> most inhibited MAO-A with an IC₅₀ value of 1.852 μM, followed by <bold>IS3</bold> (IC₅₀ = 2.385 μM). These values are more efficient than those of the halogenated pyrazoline <bold>EH8</bold> (IC₅₀ = 4.31 μM). Compound <bold>IS6</bold> had the highest selectivity index (SI) value (263.8); however, compounds <bold>IS7</bold> and <bold>IS13</bold> showed similar SI values (SI = 233.85 and 212.57, respectively) and high MAO-B inhibition. These SI values indicated that compounds <bold>IS6</bold>, <bold>IS7,</bold> and <bold>IS13</bold> are selective MAO-B inhibitors (Table ##TAB##0##1##).</p>", "<p id=\"Par29\">Structurally, compound <bold>IS7</bold> (–Br in the B-ring) showed higher MAO-B inhibition than <bold>IS6</bold> (–Cl in the B-ring), and both compounds showed 50.4 × and 33.4 × , higher inhibition than the basic compound <bold>IS5</bold> (–H in the B-ring), respectively. In the subseries, MAO-B inhibition increased in the following order: Br &gt; Cl &gt; H &gt; F at the <italic>para</italic>-position of the B-ring. In contrast, in the sub-series containing –Br in the A-ring, <bold>IS2</bold> (–Cl in the B-ring, IC₅₀ = 0.269 μM) showed a higher MAO-B inhibition than the sub-parental compound <bold>IS1</bold> (–H in B-ring, IC₅₀ = 0.420 μM), and the inhibition increased with the substituents of –Cl &gt; –H &gt; –Br &gt; –F at <italic>para</italic>-position in the B-ring in order). In the other sub-series containing –F in the A ring, <bold>IS10</bold> (–Cl in B-ring, IC₅₀ = 3.995 μM) showed higher MAO-B inhibition than the sub-parental compound <bold>IS9</bold> (–H in B), and MAO-B inhibition increased with the substituents of –Cl &gt; –F &gt; –H &gt; –Br at <italic>para</italic>-position in B-ring in order. In the sub-series containing –Cl in the A-ring, <bold>IS13</bold> (–H in the B-ring, IC₅₀ = 0.104 μM) showed the highest MAO-B inhibition, which increased with the substituents of –H &gt; –Cl &gt; –Br &gt; –F at <italic>para</italic>-position in the B-ring. In comparing substituents in A ring, MAO-B inhibition increased in order by –Cl (<bold>IS13</bold>, IC₅₀ = 0.104 μM) &gt; –Br (<bold>IS1</bold>, IC₅₀ = 0.420 μM) &gt; –H (<bold>IS5</bold>, 4.136 μM) &gt; –F (<bold>IS9</bold>, 9.094 μM), and by –H (<bold>IS7</bold>, 0.082 μM) &gt; –Cl (<bold>IS15</bold>, 0.337 μM) &gt; –Br (<bold>IS3</bold>, 0.514 μM) &gt; –F (<bold>IS11</bold>, 10.586 μM). Overall, most compounds with F substituents showed low MAO-B inhibition (Table ##TAB##0##1##, Fig. ##FIG##3##3##). <bold>IS7</bold>, <bold>IS6</bold>, and <bold>IS13</bold> were more selective (SI = 233.85, 263.80, and 212.57, respectively) towards MAO-B. The lead molecules (<bold>IS7</bold>, <bold>IS6</bold>, and <bold>IS13</bold>) were comparable to lazabemide and pargyline.</p>", "<p id=\"Par30\">In MAO-A inhibition, compound <bold>IS15</bold> (–Cl in the A-ring and –Br in the B-ring) was the highest (IC₅₀ = 1.852 μM) (Table ##TAB##0##1##, Fig. ##SUPPL##0##S53##) and showed 11.94-times higher MAO-A inhibition than <bold>IS13</bold> (–Cl in the A-ring and –H in the B-ring), and 10.35-times higher than <bold>IS7</bold> (–H in the A-ring and –Br in the B-ring). This indicates that the –Cl substituent in the A-ring contributed to an increase in MAO-A inhibition (Table ##TAB##0##1##, Fig. ##FIG##3##3##). These results suggest that compounds <bold>IS6</bold>, <bold>IS7</bold>, and <bold>IS13</bold> are potent selective MAO-B inhibitors and that compound <bold>IS15</bold> is a selective MAO-A inhibitor.</p>", "<title>Reversibility studies</title>", "<p id=\"Par31\">Reversibility tests were performed using the dialysis method. In this study, the concentration of compound <bold>IS15</bold> used for MAO-A was 1.5 × that of the IC₅₀ (3.00 μM), and those of compounds <bold>IS6</bold>, <bold>IS7</bold>, and <bold>IS13</bold> used for MAO-B were 1.5 × that of the IC₅₀ (0.18, 0.12, and 0.15 μM, respectively). Recovery patterns were compared using undialyzed (A_U) and dialyzed (A_D) relative activity after 30 min of pre-incubation. For MAO-A inhibition, compound <bold>IS15</bold> recovered from 47.16 to 78.73% (Fig. ##FIG##4##4##). The recovery of the compound was similar to that of toloxatone (from 33.76 to 87.22%), and it could be distinguished from clorgyline (from 32.32 to 39.23%). For MAO-B inhibition, compounds <bold>IS6</bold>, <bold>IS7,</bold> and <bold>IS13</bold> recovered from 42.81 to 79.52%, 28.65–72.89%, and 31.45–80.12%, respectively (Fig. ##FIG##5##5##). The recovery values of the compounds were similar to those of lazabemide (from 41.48 to 77.71%) and could be distinguished from those of pargyline (from 41.04 to 34.34%). These results indicate that <bold>IS15</bold> is a reversible inhibitor of MAO-A, whereas <bold>IS6</bold>, <bold>IS7</bold>, and <bold>IS15</bold> are reversible inhibitors of MAO-B.</p>", "<title>Enzyme kinetics</title>", "<p id=\"Par32\">The enzyme kinetics and inhibition types were analyzed at five substrate concentrations and three inhibitor concentrations. In the LB plot, <bold>IS15</bold> showed was a competitive MAO-A inhibitor (Fig. ##FIG##6##6##A), and the secondary plot revealed that the K_i value was 1.004 ± 0.171 μM (Fig. ##FIG##6##6##B). In contrast, <bold>IS6</bold>, <bold>IS7</bold>, and <bold>IS13</bold> LB plots indicated competitive MAO-B inhibitors (Fig. ##FIG##7##7##A, C, and E), and the secondary plots showed that their K_i values were 0.068 ± 0.022, 0.044 ± 0.002, and 0.061 ± 0.001 μM, respectively (Fig. ##FIG##7##7##B, D, and F). The K_i value of the inhibitor was calculated by the secondary plot constructed with each slope vs. inhibitor concentration in LB plot. The minus value of X-axis of the plot means − K_i. Though <bold>IS6</bold> and <bold>IS7</bold> were not exactly intercepted on one point of Y-axis, V_max values in the presence of the inhibitors were almost same within the experimental error range, indicating both also were competitive inhibitors. In the presence of the inhibitors <bold>IS6</bold>, <bold>IS7</bold>, <bold>IS13</bold>, and <bold>IS15</bold>, K_m values were increased and V_max values were the same as the control. These results suggest that <bold>IS15</bold> is a competitive MAO-A inhibitor, whereas <bold>IS6</bold>, <bold>IS7</bold>, and <bold>IS13</bold> are competitive MAO-B inhibitors.</p>", "<title>PAMPA assay</title>", "<p id=\"Par33\">The PAMPA assay demonstrated that isatin-based hydrazone derivatives (<bold>IS6</bold>, <bold>IS7</bold>, <bold>IS13</bold>, and <bold>IS15</bold>) had high permeability and CNS bioavailability, with <italic>P</italic>e values of &gt; 4.00 × 10^–6 cm/s (Table ##TAB##1##2##). Brain penetration is crucial for the efficient administration of CNS medication^{##REF##32315731##61##}. The effective permeability of the chemical and the equation were used to calculate the penetration rate (Log Pe). A compound is categorized as potentially permeable (CNS+), if its P<italic>e</italic> value is &gt; 4.00 × 10^–6 cm/s, and perhaps non-BBB permeable (CNS-), if &lt; 2.00 × 10^–6 cm/s. This study showed that while halogenated isatin has BBB permeability, the substitution of the phenyl ring results in greater penetration. Chloro substitution resulted in higher BBB permeability, as revealed in this study.</p>", "<title>MTT, ROS, anti-oxidant, and anti-inflammatory characteristics of the cell line-based assay</title>", "<p id=\"Par34\">Cytotoxic activity of <bold>IS6</bold>, <bold>IS7</bold>, and <bold>IS13</bold> was tested by MTT assay using the human neuroblastoma SH-SY5Y cells. The cytotoxicity level was assessed by how well the live cells converted the tetrazolium dye into formazan crystals, with the untreated cells serving as the control group. <bold>IS6</bold>, <bold>IS7</bold>, and I<bold>S13</bold> remarkably decreased cell viability at 100 μM. However, IC₅₀ values of <bold>IS6</bold>, <bold>IS7</bold>, and <bold>1S13</bold> were 75.72, 97.15, and 85.30 μM, respectively, indicating that these were cell- proliferative and non-cytotoxic in nature or working low concentration to SH-SY5Y cells (Fig. ##FIG##8##8##).</p>", "<p id=\"Par35\">The preliminary in vitro neuroprotective activity against LPS-induced inflammatory events was tested using the synthesized <bold>IS6</bold>, <bold>IS7</bold>, <bold>IS13</bold>, and <bold>IS15</bold> in SH-SY5Y neuroblastoma cell lines. Cell viability assays and ELISA measurements of the intracellular pro-inflammatory cytokines TNF-alpha, IL-6, and NF-kB were used to evaluate neuroprotective activity. A common technique is to incubate SH-SY5Y cell lines with LPS (10 ng/mL) in minimum necessary medium at 37 °C for 24 h to induce neuroinflammation. We previously discovered that a 2 h incubation period with 1 ug/ml of LPS is adequate to trigger inflammatory responses.</p>", "<p id=\"Par36\">LPS treatment significantly raised the levels of IL-6, TNF-alpha, and NF-kB in LPS-intoxicated SH-SY5Y cell lines compared to control SH-SY5Y cell lines, demonstrating the magnitude of inflammatory reactions mediated by LPS toxicity (Fig. ##FIG##9##9##). <bold>IS6</bold>, <bold>IS7</bold>, <bold>IS13</bold>, and <bold>IS15</bold> pretreatment significantly (<italic>p</italic> &lt; 0.0001) reduced TNF-alpha (Fig. ##FIG##9##9##A), IL-6 (Fig. ##FIG##9##9##B), and NF-kB (Fig. ##FIG##9##9##C) levels in comparison to the LPS-treated group, demonstrating the anti-inflammatory potential of isatin derivatives (Figs. ##SUPPL##0##S49##, ##SUPPL##0##S50##). Intriguingly, in LPS-treated cell lines, <bold>IS6</bold> and <bold>IS7</bold> significantly decreased the levels of TNF-alpha and IL-6 compared to <bold>IS13</bold> and <bold>IS15</bold>. When compared to <bold>IS6</bold> and <bold>IS7</bold>, <bold>IS15</bold> and <bold>IS13</bold> have significantly lower NF-kB expression. The lead compounds confirmed the anti-inflammatory potential of all the compounds in the human neuroblastoma model by inhibiting LPS-induced pro-inflammatory cytokine expression (IL-6, TNF-alpha, and NF-kB).</p>", "<p id=\"Par37\">To further confirm the neuroprotective and anti-oxidant effects of the lead compounds, the effect of the lead compounds on decrease of ROS production using the LPS-treated SH-SY5Y cells (Fig. ##FIG##10##10##). The compounds <bold>IS6</bold>, <bold>IS7</bold>, <bold>IS13,</bold> and <bold>IS15</bold> significantly inhibited (<italic>p</italic> &lt; 0.0001) 2′,7′-dichlorofluorescin (DCF) expression in the LPS-induced model compared to cells treated with LPS alone. The maximum concentration of test compounds that significantly inhibited LPS inflammation was considered in the ROS study. The lead compounds, <bold>IS15</bold> and <bold>IS7</bold>, at 10 µM/mL concentrations, exhibited an effective DCF intensity decrease compared to the LPS-induced cells, whereas <bold>IS6</bold> and <bold>IS13</bold>, at 10 µM concentrations, exhibited moderate DCF intensity suppression (Fig. ##SUPPL##0##S51##). LPS alone induced 65% of DCF expression. The cellular anti-oxidant assay results suggested that the lead compounds showed significant neuroprotective activity by enhancing the cellular oxidant enzymes CAT, GPx, GSH, and SOD (Figs. ##FIG##11##11##, ##FIG##12##12##, ##FIG##13##13## and ##FIG##14##14##) in the LPS-induced model, while cells treated with LPS alone effectively expressed SOD, CAT, GSH, and GPx activities. The obtained values confirmed the promising neuroprotective and neuro-inflammatory activity of <bold>IS6</bold>, <bold>IS7</bold>, <bold>IS13,</bold> and <bold>IS15</bold> compounds (at 10 µM/mL) (<italic>p</italic> &lt; 0.0001) in relation to the neuroprotective effect in the LPS-induced human neuroblastoma model by enhancing the enzyme activity and inhibiting the oxidative stress-induced apoptosis caused by LPS by determining DCF intensity.</p>", "<title>Molecular docking</title>", "<p id=\"Par38\">Molecular docking studies were performed to better understand the binding processes of lead compounds. Interactions occurred between the molecules <bold>IS3</bold>, <bold>IS6</bold>, <bold>IS7</bold>, <bold>IS13</bold>, and <bold>IS15</bold>, with MAO-B (2V5Z) and MAO-A (2Z5X). Native ligands were used to confirm docking^{##REF##23988151##57##}. The compounds were mentioned in Table ##TAB##2##3##, docking scores (XP mode) ranges were − 6.55 to − 10.85 kcal/mol. The scores of the best lead molecules <bold>IS6</bold>, <bold>IS7</bold>, and <bold>IS13</bold>, through biological evaluation, were − 9.47, − 9.88, and − 9.72 kcal/mol, respectively, while safinamide had a score that was comparable (− 10.85 kcal/mol). The docking scores showed similar affinity like biological activity (IC₅₀). When looking into the orientations, the fluorobenzyl group of safinamide was placed towards the opening of the cavity, whereas the amide side chain pointed in the direction of the FAD molecule. A similar orientation was present in the compounds <bold>IS3</bold>, <bold>IS6</bold>, <bold>IS7</bold>, <bold>IS13</bold>, <bold>IS15</bold>, and isatin (Fig. ##FIG##15##15##A), where the phenyl group was positioned towards the cavity opening and the variable isatin moiety was oriented towards FAD. The entrance and substrate cavities of the MAO-B-binding pocket (Fig. ##FIG##15##15##B) were completely occupied with all inhibitors. The best lead molecule (<bold>IS7</bold>) interacted with Tyr60, Pro102, Pro104, Trp119, Phe168, Leu171, Cys172, Tyr326, and Tyr435 were primarily hydrophobic, whereas Gln206 was in polar contact. The isatin moiety demonstrated that Pi-Pi stacking with Tyr398 provided the <bold>IS7</bold>-MAO-B protein with complex stability.</p>", "<p id=\"Par39\">For MAO-A, the docking score range is − 3.96 to − 8.00 kcal/mol (Table ##TAB##2##3##). The <bold>IS15</bold> and Harmine docking scores (XP mode) were − 8.00 and − 6.04 kcal/mol, respectively. Comparing the docking score to each compound’s IC₅₀, <bold>IS15</bold> had the best profile, followed by <bold>IS3</bold> (− 7.65 kcal/mol), <bold>IS13</bold> (− 4.93 kcal/mol), <bold>IS7</bold> (− 4.32 kcal/mol), and <bold>IS6</bold> (− 3.96 kcal/mol). Every molecule was oriented in the same way as its native ligand, and its isatin moiety was always closed to the substrate cavity (FAD) (Fig. ##FIG##16##16##A). Thorough analysis of compound <bold>IS15</bold> in the MAO-A active site revealed that it was present at the following positions: Leu97, Ala111, Ile180, Asn181, Tyr197, Ile207, Phe208, Ser209, Val210, Gln215, Cys323, Ile325, Thr326, Ile335, Leu337, Tyr407, and Tyr444 (Fig. ##FIG##16##16##B). Through hydrophobic contacts, the phenyl ring of <bold>IS15</bold> interacted with Leu97, Ala111, Val210, Ile325, and Cys323. Residues Ile180, Asn181, Tyr197, Tyr407, and Tyr444 interacted with the isatin ring through hydrophobic and polar interactions. Its similar interaction with <bold>IS15</bold> and harmine^{##REF##36844523##62##} during the binding contact demonstrated its potential to inhibit MAO-A.</p>", "<title>Molecular dynamic simulation</title>", "<p id=\"Par40\">Desmond MD simulations were used to follow the binding mode of <bold>IS7</bold> in the inhibitor-binding cavity (IBC) of MAO-B. Protein C-alpha and ligand were tracked within an acceptable range for a long simulation duration (100 ns) according to root mean square deviation (RMSD) analysis. In contrast to the protein RMSD, the ligand (red) RMSD remained steady after 25 ns. The protein RMSD ranged between 1.2 and 3.6 Å with an average of 2.54 ± 0.01 Å (Fig. ##FIG##17##17##A). The protein-specific RMSD for the simulation was constant, with the exception of a slight variation, reaching a maximum of 3.6 Å at 68–70 ns, where after it stabilized. The simulation evaluated the flexibility of the protein system by computing the RMSF for each amino acid residue of the protein. The 480–498 residues of MAO-B showed a larger fluctuation. The atoms in the benzoyl ring of the RMSF ligand (Fig. ##FIG##17##17##B) showed slight fluctuations during the binding process. The 21 amino acid residues that interacted with the ligand were Tyr60 (0.541 Å),Gly101 (1.13 Å), Pro102(1.117 Å), Pro104 (0.981 Å), Trp119 (1.15 Å), Leu167 (0.955 Å), Phe168 (0.892 Å), Leu171 (0.638 Å), Cys172 (0.706 Å), Ile198 (0.689 Å), Ile199 (0.833 Å), Ser200 (0.88 Å), Thr201 (0.94 Å), Gln206 (0.618 Å), Ile316 (0.604 Å), Tyr326 (0.544 Å), Leu328 (0.572 Å), Met341 (0.493 Å), Phe343 (0.642 Å), Tyr398 (0.97 Å), and Tyr435 (0.497 Å). Hydrogen bonds, hydrophobic contacts, and water bridges were identified in the interaction histograms of <bold>IS7</bold> and MAO-B (Fig. ##FIG##17##17##C and D). Over a trajectory of 100 ns, the number of individual interactions between the amino acids and ligand was normalized. Several significant amino acids, including Tyr326 (hydrogen bond, water bridge, and hydrophobic), Tyr398 (hydrogen bond), Leu171 (hydrophobic), Cys172 (hydrogen bond and water bridge), and Ile199 (water bridge and hydrophobic), interact with <bold>IS7</bold>. The measured fraction of interactions with Tyr326 was &gt; 1.0. As previously observed^{##UREF##12##63##,##REF##26189013##64##}, the hydrophobic interaction of Tyr326 at the active site of MAO-B was significant. Figure ##FIG##17##17##C and ##FIG##17##D## depict hydrogen bonding, water bridges, and hydrophobic stability in the ligand–protein complexes. Cys172 forms an 86% hydrogen bond with the carbonyl and NH atoms in the linker between the isatin and benzoyl rings. Tyr398 contributed 49% via hydrogen bonding with the NH atom of the isatin ring. With carbonyl and water molecules, Tyr326 is a 33% active participant in hydrogen bonding. Overall, it is estimated from the trajectory analysis and full MD simulation that the lead compound <bold>IS7</bold> will inhibit MAO-B.</p>", "<title>MM-GBSA</title>", "<p id=\"Par41\">From their MD simulation frames, the free binding energy was estimated for the best molecule <bold>IS7</bold> with the highest docking energy and activity value prediction. Total average energies of ΔG Bind, ΔG Bind H-bond, ΔG Bind Lipo, and ΔG Bind vdW was − 190.04, − 12.26, − 45.94, − 140.23 for 0–100 ns MD snapshot, respectively. Across all interactions, the ΔG Bind vdW and ΔG Bind Lipo energies exerted the most significant impact on the average binding energy (Table ##TAB##3##4##).</p>", "<p id=\"Par42\">The values ΔG Bind vdW for the interactions of <bold>IS7</bold> with protein complexes indicated the presence of stable van der Waals interaction with amino acid residues. Consequently, the MM-GBSA calculations, derived from MD simulation trajectories, aligned well with the binding energies computed from the docking results. The molecule exhibited very low free binding energy, indicating its higher binding affinity towards the receptor. Consequently, it can be inferred that <bold>IS7</bold> compound exhibited a strong affinity for the MAO-B protein.</p>" ]

[ "<title>Results and discussion</title>", "<title>Synthesis</title>", "<p id=\"Par26\">The target molecules were synthesized in two steps. In the first step, an intermediate <italic>acylhydrazide</italic> molecule was synthesized by reacting benzoic acid with hydrazine hydrate. This intermediate was then reacted with isatin and halogenated substituted isatins to obtain the final molecules (substituted acylhydrazone-based isatin derivatives: (<bold>IS1</bold>–<bold>IS16</bold>) via an acid-catalyzed nucleophilic addition reaction. All the procedures were performed using the microwave reactor. The structures of all synthesized compounds were confirmed by ¹H and ¹³C nuclear magnetic resonance ((Bruker Advance Neo 400 MHz NMR spectrometer). The de-shielded protons in all compounds were NH atoms from isatin, and the hydrazone linker exhibited ranges of 11.5–11.0 δ and 12.50–14.0 δ, respectively. Sharp de-shielded Sp² carbonyl carbons of the isatin and hydrazone linkers were observed at 163.60 δ and 141.10 δ, respectively (Supporting Information Figs. ##SUPPL##0##S1##–##SUPPL##0##S48##).</p>", "<title>MAO-A and MAO-B inhibition studies</title>", "<p id=\"Par27\">Of the 16 compounds, <bold>IS7</bold> most potently inhibited MAO-B with an IC₅₀ value of 0.082 μM, followed by <bold>IS13</bold> (IC₅₀ = 0.104 μM) (Table ##TAB##0##1##, Fig. ##SUPPL##0##S52##). Compounds <bold>IS7</bold> and <bold>IS6</bold> (<italic>para</italic>-Br and –Cl in the B-ring, respectively) showed higher MAO-B inhibition than the basic compound <bold>IS5</bold> (–H in B-ring, IC₅₀ = 4.136 μM), i.e., –Br &gt; –Cl &gt; –H &gt; –F in order. In contrast, MAO-B inhibition decreased in the order of <italic>meta</italic>-position substitution in the A-ring, that is, <bold>IS5</bold> (–H) &gt; <bold>IS13</bold> (–Cl) &gt; <bold>IS1</bold> (–Br) &gt; <bold>IS9</bold> (–F), suggesting that the <italic>meta</italic>-F substituent of the A-ring contributed to a decrease in MAO-B inhibition.</p>", "<p id=\"Par28\">These IC₅₀ values were lower than those of the aldoxime- and hydroxy-functionalized chalcones <bold>ACE7</bold> and <bold>HC6</bold> (IC₅₀ = 0.012 and 0.0046 μM, respectively)^{##UREF##11##59##}, but higher than those of the dimethoxy-halogenated chalcone <bold>DM2</bold> (IC₅₀ = 0.067 μM)^{##REF##36145373##60##}. In contrast, compound <bold>IS15</bold> most inhibited MAO-A with an IC₅₀ value of 1.852 μM, followed by <bold>IS3</bold> (IC₅₀ = 2.385 μM). These values are more efficient than those of the halogenated pyrazoline <bold>EH8</bold> (IC₅₀ = 4.31 μM). Compound <bold>IS6</bold> had the highest selectivity index (SI) value (263.8); however, compounds <bold>IS7</bold> and <bold>IS13</bold> showed similar SI values (SI = 233.85 and 212.57, respectively) and high MAO-B inhibition. These SI values indicated that compounds <bold>IS6</bold>, <bold>IS7,</bold> and <bold>IS13</bold> are selective MAO-B inhibitors (Table ##TAB##0##1##).</p>", "<p id=\"Par29\">Structurally, compound <bold>IS7</bold> (–Br in the B-ring) showed higher MAO-B inhibition than <bold>IS6</bold> (–Cl in the B-ring), and both compounds showed 50.4 × and 33.4 × , higher inhibition than the basic compound <bold>IS5</bold> (–H in the B-ring), respectively. In the subseries, MAO-B inhibition increased in the following order: Br &gt; Cl &gt; H &gt; F at the <italic>para</italic>-position of the B-ring. In contrast, in the sub-series containing –Br in the A-ring, <bold>IS2</bold> (–Cl in the B-ring, IC₅₀ = 0.269 μM) showed a higher MAO-B inhibition than the sub-parental compound <bold>IS1</bold> (–H in B-ring, IC₅₀ = 0.420 μM), and the inhibition increased with the substituents of –Cl &gt; –H &gt; –Br &gt; –F at <italic>para</italic>-position in the B-ring in order). In the other sub-series containing –F in the A ring, <bold>IS10</bold> (–Cl in B-ring, IC₅₀ = 3.995 μM) showed higher MAO-B inhibition than the sub-parental compound <bold>IS9</bold> (–H in B), and MAO-B inhibition increased with the substituents of –Cl &gt; –F &gt; –H &gt; –Br at <italic>para</italic>-position in B-ring in order. In the sub-series containing –Cl in the A-ring, <bold>IS13</bold> (–H in the B-ring, IC₅₀ = 0.104 μM) showed the highest MAO-B inhibition, which increased with the substituents of –H &gt; –Cl &gt; –Br &gt; –F at <italic>para</italic>-position in the B-ring. In comparing substituents in A ring, MAO-B inhibition increased in order by –Cl (<bold>IS13</bold>, IC₅₀ = 0.104 μM) &gt; –Br (<bold>IS1</bold>, IC₅₀ = 0.420 μM) &gt; –H (<bold>IS5</bold>, 4.136 μM) &gt; –F (<bold>IS9</bold>, 9.094 μM), and by –H (<bold>IS7</bold>, 0.082 μM) &gt; –Cl (<bold>IS15</bold>, 0.337 μM) &gt; –Br (<bold>IS3</bold>, 0.514 μM) &gt; –F (<bold>IS11</bold>, 10.586 μM). Overall, most compounds with F substituents showed low MAO-B inhibition (Table ##TAB##0##1##, Fig. ##FIG##3##3##). <bold>IS7</bold>, <bold>IS6</bold>, and <bold>IS13</bold> were more selective (SI = 233.85, 263.80, and 212.57, respectively) towards MAO-B. The lead molecules (<bold>IS7</bold>, <bold>IS6</bold>, and <bold>IS13</bold>) were comparable to lazabemide and pargyline.</p>", "<p id=\"Par30\">In MAO-A inhibition, compound <bold>IS15</bold> (–Cl in the A-ring and –Br in the B-ring) was the highest (IC₅₀ = 1.852 μM) (Table ##TAB##0##1##, Fig. ##SUPPL##0##S53##) and showed 11.94-times higher MAO-A inhibition than <bold>IS13</bold> (–Cl in the A-ring and –H in the B-ring), and 10.35-times higher than <bold>IS7</bold> (–H in the A-ring and –Br in the B-ring). This indicates that the –Cl substituent in the A-ring contributed to an increase in MAO-A inhibition (Table ##TAB##0##1##, Fig. ##FIG##3##3##). These results suggest that compounds <bold>IS6</bold>, <bold>IS7</bold>, and <bold>IS13</bold> are potent selective MAO-B inhibitors and that compound <bold>IS15</bold> is a selective MAO-A inhibitor.</p>", "<title>Reversibility studies</title>", "<p id=\"Par31\">Reversibility tests were performed using the dialysis method. In this study, the concentration of compound <bold>IS15</bold> used for MAO-A was 1.5 × that of the IC₅₀ (3.00 μM), and those of compounds <bold>IS6</bold>, <bold>IS7</bold>, and <bold>IS13</bold> used for MAO-B were 1.5 × that of the IC₅₀ (0.18, 0.12, and 0.15 μM, respectively). Recovery patterns were compared using undialyzed (A_U) and dialyzed (A_D) relative activity after 30 min of pre-incubation. For MAO-A inhibition, compound <bold>IS15</bold> recovered from 47.16 to 78.73% (Fig. ##FIG##4##4##). The recovery of the compound was similar to that of toloxatone (from 33.76 to 87.22%), and it could be distinguished from clorgyline (from 32.32 to 39.23%). For MAO-B inhibition, compounds <bold>IS6</bold>, <bold>IS7,</bold> and <bold>IS13</bold> recovered from 42.81 to 79.52%, 28.65–72.89%, and 31.45–80.12%, respectively (Fig. ##FIG##5##5##). The recovery values of the compounds were similar to those of lazabemide (from 41.48 to 77.71%) and could be distinguished from those of pargyline (from 41.04 to 34.34%). These results indicate that <bold>IS15</bold> is a reversible inhibitor of MAO-A, whereas <bold>IS6</bold>, <bold>IS7</bold>, and <bold>IS15</bold> are reversible inhibitors of MAO-B.</p>", "<title>Enzyme kinetics</title>", "<p id=\"Par32\">The enzyme kinetics and inhibition types were analyzed at five substrate concentrations and three inhibitor concentrations. In the LB plot, <bold>IS15</bold> showed was a competitive MAO-A inhibitor (Fig. ##FIG##6##6##A), and the secondary plot revealed that the K_i value was 1.004 ± 0.171 μM (Fig. ##FIG##6##6##B). In contrast, <bold>IS6</bold>, <bold>IS7</bold>, and <bold>IS13</bold> LB plots indicated competitive MAO-B inhibitors (Fig. ##FIG##7##7##A, C, and E), and the secondary plots showed that their K_i values were 0.068 ± 0.022, 0.044 ± 0.002, and 0.061 ± 0.001 μM, respectively (Fig. ##FIG##7##7##B, D, and F). The K_i value of the inhibitor was calculated by the secondary plot constructed with each slope vs. inhibitor concentration in LB plot. The minus value of X-axis of the plot means − K_i. Though <bold>IS6</bold> and <bold>IS7</bold> were not exactly intercepted on one point of Y-axis, V_max values in the presence of the inhibitors were almost same within the experimental error range, indicating both also were competitive inhibitors. In the presence of the inhibitors <bold>IS6</bold>, <bold>IS7</bold>, <bold>IS13</bold>, and <bold>IS15</bold>, K_m values were increased and V_max values were the same as the control. These results suggest that <bold>IS15</bold> is a competitive MAO-A inhibitor, whereas <bold>IS6</bold>, <bold>IS7</bold>, and <bold>IS13</bold> are competitive MAO-B inhibitors.</p>", "<title>PAMPA assay</title>", "<p id=\"Par33\">The PAMPA assay demonstrated that isatin-based hydrazone derivatives (<bold>IS6</bold>, <bold>IS7</bold>, <bold>IS13</bold>, and <bold>IS15</bold>) had high permeability and CNS bioavailability, with <italic>P</italic>e values of &gt; 4.00 × 10^–6 cm/s (Table ##TAB##1##2##). Brain penetration is crucial for the efficient administration of CNS medication^{##REF##32315731##61##}. The effective permeability of the chemical and the equation were used to calculate the penetration rate (Log Pe). A compound is categorized as potentially permeable (CNS+), if its P<italic>e</italic> value is &gt; 4.00 × 10^–6 cm/s, and perhaps non-BBB permeable (CNS-), if &lt; 2.00 × 10^–6 cm/s. This study showed that while halogenated isatin has BBB permeability, the substitution of the phenyl ring results in greater penetration. Chloro substitution resulted in higher BBB permeability, as revealed in this study.</p>", "<title>MTT, ROS, anti-oxidant, and anti-inflammatory characteristics of the cell line-based assay</title>", "<p id=\"Par34\">Cytotoxic activity of <bold>IS6</bold>, <bold>IS7</bold>, and <bold>IS13</bold> was tested by MTT assay using the human neuroblastoma SH-SY5Y cells. The cytotoxicity level was assessed by how well the live cells converted the tetrazolium dye into formazan crystals, with the untreated cells serving as the control group. <bold>IS6</bold>, <bold>IS7</bold>, and I<bold>S13</bold> remarkably decreased cell viability at 100 μM. However, IC₅₀ values of <bold>IS6</bold>, <bold>IS7</bold>, and <bold>1S13</bold> were 75.72, 97.15, and 85.30 μM, respectively, indicating that these were cell- proliferative and non-cytotoxic in nature or working low concentration to SH-SY5Y cells (Fig. ##FIG##8##8##).</p>", "<p id=\"Par35\">The preliminary in vitro neuroprotective activity against LPS-induced inflammatory events was tested using the synthesized <bold>IS6</bold>, <bold>IS7</bold>, <bold>IS13</bold>, and <bold>IS15</bold> in SH-SY5Y neuroblastoma cell lines. Cell viability assays and ELISA measurements of the intracellular pro-inflammatory cytokines TNF-alpha, IL-6, and NF-kB were used to evaluate neuroprotective activity. A common technique is to incubate SH-SY5Y cell lines with LPS (10 ng/mL) in minimum necessary medium at 37 °C for 24 h to induce neuroinflammation. We previously discovered that a 2 h incubation period with 1 ug/ml of LPS is adequate to trigger inflammatory responses.</p>", "<p id=\"Par36\">LPS treatment significantly raised the levels of IL-6, TNF-alpha, and NF-kB in LPS-intoxicated SH-SY5Y cell lines compared to control SH-SY5Y cell lines, demonstrating the magnitude of inflammatory reactions mediated by LPS toxicity (Fig. ##FIG##9##9##). <bold>IS6</bold>, <bold>IS7</bold>, <bold>IS13</bold>, and <bold>IS15</bold> pretreatment significantly (<italic>p</italic> &lt; 0.0001) reduced TNF-alpha (Fig. ##FIG##9##9##A), IL-6 (Fig. ##FIG##9##9##B), and NF-kB (Fig. ##FIG##9##9##C) levels in comparison to the LPS-treated group, demonstrating the anti-inflammatory potential of isatin derivatives (Figs. ##SUPPL##0##S49##, ##SUPPL##0##S50##). Intriguingly, in LPS-treated cell lines, <bold>IS6</bold> and <bold>IS7</bold> significantly decreased the levels of TNF-alpha and IL-6 compared to <bold>IS13</bold> and <bold>IS15</bold>. When compared to <bold>IS6</bold> and <bold>IS7</bold>, <bold>IS15</bold> and <bold>IS13</bold> have significantly lower NF-kB expression. The lead compounds confirmed the anti-inflammatory potential of all the compounds in the human neuroblastoma model by inhibiting LPS-induced pro-inflammatory cytokine expression (IL-6, TNF-alpha, and NF-kB).</p>", "<p id=\"Par37\">To further confirm the neuroprotective and anti-oxidant effects of the lead compounds, the effect of the lead compounds on decrease of ROS production using the LPS-treated SH-SY5Y cells (Fig. ##FIG##10##10##). The compounds <bold>IS6</bold>, <bold>IS7</bold>, <bold>IS13,</bold> and <bold>IS15</bold> significantly inhibited (<italic>p</italic> &lt; 0.0001) 2′,7′-dichlorofluorescin (DCF) expression in the LPS-induced model compared to cells treated with LPS alone. The maximum concentration of test compounds that significantly inhibited LPS inflammation was considered in the ROS study. The lead compounds, <bold>IS15</bold> and <bold>IS7</bold>, at 10 µM/mL concentrations, exhibited an effective DCF intensity decrease compared to the LPS-induced cells, whereas <bold>IS6</bold> and <bold>IS13</bold>, at 10 µM concentrations, exhibited moderate DCF intensity suppression (Fig. ##SUPPL##0##S51##). LPS alone induced 65% of DCF expression. The cellular anti-oxidant assay results suggested that the lead compounds showed significant neuroprotective activity by enhancing the cellular oxidant enzymes CAT, GPx, GSH, and SOD (Figs. ##FIG##11##11##, ##FIG##12##12##, ##FIG##13##13## and ##FIG##14##14##) in the LPS-induced model, while cells treated with LPS alone effectively expressed SOD, CAT, GSH, and GPx activities. The obtained values confirmed the promising neuroprotective and neuro-inflammatory activity of <bold>IS6</bold>, <bold>IS7</bold>, <bold>IS13,</bold> and <bold>IS15</bold> compounds (at 10 µM/mL) (<italic>p</italic> &lt; 0.0001) in relation to the neuroprotective effect in the LPS-induced human neuroblastoma model by enhancing the enzyme activity and inhibiting the oxidative stress-induced apoptosis caused by LPS by determining DCF intensity.</p>", "<title>Molecular docking</title>", "<p id=\"Par38\">Molecular docking studies were performed to better understand the binding processes of lead compounds. Interactions occurred between the molecules <bold>IS3</bold>, <bold>IS6</bold>, <bold>IS7</bold>, <bold>IS13</bold>, and <bold>IS15</bold>, with MAO-B (2V5Z) and MAO-A (2Z5X). Native ligands were used to confirm docking^{##REF##23988151##57##}. The compounds were mentioned in Table ##TAB##2##3##, docking scores (XP mode) ranges were − 6.55 to − 10.85 kcal/mol. The scores of the best lead molecules <bold>IS6</bold>, <bold>IS7</bold>, and <bold>IS13</bold>, through biological evaluation, were − 9.47, − 9.88, and − 9.72 kcal/mol, respectively, while safinamide had a score that was comparable (− 10.85 kcal/mol). The docking scores showed similar affinity like biological activity (IC₅₀). When looking into the orientations, the fluorobenzyl group of safinamide was placed towards the opening of the cavity, whereas the amide side chain pointed in the direction of the FAD molecule. A similar orientation was present in the compounds <bold>IS3</bold>, <bold>IS6</bold>, <bold>IS7</bold>, <bold>IS13</bold>, <bold>IS15</bold>, and isatin (Fig. ##FIG##15##15##A), where the phenyl group was positioned towards the cavity opening and the variable isatin moiety was oriented towards FAD. The entrance and substrate cavities of the MAO-B-binding pocket (Fig. ##FIG##15##15##B) were completely occupied with all inhibitors. The best lead molecule (<bold>IS7</bold>) interacted with Tyr60, Pro102, Pro104, Trp119, Phe168, Leu171, Cys172, Tyr326, and Tyr435 were primarily hydrophobic, whereas Gln206 was in polar contact. The isatin moiety demonstrated that Pi-Pi stacking with Tyr398 provided the <bold>IS7</bold>-MAO-B protein with complex stability.</p>", "<p id=\"Par39\">For MAO-A, the docking score range is − 3.96 to − 8.00 kcal/mol (Table ##TAB##2##3##). The <bold>IS15</bold> and Harmine docking scores (XP mode) were − 8.00 and − 6.04 kcal/mol, respectively. Comparing the docking score to each compound’s IC₅₀, <bold>IS15</bold> had the best profile, followed by <bold>IS3</bold> (− 7.65 kcal/mol), <bold>IS13</bold> (− 4.93 kcal/mol), <bold>IS7</bold> (− 4.32 kcal/mol), and <bold>IS6</bold> (− 3.96 kcal/mol). Every molecule was oriented in the same way as its native ligand, and its isatin moiety was always closed to the substrate cavity (FAD) (Fig. ##FIG##16##16##A). Thorough analysis of compound <bold>IS15</bold> in the MAO-A active site revealed that it was present at the following positions: Leu97, Ala111, Ile180, Asn181, Tyr197, Ile207, Phe208, Ser209, Val210, Gln215, Cys323, Ile325, Thr326, Ile335, Leu337, Tyr407, and Tyr444 (Fig. ##FIG##16##16##B). Through hydrophobic contacts, the phenyl ring of <bold>IS15</bold> interacted with Leu97, Ala111, Val210, Ile325, and Cys323. Residues Ile180, Asn181, Tyr197, Tyr407, and Tyr444 interacted with the isatin ring through hydrophobic and polar interactions. Its similar interaction with <bold>IS15</bold> and harmine^{##REF##36844523##62##} during the binding contact demonstrated its potential to inhibit MAO-A.</p>", "<title>Molecular dynamic simulation</title>", "<p id=\"Par40\">Desmond MD simulations were used to follow the binding mode of <bold>IS7</bold> in the inhibitor-binding cavity (IBC) of MAO-B. Protein C-alpha and ligand were tracked within an acceptable range for a long simulation duration (100 ns) according to root mean square deviation (RMSD) analysis. In contrast to the protein RMSD, the ligand (red) RMSD remained steady after 25 ns. The protein RMSD ranged between 1.2 and 3.6 Å with an average of 2.54 ± 0.01 Å (Fig. ##FIG##17##17##A). The protein-specific RMSD for the simulation was constant, with the exception of a slight variation, reaching a maximum of 3.6 Å at 68–70 ns, where after it stabilized. The simulation evaluated the flexibility of the protein system by computing the RMSF for each amino acid residue of the protein. The 480–498 residues of MAO-B showed a larger fluctuation. The atoms in the benzoyl ring of the RMSF ligand (Fig. ##FIG##17##17##B) showed slight fluctuations during the binding process. The 21 amino acid residues that interacted with the ligand were Tyr60 (0.541 Å),Gly101 (1.13 Å), Pro102(1.117 Å), Pro104 (0.981 Å), Trp119 (1.15 Å), Leu167 (0.955 Å), Phe168 (0.892 Å), Leu171 (0.638 Å), Cys172 (0.706 Å), Ile198 (0.689 Å), Ile199 (0.833 Å), Ser200 (0.88 Å), Thr201 (0.94 Å), Gln206 (0.618 Å), Ile316 (0.604 Å), Tyr326 (0.544 Å), Leu328 (0.572 Å), Met341 (0.493 Å), Phe343 (0.642 Å), Tyr398 (0.97 Å), and Tyr435 (0.497 Å). Hydrogen bonds, hydrophobic contacts, and water bridges were identified in the interaction histograms of <bold>IS7</bold> and MAO-B (Fig. ##FIG##17##17##C and D). Over a trajectory of 100 ns, the number of individual interactions between the amino acids and ligand was normalized. Several significant amino acids, including Tyr326 (hydrogen bond, water bridge, and hydrophobic), Tyr398 (hydrogen bond), Leu171 (hydrophobic), Cys172 (hydrogen bond and water bridge), and Ile199 (water bridge and hydrophobic), interact with <bold>IS7</bold>. The measured fraction of interactions with Tyr326 was &gt; 1.0. As previously observed^{##UREF##12##63##,##REF##26189013##64##}, the hydrophobic interaction of Tyr326 at the active site of MAO-B was significant. Figure ##FIG##17##17##C and ##FIG##17##D## depict hydrogen bonding, water bridges, and hydrophobic stability in the ligand–protein complexes. Cys172 forms an 86% hydrogen bond with the carbonyl and NH atoms in the linker between the isatin and benzoyl rings. Tyr398 contributed 49% via hydrogen bonding with the NH atom of the isatin ring. With carbonyl and water molecules, Tyr326 is a 33% active participant in hydrogen bonding. Overall, it is estimated from the trajectory analysis and full MD simulation that the lead compound <bold>IS7</bold> will inhibit MAO-B.</p>", "<title>MM-GBSA</title>", "<p id=\"Par41\">From their MD simulation frames, the free binding energy was estimated for the best molecule <bold>IS7</bold> with the highest docking energy and activity value prediction. Total average energies of ΔG Bind, ΔG Bind H-bond, ΔG Bind Lipo, and ΔG Bind vdW was − 190.04, − 12.26, − 45.94, − 140.23 for 0–100 ns MD snapshot, respectively. Across all interactions, the ΔG Bind vdW and ΔG Bind Lipo energies exerted the most significant impact on the average binding energy (Table ##TAB##3##4##).</p>", "<p id=\"Par42\">The values ΔG Bind vdW for the interactions of <bold>IS7</bold> with protein complexes indicated the presence of stable van der Waals interaction with amino acid residues. Consequently, the MM-GBSA calculations, derived from MD simulation trajectories, aligned well with the binding energies computed from the docking results. The molecule exhibited very low free binding energy, indicating its higher binding affinity towards the receptor. Consequently, it can be inferred that <bold>IS7</bold> compound exhibited a strong affinity for the MAO-B protein.</p>" ]

[ "<title>Conclusion</title>", "<p id=\"Par43\">We synthesized acylhydrazone-based isatin compounds and evaluated their ability to inhibit MAOs. <bold>IS15</bold> was a potent competitive reversible MAO-A inhibitor, whereas <bold>IS6</bold>, <bold>IS7</bold>, and <bold>IS13</bold> were potent competitive reversible and selective MAO-B inhibitors. A CNS permeability study using a PAMPA assay revealed that the lead compounds were BBB-permeable. The lead compounds also exhibit non-cytotoxic, neuroprotective and anti-inflammatory effects. The lead compounds (at a concentration of 10 µM/mL) effectively reduced DCF intensity. Additionally, a docking analysis of MAO-B and <bold>IS7</bold> revealed the stability of the complex due to the pi–pi stacking of Tyr326. The Cys172 residue participated in the interaction with the ligand at 86% during dynamic examination. Finally, MM-GBSA energy binding revealed that <bold>IS7</bold> provided strong stability to MAO-B protein. Overall, the results of this investigation suggest that the lead compounds, <bold>IS7</bold>, <bold>IS6</bold>, <bold>IS13</bold>, and <bold>IS15,</bold> may be viable therapeutic agents for the treatment of neurological disorders such as PD.</p>" ]

[ "<p id=\"Par1\">Sixteen isatin-based hydrazone derivatives (<bold>IS1</bold>–<bold>IS16</bold>) were synthesized and assessed for their ability to inhibit monoamine oxidases (MAOs). All the molecules showed improved inhibitory MAO-B activity compared to MAO-A. Compound <bold>IS7</bold> most potently inhibited MAO-B with an IC₅₀ value of 0.082 μM, followed by <bold>IS13</bold> and <bold>IS6</bold> (IC₅₀ = 0.104 and 0.124 μM, respectively). Compound <bold>IS15</bold> most potently inhibited MAO-A with an IC₅₀ value of 1.852 μM, followed by <bold>IS3</bold> (IC₅₀ = 2.385 μM). Compound <bold>IS6</bold> had the highest selectivity index (SI) value of 263.80, followed by <bold>IS7</bold> and <bold>IS13</bold> (233.85 and 212.57, respectively). In the kinetic study, the K_i values of <bold>IS6</bold>, <bold>IS7</bold>, and <bold>IS13</bold> for MAO-B were 0.068 ± 0.022, 0.044 ± 0.002, and 0.061 ± 0.001 μM, respectively, and that of <bold>IS15</bold> for MAO-A was 1.004 ± 0.171 μM, and the compounds were reversible-type inhibitors. The lead compounds were central nervous system (CNS) permeable, as per parallel artificial membrane permeability assay (PAMPA) test results. The lead compounds were examined for their cytotoxicity and potential neuroprotective benefits in hazardous lipopolysaccharide (LPS)-exposed SH-SY5Y neuroblastoma cells. Pre-treatment with lead compounds enhanced anti-oxidant levels (SOD, CAT, GSH, and GPx) and decreased ROS and pro-inflammatory cytokine (IL-6, TNF-alpha, and NF-kB) production in LPS-intoxicated SH-SY5Y cells. To confirm the promising effects of the compound, molecular docking, dynamics, and MM-GBSA binding energy were used to examine the molecular basis of the <bold>IS7</bold>-MAO-B interaction. Our findings indicate that lead compounds are potential therapeutic agents to treat neurological illnesses, such as Parkinson's disease.</p>", "<title>Subject terms</title>" ]

[ "<title>Supplementary Information</title>", "<p>\n</p>" ]

[ "<title>Supplementary Information</title>", "<p>The online version contains supplementary material available at 10.1038/s41598-024-51728-x.</p>", "<title>Acknowledgements</title>", "<p>The authors thank the Indian Council of Social Science Research under the Grant number (File No. 02/90/2022-23/RP/MN).</p>", "<title>Author contributions</title>", "<p>B.M. and S.K. planned and designed the study. S.K., J.M.O., A.A. and P.P. carried out the experiment and collected the data. B.M., H.K., and P.P. analyzed the data. H.K. provided technical support. B.M. and H.K. revised the manuscript. All authors approved the final version of the manuscript.</p>", "<title>Competing interests</title>", "<p id=\"Par44\">The authors declare no competing interests.</p>" ]

[ "<fig id=\"Fig1\"><label>Figure 1</label><caption><p>Base structure of isatin and their derivatives for FDA-approved drugs.</p></caption></fig>", "<fig id=\"Fig2\"><label>Figure 2</label><caption><p>Design strategy of acylhydrazone-based isatin derivatives.</p></caption></fig>", "<fig id=\"Sch1\"><label>Scheme 1</label><caption><p>Synthetic protocol of IS1-IS16.</p></caption></fig>", "<fig id=\"Fig3\"><label>Figure 3</label><caption><p>Structure–activity relationship of acylhydrazone-based isatin derivatives.</p></caption></fig>", "<fig id=\"Fig4\"><label>Figure 4</label><caption><p>Recovery of MAO-A inhibition by <bold>IS15</bold> using dialysis experiments.</p></caption></fig>", "<fig id=\"Fig5\"><label>Figure 5</label><caption><p>Recovery of MAO-B inhibition by <bold>IS6</bold>, <bold>IS7</bold>, and <bold>IS13</bold> using dialysis experiments.</p></caption></fig>", "<fig id=\"Fig6\"><label>Figure 6</label><caption><p>Lineweaver–Burk (LB) plots for MAO-A inhibition by <bold>IS15</bold> (<bold>A</bold>) and their respective secondary plots (<bold>B</bold>) of the slopes vs. inhibitor concentrations.</p></caption></fig>", "<fig id=\"Fig7\"><label>Figure 7</label><caption><p>Lineweaver–Burk (LB) plots for MAO-B inhibition by <bold>IS6</bold>, <bold>IS7</bold>, and <bold>IS13</bold> (<bold>A</bold>, <bold>C</bold>, and <bold>E</bold>, respectively), and their respective secondary plots (<bold>B</bold>, <bold>D</bold>, and <bold>F</bold>, respectively) of the slopes vs. inhibitor concentrations.</p></caption></fig>", "<fig id=\"Fig8\"><label>Figure 8</label><caption><p>MTT assay of the lead compounds using SH-SY5Y cell for cell viability and morphological studies under a phase-contrast microscope, exposed for 24 h. (<bold>A</bold>) <bold>IS6</bold>, (<bold>B</bold>) <bold>IS7</bold>, and (<bold>C</bold>) <bold>IS13</bold>. Values are expressed as mean ± SEM. ***, <italic>p</italic> &lt; 0.001; **, <italic>p</italic> &lt; 0.01; *, <italic>p</italic> &lt; 0.05 vs VC. VC, vehicle control; UT, untreated.</p></caption></fig>", "<fig id=\"Fig9\"><label>Figure 9</label><caption><p>Effect of the lead compounds on anti-neuroinflammation in lipopolysaccharide (LPS)-induced SH-SY5Y cells. (<bold>A</bold>) Tumor necrosis factor (TNF)-α, (<bold>B</bold>) Interleukin (IL)-6, and C) Nuclear factor (NF)-kB. Values are expressed as mean ± SEM. ####, <italic>p</italic> &lt; 0.0001 vs untreated; ****, <italic>p</italic> &lt; 0.0001 vs LPS.</p></caption></fig>", "<fig id=\"Fig10\"><label>Figure 10</label><caption><p>Effect of the lead compounds on reduction of ROS generation in SH-SY5Y cells. Values are expressed as mean ± SEM. ####, <italic>p</italic> &lt; 0.0001 vs untreated; ****, <italic>p</italic> &lt; 0.0001 vs LPS.</p></caption></fig>", "<fig id=\"Fig11\"><label>Figure 11</label><caption><p>Catalase (CAT) levels observed in different concentrations of (<bold>A</bold>) <bold>IS6</bold>, (<bold>B</bold>) <bold>IS7</bold>, (<bold>C</bold>) <bold>IS13</bold>, and (<bold>D</bold>) <bold>IS15</bold> treatment against the LPS-induced SH-SY5Y cells. Values are expressed as mean ± SEM. ####, <italic>p</italic> &lt; 0.0001 vs untreated; ****, <italic>p</italic> &lt; 0.0001; **, <italic>p</italic> &lt; 0.01 vs LPS.</p></caption></fig>", "<fig id=\"Fig12\"><label>Figure 12</label><caption><p>Glutathione peroxidase (GPx) levels observed in different concentrations of (<bold>A</bold>) <bold>IS6</bold>, (<bold>B</bold>) <bold>IS7</bold>, (<bold>C</bold>) <bold>IS13</bold>, and (<bold>D</bold>) <bold>IS15</bold> treatment against the LPS-induced SH-SY5Y cells. Values are expressed as mean ± SEM. ####, <italic>p</italic> &lt; 0.0001 vs untreated; ****, <italic>p</italic> &lt; 0.0001; ***, <italic>p</italic> &lt; 0.001; **, <italic>p</italic> &lt; 0.01; *, <italic>p</italic> &lt; 0.05 vs LPS.</p></caption></fig>", "<fig id=\"Fig13\"><label>Figure 13</label><caption><p>Glutathione levels observed in different concentrations of (<bold>A</bold>) <bold>IS6</bold>, (<bold>B</bold>) <bold>IS7</bold>, (<bold>C</bold>) <bold>IS13</bold>, and (<bold>D</bold>) <bold>IS15</bold> treatment of the LPS-induced SH-SY5Y cells. Values are expressed as mean ± SEM. ####, <italic>p</italic> &lt; 0.0001 vs untreated; ****, <italic>p</italic> &lt; 0.0001; ***, <italic>p</italic> &lt; 0.001; **, <italic>p</italic> &lt; 0.01; *, <italic>p</italic> &lt; 0.05 vs LPS.</p></caption></fig>", "<fig id=\"Fig14\"><label>Figure 14</label><caption><p>Superoxide dismutase (SOD) levels observed in different concentrations of (<bold>A</bold>) <bold>IS6</bold>, (<bold>B</bold>) <bold>IS7</bold>, (<bold>C</bold>) <bold>IS13</bold>, and (<bold>D</bold>) <bold>IS15</bold> treatment of the LPS-induced SH-SY5Y cells. Values are expressed as mean ± SEM. ####, <italic>p</italic> &lt; 0.0001 vs untreated; ****, <italic>p</italic> &lt; 0.0001; ***, <italic>p</italic> &lt; 0.001; **, <italic>p</italic> &lt; 0.01; *, <italic>p</italic> &lt; 0.05 vs LPS.</p></caption></fig>", "<fig id=\"Fig15\"><label>Figure 15</label><caption><p><bold>A</bold>) 3-D visualization of superimposed orientations of <bold>IS3</bold> (pink), <bold>IS6</bold> (red) <bold>IS7</bold> (green), <bold>IS13</bold> (blue), <bold>IS15</bold> (orange), isatin (cyan), and safinamide (violet) in the active site of MAO-B and the co-factor flavin adenine dinucleotide (FAD) (yellow). <bold>B</bold>) 2-D interaction of lead inhibitor <bold>IS7</bold> with MAO-B pocket.</p></caption></fig>", "<fig id=\"Fig16\"><label>Figure 16</label><caption><p>(<bold>A</bold>) 3-D visualization of superimposed orientations of <bold>IS3</bold> (orange), <bold>IS6</bold> (cyan) <bold>IS7</bold> (violet), <bold>IS13</bold> (blue), <bold>IS15</bold> (green), and harmine (red) in the active site of MAO-A and the co-factor FAD (yellow). (<bold>B</bold>) 2-D interaction of lead inhibitor <bold>IS15</bold> with MAO-A pocket.</p></caption></fig>", "<fig id=\"Fig17\"><label>Figure 17</label><caption><p>Desmond’s MD simulation analysis of the <bold>IS7</bold>-MAO-B complex. (<bold>A</bold>) Root mean square deviation (RMSD) (protein and <bold>IS7</bold> RMSD are shown in blue and red, respectively). (<bold>B</bold>) Individual RMSF for proteins' amino acids. (<bold>C</bold>) Diagram of the 2-D Interaction. (<bold>D</bold>) Protein–ligand contacts with number of specific contacts of amino acids with <bold>IS7</bold>.</p></caption></fig>" ]

[ "<table-wrap id=\"Tab1\"><label>Table 1</label><caption><p>Monoamine oxidase (MAO)-A and MAO-B inhibition by the 16 compounds of the IS series^a.</p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th align=\"left\" rowspan=\"2\">Compound</th><th align=\"left\" colspan=\"2\">Residual activity at 10 µM (%)</th><th align=\"left\" colspan=\"2\">IC₅₀ (µM)</th><th align=\"left\" rowspan=\"2\">SI^b</th></tr><tr><th align=\"left\">MAO-A</th><th align=\"left\">MAO-B</th><th align=\"left\">MAO-A</th><th align=\"left\">MAO-B</th></tr></thead><tbody><tr><td align=\"left\">IS1</td><td char=\".\" align=\"char\">83.08 ± 2.34</td><td char=\".\" align=\"char\">22.00 ± 2.75</td><td align=\"left\"> &gt; 40</td><td char=\".\" align=\"char\">0.420 ± 0.028</td><td align=\"left\"> &gt; 95.24</td></tr><tr><td align=\"left\">IS2</td><td char=\".\" align=\"char\">46.93 ± 2.73</td><td char=\".\" align=\"char\">1.18 ± 2.04</td><td align=\"left\">9.155 ± 0.324</td><td char=\".\" align=\"char\">0.269 ± 0.004</td><td align=\"left\">34.03</td></tr><tr><td align=\"left\">IS3</td><td char=\".\" align=\"char\">29.53 ± 0.34</td><td char=\".\" align=\"char\">1.04 ± 0.23</td><td align=\"left\">2.385 ± 0.018</td><td char=\".\" align=\"char\">0.514 ± 0.023</td><td align=\"left\">4.64</td></tr><tr><td align=\"left\">IS4</td><td char=\".\" align=\"char\">70.07 ± 1.93</td><td char=\".\" align=\"char\">42.46 ± 8.93</td><td align=\"left\">32.849 ± 0.134</td><td char=\".\" align=\"char\">7.284 ± 1.724</td><td align=\"left\">4.51</td></tr><tr><td align=\"left\">IS5</td><td char=\".\" align=\"char\">75.75 ± 7.44</td><td char=\".\" align=\"char\">38.20 ± 4.61</td><td align=\"left\"> &gt; 40</td><td char=\".\" align=\"char\">4.136 ± 0.068</td><td align=\"left\"> &gt; 9.67</td></tr><tr><td align=\"left\">IS6</td><td char=\".\" align=\"char\">62.04 ± 1.87</td><td char=\".\" align=\"char\">8.53 ± 1.71</td><td align=\"left\">32.711 ± 0.210</td><td char=\".\" align=\"char\">0.124 ± 0.015</td><td align=\"left\">263.80</td></tr><tr><td align=\"left\">IS7</td><td char=\".\" align=\"char\">50.76 ± 7.29</td><td char=\".\" align=\"char\">3.20 ± 0.77</td><td align=\"left\">19.176 ± 5.960</td><td char=\".\" align=\"char\">0.082 ± 0.010</td><td align=\"left\">233.85</td></tr><tr><td align=\"left\">IS8</td><td char=\".\" align=\"char\">69.65 ± 3.22</td><td char=\".\" align=\"char\">67.38 ± 7.80</td><td align=\"left\">35.001 ± 0.000</td><td char=\".\" align=\"char\">25.473 ± 2.034</td><td align=\"left\"> &gt; 1.37</td></tr><tr><td align=\"left\">IS9</td><td char=\".\" align=\"char\">67.38 ± 1.00</td><td char=\".\" align=\"char\">45.56 ± 6.28</td><td align=\"left\"> &gt; 40</td><td char=\".\" align=\"char\">9.094 ± 1.281</td><td align=\"left\">4.40</td></tr><tr><td align=\"left\">IS10</td><td char=\".\" align=\"char\">54.06 ± 0.78</td><td char=\".\" align=\"char\">42.64 ± 0.82</td><td align=\"left\">18.308 ± 0.729</td><td char=\".\" align=\"char\">3.995 ± 0.472</td><td align=\"left\">4.58</td></tr><tr><td align=\"left\">IS11</td><td char=\".\" align=\"char\">43.56 ± 0.42</td><td char=\".\" align=\"char\">53.24 ± 2.60</td><td align=\"left\">13.718 ± 0.396</td><td char=\".\" align=\"char\">10.586 ± 0.256</td><td align=\"left\">1.30</td></tr><tr><td align=\"left\">IS12</td><td char=\".\" align=\"char\">36.59 ± 1.60</td><td char=\".\" align=\"char\">44.29 ± 4.48</td><td align=\"left\">4.288 ± 0.552</td><td char=\".\" align=\"char\">7.099 ± 0.956</td><td align=\"left\">0.60</td></tr><tr><td align=\"left\">IS13</td><td char=\".\" align=\"char\">62.28 ± 0.30</td><td char=\".\" align=\"char\">5.66 ± 1.75</td><td align=\"left\">22.107 ± 0.063</td><td char=\".\" align=\"char\">0.104 ± 0.005</td><td align=\"left\">212.57</td></tr><tr><td align=\"left\">IS14</td><td char=\".\" align=\"char\">53.15 ± 0.91</td><td char=\".\" align=\"char\">23.00 ± 2.48</td><td align=\"left\">11.268 ± 0.414</td><td char=\".\" align=\"char\">0.234 ± 0.008</td><td align=\"left\">48.15</td></tr><tr><td align=\"left\">IS15</td><td char=\".\" align=\"char\">34.67 ± 0.71</td><td char=\".\" align=\"char\">12.47 ± 3.64</td><td align=\"left\">1.852 ± 0.085</td><td char=\".\" align=\"char\">0.337 ± 0.007</td><td align=\"left\">5.50</td></tr><tr><td align=\"left\">IS16</td><td char=\".\" align=\"char\">43.66 ± 1.65</td><td char=\".\" align=\"char\">26.10 ± 4.39</td><td align=\"left\">8.455 ± 0.395</td><td char=\".\" align=\"char\">3.848 ± 0.392</td><td align=\"left\">2.20</td></tr><tr><td align=\"left\">Toloxatone</td><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td align=\"left\">1.080 ± 0.025</td><td char=\".\" align=\"char\">–</td><td align=\"left\"/></tr><tr><td align=\"left\">Lazabemide</td><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td align=\"left\">–</td><td char=\".\" align=\"char\">0.110 ± 0.016</td><td align=\"left\"/></tr><tr><td align=\"left\">Clorgyline</td><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td align=\"left\">0.007 ± 0.0007</td><td char=\".\" align=\"char\">–</td><td align=\"left\"/></tr><tr><td align=\"left\">Pargyline</td><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td align=\"left\">–</td><td char=\".\" align=\"char\">0.140 ± 0.0059</td><td align=\"left\"/></tr></tbody></table></table-wrap>", "<table-wrap id=\"Tab2\"><label>Table 2</label><caption><p>Blood–brain barrier (BBB) assay of key compounds of isatin-based hydrazone derivatives by the parallel artificial membrane permeability assay (PAMPA) method.</p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th align=\"left\">Compound</th><th align=\"left\">Experimental <italic>P</italic>e (× 10⁻⁶ cm/s)</th><th align=\"left\">Prediction</th></tr></thead><tbody><tr><td align=\"left\">IS6</td><td char=\".\" align=\"char\">4.86 ± 0.26</td><td align=\"left\">CNS+</td></tr><tr><td align=\"left\">IS7</td><td char=\".\" align=\"char\">5.02 ± 0.19</td><td align=\"left\">CNS+</td></tr><tr><td align=\"left\">IS13</td><td char=\".\" align=\"char\">4.66 ± 0.18</td><td align=\"left\">CNS+</td></tr><tr><td align=\"left\">IS15</td><td char=\".\" align=\"char\">4.39 ± 0.30</td><td align=\"left\">CNS+</td></tr><tr><td align=\"left\">Selegiline</td><td char=\".\" align=\"char\">5.69 ± 0.04</td><td align=\"left\">CNS+</td></tr></tbody></table></table-wrap>", "<table-wrap id=\"Tab3\"><label>Table 3</label><caption><p>Docking score of MAO-A and MAO–B with IS3, IS6, IS7, IS13, IS15, and their native ligands.</p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th align=\"left\" rowspan=\"2\">Compound</th><th align=\"left\" colspan=\"2\">Docking score (kcal/mol)</th></tr><tr><th align=\"left\">MAO-A</th><th align=\"left\">MAO-B</th></tr></thead><tbody><tr><td align=\"left\">IS3</td><td align=\"left\"> − 7.65</td><td align=\"left\"> − 9.16</td></tr><tr><td align=\"left\">IS6</td><td align=\"left\"> − 3.96</td><td align=\"left\"> − 9.47</td></tr><tr><td align=\"left\">IS7</td><td align=\"left\"> − 4.32</td><td align=\"left\"> − 9.88</td></tr><tr><td align=\"left\">IS13</td><td align=\"left\"> − 4.93</td><td align=\"left\"> − 9.72</td></tr><tr><td align=\"left\">IS15</td><td align=\"left\"> − 8.00</td><td align=\"left\"> − 9.36</td></tr><tr><td align=\"left\">Isatin</td><td align=\"left\">–</td><td align=\"left\"> − 6.55</td></tr><tr><td align=\"left\">Harmine</td><td align=\"left\"> − 6.04</td><td align=\"left\">–</td></tr><tr><td align=\"left\">Safinamide</td><td align=\"left\">–</td><td align=\"left\"> − 10.85</td></tr></tbody></table></table-wrap>", "<table-wrap id=\"Tab4\"><label>Table 4</label><caption><p>Free binding energies of the molecule IS7 through MM-GBSA*.</p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th align=\"left\">MD (ns)</th><th align=\"left\">ΔG Bind</th><th align=\"left\">ΔG Bind H-bond</th><th align=\"left\">ΔG Bind Lipo</th><th align=\"left\">ΔG Bind vdW</th></tr></thead><tbody><tr><td align=\"left\">0</td><td align=\"left\"> − 181.24</td><td align=\"left\"> − 10.01</td><td align=\"left\"> − 52.45</td><td align=\"left\"> − 161.63</td></tr><tr><td align=\"left\">10</td><td align=\"left\"> − 198.99</td><td align=\"left\"> − 11.61</td><td align=\"left\"> − 55.85</td><td align=\"left\"> − 159.96</td></tr><tr><td align=\"left\">20</td><td align=\"left\"> − 185.63</td><td align=\"left\"> − 7.16</td><td align=\"left\"> − 48.34</td><td align=\"left\"> − 119</td></tr><tr><td align=\"left\">30</td><td align=\"left\"> − 208.04</td><td align=\"left\"> − 15.31</td><td align=\"left\"> − 52.77</td><td align=\"left\"> − 129.35</td></tr><tr><td align=\"left\">40</td><td align=\"left\"> − 225.93</td><td align=\"left\"> − 11.7</td><td align=\"left\"> − 54.78</td><td align=\"left\"> − 135.01</td></tr><tr><td align=\"left\">50</td><td align=\"left\"> − 211.25</td><td align=\"left\"> − 14.81</td><td align=\"left\"> − 45.46</td><td align=\"left\"> − 137.82</td></tr><tr><td align=\"left\">60</td><td align=\"left\"> − 156.03</td><td align=\"left\"> − 11.38</td><td align=\"left\"> − 34.52</td><td align=\"left\"> − 107.01</td></tr><tr><td align=\"left\">70</td><td align=\"left\"> − 190.78</td><td align=\"left\"> − 11.32</td><td align=\"left\"> − 51.15</td><td align=\"left\"> − 140.55</td></tr><tr><td align=\"left\">80</td><td align=\"left\"> − 191.13</td><td align=\"left\"> − 14.03</td><td align=\"left\"> − 34.11</td><td align=\"left\"> − 155.83</td></tr><tr><td align=\"left\">90</td><td align=\"left\"> − 200.74</td><td align=\"left\"> − 16.38</td><td align=\"left\"> − 45.74</td><td align=\"left\"> − 177.62</td></tr><tr><td align=\"left\">100</td><td align=\"left\"> − 140.75</td><td align=\"left\"> − 11.15</td><td align=\"left\"> − 30.27</td><td align=\"left\"> − 118.82</td></tr></tbody></table></table-wrap>" ]

[ "<disp-formula id=\"Equa\"><alternatives><tex-math id=\"M1\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\text{G}} = {\\text{E}}_{{{\\text{int}}}} + {\\text{E}}_{{{\\text{ele}}}} + {\\text{E}}_{{{\\text{vdw}}}} + {\\text{G}}_{{{\\text{pol}}}} + {\\text{G}}_{{{\\text{np}}}} - {\\text{TS}}$$\\end{document}</tex-math><mml:math id=\"M2\" display=\"block\"><mml:mrow><mml:mtext>G</mml:mtext><mml:mo>=</mml:mo><mml:msub><mml:mtext>E</mml:mtext><mml:mtext>int</mml:mtext></mml:msub><mml:mo>+</mml:mo><mml:msub><mml:mtext>E</mml:mtext><mml:mtext>ele</mml:mtext></mml:msub><mml:mo>+</mml:mo><mml:msub><mml:mtext>E</mml:mtext><mml:mtext>vdw</mml:mtext></mml:msub><mml:mo>+</mml:mo><mml:msub><mml:mtext>G</mml:mtext><mml:mtext>pol</mml:mtext></mml:msub><mml:mo>+</mml:mo><mml:msub><mml:mtext>G</mml:mtext><mml:mtext>np</mml:mtext></mml:msub><mml:mo>-</mml:mo><mml:mtext>TS</mml:mtext></mml:mrow></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equb\"><alternatives><tex-math id=\"M3\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\Delta {\\text{G}}\\;{\\text{Bind}} = \\left( {{\\text{G}}_{{{\\text{PL}}}} } \\right) - \\left( {{\\text{G}}_{{\\text{P}}} } \\right) - \\left( {{\\text{G}}_{{\\text{L}}} } \\right)$$\\end{document}</tex-math><mml:math id=\"M4\" display=\"block\"><mml:mrow><mml:mi mathvariant=\"normal\">Δ</mml:mi><mml:mtext>G</mml:mtext><mml:mspace width=\"0.277778em\"/><mml:mtext>Bind</mml:mtext><mml:mo>=</mml:mo><mml:mfenced close=\")\" open=\"(\"><mml:msub><mml:mtext>G</mml:mtext><mml:mtext>PL</mml:mtext></mml:msub></mml:mfenced><mml:mo>-</mml:mo><mml:mfenced close=\")\" open=\"(\"><mml:msub><mml:mtext>G</mml:mtext><mml:mtext>P</mml:mtext></mml:msub></mml:mfenced><mml:mo>-</mml:mo><mml:mfenced close=\")\" open=\"(\"><mml:msub><mml:mtext>G</mml:mtext><mml:mtext>L</mml:mtext></mml:msub></mml:mfenced></mml:mrow></mml:math></alternatives></disp-formula>" ]

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[ "<supplementary-material content-type=\"local-data\" id=\"MOESM1\"></supplementary-material>" ]

[ "<table-wrap-foot><p>^aResults are the means ± standard errors of duplicate or triplicate experiments.</p><p>^bSelectivity index (SI) values are expressed for MAO-B compared to MAO-A.</p></table-wrap-foot>", "<table-wrap-foot><p><italic>Pe</italic> (10⁻⁶ cm/s) &gt; 4.00: CNS + (high permeation); <italic>Pe</italic> (10⁻⁶ cm/s) &lt; 2.00: CNS − (low permeation); <italic>Pe</italic> (10⁻⁶ cm/s) from 2.00 to 4.00: CNS ± (BBB permeation uncertain).</p></table-wrap-foot>", "<table-wrap-foot><p>* kcal/mol.</p></table-wrap-foot>", "<fn-group><fn><p><bold>Publisher's note</bold></p><p>Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.</p></fn><fn><p>These authors contributed equally: Sunil Kumar and Jong Min Oh.</p></fn></fn-group>" ]

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[ "<media xlink:href=\"41598_2024_51728_MOESM1_ESM.docx\"><caption><p>Supplementary Information.</p></caption></media>" ]

{ "acronym": [], "definition": [] }

[ "<title>Introduction</title>", "<p id=\"Par3\">Mucosal pentraxin 2 (<italic>Mptx2</italic>) has been proposed to be a member of the pentraxin family due to the high homology of its sequences ( ~ 83% identity in amino acid sequences) to C-reactive protein (<italic>CRP</italic>) and serum amyloid P component protein (<italic>SAP</italic>) in other family members^{##REF##18850182##1##}. <italic>CRP</italic> and <italic>SAP</italic> are involved in defending against pathogenic bacteria^{##REF##24167754##2##–##REF##11739301##4##}. <italic>Mptx2</italic> is strongly regulated by dietary heme and calcium^{##REF##12832292##5##,##REF##15539406##6##}. Recently, a single-cell ribonucleic acid (RNA) sequencing of small-intestinal epithelium indicates <italic>Mptx2</italic> is a novel Paneth cell marker^{##REF##29144463##7##}, but its function is still unknown.</p>", "<p id=\"Par4\">Paneth cells are specialized intestinal epithelial cells (IECs) that reside at the bases of small-intestinal crypts and are important maintainers of intestinal homeostasis^{##REF##21423246##8##–##REF##15908936##10##}. Impaired autophagy in Paneth cells alters the expression and secretion of antimicrobial factors and intestinal immune responses^{##REF##24089213##11##–##REF##29089374##13##}. Therefore, understanding regulatory factors in Paneth cell function is critical to the design of new therapeutic approaches to diseases featuring mucosal inflammation. Autophagy is the process to degrade intracellular entities, such as damaged mitochondria, nuclear fragments, viruses, and bacteria^{##REF##28596378##14##}. In the intestinal tract, autophagy is essential to engulf and degrade the invading bacteria, thereby being importantly involved in regulating the intestinal immune response^{##REF##23128233##15##}. Indeed, the autophagy-associated genes, including the nucleotide-binding oligomerization domain containing 2 (<italic>NOD2</italic>), and autophagy-related 16 like 1 (<italic>ATG16L1</italic>), have been linked to pathogenesis of inflammatory bowel disease (IBD)^{##REF##11385576##16##,##REF##27273576##17##}.</p>", "<p id=\"Par5\">In the current study, to reveal the roles and mechanisms of <italic>Mptx2</italic> in intestinal homeostasis, we began by demonstrating that <italic>Mptx2</italic> was specifically expressed in Paneth cells. We then generated <italic>Mptx2</italic> knockout (<italic>Mptx2</italic>^{<italic>−/−</italic>}) mice to study the precise effects of this gene in intestinal inflammation and injury, and to test whether autophagy is involved in mechanisms. Our results showed that <italic>Mptx2</italic> deficiency impaired functions of Paneth cells and thus to cause intestinal inflammation and injury.</p>" ]

[ "<title>Materials and methods</title>", "<title>Generation of <italic>Mptx2</italic> knockout mice</title>", "<p id=\"Par18\"><italic>Mptx2</italic>^{<italic>−/−</italic>} mice (Δexon 1–2) were generated as in our previous study^{##REF##35609377##18##} via genome engineering mediated by clustered regularly interspaced short palindromic repeats (CRISPRs) and CRISPR-associated protein 9 (Cas9) in C57BL/6 J mice. All procedures involving mice were approved by the Institutional Animal Care and Use Committee of Xinhua Hospital School of Medicine, Shanghai Jiao Tong University (Shanghai, China; No. XHEC-C-F-2022-010). We have complied with all relevant ethical regulations for animal use.</p>", "<title>Production of recombinant <italic>Mptx2</italic> protein</title>", "<p id=\"Par19\">We produced recombinant <italic>Mptx2</italic> protein via the methods described in our previous study^{##REF##35609377##18##}. Briefly, the coding sequence of <italic>Mptx2</italic> (NM_001205011) was cloned into the pET-28 vector with an N-terminal 6-histone (6-His) tag (Genechem Co., Ltd, Shanghai, China). We induced <italic>Mptx2</italic> protein via supplementation of 1 mM isopropyl-β-d-thiogalactopyranoside (IPTG) in BL21 (DE3)–competent cells.</p>", "<title>Generation of antibody against <italic>Mptx2</italic></title>", "<p id=\"Par20\">The protein was purified using a nickel–nitrilotriacetic acid (Ni-NTA) column, a PD MidiTrap G-25 column (GE Healthcare, Chicago, IL, USA), and a Vivaspin 20 centrifugal concentrator (GE Healthcare) according to a published protocol^{##REF##35609377##18##}. The purified protein was assessed using Coomassie Brilliant Blue staining. We purified the antibody against <italic>Mptx2</italic> via antigen immunoaffinity. Reactivity was assessed using an enzyme-linked immunosorbent assay (ELISA).</p>", "<title>Dextran sulfate sodium–induced colitis</title>", "<p id=\"Par21\">We used 6-week-old <italic>Mptx2</italic>^{<italic>−/−</italic>} mice (female, <italic>n</italic> = 6; male, <italic>n</italic> = 10) and their wild-type (<italic>Wt</italic>) (female, <italic>n</italic> = 9; male, <italic>n</italic> = 6) littermates for dextran sulfate sodium (DSS)–induced colitis experiments. <italic>Mptx2</italic>^{<italic>−/−</italic>} (female, <italic>n</italic> = 8; male, <italic>n</italic> = 7) mice and <italic>Wt</italic> (female, <italic>n</italic> = 8; male, <italic>n</italic> = 8) mice were untreated as controls. Acute colitis was induced by administration of 2% DSS (36–50 kDa; MP Biomedicals, Solon, OH, USA) in drinking water for 7 days. We monitored changes in mouse body weight (BW) daily. To construct the DSS-induced colitis and recovery mouse model, C57BL/6 mice were induced by 3.5% DSS for 7 days and allowed to recover for 2 weeks (day 0, female, <italic>n</italic> = 8; male, <italic>n</italic> = 7), (day 1, female, <italic>n</italic> = 5; male, <italic>n</italic> = 5), (day 3, female, <italic>n</italic> = 5; male, <italic>n</italic> = 5), (day 7, female, <italic>n</italic> = 5; male, <italic>n</italic> = 5), (recovery 1 week, female, <italic>n</italic> = 5; male, <italic>n</italic> = 5), and (recovery 2 week, female, <italic>n</italic> = 6; male, <italic>n</italic> = 5).</p>", "<title>Lipopolysaccharide–induced systemic inflammation</title>", "<p id=\"Par22\">To induce an acute systemic inflammatory response, we injected C57BL/6 mice ~6 weeks old with lipopolysaccharide (LPS) intraperitoneally (i.p.; 5 mg/kg; #G5032; Wuhan Servicebio Technology Co., Ltd., Wuhan, China). Mice were sacrificed at the time points of 0 h (female, <italic>n</italic> = 8; male, <italic>n</italic> = 7), 3 h (female, <italic>n</italic> = 5; male, <italic>n</italic> = 5), 6 h (female, <italic>n</italic> = 5; male, <italic>n</italic> = 5), 18 h (female, <italic>n</italic> = 5; male, <italic>n</italic> = 5), 24 h (female, <italic>n</italic> = 5; male, <italic>n</italic> = 5), and 48 h (female, <italic>n</italic> = 5; male, <italic>n</italic> = 5)after LPS injection. Control mice received normal saline.</p>", "<title><italic>Methicillin-resistant Staphylococcus aureus</italic> peritoneal infection</title>", "<p id=\"Par23\">We housed <italic>Wt</italic> and <italic>Mptx2</italic>^{<italic>−/−</italic>} mice ~6 weeks old in a specific-pathogen–free (SPF) unit with access to tap water and pelleted food <italic>ad libitum</italic>. The murine-peritonitis model was established according to the previously described protocol^{##REF##35609377##18##,##REF##19223616##31##}. In brief, we infected <italic>Wt</italic> (female, <italic>n</italic> = 5; male, <italic>n</italic> = 7) and <italic>Mptx2</italic>^{<italic>−/−</italic>} mice (female, <italic>n</italic> = 4; male, <italic>n</italic> = 3) i.p. with methicillin-resistant <italic>Staphylococcus aureus</italic> (MRSA; 1.5 × 10⁷ CFU). 2 h post-infection, <italic>Wt</italic> mice received a single dose of 10 μg/ml <italic>Mptx2</italic> recombinant protein (female, <italic>n</italic> = 7; male, <italic>n</italic> = 7). Mice were sacrificed 24 h after infection, and tissues were collected for further analysis.</p>", "<title>Antibiotic treatments</title>", "<p id=\"Par24\">We treated mice with gentamicin (GM; 2 g L^{<italic>−</italic>1}; Shandong LuKang Pharmaceutical, Jining, China) or vancomycin (VCM; 500 mg L^{<italic>−</italic>1}; Eli Lilly Japan, Kobe, Japan) dissolved in autoclaved drinking water with 2% DSS for 7 days. BW was monitored afterward. GM-DSS-treated mice (<italic>Mptx2</italic>^{<italic>−/−</italic>}, female, <italic>n</italic> = 9; male, <italic>n</italic> = 5; <italic>Wt</italic>, female, n = 4; male, n = 5); VCM-DSS-treated mice (<italic>Mptx2</italic>^{<italic>−/−</italic>}, female, <italic>n</italic> = 6; male, <italic>n</italic> = 6; <italic>Wt</italic>, female, <italic>n</italic> = 6; male, <italic>n</italic> = 6).</p>", "<title>Intestinal characterization</title>", "<p id=\"Par25\">Intestinal tissues were fixed in 4% paraformaldehyde (PFA) for 24 h and sectioned (4 μm) for hematoxylin and eosin (H&amp;E) staining. We determined villus height and crypt depth using National Institutes of Health (NIH) Image software (NIH, Bethesda, MD, USA) with a microscope (Nikon, Tokyo, Japan). Villus height was measured from five well-oriented villi on each slide; five fields were analyzed per section. We counted goblet cells and quantified mucous secretions using Alcian blue/periodic acid–Schiff (AB/PAS) staining. The count of goblet cells in each villus was calculated from 10 well-oriented villi.</p>", "<title>Histological score</title>", "<p id=\"Par26\">We graded histological changes in the intestinal mucosa as previously described^{##REF##27155817##32##,##REF##10337013##33##}. Briefly, histological scores were determined blindly based on the sum of epithelial and infiltration scores. Epithelial scores were as follows: 0 = normal; 1 = loss of goblet cells in small areas; 2 = loss of goblet cells in large areas; 3 = loss of crypts in small areas; and 4 = loss of crypts in large areas. Infiltration scores were as follows: 0 = normal; 1= infiltrate around crypt base; 2 = moderate infiltrate reaching the muscularis mucosae; 3 = extensive infiltration reaching the muscularis mucosae; and 4 = infiltration of the submucosa.</p>", "<title>5-bromo-2′-deoxyuridine (BrdU) assay</title>", "<p id=\"Par27\">We performed a 5-bromo-2′-deoxyuridine (BrdU) assay according to previously described protocols^{##REF##17934449##34##,##REF##29621481##35##}. Briefly, mice were injected with BrdU (50 mg/kg; Solarbio, Beijing, China) 18 h after LPS treatment and euthanized by CO₂ inhalation 1 h afterward. The entire small intestine was removed, flushed with cold saline, fixed with 4% PFA, and embedded in paraffin. Each group mice count is four (<italic>Mptx2</italic>^{<italic>−/−</italic>}, female, <italic>n</italic> = 2; male, <italic>n</italic> = 2; <italic>Wt</italic>, female, <italic>n</italic> = 2; male, <italic>n</italic> = 2). We stained tissue sections (4 μm) via immunofluorescence (IF) using anti-BrdU antibody followed by examination under a fluorescence microscope (Leica, Wetzlar, Germany). BrdU⁺ cells per crypt were counted for 10 random fields per mouse and then averaged.</p>", "<title>Transmission electron microscopy</title>", "<p id=\"Par28\">We prepared intestinal tissues for transmission electron microscopy (TEM) examination followed protocols^{##REF##27435297##36##,##REF##29416008##37##}. Briefly, tissues from 6-week-old <italic>Mptx2</italic>^{<italic>−/−</italic>} mice and their <italic>Wt</italic> littermates were fixed with 2.5% glutaraldehyde (GLUT) at room temperature (RT). We then washed the tissues, postfixed them with 1% osmium tetroxide in 0.05 mol/L sodium cacodylate buffer (pH 7.4) at 4 °C for 2 h, stained them with saturated uranyl acetate for 3.5 h at RT, dehydrated them in graded alcohol, and embedded them in Eponate 12 resin (Ted Pella, Inc., Redding, CA, USA). Sections were then cut with a diamond knife and stained with a saturated solution of uranyl acetate in 50% ethanol and lead citrate. We examined and photographed the sections under a Philips CM120 transmission electron microscope (Philips Healthcare, Bothell, WA, USA) at 80 kV.</p>", "<title>Scanning electron microscopy</title>", "<p id=\"Par29\">We cut ~5 mm² gut mucosa from <italic>Mptx2</italic>^{<italic>−/−</italic>} and <italic>Wt</italic> littermate mice and fixed them with 2.5% GLUT overnight at 4 °C. The tissues were rinsed, dehydrated in ethyl alcohol, dried with carbon dioxide, coated with gold, and examined under a Hitachi S-4800 field emission scanning electron microscope (SEM; Hitachi, Tokyo, Japan).</p>", "<title>16S rRNA gene sequencing</title>", "<p id=\"Par30\">We extracted total microbial genomic-DNA samples from gut mucosa and feces using a DNeasy PowerSoil Kit (QIAGEN, Inc., Venlo, the Netherlands) per the manufacturer’s instructions. Polymerase chain reaction (PCR) amplification of the bacterial 16S ribosomal-RNA (rRNA) gene V4–V5 region was performed using the forward primer 515 F (5′-GTGCCAGCMGCCGCGGTAA-3′) and the reverse primer 907 R (5′-CCGTCAATTCMTTTRAGTTT-3′). We incorporated sample-specific 7-bp barcodes into primers for multiplex sequencing. PCR amplicons were purified using Agencourt AMPure Beads (Beckman Coulter, Indianapolis, IN, USA) and quantified using a PicoGreen Double-stranded Deoxyribonucleic Acid (dsDNA) Assay Kit (Invitrogen, Carlsbad, CA, USA). After the individual-quantification step, amplicons were pooled in equal amounts, and paired-end (PE) 2 × 300 bp sequencing was performed on an Illumina MiSeq platform with MiSeq Reagent Kit version 3 (Illumina, Inc., San Diego, CA, USA) at Shanghai Personal Biotechnology Co., Ltd. (Shanghai, China). We used the Quantitative Insights into Microbial Ecology (QIIME; <ext-link ext-link-type=\"uri\" xlink:href=\"https://qiime.org\">https://qiime.org</ext-link>) version 1.8.0) pipeline to process sequencing data, as previously described^{##REF##20383131##38##}. Sequence data were mainly analyzed using QIIME and R software version 3.2.0 (R Foundation for Statistical Computing, Vienna, Austria).</p>", "<title>Quantitative real-time polymerase chain reaction amplification of 16S rRNA genes</title>", "<p id=\"Par31\">After recording weights, we extracted bacterial DNA from colonic content using a QIAamp Fast DNA Mini Kit (QIAGEN). Quantitative real-time PCR (qRT-PCR) was performed on an ABIViiA 7 system (Applied Biosystems [Thermo Fisher Scientific, Waltham, MA, USA]) using an SYBR Green Universal Master Mix Kit (Thermo Fisher). We used the following primers, which were modified from a previous study’s sets^{##REF##29259722##39##}: <italic>all bacteria</italic>, F-5′-CGGTGAATACGTTCCCGG-3′ and R-5′-TACGGCTACCTTGTTACGACTT-3′; <italic>Bifidobacterium</italic>, F-5′-CTCCTGGAAACGGGTGG-3′ and R-5′-GGTGTTCTTCCCGATATCTACA-3′; <italic>Lactobacillus</italic>, F-5′-TGGAAACAGRTGCTAATACCG-3′ and R-5′-GTCCATTGTGGAAGATTCCC-3′; <italic>Bacteroides</italic>, F-5′-GAGAGGAAGGTCCCCCAC-3′ and R-5′-CGCTACTTGGCTGGTTCAG-3′; <italic>Prevotella</italic>, F-5′-CACRGTAAACGATGGATGCC-3′ and R-5′-GGTCGGGTTGCAGACC-3′; <italic>Escherichia/Shigella</italic>, F-5′-GAGTAAAGTTAATACCTTTGCTCATTG-3′ and R-5′-GAGACTCAAGCTKRCCAGTATCAG-3′; <italic>Helicobacter</italic>, F-5′-CTATGACGGGTATCCGGCC-3′ and R-5′-TCGCCTTCGCAATGAGTATT-3′; <italic>Staphylococcus</italic>, F-5′-TTTGGGCTACACACGTGCTACAATGGACAA-3′ and R-5′-AACAACTTTATGGGATTTGCWTGA-3′; Commensal segmented filamentous bacteria (Com-SFB), F-5′-AGGAGGAGTCTGCGGCACATTAGC-3′ and R-5′-CGCATCCTTTACGCCCAGTTATTC-3′; and murine SFB (Mus-SFB), F-5′-TGAGCGGAGATATATGGAGC-3’ and R-5′-CATGCAACTATATAGCTATATGCGG-3′.</p>", "<title>Quantitative real-time polymerase chain reaction</title>", "<p id=\"Par32\">Total RNA was extracted from intestinal-mucosal tissues using an RNeasy kit (QIAGEN) per the manufacturer’s protocol. We quantified RNA using a NanoDrop spectrophotometer (Applied Biosystems). A High Capacity Complementary DNA (cDNA) Reverse Transcription Kit (Applied Biosystems) was employed for reverse transcription using 2 μg RNA. Subsequently, we performed real-time PCR reactions using a ViiA 7 Real-Time PCR System with PowerUp SYBR Green Master Mix Kit (both, Applied Biosystems). PCR reactions were incubated in a 384-well plate at 95 °C for 10 min, followed by 40 cycles at 95 °C for 15 s and 60 °C for 1 min. All samples were assayed in triplicate, and data were normalized to endogenous control β-actin. We calculated relative RNA expression levels using the ^ΔΔCt method. Primers, which were modified from previous studies^{##REF##27027293##40##–##REF##35061945##42##} and synthesized by Invitrogen (Shanghai, China), were as follows in Supplementary Table ##SUPPL##0##1##.</p>", "<title>Western blotting</title>", "<p id=\"Par33\">For Western blotting (WB), we homogenized ~50 mg tissue in 500 μL radioimmunoprecipitation assay (RIPA) buffer (Invitrogen, Carlsbad, CA, USA) supplemented with a protease inhibitor cocktail (Servicebio). Bicinchoninic acid (BCA) reagent (Pierce Biotechnology [Thermo Fisher]) was used to determine the protein concentration. Next, we separated equal amounts of protein onto 10% NuPAGE Bis-Tris gels (Invitrogen) and transferred them to polyvinylidene difluoride (PVDF) membranes (MilliporeSigma, Burlington, MA, USA). After blocking in 5% nonfat milk, membranes were incubated with primary antibodies overnight at 4 °C. Information for primary antibodies used in this study was showed Supplementary Table ##SUPPL##0##2##. We then washed the membranes three times with tris-buffered saline containing 0.1% Polysorbate 20 (TBST) and incubated them with secondary antibodies. After final washes of the tissues with TBST, we detected signals using an Electrochemiluminescence (ECL) Reagent Kit (Pierce). All of original WB bands were provided in ##SUPPL##0##Supplementary information##.</p>", "<title>Immunofluorescence (IF) assay</title>", "<p id=\"Par34\">Immunofluorescence (IF) assay was performed as we described previously^{##REF##34168124##43##}. Briefly, the intestinal tissues from were immediately fixed in 4% paraformaldehyde for 24 h and went through dehydration, clearing and paraffin embedding. Sections were mounted on positively charged slides after cutting at 4 μm thick, were then incubated with xylol and descending concentrations of ethanol. After antigen retrieval, blocking was performed using 5% bovine serum albumin for 30 min at room temperature. The primary antibodies were incubated in a humid chamber for overnight at 4 °C. The following day, the slides were incubated with the secondary antibody for 50 min at room temperature away from light after washing with phosphate-buffered saline (PBS). The information for primary antibodies used is listed in Supplementary Table ##SUPPL##0##2##.</p>", "<title>Statistics and reproducibility</title>", "<p id=\"Par35\">Numerical source data for all charts are provided in Methods and Figure legends. Statistical tests were performed using GraphPad Prism 8 Software (GraphPad, San Diego, CA) via two–tailed unpaired t test between two groups and one-way ANOVA for multiple comparisons. Each mouse was assessed as an individual sample. All data were obtained by performing at least 3 independent experiments with representative data shown and expressed as the mean ± standard error of the mean (SEM). <italic>P</italic> values &lt; 0.05 were considered statistically significant. Significance levels were split further as to **<italic>P</italic> &lt; 0.01, ***<italic>P</italic> &lt; 0.001, ****<italic>P</italic> &lt; 0.0001.</p>", "<title>Reporting summary</title>", "<p id=\"Par36\">Further information on research design is available in the ##SUPPL##3##Nature Portfolio Reporting Summary## linked to this article.</p>" ]

[ "<title>Results</title>", "<title><italic>Mptx2</italic> was specifically expressed in Paneth cells</title>", "<p id=\"Par6\">As shown in Fig. ##FIG##0##1a##, <italic>Mptx2</italic> messenger RNA (mRNA) was synthesized in the intestines of <italic>Wt</italic> mice from embryonic day 12.5 (E12.5). The levels <italic>Mptx2</italic> mRNA increased significantly starting on postnatal day 0 (P0; Fig. ##FIG##0##1a##). Similarly, expression of the Paneth cells marker <italic>Lyz1</italic> followed an analogous pattern at the indicated time (Fig. ##FIG##0##1a##). Under normal conditions, expression of Mptx2 and Lysozyme was higher in the mucosa of mouse middle (mid) and distal (dis) small intestine than in that of proximal (pro) small intestine or colon, but it was not for intestinal stem cell marker Lgr5 (Fig. ##FIG##0##1b, c## and Supplementary Fig. ##SUPPL##0##1##). Consistently, immunofluorescence (IF) staining showed that <italic>Mptx2</italic> protein was exclusively expressed in the intestinal mucosa, especially in those of the mid and dis small intestine (Fig. ##FIG##0##1d, e##). IF analysis also showed that <italic>Mptx2</italic> protein was mainly co-localized with the Paneth cell marker <italic>Lyz1</italic> in crypt basements (Fig. ##FIG##0##1d, e##).</p>", "<title><italic>Mptx2</italic>^{<italic>−/−</italic>} mice were predisposed to intestinal inflammation</title>", "<p id=\"Par7\">To examine whether <italic>Mptx2</italic> directly affected intestinal homeostasis, we first generated mice lacking the <italic>Mptx2</italic> gene (<italic>Mptx2</italic>^{<italic>−/−</italic>}, Supplementary Fig. ##SUPPL##0##2a##). Average villus height and crypt depth in <italic>Mptx2</italic>^{<italic>−/−</italic>} mice did not differ from those of their <italic>Wt</italic> littermates (Fig. ##FIG##1##2a## and Supplementary Fig. ##SUPPL##0##2b##). <italic>Mptx2</italic>^{<italic>−/−</italic>} mice had an elevated number of lymphoid structures at the distal part of the small intestines compared to that of <italic>Wt</italic> mice (Fig. ##FIG##1##2a## and Supplementary Fig. ##SUPPL##0##2c##). In agreement with histological findings, inflammatory genes, including <italic>Il1b</italic>, <italic>Ifng</italic>, <italic>Cxcl2</italic>, <italic>Cxcl3</italic>, <italic>Cxcr3</italic>, and <italic>Cxcl12</italic>, were significantly increased in the small-intestinal mucosa of <italic>Mptx2</italic>^{<italic>−/−</italic>} mice compared with their <italic>Wt</italic> littermates (Fig. ##FIG##1##2b##). Transmission electron microscope (TEM) analysis showed abnormalities in intestinal epithelial intercellular junctions and irregular distribution of microvilli in <italic>Mptx2</italic>^{<italic>−/−</italic>} mice compared with their <italic>Wt</italic> littermates (Fig. ##FIG##1##2c##).</p>", "<p id=\"Par8\">As shown Fig. ##FIG##2##3##, we observed the autophagy-associated molecules <italic>Atg12</italic>, <italic>Atg16l1</italic>, and <italic>Becn1</italic> mRNA levels significantly decreased in pro small intestines of <italic>Mptx2</italic>^{<italic>−/−</italic>} mice in relation to <italic>Wt</italic> littermates, but the difference of autophagy marker LC3 (<italic>Map1lc3a</italic>) did not arrive at significant level. (Fig. ##FIG##2##3##). Similarly, ER stress sensor activating transcription factor 4 and 6 (<italic>ATF4</italic>, <italic>ATF6</italic>) mRNA levels especially decreased in the pro small intestines of <italic>Mptx2</italic>^{<italic>−/−</italic>} mice compared with their <italic>Wt</italic> littermates (Fig. ##FIG##2##3##). Subsequently, downstream markers, including spliced <italic>Xbp1</italic> (<italic>sXbp1</italic>), <italic>Chop</italic> (<italic>Ditt3</italic>), <italic>ERdj4</italic>, and <italic>BiP</italic>, were also reduced evidently in the pro small intestines of <italic>Mptx2</italic>^{<italic>−/−</italic>} mice compared with their <italic>Wt</italic> littermates (Fig. ##FIG##2##3##). Interestingly, the levels of these mRNAs did not altered evidently in the mid, dis small intestines or in the colon (Fig. ##FIG##2##3##).</p>", "<title><italic>Mptx2</italic> deficiency altered intestinal bacteria composition in mice</title>", "<p id=\"Par9\">Scanning electron microscopy (SEM) analysis showed that an increased number of invading bacteria attached to and aggregated over the epithelial surface in the small intestines and colons of <italic>Mptx2</italic>^{<italic>−/−</italic>} mice (Fig. ##FIG##3##4a##). We next used 16S rRNA sequencing analysis to explore intestinal bacteria composition in <italic>Mptx2</italic>^{<italic>−/−</italic>} and <italic>Wt</italic> mice (Supplementary Fig. ##SUPPL##0##3##). In comparison with their <italic>Wt</italic> littermates, <italic>Mptx2</italic>^{<italic>−/−</italic>} mice showed greater numbers of operational taxonomic units (OTUs) in the genera <italic>Lactobacillus</italic>, <italic>Bifidobacterium, Akkermansia, Bacteroides</italic>, and <italic>Prevotella</italic> in intestines, but reduced abundances of <italic>Staphylococcus</italic>, <italic>Megasphaera</italic>, <italic>Coprococcus</italic>, and <italic>Pseudomonas</italic> in feces (Fig. ##FIG##3##4b##). Numbers of OTUs in the genera <italic>Staphylococcus, Bacteroides, Lactobacillus</italic>, and increased in the small-intestinal and colonic mucosa in <italic>Mptx2</italic>^{<italic>−/−</italic>} mice (Fig. ##FIG##3##4b##). We further isolated DNA from the small-intestinal mucosa, colonic mucosa, and fecal content and then analyzed it for the presence of bacteria using PCR with bacterial-genus–specific primers. It showed that the numbers of several bacterial genera increased in the intestinal-mucosa of <italic>Mptx2</italic>^{<italic>−/−</italic>} mice compared to that of <italic>Wt</italic> mice, particularly <italic>all bacteria</italic>, <italic>Bacteroides</italic>, and <italic>Prevotella</italic> (Supplementary Fig. ##SUPPL##0##4##).</p>", "<title><italic>Mptx2</italic> knockout aggravated <italic>MRSA</italic> infection with impairing autophagy in Paneth cell</title>", "<p id=\"Par10\">Alcian blue-periodic acid Schiff (AB-PAS) staining showed that <italic>Mptx2</italic>^{<italic>−/−</italic>} mice presented Paneth cell loss compared with <italic>Wt</italic> littermates (Fig. ##FIG##4##5a##). Immunofluorescence (IF) staining-based detection of Lyz1-expressing cells confirmed that Paneth cell count was reduced in the small-intestinal mucosa of <italic>Mptx2</italic>^{<italic>−/−</italic>} mice compared with those of their <italic>Wt</italic> littermates (Fig. ##FIG##4##5b##). Additionally, quantitative teal-time PCR (qRT-PCR) analysis indicated that representative Paneth cell antimicrobial peptides (AMPs), including <italic>Lyz1</italic> and <italic>Reg3g</italic>, decreased in dis small intestinal mucosa of <italic>Mptx2</italic>^{<italic>−/−</italic>} mice, but it did not reach significant level (Fig. ##FIG##4##5c##). Representative electron microscope images further showed that <italic>Mptx2</italic>^{<italic>−/−</italic>} mice had more eosinophilic granules (Fig. ##FIG##4##5d##).</p>", "<p id=\"Par11\">We previously found that administration of recombinant <italic>Mptx2</italic> protein (<italic>rMptx2</italic>) could directly reduce methicillin-resistant <italic>Staphylococcus aureus</italic> (<italic>MRSA</italic>) load in the bloodstream, peritoneal lavage, liver, kidney, spleen, and ileum^{##REF##35609377##18##}. In the current study, we showed that <italic>Mptx2</italic>^{<italic>−/−</italic>} mice were more vulnerable to the <italic>MRSA</italic> infection in distal small intestine than <italic>Wt</italic> mice (Fig. ##FIG##5##6a, b##). Messenger RNA expression of Paneth cell–derived AMPs <italic>Lyz1</italic>, <italic>Reg3g</italic>, and <italic>Defa</italic> was significantly reduced in the distal small intestinal mucosa of <italic>Mptx2</italic>^{<italic>−/−</italic>} mice compared with those of <italic>Wt</italic> mice following <italic>MRSA</italic> infection (Fig. ##FIG##5##6c##). <italic>Lyz1</italic> and <italic>PCNA</italic> protein levels were also reduced in the distal-small intestinal mucosa of <italic>Mptx2</italic>^{<italic>−/−</italic>} mice <italic>versus</italic> those of <italic>Wt</italic> mice (Fig. ##FIG##5##6d, e##). <italic>MAP1LC3</italic> (LC3) is an autophagy marker that can capture and eliminate invading bacteria^{##REF##23768496##19##}. IF staining indicated that <italic>MRSA</italic> infection increased LC3 puncta in distal-small intestinal mucosa. LC3/Lysozyme colocalization puncta was lower in <italic>Mptx2</italic>^{<italic>−/−</italic>} mice compared to those of <italic>Wt</italic> mice (Fig. ##FIG##5##6f, g##). Western blot (WB) confirmed that expression of <italic>LC3</italic> was significantly reduced in the distal-small intestinal mucosa of <italic>Mptx2</italic>^{<italic>−/−</italic>} mice compared to those of <italic>Wt</italic> mice (Fig. ##FIG##5##6h, i##). P62/sequestosome 1 (SQSTM1), a autophagosomal cargo for degradation^{##REF##20144757##20##}, increased in distal-small intestinal mucosa of <italic>Mptx2</italic>^{<italic>−/−</italic>} mice (Fig. ##FIG##5##6h, i##).</p>", "<title><italic>Mptx2</italic>^{<italic>−/−</italic>} mice increased susceptibility to lipopolysaccharide (LPS)-induced intestinal injury</title>", "<p id=\"Par12\">In our LPS-induced mouse sepsis model, <italic>Mptx2</italic> mRNA peaked at 18 h and was gradually silenced by 48 h in the small-intestinal mucosa (Supplementary Fig. ##SUPPL##0##5a##). Nineteen hours after LPS injection, we observed more-severe mucosal injury in the small intestines of <italic>Mptx2</italic>^{<italic>−/−</italic>} mice than in those of their <italic>Wt</italic> littermates (Supplementary Fig. ##SUPPL##0##5b, c##). In addition, we observed that crypt proliferation was impaired in <italic>Mptx2</italic>^{<italic>−/−</italic>} mice <italic>versus</italic> their <italic>Wt</italic> littermates after LPS administration (Supplementary Fig. ##SUPPL##0##6a, b##). Proliferative marker <italic>Yap1</italic>, but not <italic>Lgr5</italic>, decreased in the small-intestinal mucosa of <italic>Mptx2</italic>^{<italic>−/−</italic>} mice (Supplementary Fig. ##SUPPL##0##6c##). WB analysis first showed the tight junction proteins E-cadherin and ZO-1 reduced in the pro and dis small intestines of <italic>Mptx2</italic>^{<italic>−/−</italic>} mice compared to their control littermates, but it failed to reach a significant difference (Fig. ##FIG##6##7a, b## and Supplementary Fig. ##SUPPL##0##7##). The apoptotic marker cleaved caspase-3 increased in the pro and dis small intestines of <italic>Mptx2</italic>^{<italic>−/−</italic>} mice compared to their littermates (Fig. ##FIG##6##7a, b##). It also showed that loss of <italic>Mptx2</italic> impaired the process of the autophagy featured with <italic>Atg5</italic>, <italic>Atg12-Atg5</italic>, and <italic>LC3</italic> proteins were reduced in the pro and dis small intestinal mucosa of <italic>Mptx2</italic>^{<italic>−/−</italic>} mice compared to those of <italic>Wt</italic> mice after LPS-treatment, but the <italic>p62/ SQSTM1</italic> increased in that of <italic>Mptx2</italic>^{<italic>−/−</italic>} mice (Fig. ##FIG##6##7a, b##).</p>", "<title>Loss of <italic>Mptx2</italic> worsened dextran sulfate sodium (DSS)-induced colitis in mice</title>", "<p id=\"Par13\">In our DSS-induced colitis and recovery mouse model, high <italic>Mptx2</italic> expression occurred in colonic mucosa during the acute colitis and recovery phases (Fig. ##FIG##7##8a##). During the process of DSS-induced colitis, <italic>Mptx2</italic>^{<italic>−/−</italic>} mice slightly greater body weight loss than their <italic>Wt</italic> littermates (Supplementary Fig. ##SUPPL##0##8a##). The length of colons did not altered evidently between <italic>Mptx2</italic>^{<italic>−/−</italic>} mice and their <italic>Wt</italic> littermates (Supplementary Fig. ##SUPPL##0##8b, c##). Histologically, <italic>Mptx2</italic>^{<italic>−/−</italic>} mice had greater colonic-mucosal damage and more inflammatory infiltration than DSS-treated <italic>Wt</italic> mice (Fig. ##FIG##7##8b, c##). In agreement with histological findings, expression of inflammatory genes, including <italic>Ifng</italic> and <italic>Cxcl2</italic>, was increased in the colonic mucosa of <italic>Mptx2</italic>^{<italic>−/−</italic>} mice after DSS treatment <italic>versus</italic> their DSS-treated <italic>Wt</italic> littermates, but the difference of neither <italic>Tnfa</italic> nor <italic>Cxcl12</italic> arrive at significant level (Supplementary Fig. ##SUPPL##0##9##). Moreover, <italic>Mptx2</italic>^{<italic>−/−</italic>} mice had fewer goblet cells in their colons than <italic>Wt</italic> mice in the presence of DSS-treatment (Supplementary Fig. ##SUPPL##0##10##). We also found that oral antibiotics (gentamicin, GM or vancomycin, VCM) affected colonic injury and inflammation in <italic>Mptx2</italic>^{<italic>−/−</italic>} mice after DSS treatment (Fig. ##FIG##7##8b, c## and Supplementary Fig. ##SUPPL##0##10a, b##).</p>" ]

[ "<title>Discussion</title>", "<p id=\"Par14\">To the best of our knowledge, this study confirms <italic>Mptx2</italic> acts as a novel marker of Paneth cells, implying that it might play important roles in intestinal inflammation and homeostasis. We firstly showed that <italic>Mptx2</italic> protein was selectively expressed in Paneth cells in the normal crypt base and that it increased in response to LPS treatment or MRSA infection. <italic>Mptx2</italic> deficiency increased susceptibility to intestinal inflammation and injury might via impairing the autophagy process in Paneth cells.</p>", "<p id=\"Par15\">In the normal intestine, we found that <italic>Mptx2</italic> was exclusively expressed in the mucosa and at higher levels in the small intestine than in the colon. We recently found that <italic>Mptx2</italic> mRNA is also expressed in bone marrow and the spleen^{##REF##35609377##18##}. Taken together, these findings suggested that <italic>Mptx2</italic> might be involved in the immune response. In a previous study, a single-cell survey of the small-intestinal epithelium revealed that <italic>Mptx2</italic> might be a new marker for Paneth cells^{##REF##29144463##7##}. Indeed, in this study, we confirmed <italic>Mptx2</italic> protein was mainly localized in Paneth cells. Paneth cells are specialized small-intestinal epithelial cells that reside at the bases of crypts and protect the small intestine from enteropathogens by constitutively secreting a broad spectrum of AMPs and bactericidal proteins^{##REF##21423246##8##,##REF##23398152##21##}. It is reported that autophagy deficiency within the intestinal leads to an aberrant morphology of Paneth cells^{##REF##24089213##11##,##REF##18849966##12##,##REF##20602997##22##}. Our study indicated Mptx2 loss resulted in susceptibility to intestinal inflammation may via the Paneth cells in mice.</p>", "<p id=\"Par16\"><italic>Salmonella</italic> infection has been found to induce expression of AMPs and <italic>Mptx2</italic> in Paneth cells^{##REF##29144463##7##}. In the current study, <italic>Mptx2</italic> mRNA increased in the small-intestinal mucosa shortly after LPS treatment. Therefore, <italic>Mptx2</italic> might defend the gut from bacterial infection by modulating Paneth cell. Indeed, we found that loss of <italic>Mptx2</italic> not only reduced Paneth cell count but also inhibited expression of AMPs such as <italic>Lyz1</italic> and <italic>Reg3g</italic>. Subsequently, SEM indicated that <italic>Mptx2</italic>^{<italic>−/−</italic>} mice had greater numbers of invading bacteria that attached to and aggregated over the epithelial surface of the intestine. 16S rRNA sequencing showed that loss of <italic>Mptx2</italic> altered bacteria composition and caused bacteria including the <italic>Staphylococcus</italic>, <italic>Bacteroides</italic>, and <italic>Enterococcus</italic> increased in the intestinal mucosa. Therefore, it is very likely that <italic>Mptx2</italic> defends against invading bacteria via its bactericidal activity and/or by modulating Paneth cell functions. Indeed, we previously found that <italic>Mptx2</italic> exerted bactericidal activity against methicillin-resistant <italic>Staphylococcus aureus</italic> (<italic>MRSA</italic>) both in vitro and in vivo^{##REF##35609377##18##}. In the colon, <italic>Mptx2</italic>^{<italic>−/−</italic>} mice aggravated DSS-induced colitis that was ameliorated by GM or VCM treatment, indicating <italic>Mptx2</italic> maintain bacteria homeostasis may via its bactericidal activity. It has been reported that Paneth cells secrete lysozyme to counteract bacterial infection via secretory autophagy^{##REF##28751470##23##}. In this study, we found that loss of <italic>Mptx2</italic> aggravated <italic>MRSA</italic> infection with inhibiting the autophagy process in Paneth cells. An impaired ER stress/autophagy crosstalk has been strongly linked to inflammatory bowel disease (IBD)^{##REF##23964099##24##–##REF##21699776##27##}. Conditional deletion of intestinal ER stress-marker <italic>Xbp1</italic> leads to a spontaneous enteritis in mice^{##REF##18775308##25##}. We observed in the current study that <italic>Mptx2</italic> deficiency exaggerated LPS-induced intestinal injury with reducing <italic>Xbp1</italic> expression and autophagy process. Paneth cells are intercalated between active intestinal stem cells (ISCs) in the small intestine (SI) of mice and humans^{##REF##24326621##28##}. Other studies suggest that Paneth cells can constitute a niche for intestinal stem cells in crypts and modulate the regeneration of the intestinal epithelium^{##REF##21113151##29##,##REF##19075245##30##}. Therefore, Paneth cells might produce <italic>Mptx2</italic> to promote the regeneration of the intestinal epithelium via constituting the niche. Therefore, we suggest that <italic>Mptx2</italic> might maintain intestinal homeostasis in three ways: (1) acting as an AMP to kill invading bacteria directly, (2) regulating the secretory functions of beyond regulating the microbiota via secretion of AMPs, and (3) contributing to the intestinal repaired via regulating the autophagy/ER-stress.</p>" ]

[ "<title>Conclusions</title>", "<p id=\"Par17\">Our findings revealed that <italic>Mptx2</italic> was a novel marker of Paneth cells. <italic>Mptx2</italic> deficiency triggered microbiota dysbiosis and increased epithelial invasion by bacteria, leading to greater susceptibility to intestinal inflammation with reducing secretory autophagy. In addition, Paneth cells, which can produce <italic>Mptx2</italic>, contributed to the regeneration of the intestinal epithelium. These findings suggested that <italic>Mptx2</italic> was essential to the functions of Paneth cells and maintained intestinal homeostasis.</p>" ]

[ "<p id=\"Par1\">A recent single-cell survey of the small-intestinal epithelium suggests that mucosal pentraxin 2 (<italic>Mptx2</italic>) is a new Paneth cell marker, but its function and involved mechanism in the Paneth cell are still unknown. Therefore, we create <italic>Mptx2</italic> knockout (<italic>Mptx2</italic>^{<italic>−/−</italic>}) mice to investigate its precise effects on intestinal homeostasis using models of lipopolysaccharide (LPS), methicillin-resistant <italic>Staphylococcus aureus</italic> (<italic>MRSA</italic>) peritoneal infection, and dextran sulfate sodium (DSS)–induced intestinal injury and inflammation. We here find that <italic>Mptx2</italic> is selectively expressed in Paneth cells in the small intestines of mice. <italic>Mptx2</italic>^{<italic>−/−</italic>} mice have increased susceptibility to intestinal inflammation and injured. Mptx2 deficiency reduces Paneth cell count and expression of antimicrobial factors, leading to altered intestinal bacteria composition. Loss of <italic>Mptx2</italic> aggravates MRSA infection–induced damage in the intestine while decreasing autophagy in Paneth cells. <italic>Mptx2</italic>^{<italic>−/−</italic>} mice are more vulnerable to LPS-induced intestinal possibly due to inhibition of the autophagy/endoplasmic reticulum (ER) stress pathway. <italic>Mptx2</italic>^{<italic>−/−</italic>} mice are susceptible to DSS-induced colitis that could be ameliorated by treatment with gentamicin or vancomycin antibiotics. In conclusion, Mptx2 is essential to maintain intestinal homeostasis potentially via regulation of autophagy in Paneth cells.</p>", "<p id=\"Par2\">Mucosal pentraxin 2 (Mptx2) is identified as a marker of Paneth cells, and its deficiency is shown to trigger microbiota dysbiosis and increased epithelial invasion by bacteria while also reducing secretory autophagy.</p>", "<title>Subject terms</title>" ]

[ "<title>Supplementary information</title>", "<p>\n\n\n\n\n</p>" ]

[ "<title>Supplementary information</title>", "<p>The online version contains supplementary material available at 10.1038/s42003-024-05785-7.</p>", "<title>Acknowledgements</title>", "<p>This work was supported by the Natural Science Foundation of Shanghai (22ZR1480600), Shanghai Key Laboratory of Pediatric Gastroenterology and Nutrition (17DZ2272000). We thank Drs. Hui Cai, Jie Zhou, Jing Zhu and Xinbei Tian from Xin Hua hospital, School medicine of Shanghai Jiao Tong University for their technical supports.</p>", "<title>Author contributions</title>", "<p>Y.W., C.S., L.Y., Y.Y., and W.W.: Investigation, Methodology, Visualization. W.Y., W. B., P.S., and D.J.: Software, and Validation. C. W. and X.Y.: Conceptualization, Data curation, Project administration, Resources, Funding acquisition, Supervision, Writing—original draft, Writing—review &amp; editing.</p>", "<title>Peer review</title>", "<title>Peer review information</title>", "<p id=\"Par37\"><italic>Communications Biology</italic> thanks Relber Gonçales and the other, anonymous, reviewer(s) for their contribution to the peer review of this work. Primary Handling Editors: Si Ming Man and David Favero.</p>", "<title>Data availability</title>", "<p>The data generated or analyzed during this study are available from the corresponding author upon reasonable request. Source data, as well as statistical analysis for all graphs, are provided in the Excel file Supplementary Data ##SUPPL##2##1##. Source images for representative Western blots shown in figures are provided in Supplementary Figure ##SUPPL##0##11## in ##SUPPL##0##Supplementary information##. Source data of 16S rRNA gene sequencing is deposited in National Center for Biotechnology Information BioProject (Accession: PRJNA1051373).</p>", "<title>Competing interests</title>", "<p id=\"Par38\">The authors declare no competing interests.</p>" ]

[ "<fig id=\"Fig1\"><label>Fig. 1</label><caption><title><italic>Mptx2</italic> was selectively expressed in the Paneth cells.</title><p><bold>a</bold> Alteration of <italic>Mptx2</italic> mRNA and Lysozyme (<italic>Lyz1</italic>) mRNA from the embryonic stages (E12.5 – 17.5) to the postnatal time (P0 – 13.5) (each group, <italic>n</italic> = 4). <bold>b</bold> Western blot (WB) analysis for Mptx2 and Lysozyme in mouse proximal (pro), middle (mid), distal (dis) small bowel and colon (each group, <italic>n</italic> = 3). Independent experiments at least two times. <bold>c</bold> Quantification of <bold>b</bold>. <bold>d</bold> Quantification of Mptx2 and Lysozyme positive cells in the different segments of mice intestine in <bold>e</bold>. <bold>e</bold> Representative images of immunofluorescence (IF) staining for Mptx2 and Lysozyme in mouse proximal (pro), middle (mid), distal (dis) small bowel and colon (each group, <italic>n</italic> = 4). Unpaired two-tailed Student’s <italic>t</italic> test with or without Welch’s correction analysis for D.*<italic>p</italic> &lt; 0.05, *** <italic>p</italic> &lt; 0.001,**** <italic>p</italic> &lt; 0.0001.</p></caption></fig>", "<fig id=\"Fig2\"><label>Fig. 2</label><caption><title><italic>Mptx2</italic> deficiency triggered intestinal inflammation.</title><p><bold>a</bold> Representative images of hematoxylin and eosin (H&amp;E) staining for the proximal (pro), middle (mid), distal (dis) small bowel and colon from both <italic>Mptx2</italic>^{<italic>−/−</italic>} mice (<italic>n</italic> = 12) and <italic>Wt</italic> mice (<italic>n</italic> = 12). <bold>b</bold> qRT-PCR analysis of inflammatory genes mRNA expression in distal (dis) small bowel from both <italic>Mptx2</italic>^{<italic>−/−</italic>} mice (<italic>n</italic> = 5) and <italic>Wt</italic> mice (<italic>n</italic> = 7). <bold>c</bold> Representative images of transmission electron microscopy (TEM) in proximal (pro), distal (dis) small bowel and colon from <italic>Mptx2</italic>^{<italic>−/−</italic>} mice (<italic>n</italic> = 3) and <italic>Wt</italic> mice (<italic>n</italic> = 3). Unpaired two-tailed Student’s <italic>t</italic> test with or without Welch’s correction analysis for <bold>b</bold>.*** <italic>p</italic> &lt; 0.001,**** <italic>p</italic> &lt; 0.0001.</p></caption></fig>", "<fig id=\"Fig3\"><label>Fig. 3</label><caption><title>Loss of <italic>Mptx2</italic> impaired the autophagy/endoplasmic reticulum (ER) stress in mice intestine.</title><p>qRT-PCR analysis of endoplasmic reticulum (ER) stress-autophagy genes mRNA expression in mouse proximal (pro), middle (mid), distal (dis) small bowel and colon from both <italic>Mptx2</italic>^{<italic>−/−</italic>} mice and <italic>Wt</italic> mice (each group, <italic>n</italic> = 4-6). Unpaired two-tailed Student’s <italic>t</italic> test with or without Welch’s correction analysis. ns not significant, *<italic>p</italic> &lt; 0.05, **<italic>p</italic> &lt; 0.01, *** <italic>p</italic> &lt; 0.001.</p></caption></fig>", "<fig id=\"Fig4\"><label>Fig. 4</label><caption><title><italic>Mptx2</italic> deficiency altered the intestinal microbiota composition.</title><p><bold>a</bold> Representative images of scanning electron microscopy (SEM) analysis for the proximal (pro), distal (dis) small bowel, and colon from both <italic>Mptx2</italic>^{<italic>−/−</italic>} mice (<italic>n</italic> = 3) and <italic>Wt</italic> mice (<italic>n</italic> = 3). <bold>b</bold> The relative abundance of the top bacteria (genus) in the intestinal mucosa and feces of <italic>Mptx2</italic>^{<italic>−/−</italic>} mice and <italic>Wt</italic> mice. WTp: <italic>Wt</italic> mice proximal intestine; MPp: <italic>Mptx2</italic> KO mice proximal intestine; WTm: <italic>Wt</italic> mice middle intestine; MPm: <italic>Mptx2</italic> KO mice middle intestine; WTd: <italic>Wt</italic> mice distal intestine; MPd: <italic>Mptx2</italic> KO mice distal intestine; WTc: <italic>Wt</italic> mice colon; MPc: <italic>Mptx2</italic> KO mice colon; WTf <italic>Wt</italic> mice feces, MPf <italic>Mptx2</italic> KO mice feces (Each group, <italic>n</italic> = 4–6).</p></caption></fig>", "<fig id=\"Fig5\"><label>Fig. 5</label><caption><title><italic>Mptx2</italic> loss decreased the Paneth cells.</title><p><bold>a</bold> Representative images of Alcian blue/periodic acid Schiff base (AB-PAS) staining for the proximal (pro), middle (mid), and distal (dis) small bowel from both <italic>Mptx2</italic>^{<italic>−/−</italic>} mice (<italic>n</italic> = 5) and <italic>Wt</italic> mice (<italic>n</italic> = 6). Quantification of Paneth cells number in both <italic>Mptx2</italic>^{<italic>−/−</italic>} mice (<italic>n</italic> = 5) and <italic>Wt</italic> mice (<italic>n</italic> = 6). <bold>b</bold> Representative images of immunofluorescence (IF) staining of lysozyme for the proximal (pro), middle (mid), and distal (dis) small bowel from both <italic>Mptx2</italic>^{<italic>−/−</italic>} mice (<italic>n</italic> = 3) and <italic>Wt</italic> mice (<italic>n</italic> = 3); Lysosome (green) and DAPI (blue). Quantification of lysozyme-positive cells. <bold>c</bold> Quantitative real-time PCR (qRT-PCR) of <italic>Lyz1</italic> and <italic>Reg3g</italic> in the mucosa of distal (dis) small bowel. <bold>d</bold> Transmission electron microscopy (TEM) analysis for Paneth cells. Arrows indicated granules. Unpaired two-tailed Student’s <italic>t</italic> test with or without Welch’s correction analysis for <bold>a</bold>–<bold>c</bold>. ns not significant, *<italic>p</italic> &lt; 0.05, **<italic>p</italic> &lt; 0.01, ***<italic>p</italic> &lt; 0.001.</p></caption></fig>", "<fig id=\"Fig6\"><label>Fig. 6</label><caption><title><italic>Mptx2</italic> deficiency worsened the MRSA-infection and disrupted the secretory autophagy in Paneth cells.</title><p><bold>a</bold> Schematic of the MRSA-infected mice model. <italic>Mptx2</italic>^{<italic>−/−</italic>} mice (<italic>n</italic> = 7) and <italic>Wt</italic> mice (<italic>n</italic> = 12) mice were infected intraperitoneally with a dose of 1.5 × 10⁷ CFU (colony forming units) of MRSA. This image and every element of this image created by author Dr. X.Y. <bold>b</bold> Representative images of histology in small intestine sections from <italic>Mptx2</italic>^{<italic>−/−</italic>} mice and <italic>Wt</italic> mice (each group, <italic>n</italic> = 4–6). <bold>c</bold> Quantitative real-time PCR (qRT-PCR) of <italic>Lyz1</italic>, <italic>Reg3g</italic>, and defensin alpha (<italic>Defa</italic>) in the mucosa of distal (dis) small bowel followed the MRSA-infection in <italic>Mptx2</italic>^{<italic>−/−</italic>} mice (<italic>n</italic> = 6) and <italic>Wt</italic> mice (<italic>n</italic> = 6). <bold>d</bold> The western blotting (WB) analysis was used to determine the expression levels of PCNA and Lysozyme in the distal mucosa of <italic>Mptx2</italic>^{<italic>−/−</italic>} mice and <italic>Wt</italic> mice. Independent experiments at least two times. <bold>e</bold> Quantification of them against β-actin (each group, <italic>n</italic> = 3). <bold>f</bold>, <bold>g</bold> Representative images of Immunofluorescence analysis for LC3 and Lysosome in the sections of small intestine of <italic>Mptx2</italic>^{<italic>−/−</italic>} mice and <italic>Wt</italic> mice and quantification of them (each group, <italic>n</italic> = 4). <bold>h</bold>, <bold>i</bold> The western blotting (WB) analysis for P62 and LC3 protein in the distal mucosa of <italic>Mptx2</italic>^{<italic>−/−</italic>} mice and <italic>Wt</italic> mice. Quantification of them against β-actin (each group, <italic>n</italic> = 3). These bands from different membranes that have the same protein loading and have their own β-actin as housekeeping protein. Independent experiments at least two times. Unpaired two-tailed Student’s <italic>t</italic> test with or without Welch’s correction analysis for <bold>c</bold>, <bold>e</bold>, <bold>i</bold>. Ordinary One-way ANOVA analysis for F. ns not significant, *<italic>p</italic> &lt; 0.05, **<italic>p</italic> &lt; 0.01, ***<italic>p</italic> &lt; 0.001.</p></caption></fig>", "<fig id=\"Fig7\"><label>Fig. 7</label><caption><title><italic>Mptx2</italic> deficiency increased LPS-induced small intestinal injury with inhibiting the ER stress-autophagy.</title><p><bold>a</bold> Representative images of western blotting (WB) analysis for E-cadherin, Atg5, Atg12, LC3,ZO-1,Cleaved-Caspase3 and P62 proteins in small intestines of <italic>Mptx2</italic>^{<italic>−/−</italic>} mice and <italic>Wt</italic> mice with or without LPS treatment. <bold>b</bold> The qualification of WB results in <bold>a</bold>. These bands from different membranes that have the same protein loading and have their own β-actin as housekeeping protein. Independent experiments at least two times. Ordinary One-way ANOVA analysis for B. Aspect ratio of Atg12-Atg5 bands has been adjusted compared to the original image because of limited space. The original images were provided in ##SUPPL##0##Supplementary Information##. Each group, <italic>n</italic> = 3, ns, not significant, *<italic>p</italic> &lt; 0.05, **<italic>p</italic> &lt; 0.01, ***<italic>p</italic> &lt; 0.001, ****<italic>p</italic> &lt; 0.0001.</p></caption></fig>", "<fig id=\"Fig8\"><label>Fig. 8</label><caption><title><italic>Mptx2</italic> deficiency exaggerated DSS-induced colitis.</title><p><bold>a</bold> qRT-PCR analysis of <italic>Mptx2</italic> mRNA expression in colons of mice (<italic>n</italic> = 10–14) subjected to DSS-induced colitis. <bold>b</bold> Quantification of pathological scores of <bold>c</bold>. <bold>c</bold> Representative images of haematoxylin &amp; eosin (H&amp;E) staining on the colon of mice. Ordinary One-way ANOVA analysis for <bold>b</bold>. each group, <italic>n</italic> = 6–10, ns not significant, *<italic>p</italic> &lt; 0.05, ****<italic>p</italic> &lt; 0.0001.</p></caption></fig>" ]

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[ "<supplementary-material content-type=\"local-data\" id=\"MOESM1\"></supplementary-material>", "<supplementary-material content-type=\"local-data\" id=\"MOESM2\"></supplementary-material>", "<supplementary-material content-type=\"local-data\" id=\"MOESM3\"></supplementary-material>", "<supplementary-material content-type=\"local-data\" id=\"MOESM4\"></supplementary-material>" ]

[ "<fn-group><fn><p><bold>Publisher’s note</bold> Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.</p></fn><fn><p>These authors contributed equally: Weihui Yan, Shanshan Chen.</p></fn></fn-group>" ]

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[ "<media xlink:href=\"42003_2024_5785_MOESM1_ESM.pdf\"><caption><p>Supplementary Information</p></caption></media>", "<media xlink:href=\"42003_2024_5785_MOESM2_ESM.pdf\"><caption><p>Description of Additional Supplementary Files</p></caption></media>", "<media xlink:href=\"42003_2024_5785_MOESM3_ESM.xlsx\"><caption><p>Supplementary data 1</p></caption></media>", "<media xlink:href=\"42003_2024_5785_MOESM4_ESM.pdf\"><caption><p>Reporting Summary</p></caption></media>" ]

{ "acronym": [], "definition": [] }

[ "<title>Introduction</title>", "<p id=\"Par3\">The recent development of mapping technologies such as Hi–C^{##REF##19815776##1##} that probes the 3D genome organization reveals that a chromosome is divided into topologically associating domains (TADs)^{##REF##22495300##2##,##REF##22495304##3##}. TADs are genomic regions where chromatin loci more frequently interact with other chromatin loci within the TAD than with those from outside of the TAD. TADs are functional units for transcriptional regulation by constraining interactions between enhancers and promoters^{##REF##31887284##4##}, for example. Although TADs are stable between cell types as revealed by earlier studies^{##REF##22495300##2##,##REF##25693564##5##}, there is growing evidence for TAD reorganization in diseases^{##REF##26940867##6##–##REF##36289338##8##}, cell differentiation^{##REF##25693564##5##,##REF##26971819##9##,##REF##28709003##10##}, somatic cellular reprogramming^{##REF##34038708##11##}, between neuronal cell types^{##REF##34789882##12##}, and between species^{##REF##33203573##13##,##REF##36179666##14##}. For example, extensive reorganizations of TADs are observed during somatic cell reprogramming, associating with dynamics of transcriptional regulation and changes in cellular identity^{##REF##34038708##11##}. TADs are also variable among individual cells, as revealed by single-cell studies^{##REF##24067610##15##–##REF##33077913##20##} and live-cell imaging^{##REF##35420890##21##}. Thus, it is important to identify reorganized TADs through comparative analysis to further understand the functional relevance of 3D genome organization, a major priority of current work in the field^{##REF##32976797##22##}.</p>", "<p id=\"Par4\">The majority of current methods call a reorganized TAD if at least one of its two boundaries changed between two conditions^{##REF##34038708##11##,##REF##34789882##12##,##REF##29949963##23##–##REF##35637517##28##}. These methods enable easy integration with other analysis pipelines and identify reorganized TADs with clear biological interpretation. However, they fail to identify reorganized TADs without changes in boundaries, in addition to lacking statistical tests to differentiate random perturbations and significant structural reorganization of a TAD. Only a few nonparametric statistical methods are proposed to call TAD reorganization^{##UREF##1##29##–##REF##36869353##32##}. These methods define the structural similarity of a TAD by statistics from two Hi–C contact matrices, such as the stratum-adjusted correlation coefficient used by DiffGR^{##UREF##2##30##}. Distributions of the statistics on pairs of simulated Hi–C matrices are then used to compute empirical <italic>P</italic> values. However, these nonparametric statistical methods are conservative (see our own comparison later). TADs in high-resolution Hi–C data are relatively small. The median size of TADs is 185 kb^{##REF##25497547##33##}. The small size feature of TADs poses another computational challenge for identifying structurally rewired TADs using low-resolution Hi–C data. Importantly, identifying reorganized TADs using emerging single-cell Hi–C (scHi–C) data is largely under-explored. Other methods are developed for comparing Hi–C matrices at different scales and for different purposes: quantifying similarities of genome-wide Hi–C contact matrices^{##REF##28855260##34##,##REF##33077914##35##}, identifying differential A/B compartments^{##REF##36369226##36##}, and identifying differential chromatin interactions^{##REF##30668639##37##–##REF##31510653##39##}. However, these methods are not tailored to compare Hi–C contact matrices at the TAD level, which is not optimal for identifying reorganized TADs (see our own comparison later). Therefore, new algorithms are needed to fill these gaps.</p>", "<p id=\"Par5\">Here, we develop DiffDomain, a new parametric statistical method for identifying reorganized TADs. Its inputs are two Hi–C contact matrices from two biological conditions and a set of TADs called in biological condition 1. This setting enables straightforward integration of DiffDomain with other analysis pipelines of Hi–C data, such as TAD calling and integrative analysis of multi-omics data. For each TAD, DiffDomain directly computes a difference matrix and then normalizes it properly, skipping the challenging normalization steps for individual Hi–C contact matrices. DiffDomain then borrows well-established theoretical results in random matrix theory to compute a <italic>P</italic> value. We show that the assumptions of DiffDomain are reasonable. Method comparisons on real data reveal that DiffDomain has substantial advantages over alternative methods in false positive rates and accuracy in identifying truly reorganized TADs. Reorganized TADs identified by DiffDomain are biologically relevant in different human cell lines and disease states. Application to scHi–C data reveals that DiffDomain can identify reorganized TADs between cell types and TADs with differential variabilities among individual cells within the same cell type. Moreover, DiffDomain can quantify cell-to-cell variability of TADs between individual cells. Together, these analyses demonstrate the power of DiffDomoain for better identification of structurally reorganized TADs using both bulk Hi–C and single-cell Hi–C data.</p>" ]

[ "<title>Methods</title>", "<p id=\"Par30\">In this section, we first introduce the first part of DiffDomain: a model-based method to identify reorganized TADs. We state the model assumptions and their verification using real Hi–C data before reporting the second part of DiffDomain: classification of reorganized TADs into six subtypes. Last is missing value imputation.</p>", "<title>Model-based method to identify reorganized TADs</title>", "<p id=\"Par31\">In this paper, our aim is to identify a subset of TADs that are reorganized between two biological conditions, such as a pair of healthy and diseased cell lines/tissues. Specifically, given a set of TADs identified in one biological condition, we aim to identify the subset of TADs that are reorganized in another biological condition. To achieve this goal, we develop DiffDomain that takes a set of TADs and their Hi–C contact matrices as the input. The TADs are identified using the Arrowhead method^{##REF##25497547##33##} (Supplementary Method ##SUPPL##0##1),## and Hi–C contact matrices specific to each TAD region are extracted from the genome-wide KR-normalized Hi–C contact maps unless specified otherwise. The core of DiffDomain is converting the comparison of Hi–C contact matrices into a hypothesis-testing problem on their difference matrix. This difference matrix is modeled as a symmetric random matrix, enabling DiffDomain to borrow well-established theoretical results in high-dimensional random matrix theory.</p>", "<p id=\"Par32\">Before explaining the hypothesis testing problem, we first introduce some mathematical notations and normalization operations. For each TAD in biological condition 1, let <italic>N</italic> denote the number of consecutive and equal-length chromosome bins within the genomic region covered by the TAD. Let represent the symmetric KR-normalized Hi–C contact matrix, where represents the non-negative Hi–C contact frequency between chromosome bins <italic>i</italic> and <italic>j</italic> (1 ≤ <italic>i</italic>, <italic>j</italic> ≤ <italic>N</italic>) in the TAD region in condition 1. In other words, <bold><italic>A</italic></bold>₁ serves as the Hi–C contact matrix specific to the TAD region in biological condition 1, forming a submatrix within the genome-wide Hi–C contact matrix. Similarly, denotes the KR-normalized Hi–C contact matrix corresponding to the same TAD region but in biological condition 2. It is well-known that the Hi–C contact frequency <italic>A</italic>_{<italic>i</italic><italic>j</italic>} exponentially decreases with an increased linear distance between bins <italic>i</italic> and <italic>j</italic>. We first log-transform the Hi–C contact matrices <bold><italic>A</italic></bold>₁ and <bold><italic>A</italic></bold>₂ and compute their entry-wise differences, denoted by <bold><italic>D</italic></bold>, as shown in Eq. (##FORMU##11##2##).Values in Hi–C contact matrices <bold><italic>A</italic></bold>₁ and <bold><italic>A</italic></bold>₂ could have large differences because of variations in reading depths. For example, the GM12878 Hi–C experiment has 4.76 times more Hi–C contacts than the K562 Hi–C experiment (Supplementary Table ##SUPPL##0##2)##. Among the 889 GM12878 TADs on Chromosome 1, the averages in the 889 <bold><italic>D</italic></bold>s range from 1.479 to 2.611, with a median at 2.229. To adjust for the differences due to variations in read depths, we normalize by standardizing each of its <italic>k</italic>-off diagonal blocks bywhere <italic>k</italic> = <italic>j</italic> − <italic>i</italic>, 2 − <italic>N</italic>≤<italic>k</italic>≤<italic>N</italic> − 2, and are the sample mean and standard deviation of . Here, without abuse of notations, we continue to use <bold><italic>D</italic></bold> to denote the resulted normalized difference matrix. Note that the normalization is TAD-specific because two different TADs most likely have different and , 2 − <italic>N</italic> ≤ <italic>k</italic> <italic>N</italic> − 2. Besides visualization in Fig. ##FIG##0##1##a–c, the effects of the above procedures for both bulk and single-cell Hi–C matrices from the same TAD are also visualized in Supplementary Fig. ##SUPPL##0##1##.</p>", "<p id=\"Par33\">Intuitively, if a TAD does not undergo structural reorganization from biological condition 1 to biological condition 2, the differences between <bold><italic>A</italic></bold>₁ and <bold><italic>A</italic></bold>₂ are caused by multiple factors, including variations in read depths and random perturbations of 3D genome organization. Thus, we assume that entries in <bold><italic>D</italic></bold> follow a standard Gaussian distribution, resulting in <bold><italic>D</italic></bold> being a symmetric random noise matrix with entries that follow a standard Gaussian distribution. Scaling <bold><italic>D</italic></bold> by , where <italic>N</italic> represents the number of bins in the TAD, results in exhibiting characteristics typical of a well-studied random matrix known as a generalized Wigner matrix. With the justifications presented in the next subsection, we assume that is a generalized Wigner matrix. The problem of identifying reorganized TADs is reformulated as the following hypothesis testing problem:The largest eigenvalue of , denoted by <italic>λ</italic>_{<italic>N</italic>}, converges to 2 with increased <italic>N</italic>^{##UREF##5##67##}. This result helps us to reformulate the hypothesis testing problem as the following:Under <italic>H</italic>₀, <italic>θ</italic>_{<italic>N</italic>} = <italic>N</italic>^2/3(<italic>λ</italic>_{<italic>N</italic>} − 2), a normalized <italic>λ</italic>_{<italic>N</italic>}, converges in distribution to Tracy-Widom distribution with index <italic>β</italic> = 1, denoted as <italic>T</italic><italic>W</italic>₁^{##UREF##6##68##}.In other words, under <italic>H</italic>₀, a TAD does not undergo structural reorganization in biological condition 2. Then, the fluctuations of <italic>θ</italic>_{<italic>N</italic>} is governed by Tracy–Widom distribution <italic>T</italic><italic>W</italic>₁. Thus, we choose <italic>θ</italic>_{<italic>N</italic>} as the test statistic and compute a one-sided <italic>P</italic> value byA smaller <italic>P</italic> value means that the TAD is more likely to be reorganized in condition 2.</p>", "<p id=\"Par34\">For a set of TADs, <italic>P</italic> values are adjusted for multiple comparisons by a few methods, with the BH method as the default. The pseudocode of our DiffDomain algorithm is presented in Supplementary Method ##SUPPL##0##2##.</p>", "<title>Model assumptions and their verifications</title>", "<p id=\"Par35\">Given two KR-normalized Hi–C contact matrices, DiffDomain computes the normalized difference matrix <bold><italic>D</italic></bold>, bypassing complicated further normalization of individual Hi–C contact matrices^{##REF##27580841##69##}. Thus, DiffDomain makes no explicit assumptions on the individual Hi–C contact matrices. DiffDomain only makes assumptions on the normalized difference matrix <bold><italic>D</italic></bold>. First, under <italic>H</italic>₀, DiffDomain assumes that is a generalized Wigner matrix: a symmetric random matrix with independent mean zero upper diagonal entries. Symmetry is satisfied by because Hi–C contact matrices are symmetric. The independence assumption on the upper diagonal entries is violated by considering the well-known fact that Hi–C contact frequencies positively correlate with each other among nearby chromosome bins. However, the violation of the independence assumption does not substantially alter the properties of and DiffDomain (Supplementary Note ##SUPPL##0##1)##. Briefly, the empirical properties of and DiffDomain resemble the following theoretical properties: (i) empirical spectral distribution of a generalized Wigner matrix converging to the well-established semicircle law^{##UREF##7##70##}, (ii) <italic>λ</italic>_{<italic>N</italic>} → 2^{##UREF##5##67##}, (iii) unadjusted <italic>P</italic> values following a uniform distribution when <italic>H</italic>₀ is true and model assumptions are satisfied (Supplementary Note ##SUPPL##0##1##, Supplementary Fig. ##SUPPL##0##3)##. The key result (##FORMU##25##5##) requires one more assumption. It holds under the condition that the distributions of entries in generalized Wigner matrices have vanishing third-moments as <italic>N</italic> tends to infinity^{##UREF##8##71##}. After the standardization procedure (##FORMU##13##3##), approximately follows a Gaussian distribution <italic>N</italic>(0, 1/<italic>N</italic>) whose third moment is 0, satisfying the vanishing third-moment assumption. Taken together, assumptions of DiffDomain are appropriate. For additional references on the generalized Wigner matrix, please refer to the comprehensive books authored by Bai and Silverstein^{##UREF##9##72##} and Couillet and Liao^{##UREF##10##73##}.</p>", "<title>Reorganized TAD classification</title>", "<p id=\"Par36\">Once a subset of reorganized TADs is identified, the classification of reorganized TADs is critical to interpreting TAD reorganization and linking them to the dynamics of genome functions. Motivated by classifications in previous studies^{##REF##28709003##10##,##UREF##2##30##,##REF##32211023##31##}, DiffDomain classifies reorganized TADs into six subtypes: <italic>strength-change</italic>, <italic>loss</italic>, <italic>split</italic>, <italic>merge</italic>, <italic>zoom</italic>, and <italic>complex</italic>. TADs are hierarchically organized, as identified by methods such as Arrowhead. Large TADs can subdivide into smaller TADs, and a genomic region may belong to multiple TADs, complicating reorganized TAD classification. To address this, we compare the TAD list in condition 1 with the TAD list in condition 2, utilizing combinations of identical TADs and overlapping TADs between the two conditions to distinguish the distinct reorganized TAD subtypes (Fig. ##FIG##0##1##f, Supplementary Method ##SUPPL##0##3)##. A brief description of the subtypes is provided below.<list list-type=\"order\"><list-item><p id=\"Par37\"><italic>Strength-change</italic> represents that the boundaries of the reorganized TAD are the same in both conditions. Specifically, the reorganized TAD in condition 1 has a one-to-one identical relationship with a TAD in condition 2.</p></list-item><list-item><p id=\"Par38\"><italic>Loss</italic> represents that the reorganized TAD in condition 1 does not overlap with or be identical to any TADs in condition 2.</p></list-item><list-item><p id=\"Par39\"><italic>Split</italic> represents that a reorganized TAD in condition 1 is split into at least two TADs in biological condition 2. Specifically, the reorganized TAD has either a one-to-many identical relationship or a one-to-many overlapping relationship with TADs in condition 2.</p></list-item><list-item><p id=\"Par40\"><italic>Merge</italic> represents that the reorganized TAD has a many-to-one identical or a many-to-one overlapping relationship with a TAD in condition 2. Specifically, the reorganized TAD and at least one of its adjacent/overlapping TADs in condition 1 are identical to or overlap with a single TAD in condition 2. <italic>Merge</italic> is the opposite of <italic>split</italic> when condition 2 is treated as condition 1.</p></list-item><list-item><p id=\"Par41\"><italic>Zoom</italic> represents that the reorganized TAD in condition 1 has a one-to-one overlapping relationship with a TAD in condition 2, addition to not being identical to any TADs in condition 2.</p></list-item><list-item><p id=\"Par42\"><italic>Complex</italic> represents the other reorganized TADs.</p></list-item></list></p>", "<p id=\"Par43\">After the classification of reorganized TADs into the six subtypes, the <italic>strength-change</italic> TADs are further subdivided into two categories. Within a <italic>strength-change</italic> TAD, the Hi–C contact frequencies either increase or decrease in biological condition 2, after proper normalization on the differences in total sequenced reads. Subsequently, a <italic>strength-change</italic> TAD can be classified into a <italic>strength-change up</italic> TAD or a <italic>strength-change down</italic> TAD. Before explaining the classification, a few mathematical notions are introduced. Given a <italic>strength-change</italic> TAD, let <italic>m</italic>₁ be the median value of KR-normalized Hi–C contact frequencies within the strength-change TAD in condition 1, <italic>m</italic>₂ be the median value of the KR-normalized Hi–C contact frequencies within the same TAD region but in condition 2. Let <italic>s</italic>₁ be the sum of the KR-normalized Hi–C contact frequencies across all condition 1 TADs, <italic>s</italic>₂ be the sum of the KR-normalized Hi–C contact frequencies across all condition 2 TADs. If the <italic>strength-change</italic> TAD satisfies , it is classified as a <italic>strength-change up</italic> TAD. Otherwise, the <italic>strength-change</italic> TAD is classified as a <italic>strength-change down</italic> TAD.</p>", "<title>Missing value imputation</title>", "<p id=\"Par44\">Missing values may exist in Hi–C contact matrices <bold><italic>A</italic></bold>₁ or <bold><italic>A</italic></bold>₂ for a specific TAD region, and their origin can vary. DiffDomain distinguishes between missing values caused by SVs and those caused by other factors, such as low sequencing depth. When SVs are present, say in condition 2, DiffDomain first checks if an <italic>m</italic> × <italic>m</italic> submatrix, <italic>m</italic>≥3, of <bold><italic>A</italic></bold>₂ contains exclusively missing values. In such cases, the <italic>m</italic> × <italic>m</italic> submatrix is imputed with a constant, with the default value of 1. Otherwise, if any row and column with a proportion of missing values greater than a given threshold, with a default value of 0.5, DiffDomain removes the corresponding row/column from both <bold><italic>A</italic></bold>₁ and <bold><italic>A</italic></bold>₂. Subsequently, the remaining missing values are imputed by the median contact frequency of interactions at the same distance within the corresponding contact matrix.</p>", "<title>Reporting summary</title>", "<p id=\"Par45\">Further information on research design is available in the ##SUPPL##2##Nature Portfolio Reporting Summary## linked to this article.</p>" ]

[ "<title>Results</title>", "<title>Overview of DiffDomain</title>", "<p id=\"Par6\">The workflow of DiffDomain is illustrated in Fig. ##FIG##0##1##. Its input is a set of TADs called in biological condition 1 and their corresponding Hi–C contact matrices from biological conditions 1 and 2 (Fig. ##FIG##0##1##a). In this paper, TADs are called by Arrowhead^{##REF##25497547##33##} and Hi–C contact matrices are KR-normalized, unless otherwise stated (Supplementary Method ##SUPPL##0##1##). Our goal is to test if each TAD identified in biological condition 1 has significant structural reorganization in biological condition 2. The core of DiffDomain is formulating the problem as a hypothesis testing problem, where the null hypothesis is that the TAD doesn’t undergo significant structural reorganization in condition 2. To achieve this goal, for each TAD with <italic>N</italic> bins, DiffDomain extracts the <italic>N</italic> × <italic>N</italic> KR-normalized Hi–C contact matrices specific to the TAD region from the two biological conditions, which are denoted as <bold><italic>A</italic></bold>₁ and <bold><italic>A</italic></bold>₂ (Fig. ##FIG##0##1##a). Note that <bold><italic>A</italic></bold>₁ and <bold><italic>A</italic></bold>₂ are <italic>N</italic> × <italic>N</italic> submatrices of the genome-wide Hi–C contact matrices. DiffDomain first log-transform them to adjust for the exponential decay of Hi–C contacts with increased 1D distances between chromosome bins. Their difference is calculated and denoted by <bold><italic>D</italic></bold> (Fig. ##FIG##0##1##b). <bold><italic>D</italic></bold> is further normalized by a 1D distance-stratified standardization procedure, similar to the procedures in HiC-DC+^{##REF##34099725##38##} and SnapHiC^{##REF##34446921##40##}. Specifically, each <italic>d</italic>-off diagonal part of <bold><italic>D</italic></bold> is subtracted by its sample mean and divided by its sample standard deviation (Fig. ##FIG##0##1##c), −<italic>N</italic> + 2 ≤ <italic>d</italic> ≤ <italic>N</italic> − 2, reducing 1D distance-dependence of values in <bold><italic>D</italic></bold> and differences caused by variation in read depths between two biological conditions (see Supplementary Fig. ##SUPPL##0##1## for two more detailed visualization). Intuitively, if a TAD is not significantly reorganized, normalized <bold><italic>D</italic></bold> would resemble a white noise random matrix, enabling us to borrow theoretical results in random matrix theory. Under the null hypothesis, DiffDomain assumes that is a generalized Wigner matrix (Fig. ##FIG##0##1##d), a well-studied random matrix model. Its largest eigenvalue <italic>λ</italic>_{<italic>N</italic>} is proved to be fluctuating around 2. Armed with this fact, DiffDomain reformulates the reorganized TAD identification problem into the hypothesis testing problem:The key theoretical results empowering DiffDomain is that <italic>θ</italic>_{<italic>N</italic>} = <italic>N</italic>^2/3(<italic>λ</italic>_{<italic>N</italic>} − 2), a normalized <italic>λ</italic>_{<italic>N</italic>}, asymptotically follows a Tracy-Widom distribution with <italic>β</italic> = 1, denoted as <italic>T</italic><italic>W</italic>₁. Thus, <italic>θ</italic>_{<italic>N</italic>} is chosen as the test statistic, and the one-sided <italic>P</italic> value is calculated as . <italic>H</italic>₀ is rejected if the <italic>P</italic> value is less than a predefined significant level <italic>α</italic>, which is 0.05 in this paper (Fig. ##FIG##0##1##e). The pseudocode is shown in Supplementary Method ##SUPPL##0##2##. For a set of TADs, <italic>P</italic> values are adjusted for multiple comparisons using the Benjamini–Hochberg (BH) method as the default. Once DiffDomain identifies the subset of reorganized TADs, it further classifies them into six subtypes based on changes in their boundaries, which is beneficial for downstream biological analyses and interpretations (Fig. ##FIG##0##1##f, Supplementary Method ##SUPPL##0##3)##. TAD reorganization subtypes are verified by aggregation peak analyses (APA) on multiple real datasets (Supplementary Fig. ##SUPPL##0##2)##. A few reorganized TADs in real Hi–C data are shown in Fig. ##FIG##0##1##g. Details are described in the Methods section.</p>", "<p id=\"Par7\">Note that although each biological condition may have multiple Hi–C replicates, DiffDomain takes the combined Hi–C contact matrix from the replicates as the input, which is a common practice to generate a large number of Hi–C interactions^{##REF##25497547##33##}. Correlations among Hi–C interactions lead to correlations among entries in the matrix, violating the independent assumption among upper diagonal entries of generalized Wigner matrix. However, the violation of the independence assumption does not substantially alter the properties of and DiffDomain based on empirical analysis results, suggesting that assumptions of DiffDomain are appropriate (see Methods section, Supplementary Note ##SUPPL##0##1##, Supplementary Fig. ##SUPPL##0##3)##. When <italic>N</italic> = 10, the Tracy-Widom distribution <italic>T</italic><italic>W</italic>₁ is an adequate approximation of the exact distribution of <italic>θ</italic>_{<italic>N</italic>}^{##REF##23667298##41##}. Under common 10 kb resolution Hi–C data, <italic>N</italic> = 10 refers to TADs with 100 kb in length, much smaller than the median TAD length of 185 kb^{##REF##25497547##33##}. Thus DiffDomain only computes the <italic>P</italic> value for TADs with at least 10 chromosome bins, a practical constraint. DiffDomain is robust to a varied number of sequencing reads, Hi–C resolution, and different TAD callers (Supplementary Note ##SUPPL##0##2##, Supplementary Figs. ##SUPPL##0##4## and ##SUPPL##0##5)##.</p>", "<title>DiffDomain consistently outperforms alternative methods in multiple aspects</title>", "<p id=\"Par8\">First, we assess the false positive rate (FPR) which is the ratio of the number of false positives to the number of true negatives. A smaller FPR means that the identified significantly reorganized TADs are more likely to be true. Due to the lack of gold-standard data, we resort to analyzing the proportions of significantly reorganized TADs between five different Hi–C replicates from the GM12878 cell line. These Hi–C replicates are generated by different experimental procedures and have a highly varied total number of Hi–C contacts (Supplementary Table ##SUPPL##0##1)##. However, the TADs are expected to have few structural changes between these Hi–C replicates. The proportion of identified reorganized TADs is treated as an estimate of FPR (Supplementary Method ##SUPPL##0##4)## and is expected to be small. The Hi–C resolution is chosen as 10 kb. Comparing GM12878 Hi–C replicates <italic>primary</italic> and <italic>replicate</italic>, we find that DiffDomain, TADCompare^{##REF##32211023##31##}, and HiCcompare^{##UREF##3##42##} have FPRs that are close to the given significant level of 0.05, suggesting good controls of FPR. In contrast, DiffGR^{##UREF##2##30##}, DiffTAD^{##UREF##1##29##}, and HiC−DC+^{##REF##34099725##38##} have inflated FPRs (more than two-fold higher than 0.05), indicating poor controls of FPR (Fig. ##FIG##1##2##a). Similar results are observed by repeating the above analysis to other GM12878 Hi–C replicates and 25 kb resolution Hi–C data (Supplementary Fig. ##SUPPL##0##6)##.</p>", "<p id=\"Par9\">Good control in FPR does not necessarily represent high power in detecting reorganized TADs between biological conditions. To investigate this, we compare TADs between two blood cell lines, GM12878 and K562. DiffDomain identifies that 30.771% of GM12878 TADs are reorganized in K562. In contrast, TADCompare, HiCcompare, and HiC-DC+ only identify ≤8.256% of GM12878 TADs that are reorganized in K562 (Fig ##FIG##1##2##b), suggesting that they are too conservative in identifying reorganized TADs between biological conditions. Similar results are observed by repeating the above analysis to other human cell lines and 25 kb resolution Hi–C data (Supplementary Table ##SUPPL##0##2##, Supplementary Fig. ##SUPPL##0##7)##, demonstrating the robustness of the observations. Conservation of TADCompare is because it is designed to scan every chromatin loci for potential reorganized TAD boundaries. But this analysis uses a given list of TADs, a common practice in Hi–C data analysis, which sharply narrows down the search space of TADCompare. Conservations of HiCcompare and HiC−DC+ are because they are designed for detecting differential chromatin interactions, not specifically tailored for identifying reorganized TADs.</p>", "<p id=\"Par10\">Compared with TADsplimer which specifically identifies <italic>split</italic> and <italic>merge</italic> TADs^{##REF##32241291##25##}, DiffDomain identifies similar numbers of <italic>split</italic> and <italic>merge</italic> TADs between multiple pairs of human cell lines (Supplementary Fig. ##SUPPL##0##8)##. Importantly, DiffDomain identifies that the majority (minimum 43.137%, median 81.357%, maximum 98.022%) of the identified reorganized TADs are the other four subtypes (Supplementary Fig. ##SUPPL##0##9)##, which can not be detected by TADsplimer. For example, among the GM12878 TADs that are identified as reorganized in K562 by DiffDomain, <italic>strength-change</italic> is the leading subtype of reorganized TADs, consistent with the fact that both GM12878 and K562 are blood cell lines (Fig ##FIG##1##2##c). These results demonstrate that DiffDomain has substantial improvements over TADsplimer.</p>", "<p id=\"Par11\">We next investigate the true positive rate (TPR). A higher TPR means that more truly reorganized TADs are correctly identified as reorganized TADs. Through an extensive literature search, we collect 65 TADs that are reorganized between 146 pairs of biological conditions in 15 published papers (Supplementary Table ##SUPPL##0##3##, Supplementary Method ##SUPPL##0##5)##. We use these TADs as the gold standard data to compute the TPR (Supplementary Method ##SUPPL##0##4)## and also call these TADs truly reorganized TADs. Four truly reorganized TADs, either correctly identified or missed by DiffDomain, are shown in Fig. ##FIG##1##2##d. HiCcompare and HiC−DC+, designed for identifying differential chromatin interactions, are not directly applicable to the only testing reorganization of one single TAD and thus are excluded from the analysis. We find that the TPR of DiffDomain is 68.493%, which is 1.639, 2.703, and 16.665 times higher than that of alternative methods DiffGR, DiffTAD, and TADCompare, respectively (Fig. ##FIG##1##2##e). Compared with DiffDomain, DiffGR, DiffTAD, and TADCompare only uniquely identify 11, 10, and 1 truly reorganized TADs, respectively (Supplementary Fig. ##SUPPL##0##10)##. Closer examination shows that DiffDomain has much smaller <italic>P</italic> values than other methods (Wilcoxon rank-sum test, <italic>P</italic> ≤ 2.31 × 10⁻⁸, Fig. ##FIG##1##2##f), demonstrating that DiffDomain has stronger statistical evidence in favor of truly reorganized TADs. Based on the depictions of TAD changes reported in the publications, the truly reorganized TADs are broadly categorized into three groups: domain-level change, boundary-level change, and loop-level change (Supplementary Method ##SUPPL##0##5)##. These groups have decreased reorganization levels with increased stratum-adjusted correlation coefficient (SCC) scores^{##REF##28855260##34##} between biological conditions (Supplementary Fig. ##SUPPL##0##11a)##. Across the groups, DiffDomain consistently achieves the highest TPRs, while the second-best method varies (Supplementary Fig. ##SUPPL##0##11b)##, further demonstrating the advantages of DiffDomain over alternative methods. DiffDomain still misses 31.507% of possible pairwise comparisons of truly reorganized TADs. One reason is that some of the missed truly reorganized TADs have highly similar Hi–C contact matrices between biological conditions. For example, the missed truly reorganized TAD, chr11:1500000–2200000 (Fig. ##FIG##1##2##d), has the SCC score at 0.998. Generally, missed truly reorganized TADs have significantly (<italic>P</italic> = 8.26 × 10⁻⁶) higher SCC scores than those correctly identified reorganized TADs by DiffDomain (Fig. ##FIG##1##2##g). Similar results are observed when stratifying by the groups of truly reorganized TADs (Supplementary Fig. ##SUPPL##0##11c)##. Because DiffGR uses SCC as the test statistic, these results also partially explain the low TPR (41.781%) of DiffGR and highlight that SCC alone is not optimal for identifying reorganized TADs. Another reason is that the resolutions of some Hi–C data are low since <italic>P</italic> values are moderately negatively associated with the maximum values of Hi–C contact matrices (Spearman’s rank correlation coefficient <italic>ρ</italic> = − 0.534).</p>", "<p id=\"Par12\">Additionally, DiffDomain is efficient in memory usage and acceptable in computation time compared with alternative methods (Supplementary Note ##SUPPL##0##3##, Supplementary Fig. ##SUPPL##0##12)##.</p>", "<p id=\"Par13\">In summary, compared with alternative methods, DiffDomain has multiple improvements, including FPRs, proportions of identified reorganized TADs between different biological conditions, subtypes of reorganized TADs, and TPRs.</p>", "<title>Reorganized TADs are associated with epigenomic changes</title>", "<p id=\"Par14\">Armed with the advantages of DiffDomain over alternative methods, we explore the connections between TAD reorganization and epigenomic dynamics. We first showcase a GM12878 TAD that is significantly reorganized in K562 and classified as a <italic>strength-change</italic> TAD by DiffDomain (Fig. ##FIG##2##3##). The TAD covers a 445 kb region on chromosome 6. The TAD structural changes involve the vascular endothelial growth factor gene <italic>VEGFA</italic>, which is a major tumor angiogenic gene that is over-expressed in leukemia (see reviews^{##REF##16633338##43##,##REF##18463380##44##} for more details), consistent with the fact that K562 cells are chronic myelogenous leukemia cells. We find that the reorganized TAD has K562-specific functional annotations. The genomic region covered by the TAD is more accessible (1.71 times higher DNase peak coverage) in K562 than in GM12878 (Fig. ##FIG##2##3##b). The H3K27ac and H3K4me1 peak coverages of the TAD region in K562 are 3.24 and 3.28 times higher than the coverages in GM12878, respectively. In contrast, the H3K4me3 and H3K36me3 peak coverages of the TAD in K562 are only 1.31 and 1.25 times higher than the coverages in GM12878, respectively. Four regions in the TAD are annotated as super-enhancers^{##REF##24119843##45##} only in K562 (Fig. ##FIG##2##3##b). Note that the TAD region is in A compartments in both cell types, suggesting that the A/B compartments switch is not the reason for the gain in accessibility and histone modifications that are associated with gene activation. The normalized difference matrix <bold><italic>D</italic></bold> between Hi–C contact matrices of the TAD highlights that super-enhancer SE2 has increased Hi–C contacts with the <italic>VEGFA</italic> gene in K562 (Fig. ##FIG##2##3##c). To gain further insights into structural differences of the TAD in the two cell types, we compare the 3D structural representations of the TAD region. We run Chrom3D^{##REF##29700484##46##} 100 times to construct 100 possible 3D structures in each cell type for statistical comparisons. Two possible 3D structures with each per cell type illustrate the 3D structural differences of the TAD between GM12878 and K562 (Fig. ##FIG##2##3##d, e). Overall, the super-enhancer SE2, but not SE3 (Fig. ##FIG##2##3##b), is much spatially closer (<italic>P</italic> &lt; 2.22 × 10⁻¹⁶) to VEGFA in K562 than in GM12878 (Supplementary Fig. ##SUPPL##0##13)##. These results show that the reorganized GM12878 TAD in K562 has K562-specific chromatin organization and potential biological functions.</p>", "<p id=\"Par15\">Generally, comparative analyses across multiple pairs of human normal and disease cell lines reveal that <italic>strength-change</italic> reorganized TADs with increased contact frequencies have significant increase in the number of CTCF binding sites at TAD boundaries compared with other TADs, whereas lost TAD boundaries associated with <italic>loss</italic>, <italic>zoom</italic>, and <italic>merge</italic> TADs have significantly fewer number of CTCF binding sites, consisting with enrichment of CTCF binding sites in TAD boundaries (Supplementary Note ##SUPPL##0##4##, Supplementary Fig. ##SUPPL##0##14)##. Across diverse human cell lines, while TADs remain relatively stable, their proportions of reorganized TADs vary depending on the cell type, and these variations can cluster cell lines with similar cell identities (Supplementary Note ##SUPPL##0##5##, Supplementary Figs. ##SUPPL##0##7## and ##SUPPL##0##15)##. GM12878 (normal lymphoblastoid cell line) TADs that are reorganized in K562 (chronic myeloid leukemia cell lines) are enriched (<italic>P</italic> = 0.01) in disease genes in chronic myelogenous leukemia. Across pairs of cell types, reorganized TADs tend to gain in chromatin accessibility and active transcription signals H3K27ac and K3K4me1. Particularly, TAD reorganization subtypes have distinct associations with chromatin accessibility as well as histone modifications. Specifically, TAD reorganization subtypes <italic>strength-change-up</italic>, <italic>zoom</italic>, <italic>split</italic>, and <italic>complex</italic> are associated with increased chromatin accessibility and histone modification signals marking active transcription activities. Conversely, TAD reorganization subtypes <italic>loss</italic>, <italic>strength-change-down</italic>, and <italic>merge</italic> are associated with decreased histone modifications signals marking active transcription activities, emphasizing the importance of TAD reorganization subtypes in investigating genome activity and functionality (Supplementary Note ##SUPPL##0##6##, Supplementary Figs. ##SUPPL##0##16##–##SUPPL##0##18)##. Compared to normal human astrocytes (NHA), patient-derived diffuse intrinsic pontine glioma cell lines DIPG007 and DIPGXIII share a substantial proportion (73.46%) of reorganized TADs, some harboring potential oncogenes and super-enhancers, while dBET6 treatment demonstrates a stronger effect on TAD reorganization than BRD4 inhibition (Supplementary Note ##SUPPL##0##7##, Supplementary Figs. ##SUPPL##0##19##–##SUPPL##0##21##, Supplementary Table ##SUPPL##0##4)##. Together, these results demonstrate the functional relevance of reorganized TADs in multiple human normal and disease cell lines.</p>", "<title>Reorganized TADs are enriched with structural variations (SVs)</title>", "<p id=\"Par16\">SVs can contribute to diseases by rewiring 3D genome organization. To further demonstrate the biological relevance of reorganized TADs, we systematically investigate the associations between SVs and reorganized TADs. High-resolution SVs, including deletions and duplications, from erythroleukemia (K562 cell line) and pediatric high-grade glioma (DIPG007 and DIPGXIII cell lines) are downloaded from Wang et al.^{##REF##35704579##47##}. Because Arrowhead TADs does not necessarily cover the whole genome, SVs are filtered by keeping only those with their genomic regions overlapping with TADs (illustrated by two examples in Fig. ##FIG##3##4##a). The number of SVs and paired normal Hi–C data are summarized in Supplementary Table ##SUPPL##0##5##.</p>", "<p id=\"Par17\">If an SV region overlaps with one reorganized TAD, we consider the SV to have an associated reorganized TADs. We find that the majority (72.2%) of the SVs have such associations (Fig ##FIG##3##4##b), with proportions significantly higher than those of randomly sampled, equal-numbered reorganized TADs (Supplementary Fig. ##SUPPL##0##22)##. SVs are associated with distinct abnormal patterns in Hi–C contact maps and are categorized into four types: deletions and duplications with 5′ to 3′ fusion, 5′ to 5′ fusion, and 3′ to 3′ fusion^{##REF##35704579##47##}. When stratified by SV types, the majority (&gt;55%) of SVs with the same type also have associated reorganized TADs (Fig ##FIG##3##4##b). However, each type of SVs has distinct associations with the subtypes of reorganized TADs. For example, in the comparison between GM12878 and K562 cell lines, the reorganized TADs associated with the four types of K562 SVs have differential distributions over their subtypes (Fig. ##FIG##3##4##c). The leading subtype of reorganized TADs, <italic>strength-change</italic>, is consistently observed across the four types of SVs. However, the second leading subtype of reorganized TADs varies among the four types of SVs (Fig. ##FIG##3##4##c). This observation is further emphasized by evident differences in the APA plots (Fig. ##FIG##3##4##d). Importantly, the association between SV type and TAD reorganization subtype is disease-specific, supported by the clear distinctions in both the APA plots and the subtype distributions of reorganized TADs across K562, DIPG007, and DIPGXIII cell lines (Fig. ##FIG##3##4##c, d). Particularly, leading reorganized TAD subtypes associated with SVs vary among cell types, with <italic>strength-change</italic> and <italic>loss</italic> in K562; <italic>strength-change</italic>, <italic>zoom</italic>, and <italic>split</italic> in DIPG007; and <italic>loss</italic> in DIPGXIII (Fig. ##FIG##3##4##c). This variability may be due to the substantial differences in SV lengths among these cell types (Fig. ##FIG##3##4##e). Notably, small proportions (17.4–27.7%, 8 in K562, 10 in DIPG007, and 4 in DIPGXIII) of SVs lack associated reorganized TADs (Fig. ##FIG##3##4##b). Upon visual examination through the Nucleome Browser, in total, 13 SVs have reorganized TADs that are not detected by DiffDomain, implying that the remaining 9 SVs in the three cell types may lack associated TAD reorganization (Supplementary Figs. ##SUPPL##0##23##–##SUPPL##0##25)##. Nevertheless, these results significantly enhance our understanding of the relationship between SVs and TADs compared to the previous study^{##REF##35704579##47##}, further highlighting the biological relevance of reorganized TADs.</p>", "<title>DiffDomain improves profiling of TAD reorganization related to SARS-CoV-2 infection</title>", "<p id=\"Par18\">Severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) caused over 640 million confirmed coronavirus disease 2019 (COVID-19) cases, including over 6.6 million deaths, worldwide as of December 2, <ext-link ext-link-type=\"uri\" xlink:href=\"https://covid19.who.int\">2022</ext-link>, posing a huge burden to global public health. Wang et al.^{##REF##36959507##48##} is the first Hi–C study into the effects of SARS-CoV-2 infection on host 3D genome organization, finding a global pattern of TAD weakening after SARS-CoV-2 infection. However, the analysis uses aggregation domain analyses which cannot directly identify individual weakened TADs, in addition to missing other subtypes of TAD reorganization.</p>", "<p id=\"Par19\">To further demonstrate the biological applications of DiffDomain, we reanalyze the data. We find that 20.58% (840 in 4082) mock-infected A549-ACE2 TADs are reorganized in SARS-CoV-2-infected A549-ACE2 cells. Among the reorganized TADs, <italic>strength-change</italic> TADs are the leading subtype (64.64%) (Fig. ##FIG##4##5##a), which is consistent with the global pattern of TAD weakening^{##REF##36959507##48##}, verifying the reorganized TADs identified by DiffDomain. The following most frequent subtypes are <italic>merge</italic> TADs and <italic>complex</italic> TADs (18.45% and 10.95%, Fig. ##FIG##4##5##a), refining the characterization of TAD reorganization after SARS-CoV-2 infection. These reorganized TADs also enable refined profiling of transcriptional regulation in response to SARS-CoV-2 infection. Compared with the other TADs, the reorganized TADs have significantly higher numbers of upregulated genes and downregulated genes (<italic>P</italic> ≤ 1.27 × 10⁻⁴, Fig. ##FIG##4##5##b, Supplementary Fig. ##SUPPL##0##26a)##. Similar significant patterns are observed comparing <italic>strength-change</italic> TADs and <italic>split</italic> TADs with the other TADs (<italic>P</italic> ≤ 8.07 × 10⁻³, Supplementary Fig. ##SUPPL##0##26b)##, highlighting that the two subtypes have stronger connections with differentially expressed genes than other subtypes of reorganized TADs. In contrast, compared to the other TADs, the six subtypes of reorganized TADs have significantly higher numbers of both enhanced and weakened peaks of H3K27ac, SMC3, and RAD21 where H3K27ac is a marker for active enhancers and SMC3 and RAD21 are two critical cohesin subunits that regulate 3D genome organization (Supplementary Note ##SUPPL##0##8##, Fig. ##FIG##4##5##c–e, Supplementary Fig. ##SUPPL##0##26c–e)##. Gene-centric analysis shows that differentially expressed genes in reorganized TADs have stronger connections with differential chromatin interactions than in other TADs. In particular, the <italic>strength-change</italic> subtype has a 3-fold higher proportion (9.73%) of downregulated genes with both enhanced and weakened chromatin interactions compared to other TADs (Supplementary Note ##SUPPL##0##9##, Supplementary Fig. ##SUPPL##0##27)##. These results suggest that, after SARS-CoV-2 infection, the subtypes of reorganized TADs all have strong associations with epigenome reprogram, and <italic>strength-change</italic> TADs and <italic>split</italic> TADs have strong associations with deregulation of gene expression, highlighting the importance of subtypes of reorganized TADs identified by DiffDomain.</p>", "<title>DiffDomain characterizes cell-to-population and cell-to-cell variability of TADs using scHi–C data</title>", "<p id=\"Par20\">Recent advances in scHi–C sequencing methods enable profiling of 3D genome organization in individual cells, revealing intrinsic cell-to-cell variability of TADs among individual cells. However, quantifying the variability is challenging due to the properties of scHi–C data, such as high sparsity, low genome coverage, and heterogeneity^{##REF##34465168##49##,##REF##34406348##50##}. As a proof-of-concept, we apply DiffDomain to a moderate-sized scHi–C dataset from mouse neuronal development (median number of contacts per cell at 400,000)^{##REF##33484631##51##}.</p>", "<p id=\"Par21\">We first ask how many individual cells are sufficient to identify reorganized TADs between cell types with high reproducibility using pseudo-bulk Hi–C data (Supplementary Method ##SUPPL##0##6.1)##. To do this, we design a sampling experiment to gradually increase the number of used individual cells, and the reproducibility in identified reorganized TADs is quantified using the Jaccard index (Supplementary Method ##SUPPL##0##6.2)##. We find that DiffDomain can identify reorganized TADs between cell types with reasonable reproducibility (average Jaccard index ≥ 0.104) using as few as one hundred sampled cells (Fig. ##FIG##5##6##b, c). For example, DiffDomain consistently identifies that a neuronal TAD, harboring neuronal marker genes <italic>GM24071</italic>, <italic>LRFN2</italic>, <italic>MOCS1</italic>, and <italic>1700008K24RIK</italic>^{##REF##33484631##51##}, is reorganized in oligodendrocytes with numbers of cells starting at 100 (Fig. ##FIG##5##6##a). Consistent of DiffDomain on other example genomic regions are shown in Supplementary Fig. ##SUPPL##0##28##. On average, DiffDomain identifies that 19.25% neuronal TADs are reorganized in oligodendrocytes using only 250 sampled cells from each cell type, consistent (average Jaccard index at 0.49) with the identified reorganized TADs using all available cells in both cell types (Fig. ##FIG##5##6##b). Similar results are observed when identifying neonatal neuron 1 (the youngest structure type) TADs that are reorganized in cortical L2–5 pyramidal cells (adult type) (Fig. ##FIG##5##6##c, d). Jointly increasing the numbers of sampled cells in both cell types improves the performance of DiffDomain, as expected (Fig. ##FIG##5##6##b). In contrast, only increasing the number of sampled cells in one cell type has a limited boost in performance. For example, oligodendrocytes have only 257 cells, but neurons have 1380 cells. Further increasing the number of sampled neurons from 250 to 500 has a slight performance improvement (Fig. ##FIG##5##6##b). The observation is further confirmed when comparing neuronal subtypes neonatal neuron 1 and neonatal cortical L2–5 pyramidal cells, in which the number of sampled cells in the latter subtype is no more than 150 (Fig. ##FIG##5##6##c) or 228 (Fig. ##FIG##5##6##d). Repeating the analysis by using bulk Hi–C data^{##REF##28671686##52##,##REF##24185899##53##} to create gold-standard reorganized TADs, we observed similar patterns in neuronal TADs that are reorganized in astrocytes. For example, sampling 150 cells in each cell type identifies 12.40% neuronal TADs that are reorganized in astrocytes on average. Among the reorganized TADs, 62.55% are also identified as reorganized TADs when bulk Hi–C data are used (Fig. ##FIG##5##6##e). Considering the median number of contacts per cell at 400000, the merged Hi–C data from hundreds of cells are ultra-sparse pseudo-bulk Hi–C data. These results demonstrate that DiffDomain can work with ultra-sparse Hi–C data.</p>", "<p id=\"Par22\">Next, we move to quantify the cell-to-population variability of TADs, that is, comparing TADs in individual cells to the population average. To do this, scHi–C data with 50 kb resolution from neonatal neuron 1 cells is imputed by scHiCluster^{##REF##31235599##54##}. For each TAD, DiffDomain compares the imputed Hi–C contact map of the TAD in each cell to the pseudo-bulk Hi–C contact map. Resulted <italic>P</italic> values reflect cell-to-population variability of TADs and thus are used by hierarchical clustering to divide TADs into three categories: high, median, and low cell-to-population variational TADs (Fig. ##FIG##6##7##a). We find that TADs have clear differential cell-to-population variability. One example high cell-to-population variational TAD and its adjacent median cell-to-population variational TADs in 9 cells are shown in Fig. ##FIG##6##7##c. Among the 2146 neonatal neuron 1 TADs, 8.90% (191) are high cell-to-population variational TADs, 7.50% (161) and 83.60% (1794) are median and low cell-to-population variational TADs (Fig. ##FIG##6##7##b). They are distributed across chromosomes (Fig. ##FIG##6##7##d). Similar results are observed in other cell types (Supplementary Fig. ##SUPPL##0##29)##. These results demonstrate that TADs have clear differential variability between individual cells and the population average, consistent with earlier observations^{##REF##34635838##26##,##REF##28289288##55##}.</p>", "<p id=\"Par23\">Next, we move to investigate the cell-to-cell variability of TADs. Requiring only one Hi–C contact matrix from each condition, DiffDomain can directly quantify cell-to-cell variability of TADs between individual cells using imputed scHi–C data. Note that, similar to other methods, pairwise comparison of TADs using scHi–C data from thousands of individual cells leads to exponential growth in runtime and thus is computationally expensive^{##REF##34465168##49##}. As a proof of concept, we apply DiffDomain to scHi–C data from randomly selected 50 cortical L2–5 pyramidal cells and 50 adult astrocytes. We find that the cell-to-cell variability of TADs is heterogeneous. The heterogeneity is consistent among the pairwise comparisons but quite different from those from random scenarios in which equal-numbered reorganized TADs are randomly assigned in pairwise comparisons. The proportion of reorganized TADs is consistent among the 2500 pairs of individual cells, ranging from 46.0% to 75.7% (Fig ##FIG##6##7##e). Across the 50 cortical L2–5 pyramidal cells, the proportions of TADs that are reorganized in a varied number of adult astrocytes are fairly consistent (Fig. ##FIG##6##7##f). Moreover, the proportions of TADs that are either low in cell-to-cell variability (reorganized in no more than 10 adult astrocytes) or high in cell-to-cell variability (reorganized in more than 40 adult astrocytes) are much higher than those from random scenarios (Fig. ##FIG##6##7##f, Supplementary Fig. ##SUPPL##0##30)##, consistent with differential cell-to-population variability of TADs as reported in the previous paragraph. This observation is also in concordance with the randomized placement of TAD-like blocks in individual cells but with a strong preference for TAD boundaries observed in bulk Hi–C data^{##REF##30361340##18##}, further demonstrating the utilization of DiffDomain.</p>", "<p id=\"Par24\">In summary, DiffDomain works on scHi–C data to identify reorganized TADs between cell types, identify TADs with differential cell-to-population variability, and characterize cell-to-cell variability of TADs.</p>" ]

[ "<title>Discussion</title>", "<p id=\"Par25\">In this work, we present a statistical method, DiffDomain, for comparative analysis of TADs using a pair of Hi–C datasets. Extensive evaluation using real Hi–C datasets demonstrates clear advantages of DiffDomain over alternative methods for controlling false positive rates and identifying truly reorganized TADs with much higher accuracy. Applications of DiffDomain to Hi–C datasets from different cell lines and disease states demonstrate that reorganized TADs are enriched with structural variations and associated with CTCF binding site changes and other epigenomic changes, revealing their condition-specific biological relevance. By applying it to a scHi–C dataset from mouse neuronal development, DiffDomain can identify reorganized TADs between cell types with considerable reproducibility using pseudo-bulk Hi–C data from as few as a hundred cells. Moreover, DiffDomain can reliably characterize the cell-to-population and cell-to-cell variability of TADs using scHi–C data.</p>", "<p id=\"Par26\">The major methodological contribution of DiffDomain is directly characterizing the differences between Hi–C contact matrices using high-dimensional random matrix theory. First, DiffDomain makes no explicit assumption on the input chromatin contact matrices, directly applicable to both bulk and single-cell Hi–C data. Second, DiffDomain computes the largest eigenvalue <italic>λ</italic>_{<italic>N</italic>} of a properly normalized difference contact matrix <bold><italic>D</italic></bold>, enabling the quantification of the differences of a TAD using all chromatin interactions within the TAD. Third, leveraging the asymptotic distribution of <italic>λ</italic>_{<italic>N</italic>}, DiffDomain computes theoretical <italic>P</italic> values, which is much faster in computation than simulation methods used in alternative methods. The model assumptions are realistic (Methods). Last but not least, the normalized difference contact matrix <bold><italic>D</italic></bold> can help pinpoint genomic regions with increased or decreased chromatin interactions within the reorganized TAD, enabling model interpretation and refined integrative analysis with other genomic and epigenomic data.</p>", "<p id=\"Par27\">There is room for improvement. First, DiffDomain has the highest accuracy in detecting truly reorganized TADs, but it misses some truly reorganized TADs that only show subtle structural changes (see example in Fig. ##FIG##1##2##d, g). Developing more powerful model-based methods is future work. The manually created list of gold-standard reorganized TADs is deposited in the GitHub repository that hosts the source code, which would benefit the research community for better method development. Second, because of the hierarchy of TADs and sub-TADs^{##REF##22495300##2##,##REF##25497547##33##}, generalizations of DiffDomain to explicitly consider dependencies among TADs to further refine reorganized TADs identification and classification is future work. Third, it would be desirable to generalize DiffDomain to compare other TAD-like domains^{##REF##33397451##56##–##REF##31925403##61##}. scHi–C data is imputed by scHiCluster^{##REF##31235599##54##} for the characterization of cell-to-population and cell-to-cell variability of TADs. Benchmarking the effects of different imputation algorithms, including Higashi^{##REF##34635838##26##,##REF##36265466##62##} and scVI-3D^{##REF##34980209##63##}, on quantifying cell-to-population and cell-to-cell variability of TADs is future work.</p>", "<p id=\"Par28\">As a subset of TADs, the identified reorganized TADs could be a critical unit for refined integrative analyses of multi-omics data. We demonstrate this type of application on different human cell lines and disease states, including SARS-CoV-2-infected A549-ACE2 cells. Applying DiffDomain to investigate the connections between TAD reorganization and changes in H3K27me3 modification, a marker recently implicated in development and disease^{##REF##35617427##64##,##REF##31837995##65##}, is a valuable future work. Notably, future work integrating multiple types of omics data and functional perturbation experiments^{##REF##33558716##66##} is necessary to elucidate the causal relationships between TAD reorganization and disease.</p>", "<p id=\"Par29\">DiffDomain is an interpretable statistical method for enhanced comparative analysis of TADs and it works for both bulk and single-cell Hi–C data. The accelerated application of Hi–C and scHi–C mapping technologies would generate ever-growing numbers of bulk and single-cell Hi–C data from different health and disease states. DiffDomain and its future generalizations would be an essential part of the Hi–C analysis toolkit for the emerging comparative analysis of TADs, which in turn would advance understanding of the genome’s structure-function relationship in health and disease.</p>" ]

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[ "<p id=\"Par1\">Topologically associating domains (TADs) are critical structural units in three-dimensional genome organization of mammalian genome. Dynamic reorganizations of TADs between health and disease states are associated with essential genome functions. However, computational methods for identifying reorganized TADs are still in the early stages of development. Here, we present DiffDomain, an algorithm leveraging high-dimensional random matrix theory to identify structurally reorganized TADs using high-throughput chromosome conformation capture (Hi–C) contact maps. Method comparison using multiple real Hi–C datasets reveals that DiffDomain outperforms alternative methods for false positive rates, true positive rates, and identifying a new subtype of reorganized TADs. Applying DiffDomain to Hi–C data from different cell types and disease states demonstrates its biological relevance. Identified reorganized TADs are associated with structural variations and epigenomic changes such as changes in CTCF binding sites. By applying to a single-cell Hi–C data from mouse neuronal development, DiffDomain can identify reorganized TADs between cell types with reasonable reproducibility using pseudo-bulk Hi–C data from as few as 100 cells per condition. Moreover, DiffDomain reveals differential cell-to-population variability and heterogeneous cell-to-cell variability in TADs. Therefore, DiffDomain is a statistically sound method for better comparative analysis of TADs using both Hi–C and single-cell Hi–C data.</p>", "<p id=\"Par2\">Topologically associating domains (TADs) are critical structural units in 3D genome organization, and their reorganization between health and disease states is associated with essential genome functions. However, computational methods for identifying reorganized TADs are still in the early stages of development. Here, the authors present an algorithm leveraging random matrix theory to identify reorganized TADs.</p>", "<title>Subject terms</title>" ]

[ "<title>Supplementary information</title>", "<p>\n\n\n\n</p>" ]

[ "<title>Supplementary information</title>", "<p>The online version contains supplementary material available at 10.1038/s41467-024-44782-6.</p>", "<title>Acknowledgements</title>", "<p>This work was supported by the National Natural Science Foundation of China grant 12271536 (D.T. and Z.B.), National Key Research and Development Program of China grant 2021YFC2300102 (D.T.), GuangDong Basic and Applied Basic Research Foundation grant 2022A1515010043 (D.T.), Shenzhen Sustainable Research grant KCXFZ20211020172545006 (D.T.), National Natural Science Foundation of China grant 12171198 (Z.B.), and Jilin Provincial Foundation grant 20210101147JC (Z.B.). We thank Jiang Hu for the helpful discussion on the theoretical properties of the proposed method, Jun Ding and Yang Zhang for the helpful discussion that improved the paper Jian Ma for helpful comments to improve the paper.</p>", "<title>Author contributions</title>", "<p>Conceptualization: X. Zhu and D.T.; Methodology: M.G., X. Zhang, and D.T.; Software: M.G., D.H., X. Zhang, and D.T.; Investigation: D.H., M.G., X. Zhang, Y.D., H.X., and D.T.; Writing-Original Draft: D.T.; Writing-Review and Editing: L.Q., X.D., Z.B., X. Zhu, and D.T.; Funding Acquisition: Z.B. and D.T.</p>", "<title>Peer review</title>", "<title>Peer review information</title>", "<p id=\"Par46\"><italic>Nature Communications</italic> thanks Daniele Tavernari and the other anonymous reviewer(s) for their contribution to the peer review of this work. A peer review file is available.</p>", "<title>Data availability</title>", "<p>All datasets used in this study are publicly available. Hi–C data of multiple human cell lines and replicates of GM12878 cell line are downloaded from the Gene Expression Omnibus (GEO) database under accession code GSE63525 [<ext-link ext-link-type=\"uri\" xlink:href=\"https://www.ncbi.nlm.nih.gov/geo/query/acc.cgi?acc=GSE63525\">https://www.ncbi.nlm.nih.gov/geo/query/acc.cgi?acc=GSE63525</ext-link>]^{##REF##25497547##33##}. Hi–C data of patient-derived DIPG, NHA, and GBM cell lines and DIPG frozen tissue specimens are downloaded from the GEO database under accession code GSE162976 [<ext-link ext-link-type=\"uri\" xlink:href=\"https://www.ncbi.nlm.nih.gov/geo/query/acc.cgi?acc=GSE162976\">https://www.ncbi.nlm.nih.gov/geo/query/acc.cgi?acc=GSE162976</ext-link>]^{##REF##34078608##74##}. Hi–C data of mock-infected and SARS-CoV-2-infected A549-ACE2 cells are downloaded from the GEO database under accession code GSE179184 [<ext-link ext-link-type=\"uri\" xlink:href=\"https://www.ncbi.nlm.nih.gov/geo/query/acc.cgi?acc=GSE179184\">https://www.ncbi.nlm.nih.gov/geo/query/acc.cgi?acc=GSE179184</ext-link>]^{##REF##36959507##48##}. Processed single-cell Dip-C data of multiple cell types in mouse brains are downloaded from the GEO database under accession code GSE162511 [<ext-link ext-link-type=\"uri\" xlink:href=\"https://www.ncbi.nlm.nih.gov/geo/query/acc.cgi?acc=GSE162511\">https://www.ncbi.nlm.nih.gov/geo/query/acc.cgi?acc=GSE162511</ext-link>]^{##REF##33484631##51##}. The other Hi–C data, DNase-seq data, super-enhancers, and cancer genes are downloaded from the GEO, 4DN, OncoKB, and GeneCards databases, and the data sources are listed in Supplementary Method ##SUPPL##0##5##. All relevant analyzed data is available upon request.</p>", "<title>Code availability</title>", "<p>The software is published under the GNU GPL v3.0 license. The source code of DiffDomain is available at <ext-link ext-link-type=\"uri\" xlink:href=\"https://github.com/Tian-Dechao/diffDomain\">https://github.com/Tian-Dechao/diffDomain</ext-link> or at this 10.5281/zenodo.10205208^{##UREF##11##75##}.</p>", "<title>Competing interests</title>", "<p id=\"Par47\">The authors declare no competing interests.</p>" ]

[ "<fig id=\"Fig1\"><label>Fig. 1</label><caption><title>DiffDomain workflow and example outputs.</title><p><bold>a</bold> Input are a TAD in condition 1 and its two Hi–C contact matrices (<bold><italic>A</italic></bold>₁ and <bold><italic>A</italic></bold>₂) in two biological conditions 1 and 2. <bold>b</bold> Difference between log-transformed <bold><italic>A</italic></bold>₁ and <bold><italic>A</italic></bold>₂, which is denoted as <bold><italic>D</italic></bold>. <bold>c</bold> Normalization of <bold><italic>D</italic></bold> by a 1D distance-stratified standardization procedure. Its <italic>d</italic>-off diagonal part is normalized by <italic>d</italic>-off diagonal part-specific sample mean and sample standard deviation. <bold>d</bold> <bold><italic>D</italic></bold> is transformed by dividing . Under the null hypothesis, it is assumed to be a generalized Wigner matrix. <bold>e</bold> One-sided <italic>P</italic> value is calculated based on the fact that <italic>θ</italic>_{<italic>N</italic>}, normalized largest eigenvalue of <bold><italic>D</italic></bold>, follows Tracy-Widom distribution with <italic>β</italic> = 1 (denoted as <italic>T</italic><italic>W</italic>₁ distribution). A TAD is identified as a reorganized TAD if <italic>P</italic> value ≤0.05. <bold>f</bold> Reorganized TADs are classified into six subtypes based on changes in TAD boundaries. The heatmap diagram illustrates TADs in condition 1 (<italic>Top</italic>) and condition 2 (<italic>Bottom</italic>), with lines representing TAD regions of the same condition sharing the same color. <bold>g</bold> Example of the subtypes of reorganized TADs. Data are from two studies^{##REF##25497547##33##,##REF##34078608##74##}. <italic>Top:</italic> upper and lower triangular matrices represent Hi–C data in conditions 1 and 2, with blue triangles representing TADs and yellow triangles representing reorganized TADs; <italic>Middle:</italic> TAD regions from the same condition are represented by lines of the same color; <italic>Bottom:</italic> upper triangular section of the normalized difference matrix computated from the two Hi–C matrices in the <italic>Top</italic> section. Red and blue boxes represent the maximum and minimum values in the visualized matrices. The score and one-sided <italic>P</italic> value are computed by DiffDomain. <italic>P</italic> values smaller than 2.22 × 10⁻¹⁶ are denoted by <italic>P</italic> &lt; 2.22 × 10⁻¹⁶.</p></caption></fig>", "<fig id=\"Fig2\"><label>Fig. 2</label><caption><title>Benchmarking DiffDomain against alternative methods.</title><p><bold>a</bold> FPRs of DiffDomain and alternative methods in comparing two GM12878 Hi–C replicates (<italic>primary</italic> and <italic>replicate</italic>). FPRs in comparing other Hi–C replicates of GM12878 are shown in Supplementary Fig. ##SUPPL##0##6##. FPR equals the ratio of the number of identified reorganized TADs to the number of TADs in GM12878. <bold>b</bold> Proportions of identified reorganized TADs by DiffDomain and alternative methods when comparing blood-related cell lines GM12878 and K562. TADs are GM12878 TADs. Results on other pairs of human cell lines are reported in Supplementary Fig. ##SUPPL##0##7##. <bold>c</bold> Percentages of the subtypes of reorganized TADs when comparing GM12878 and K562. Results on other pairs of human cell lines are reported in Supplementary Fig. ##SUPPL##0##9##. <bold>d</bold> Heatmaps showing Hi–C contact matrices of four truly reorganized TADs. Upper and lower triangular matrices represent Hi–C data in conditions 1 and 2. The scores and unadjusted one-sided <italic>P</italic> value below the heatmap are computed by DiffDomain. Among them, three are correctly identified as such by DiffDomain (unadjusted <italic>P</italic> ≤ 0.05, true positives), and one is missed by DiffDomain (unadjusted <italic>P</italic> &gt; 0.05, false negative). Truly reorganized TADs are manually collected and treated as the gold standard positives (see Supplementary Method ##SUPPL##0##5## for more details). <bold>e</bold> Barplot showing TPRs of DiffDomain and alternative methods. TPR equals the ratio of the number of reorganized TADs that are identified as such (true positives) to the number of reorganized TADs (positives). <bold>f</bold> Heatmap showing unadjusted one-sided <italic>P</italic> values computed by DiffDomain for testing truly reorganized TADs. The purple dot with a white border represents <italic>P</italic> &lt; 0.05. <bold>g</bold> Scatter points of unadjusted one-sided <italic>P</italic> values by DiffDomain (<italic>y</italic>-axis) and SCCs^{##REF##28855260##34##} (<italic>x</italic>-axis) when testing the truly reorganized TADs. Detailed information, including reference, TAD region, data accession number, and species for the truly reorganized TADs (Fig. 2e, f), is presented in Supplementary Method ##SUPPL##0##5## and Supplementary Table ##SUPPL##0##3##. <italic>P</italic> values smaller than 2.22 × 10⁻¹⁶ are denoted by <italic>P</italic> &lt; 2.22 × 10⁻¹⁶. Abbreviation: SCC stratum-adjusted correlation coefficient.</p></caption></fig>", "<fig id=\"Fig3\"><label>Fig. 3</label><caption><title>A reorganized TAD involves the angiogenic gene <italic>VEGFA</italic>.</title><p>Hi–C contact matrices and 1D genomic tracks of the TAD region (Chr6:43605000-44050000) in GM12878 <bold>a</bold> and K562 <bold>b</bold> cell lines that show differences in both Hi–C data and epigenomics profiles involving <italic>VEGFA</italic> gene. <bold>c</bold> The normalized difference matrix <bold><italic>D</italic></bold> between the two cell lines highlights the differences in Hi–C contact maps. Rectangular boxes highlight the increased Hi–C interactions in K562. Rectangular boxes in panels <bold>a</bold>, <bold>b</bold> highlight corresponding sections in Hi–C contact maps. <bold>d</bold>, <bold>e</bold> Potential 3D structures of the TAD in GM12878 and K562. They are estimated by Chrom3D^{##REF##29700484##46##} and demonstrate the TAD structural differences between GM12878 and K562.</p></caption></fig>", "<fig id=\"Fig4\"><label>Fig. 4</label><caption><title>Associations between SVs and reorganized TADs in cell lines with disease.</title><p><bold>a</bold> Heatmaps showing two example SVs with associated reorganized TADS. <italic>Left</italic> two heatmaps showing the Hi–C contact maps from GM12878 and K562 for a specific K562 SV region, while <italic>Right</italic> two heatmaps showing the Hi–C contact maps from NHA and DIPGXIII for a specific DIPGXIII SV region. The first track below the heatmaps outlines the reorganized TAD regions, and the second track shows the SV regions. <bold>b</bold> Barplot showing the proportions of SVs with associated reorganized TADs across the three cell lines. <italic>Left</italic> barplot includes all SVs; <italic>Right</italic> barplots are stratified by the four types of SV. <bold>c</bold> Stacked barplot showing the proportions of reorganized TAD subtypes associated with each SV type. <bold>d</bold> APA plot summarizing the aggregated changes in reorganized TADs associated with each type of SV. <italic>First column</italic> APA plot summarizing the other TADs (not reorganized) using Hi–C data from condition 2 as a control. <italic>Second column</italic> APA plot summarizing reorganized TADs associated with deletion SVs using Hi–C data from condition 2. <italic>Third column</italic> Heatmap showcasing log2-transformed fold-change of APA matrices, using APA matrices from the <italic>second column</italic> and the <italic>first column</italic>. Subsequent columns are APA plots summarizing reorganized TADs associated with 5′ to 3′ fusion (<italic>fourth and fifth column</italic>), 5′ to 5′ fusion, and 3′ to 3′ fusion. Rows represent comparisons including GM12878 vs. K562, NHA vs. DIPG007, and NHA vs. DIPGXIII. <bold>e</bold> Jitterplot showing the length of SVs associated with each reorganized TAD subtype across the three cell lines. The APA matrices are based on 25 kb resolution Hi–C data produced by FAN-C using the command ‘fanc aggregate -m -p –pixels 90 -r -e –rescale’. The APA plots are generated using python function ‘sns.heatmap’. Abbreviations: ‘+−’, deletion; ‘−+’, 5′ to 3′ fusion; ‘−−’, 5′ to 5′ fusion; ‘++’, 3′ to 3′ fusion. NHA, normal human astrocytes; DIPG007 and DIPGXIII, pediatric high-grade glioma cell lines.</p></caption></fig>", "<fig id=\"Fig5\"><label>Fig. 5</label><caption><title>Mock-infected A549-ACE2 TADs that are reorganized in SARS-CoV-2-infected A549-ACE2.</title><p><bold>a</bold> Pie chart showing the percentages of subtypes of reorganized TADs in A549-ACE3 after SARS-CoV-2 infection. <italic>Strength-change</italic> TADs are the leading subtype. <bold>b</bold> Barplot comparing the number of upregulated genes in the reorganized TADs and the other TADs. <italic>x</italic>-axis represents TADs categorized based on the number of upregulated genes located within them, <italic>y</italic>-axis represents the proportion of TADs. Boxplots comparing the numbers of enhanced H3K27ac peaks <bold>c</bold> enhanced SMC3 peaks <bold>d</bold> and weakened RAD21 peaks <bold>e</bold> per 100 kb. <italic>Left</italic>: comparing reorganized TADs with the other TADs (<italic>x</italic>-axis); <italic>right</italic>: comparison stratified by the subtypes of reorganized TADs (<italic>x</italic>-axis). <italic>y</italic>-Axis represents the number of differential peaks per 100 kb within TADs. <italic>P</italic> values are computed using one-sided Mann–Whitney <italic>U</italic> test, a nonparametric test dealing with asymmetric distributions. The number (<italic>n</italic>) of reorganized TADs in each subtype is presented in (<bold>a</bold>). In the box plots, the middle line represents the median; the lower and upper lines correspond to the first and third quartiles; and the upper and lower whiskers extend to values no farther than 1.5 × IQR. <italic>P</italic> values smaller than 2.22 × 10⁻¹⁶ are denoted by <italic>P</italic> &lt; 2.22 × 10⁻¹⁶.</p></caption></fig>", "<fig id=\"Fig6\"><label>Fig. 6</label><caption><title>Application of DiffDomain on scHi–C data to identify reorganized TADs between cell types.</title><p><bold>a</bold> Visualization of the pseudo-bulk Hi–C contact maps and the identified neuronal TADs that are reorganized in oligodendrocytes (dark pink horizontal bars) using varied numbers of randomly sampled individual cells. Gene track shows four neuronal marker genes. <bold>b</bold> Scatter plot showing the proportions of neuronal TADs that are reorganized in oligodendrocytes using varied numbers (<italic>k</italic>₁, <italic>k</italic>₂) of randomly sampled individual cells from the two cell types. Their agreements with the set of reorganized TADs identified using all cells in each cell type are quantified by the Jaccard index (<italic>x-</italic>axis). The vertical dashed line is <italic>J</italic><italic>I</italic> = 1/3, representing that two equal-sized sets share half of reorganized TADs. <bold>c</bold>, <bold>d</bold> Scatter plots showing the proportions of neonatal neuron 1 TADs that are reorganized in cortical L2–5 pyramidal cells. Up to 228 and 150 cortical L2–5 pyramidal cells are randomly sampled, respectively. <bold>e</bold> Scatter plot showing the agreements between sets of reorganized TADs that are identified using (1) pseudo-bulk Hi–C data and (2) bulk Hi–C data.</p></caption></fig>", "<fig id=\"Fig7\"><label>Fig. 7</label><caption><title>Application of DiffDomain on scHi–C data to characterize cell-to-population and cell-to-cell variability of TADs.</title><p><bold>a</bold> Heatmap showing high, median, and low cell-to-population variational TADs. <italic>P</italic> value is from comparing scHi–C contact map of a TAD in an individual cell to the pseudo-bulk Hi–C contact map that represents the population average. Classification of TADs is done by hierarchical clustering. <bold>b</bold> Pie chart showing the percentages of the high, median, and low cell-to-population variational TADs. <bold>c</bold> Heatmaps visualizing one high cell-to-population variational TAD (middle red rectangular box) and two median cell-to-population variational TADs (top-left and bottom-right blue rectangular boxes). <bold>d</bold> Chromosome map showing the genomic locations of high, median, and low cell-to-population variational TADs. <bold>e</bold> Histogram showing the percentages of reorganized TADs in 2500 pairwise comparisons of 50 cortical L2–5 pyramidal cells and 50 adult astrocytes. <bold>f</bold> Stacked bar graph showing the number (the percentage) of the cortical L2–5 pyramidal TADs that are reorganized in a varied number (0–50) of adult astrocytes. <italic>y</italic>-Axis represents the selected 50 cortical L2–5 pyramidal cells, indexed from 1 to 50. When comparing a specific cortical L2–5 pyramidal cell, such as cell 1, with 50 adult astrocytes, each cortical L2–5 pyramidal TAD is reorganized in a different number of adult astrocytes. <italic>x</italic>-Axis (top of the plot) is the number (proportion) of the cortical L2–5 pyramidal TADs that are reorganized in 0–5 adult astrocytes, 6–10 adult astrocytes, and subsequent ranges (legend). Averaging the data in the stacked bar graph across the 50 cortical L2–5 pyramidal cells (<italic>y</italic>-axis) results in the first stacked bar graph below, labeled as “Average” on the right. The subsequent stacked bar graph below is the average computed in random scenarios (Supplementary Fig. ##SUPPL##0##30)##, labeled as “Average in random scenarios\" on the right.</p></caption></fig>" ]

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[ "<inline-formula id=\"IEq1\"><alternatives><tex-math id=\"M1\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\log ({{{{{{{{\\boldsymbol{A}}}}}}}}}_{1})-\\log ({{{{{{{{\\boldsymbol{A}}}}}}}}}_{2})$$\\end{document}</tex-math><mml:math id=\"M2\"><mml:mi>log</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mrow><mml:mi mathvariant=\"bold-italic\">A</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>−</mml:mo><mml:mi>log</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mrow><mml:mi mathvariant=\"bold-italic\">A</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq2\"><alternatives><tex-math id=\"M3\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${{{{{{{\\boldsymbol{D}}}}}}}}/\\sqrt{N}$$\\end{document}</tex-math><mml:math id=\"M4\"><mml:mi mathvariant=\"bold-italic\">D</mml:mi><mml:mo>/</mml:mo><mml:msqrt><mml:mrow><mml:mi>N</mml:mi></mml:mrow></mml:msqrt></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ1\"><label>1</label><alternatives><tex-math id=\"M5\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${H}_{0}:{\\lambda }_{N}=2\\,{{\\mbox{vs.}}}\\;{H}_{1}:{\\lambda }_{N} \\, &gt; \\, 2.$$\\end{document}</tex-math><mml:math id=\"M6\"><mml:msub><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub><mml:mo>:</mml:mo><mml:msub><mml:mrow><mml:mi>λ</mml:mi></mml:mrow><mml:mrow><mml:mi>N</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mn>2</mml:mn><mml:mspace width=\"0.25em\"/><mml:mstyle><mml:mtext>vs.</mml:mtext></mml:mstyle><mml:mspace width=\"0.16em\"/><mml:msub><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>:</mml:mo><mml:msub><mml:mrow><mml:mi>λ</mml:mi></mml:mrow><mml:mrow><mml:mi>N</mml:mi></mml:mrow></mml:msub><mml:mspace width=\"0.25em\"/><mml:mo>&gt;</mml:mo><mml:mspace width=\"0.25em\"/><mml:mn>2</mml:mn><mml:mo>.</mml:mo></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq3\"><alternatives><tex-math id=\"M7\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${P}_{T{W}_{1}}({\\theta }_{N}\\ge x)$$\\end{document}</tex-math><mml:math id=\"M8\"><mml:msub><mml:mrow><mml:mi>P</mml:mi></mml:mrow><mml:mrow><mml:mi>T</mml:mi><mml:msub><mml:mrow><mml:mi>W</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mrow><mml:mi>θ</mml:mi></mml:mrow><mml:mrow><mml:mi>N</mml:mi></mml:mrow></mml:msub><mml:mo>≥</mml:mo><mml:mi>x</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq4\"><alternatives><tex-math id=\"M9\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\sqrt{N}$$\\end{document}</tex-math><mml:math id=\"M10\"><mml:msqrt><mml:mrow><mml:mi>N</mml:mi></mml:mrow></mml:msqrt></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq5\"><alternatives><tex-math id=\"M11\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${{{{{{{\\boldsymbol{D}}}}}}}}/\\sqrt{N}$$\\end{document}</tex-math><mml:math id=\"M12\"><mml:mi mathvariant=\"bold-italic\">D</mml:mi><mml:mo>/</mml:mo><mml:msqrt><mml:mrow><mml:mi>N</mml:mi></mml:mrow></mml:msqrt></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq6\"><alternatives><tex-math id=\"M13\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${{{{{{{\\boldsymbol{D}}}}}}}}/\\sqrt{N}$$\\end{document}</tex-math><mml:math id=\"M14\"><mml:mi mathvariant=\"bold-italic\">D</mml:mi><mml:mo>/</mml:mo><mml:msqrt><mml:mrow><mml:mi>N</mml:mi></mml:mrow></mml:msqrt></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq7\"><alternatives><tex-math id=\"M15\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${{{{{{{\\boldsymbol{D}}}}}}}}/\\sqrt{N}$$\\end{document}</tex-math><mml:math id=\"M16\"><mml:mi mathvariant=\"bold-italic\">D</mml:mi><mml:mo>/</mml:mo><mml:msqrt><mml:mrow><mml:mi>N</mml:mi></mml:mrow></mml:msqrt></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq10\"><alternatives><tex-math id=\"M17\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${{{{{{{{\\boldsymbol{A}}}}}}}}}_{1}=({A}_{ij}^{(1)})\\in {R}_{\\ge 0}^{N\\times N}$$\\end{document}</tex-math><mml:math id=\"M18\"><mml:msub><mml:mrow><mml:mi mathvariant=\"bold-italic\">A</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msubsup><mml:mrow><mml:mi>A</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:msubsup></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>∈</mml:mo><mml:msubsup><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:mo>≥</mml:mo><mml:mn>0</mml:mn></mml:mrow><mml:mrow><mml:mi>N</mml:mi><mml:mo>×</mml:mo><mml:mi>N</mml:mi></mml:mrow></mml:msubsup></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq11\"><alternatives><tex-math id=\"M19\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${A}_{ij}^{(1)}$$\\end{document}</tex-math><mml:math id=\"M20\"><mml:msubsup><mml:mrow><mml:mi>A</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:msubsup></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq12\"><alternatives><tex-math id=\"M21\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${{{{{{{{\\boldsymbol{A}}}}}}}}}_{2}=({A}_{ij}^{(2)})\\in {R}_{\\ge 0}^{N\\times N}$$\\end{document}</tex-math><mml:math id=\"M22\"><mml:msub><mml:mrow><mml:mi mathvariant=\"bold-italic\">A</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msubsup><mml:mrow><mml:mi>A</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mn>2</mml:mn></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:msubsup></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>∈</mml:mo><mml:msubsup><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:mo>≥</mml:mo><mml:mn>0</mml:mn></mml:mrow><mml:mrow><mml:mi>N</mml:mi><mml:mo>×</mml:mo><mml:mi>N</mml:mi></mml:mrow></mml:msubsup></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ2\"><label>2</label><alternatives><tex-math id=\"M23\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${{{{{{{\\boldsymbol{D}}}}}}}}=\\log ({{{{{{{{\\boldsymbol{A}}}}}}}}}_{1})-\\log ({{{{{{{{\\boldsymbol{A}}}}}}}}}_{2}).$$\\end{document}</tex-math><mml:math id=\"M24\"><mml:mi mathvariant=\"bold-italic\">D</mml:mi><mml:mo>=</mml:mo><mml:mi>log</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mrow><mml:mi mathvariant=\"bold-italic\">A</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>−</mml:mo><mml:mi>log</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mrow><mml:mi mathvariant=\"bold-italic\">A</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>.</mml:mo></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq13\"><alternatives><tex-math id=\"M25\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${{{{{{{\\boldsymbol{D}}}}}}}}={({D}_{ij})}_{i,j=1}^{N}$$\\end{document}</tex-math><mml:math id=\"M26\"><mml:mi mathvariant=\"bold-italic\">D</mml:mi><mml:mo>=</mml:mo><mml:msubsup><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mrow><mml:mi>D</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mo>,</mml:mo><mml:mi>j</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mi>N</mml:mi></mml:mrow></mml:msubsup></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ3\"><label>3</label><alternatives><tex-math id=\"M27\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$({D}_{ij}-{\\hat{\\mu }}_{k})/{\\hat{\\sigma }}_{k},$$\\end{document}</tex-math><mml:math id=\"M28\"><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mrow><mml:mi>D</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo>−</mml:mo><mml:msub><mml:mrow><mml:mover accent=\"true\"><mml:mrow><mml:mi>μ</mml:mi></mml:mrow><mml:mrow><mml:mo>^</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mi>k</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>/</mml:mo><mml:msub><mml:mrow><mml:mover accent=\"true\"><mml:mrow><mml:mi>σ</mml:mi></mml:mrow><mml:mrow><mml:mo>^</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mi>k</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq14\"><alternatives><tex-math id=\"M29\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\hat{\\mu }}_{k}$$\\end{document}</tex-math><mml:math id=\"M30\"><mml:msub><mml:mrow><mml:mover accent=\"true\"><mml:mrow><mml:mi>μ</mml:mi></mml:mrow><mml:mrow><mml:mo>^</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mi>k</mml:mi></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq15\"><alternatives><tex-math id=\"M31\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\hat{\\sigma }}_{k}$$\\end{document}</tex-math><mml:math id=\"M32\"><mml:msub><mml:mrow><mml:mover accent=\"true\"><mml:mrow><mml:mi>σ</mml:mi></mml:mrow><mml:mrow><mml:mo>^</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mi>k</mml:mi></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq16\"><alternatives><tex-math id=\"M33\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${({D}_{mn})}_{1\\le m,n\\le N,n-m=k}$$\\end{document}</tex-math><mml:math id=\"M34\"><mml:msub><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mrow><mml:mi>D</mml:mi></mml:mrow><mml:mrow><mml:mi>m</mml:mi><mml:mi>n</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mrow><mml:mn>1</mml:mn><mml:mo>≤</mml:mo><mml:mi>m</mml:mi><mml:mo>,</mml:mo><mml:mi>n</mml:mi><mml:mo>≤</mml:mo><mml:mi>N</mml:mi><mml:mo>,</mml:mo><mml:mi>n</mml:mi><mml:mo>−</mml:mo><mml:mi>m</mml:mi><mml:mo>=</mml:mo><mml:mi>k</mml:mi></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq17\"><alternatives><tex-math id=\"M35\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\hat{\\mu }}_{k}$$\\end{document}</tex-math><mml:math id=\"M36\"><mml:msub><mml:mrow><mml:mover accent=\"true\"><mml:mrow><mml:mi>μ</mml:mi></mml:mrow><mml:mrow><mml:mo>^</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mi>k</mml:mi></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq18\"><alternatives><tex-math id=\"M37\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\hat{\\sigma }}_{k}$$\\end{document}</tex-math><mml:math id=\"M38\"><mml:msub><mml:mrow><mml:mover accent=\"true\"><mml:mrow><mml:mi>σ</mml:mi></mml:mrow><mml:mrow><mml:mo>^</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mi>k</mml:mi></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq19\"><alternatives><tex-math id=\"M39\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\sqrt{N}$$\\end{document}</tex-math><mml:math id=\"M40\"><mml:msqrt><mml:mrow><mml:mi>N</mml:mi></mml:mrow></mml:msqrt></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq20\"><alternatives><tex-math id=\"M41\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${{{{{{{\\boldsymbol{D}}}}}}}}/\\sqrt{N}$$\\end{document}</tex-math><mml:math id=\"M42\"><mml:mi mathvariant=\"bold-italic\">D</mml:mi><mml:mo>/</mml:mo><mml:msqrt><mml:mrow><mml:mi>N</mml:mi></mml:mrow></mml:msqrt></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq21\"><alternatives><tex-math id=\"M43\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${{{{{{{\\boldsymbol{D}}}}}}}}/\\sqrt{N}$$\\end{document}</tex-math><mml:math id=\"M44\"><mml:mi mathvariant=\"bold-italic\">D</mml:mi><mml:mo>/</mml:mo><mml:msqrt><mml:mrow><mml:mi>N</mml:mi></mml:mrow></mml:msqrt></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equa\"><alternatives><tex-math id=\"M45\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${H}_{0}\t:{{{{{{{\\boldsymbol{D}}}}}}}}/\\sqrt{N}\\,{{{{{{{\\rm{resembles}}}}}}}}\\,{{{{{{{\\rm{a}}}}}}}}\\,{{{{{{{\\rm{generalized}}}}}}}}\\,{{{{{{{\\rm{Wigner}}}}}}}}\\,{{{{{{{\\rm{matrix,}}}}}}}}\\\\ {H}_{1}\t:{{{{{{{\\boldsymbol{D}}}}}}}}/\\sqrt{N}\\,{{{{{{{\\rm{does}}}}}}}}\\,{{{{{{{\\rm{not}}}}}}}}\\,{{{{{{{\\rm{resemble}}}}}}}}\\,{{{{{{{\\rm{a}}}}}}}}\\,{{{{{{{\\rm{generalized}}}}}}}}\\,{{{{{{{\\rm{Wigner}}}}}}}}\\,{{{{{{{\\rm{matrix.}}}}}}}}$$\\end{document}</tex-math><mml:math id=\"M46\"><mml:msub><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub><mml:mo>:</mml:mo><mml:mi mathvariant=\"bold-italic\">D</mml:mi><mml:mo>/</mml:mo><mml:msqrt><mml:mrow><mml:mi>N</mml:mi></mml:mrow></mml:msqrt><mml:mspace width=\"0.25em\"/><mml:mi mathvariant=\"normal\">resembles</mml:mi><mml:mspace width=\"0.25em\"/><mml:mi mathvariant=\"normal\">a</mml:mi><mml:mspace width=\"0.25em\"/><mml:mi mathvariant=\"normal\">generalized</mml:mi><mml:mspace width=\"0.25em\"/><mml:mi mathvariant=\"normal\">Wigner</mml:mi><mml:mspace width=\"0.25em\"/><mml:mi mathvariant=\"normal\">matrix,</mml:mi><mml:msub><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>:</mml:mo><mml:mi mathvariant=\"bold-italic\">D</mml:mi><mml:mo>/</mml:mo><mml:msqrt><mml:mrow><mml:mi>N</mml:mi></mml:mrow></mml:msqrt><mml:mspace width=\"0.25em\"/><mml:mi mathvariant=\"normal\">does</mml:mi><mml:mspace width=\"0.25em\"/><mml:mi mathvariant=\"normal\">not</mml:mi><mml:mspace width=\"0.25em\"/><mml:mi mathvariant=\"normal\">resemble</mml:mi><mml:mspace width=\"0.25em\"/><mml:mi mathvariant=\"normal\">a</mml:mi><mml:mspace width=\"0.25em\"/><mml:mi mathvariant=\"normal\">generalized</mml:mi><mml:mspace width=\"0.25em\"/><mml:mi mathvariant=\"normal\">Wigner</mml:mi><mml:mspace width=\"0.25em\"/><mml:mi mathvariant=\"normal\">matrix.</mml:mi></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq22\"><alternatives><tex-math id=\"M47\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${{{{{{{\\boldsymbol{D}}}}}}}}/\\sqrt{N}$$\\end{document}</tex-math><mml:math id=\"M48\"><mml:mi mathvariant=\"bold-italic\">D</mml:mi><mml:mo>/</mml:mo><mml:msqrt><mml:mrow><mml:mi>N</mml:mi></mml:mrow></mml:msqrt></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ4\"><label>4</label><alternatives><tex-math id=\"M49\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${H}_{0}:{\\lambda }_{N}=2\\,{{{{{{{\\rm{vs.}}}}}}}}\\;{H}_{1}:{\\lambda }_{N} \\, &gt; \\, 2.$$\\end{document}</tex-math><mml:math id=\"M50\"><mml:msub><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub><mml:mo>:</mml:mo><mml:msub><mml:mrow><mml:mi>λ</mml:mi></mml:mrow><mml:mrow><mml:mi>N</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mn>2</mml:mn><mml:mspace width=\"0.25em\"/><mml:mi mathvariant=\"normal\">vs.</mml:mi><mml:mspace width=\"0.16em\"/><mml:msub><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>:</mml:mo><mml:msub><mml:mrow><mml:mi>λ</mml:mi></mml:mrow><mml:mrow><mml:mi>N</mml:mi></mml:mrow></mml:msub><mml:mspace width=\"0.25em\"/><mml:mo>&gt;</mml:mo><mml:mspace width=\"0.25em\"/><mml:mn>2</mml:mn><mml:mo>.</mml:mo></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ5\"><label>5</label><alternatives><tex-math id=\"M51\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\theta }_{N}\\mathop{\\to }\\limits^{d}T{W}_{1}.$$\\end{document}</tex-math><mml:math id=\"M52\"><mml:msub><mml:mrow><mml:mi>θ</mml:mi></mml:mrow><mml:mrow><mml:mi>N</mml:mi></mml:mrow></mml:msub><mml:mover accent=\"true\"><mml:mrow><mml:mo>→</mml:mo></mml:mrow><mml:mrow><mml:mi>d</mml:mi></mml:mrow></mml:mover><mml:mi>T</mml:mi><mml:msub><mml:mrow><mml:mi>W</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>.</mml:mo></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ6\"><label>6</label><alternatives><tex-math id=\"M53\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$P\\,{{{{{{{\\rm{value}}}}}}}}={P}_{T{W}_{1}}({\\theta }_{N} \\, \\ge \\, x).$$\\end{document}</tex-math><mml:math id=\"M54\"><mml:mi>P</mml:mi><mml:mspace width=\"0.25em\"/><mml:mi mathvariant=\"normal\">value</mml:mi><mml:mo>=</mml:mo><mml:msub><mml:mrow><mml:mi>P</mml:mi></mml:mrow><mml:mrow><mml:mi>T</mml:mi><mml:msub><mml:mrow><mml:mi>W</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mrow><mml:mi>θ</mml:mi></mml:mrow><mml:mrow><mml:mi>N</mml:mi></mml:mrow></mml:msub><mml:mspace width=\"0.25em\"/><mml:mo>≥</mml:mo><mml:mspace width=\"0.25em\"/><mml:mi>x</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>.</mml:mo></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq23\"><alternatives><tex-math id=\"M55\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${{{{{{{\\boldsymbol{D}}}}}}}}/\\sqrt{N}$$\\end{document}</tex-math><mml:math id=\"M56\"><mml:mi mathvariant=\"bold-italic\">D</mml:mi><mml:mo>/</mml:mo><mml:msqrt><mml:mrow><mml:mi>N</mml:mi></mml:mrow></mml:msqrt></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq24\"><alternatives><tex-math id=\"M57\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${{{{{{{\\boldsymbol{D}}}}}}}}/\\sqrt{N}$$\\end{document}</tex-math><mml:math id=\"M58\"><mml:mi mathvariant=\"bold-italic\">D</mml:mi><mml:mo>/</mml:mo><mml:msqrt><mml:mrow><mml:mi>N</mml:mi></mml:mrow></mml:msqrt></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq25\"><alternatives><tex-math id=\"M59\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${{{{{{{\\boldsymbol{D}}}}}}}}/\\sqrt{N}$$\\end{document}</tex-math><mml:math id=\"M60\"><mml:mi mathvariant=\"bold-italic\">D</mml:mi><mml:mo>/</mml:mo><mml:msqrt><mml:mrow><mml:mi>N</mml:mi></mml:mrow></mml:msqrt></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq26\"><alternatives><tex-math id=\"M61\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${{{{{{{\\boldsymbol{D}}}}}}}}/\\sqrt{N}$$\\end{document}</tex-math><mml:math id=\"M62\"><mml:mi mathvariant=\"bold-italic\">D</mml:mi><mml:mo>/</mml:mo><mml:msqrt><mml:mrow><mml:mi>N</mml:mi></mml:mrow></mml:msqrt></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq27\"><alternatives><tex-math id=\"M63\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${{{{{{{\\boldsymbol{D}}}}}}}}/\\sqrt{N}$$\\end{document}</tex-math><mml:math id=\"M64\"><mml:mi mathvariant=\"bold-italic\">D</mml:mi><mml:mo>/</mml:mo><mml:msqrt><mml:mrow><mml:mi>N</mml:mi></mml:mrow></mml:msqrt></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq28\"><alternatives><tex-math id=\"M65\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${D}_{ij}/\\sqrt{N}$$\\end{document}</tex-math><mml:math id=\"M66\"><mml:msub><mml:mrow><mml:mi>D</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo>/</mml:mo><mml:msqrt><mml:mrow><mml:mi>N</mml:mi></mml:mrow></mml:msqrt></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq29\"><alternatives><tex-math id=\"M67\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\frac{{m}_{1}}{{m}_{2}}\\times \\frac{{s}_{2}}{{s}_{1}}\\ge 1$$\\end{document}</tex-math><mml:math id=\"M68\"><mml:mfrac><mml:mrow><mml:msub><mml:mrow><mml:mi>m</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>m</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:mfrac><mml:mo>×</mml:mo><mml:mfrac><mml:mrow><mml:msub><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:mfrac><mml:mo>≥</mml:mo><mml:mn>1</mml:mn></mml:math></alternatives></inline-formula>" ]

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[ "<supplementary-material content-type=\"local-data\" id=\"MOESM1\"></supplementary-material>", "<supplementary-material content-type=\"local-data\" id=\"MOESM2\"></supplementary-material>", "<supplementary-material content-type=\"local-data\" id=\"MOESM3\"></supplementary-material>" ]

[ "<fn-group><fn><p><bold>Publisher’s note</bold> Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.</p></fn><fn><p>These authors contributed equally: Dunming Hua, Ming Gu, Xiao Zhang, Yanyi Du.</p></fn></fn-group>" ]

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[ "<media xlink:href=\"41467_2024_44782_MOESM1_ESM.pdf\"><caption><p>Supplementary Information</p></caption></media>", "<media xlink:href=\"41467_2024_44782_MOESM2_ESM.pdf\"><caption><p>Peer Review File</p></caption></media>", "<media xlink:href=\"41467_2024_44782_MOESM3_ESM.pdf\"><caption><p>Reporting Summary</p></caption></media>" ]

{ "acronym": [], "definition": [] }

[ "<title>Background &amp; Summary</title>", "<p id=\"Par2\">GRACE (Gravity Recovery And Climate Experiment) satellites are designed to monitor spatiotemporal variations of the Earth’s gravitational field to improve our understanding of the changes in the global climate system, with the primary goal of properly mapping mass variations, including terrestrial water cycle, ice sheet and glacier mass balance, sea level change, and ocean bottom pressure variations. Data have been acquired for fifteen years, exceeding the anticipated five-year mission span from March 17 2002 through October 2017^{##UREF##0##1##,##UREF##1##2##}. The collection of science mission data ended in October 2017 because of the age-related battery issue on GRACE-B in September 2017. GRACE-FO was launched in May 2018 as a successor mission to GRACE in order to ensure the mission’s continuity^{##UREF##2##3##,##UREF##3##4##}. 11 consecutive months of data gap exist between GRACE and GRACE-FO missions. In addition, some of the monthly solutions are missing due to improper retracked orbit issues throughout the lifetime of satellites^{##UREF##4##5##}. In recent years, there have been studies using different methods to fill this gap. While mostly focusing only on terrestrial water storage and excluding mass changes over oceans, few studies have also reconstructed long-term simulations of total water storage anomaly, i.e., the climate-induced mass anomaly, before the GRACE-era using different approaches and spaceborne data.</p>", "<p id=\"Par3\">For instance, Humphrey and Gudmundsson^{##UREF##5##6##} simulated six different forms of mass anomaly from 1901 to 2019 using a statistical approach with three different land surface temperature (TEMP) and two different precipitation (PPT) data products as meteorological forcing datasets, and two different GRACE mascon solutions. Li <italic>et al</italic>.^{##UREF##6##7##}, first separated both input (PPT, TEMP, Sea Surface Temperature (SST), and 17 other climate indices) and output (GRACE mascon mass anomaly) into spatial patterns and temporal modes using independent component /principal component analysis techniques. Then, the temporal modes were further decomposed using least squares and seasonal-trend decomposition to obtain trend, seasonal, inter-annual and residual components. Excluding the trend, each decomposed component is used in Artificial Neural Networks (ANN), AutoRegressive Exogenous (ARX) and Multiple Linear Regression (MLR) approaches independently to simulate/predict temporal modes of each component at grid cell scale. Finally, GRACE-estimated trend and spatial patterns are restored by adding them back to simulated modes. Thus, the long-term mass anomaly simulations from 1979 to 2020 are obtained. Differently from the two studies mentioned above, Löcher and Kusche^{##UREF##7##8##} calculated the monthly global gravity field by combining the low-degree gravity solution estimated from Satellite Laser Ranging (SLR) observations with the decomposed spatial patterns retrieved from the available monthly GRACE gravity field solutions using Empirical Orthogonal Functions (EOF). In this way, the hybrid monthly spherical harmonic gravity field models with GRACE-like spatial resolution, i.e., degree/order (d/o) 60 models from 1992 to 2019 are obtained though the solutions before 1994 are dominated by very large noise due to worse constellation of the SLR satellites prior to 1994.</p>", "<p id=\"Par4\">In this study, we used an in-house developed hybrid deep learning architecture, namely Residual Deep Convolutional Autoencoder (ResDCAE), to simulate long-term high resolution (at monthly temporal and 1° × 1° spatial resolution) mass anomaly from 1994 to 2021. ResDCAE is based on the concept of residual learning and utilizes stacked autoencoders to increase learning efficiency and is developed considering the TensorFlow^{##UREF##8##9##} and Keras^{##UREF##9##10##} libraries. No prior detrending, deseasoning, or decomposing processes either to the input or to the output datasets are applied. Thus, the simulations avoid possible biasing or aliasing of long-term climate signals. In order to successfully simulate trend, interannual, and seasonal signals, we included both SLR-based coarse resolution mass anomaly and normalized Day of Year (nDOY) as additional input, where the latter is computed by dividing the DOY of the mid-day of that month by 365 (or 366). For this purpose, the monthly SLR-only spherical harmonic gravity field models (up to d/o 10)^{##UREF##7##8##} are used to effectively simulate the long-term trend, since the long-wavelength component of gravitational signals can be derived from SLR-only temporal gravity solutions. Interannual and seasonal signals, on the other hand, are simulated more accurately, thanks to nDOY. Because all geophysical signals are functions of time, using time epoch (nDOY) as an input acts as a constraint to obtain more realistic simulations. These novel ideas have already been tackled in recent study by Uz <italic>et al</italic>.^{##UREF##10##11##} comprehensively, but Swarm-derived mass anomaly instead of SLR mass anomaly is used to obtain the long-wavelength component of gravitational signals between January 2014 and January 2021. Here, we focused on longer-term simulation considering a similar strategy and provided global simulations including both continents and oceans. The simulated mass anomalies are validated using the internal and external validation data. Furthermore, each monthly mass anomaly simulations are also converted to global geopotential field models expressed in spherical harmonics complete to degree and order 200.</p>" ]

[ "<title>Methods</title>", "<title>Residual deep convolutional autoencoders</title>", "<p id=\"Par5\">Convolutional Neural Networks (CNNs) are special types of neural networks and are useful for processing data with a grid-like architecture, such as images or time series data^{##REF##7370364##12##,##UREF##11##13##}. In particular, CNNs utilize the convolutional layers, which are linear operators and convolve the input with the set of filters. Therefore, the CNNs can be considered as spatial feature extractors by their layered structure. For this reason, CNNs have been widely used for problems such as filling the data gap in remote sensing^{##UREF##12##14##}, land surface temperature reconstruction^{##UREF##13##15##}, etc. In this manner, the relationship between the output vectors <bold>a</bold> of the consecutive layers of CNNs is represented as:where ∗ denotes convolution operator, σ(·) is the activation function, with weight matrix <bold>W</bold> and bias vector <bold>b</bold> while the superscript indicates the layer ID. In addition, CNNs mainly comprise three types of layers: convolutional layer, pooling layer, and fully connected layer. Deep Convolutional AutoEncoder (DCAE) is a deep learning architecture that may be considered as a combination of two neural networks, namely encoder and decoder^{##UREF##14##16##–##UREF##16##18##}. In particular, the encoder maps the input space into a lower-dimensional latent space by h = f (<bold>x</bold>), while the decoder maps the latent space into the reconstruction space by r = g (h) and here x is the input vector. By this way, the network learns the representations of input data by reducing the dimensionality of data in either a supervised or an unsupervised manner. Basically, the high-abstraction features are learned while mapping through an internal representation, or code, h, in the intermediate layer. In addition, the distinction between standard AutoEncoder (AE) and DCAE is the utilization of convolutional layers. Accordingly, DCAE takes input data and maps it to h,where σ denotes the activation function, <bold>W</bold> and <bold>b</bold> are the weight matrix and bias vector of the encoder. Accordingly, the output of the decoder is given as follows:</p>", "<p id=\"Par6\">One of the common themes in deep learning architectures is that the deeper the network, the more advantageous and the better the modelling performance. However, there is the problem of vanishing/exploding gradients due to the successive calculation of gradients with respect to the gradient from the previous layer. To overcome this problem, a residual neural network was proposed by He <italic>et al</italic>.^{##UREF##17##19##}, which led to an effective strategy for developing deeper neural networks. The main reason for this is the fact that instead of calculating gradients over F(x) which represents the mapping, gradients are calculated over F(x)+x, by introducing the skip connections between layers. Accordingly, output y is obtained by the combination of the input and output of the earlier layer as follows.where <italic>F</italic>(<italic>x,W</italic>_{<italic>i</italic>}) is the residual mapping and <italic>W</italic>_{<italic>i</italic>} corresponds the i-th weight matrix in the hidden layer weighted value of the layer. It should also be noted that the dimensions of the x and y must be equal. The residual learning strategy is successfully applied to problems such as classification^{##UREF##18##20##}, the spatiotemporal estimation of citywide crowd flows^{##UREF##19##21##} and influenza trends^{##UREF##20##22##}.</p>", "<p id=\"Par7\">The proposed architecture is given in Supplementary Fig. ##SUPPL##0##S1## and based on the combination of DCAEs and CNNs with the concept of residual learning. The main reason for this implementation is that it improves learning efficiency by developing deeper structures with the help of residual learning. Accordingly, the structure of the proposed network may be divided into three parts as follows:<list list-type=\"order\"><list-item><p id=\"Par8\">Convolutional building block: The developed architecture is based on the use of the CNN which consists of two convolutional layers with 126 and 63 filters in each layer, respectively, followed by a dense layer with 21 neurons. It should also be emphasized that the size of the filters and neurons is chosen according to the channel size of the input. In addition, for each convolutional and dense layer, an Exponential Linear Unit (ELU) activation function is used throughout the network, and each convolutional layer also has a regularizer to prevent overfitting. Besides, a single convolution layer with 21 filter sizes is utilized as the residual connection. In the stage of selecting hyperparameters of CNN, we have considered the lowest generalization error subject to runtime and memory constraints. In addition, elastic net is utilized as a regularization method which combines the lasso and ridge regulators^{##UREF##21##23##}, with penalty value of 10⁻⁴. It is also worth noting that the reason for adopting the ELU as an activation function is that it allows to negative outputs which leads to adjusting weights and biases in the correct direction during the iterative optimization process.</p></list-item><list-item><p id=\"Par9\">DCAE building block: The structure of the DCAE model consists of five convolutional layers with a gradually increasing filter size from 21 to 316. Each convolutional layer is followed by a maxpooling layer with a pool size 2 × 2 and 3 × 3 respectively. In the intermediate layer, which is also known as latent space, a flattened layer and a fully connected layer with 400 neurons are used. In this manner, in the decoding part, the fully connected layer of 400 neurons is followed by the four transposed convolutional layers, symmetric to the encoder part. In addition, the ELU activation function is used in each layer of the network.</p></list-item><list-item><p id=\"Par10\">Regression building block: To complete the end-to-end image-to-image regression task, a CNN-based structure is employed, which consists of two convolutional layers with 21 and 14 filters, respectively, followed by a dense layer with 1 neuron. Furthermore, the ELU activation function is applied to each convolutional layer except the dense layer.</p></list-item></list></p>", "<p id=\"Par11\">Accordingly, the input features first pass through the DCAE building block and are concatenated to the output features of this block. Further, this serves as the input features to the convolutional building block, and before the concatenation, input features are convolved. In order to complete image-to-image regression, regression building blocks are employed as the final layers of the model. Therefore, the model has 6 consecutive DCAE and convolutional blocks and 1 regression block. Regarding the implementation of the network, the Adamax optimizer is utilized with the initial learning rate equal to 10⁻³ and a batch size of 27. In addition, the learning rate is reduced by a factor of 0.8 when learning stagnates by monitoring the validation loss. Furthermore, early-stopping is implemented to mitigate overwriting. According to this, validation loss is selected as the monitored metric, and the training procedure is stopped if no improvement is seen for 25 epochs. The Huber function has been selected as the training loss function since it is robust against outliers and has fast convergence to near negligible loss. The main reason for this implementation is that the input consists of various earth observations that have different characteristics. The training is performed on a single NVIDIA Tesla P100-PCIE-16GB GPU. It should also be noted that the full memory capacity of the graphic card is used, and the running time of the model is about 45 minutes.</p>", "<title>Mitigating trend error in backwards extrapolation</title>", "<p id=\"Par12\">Our objective is to provide monthly gravity field data products similar to those within the GRACE era, even for the pre-GRACE period. Achieving this goal involves extrapolating data backwards in time, which is a complex task. Extrapolation should be performed with caution, especially when dealing with non-stationary processes, as is the case in our study. Non-stationarity in Earth and environmental systems, such as spatiotemporal changes in Earth’s water mass, primarily results from the inherent secular trend signal. This signal alters the mean rather than the signal variance and may or may not follow a linear pattern. Factors like climate change, human interventions, and low-frequency internal variability, such as the Atlantic multidecadal oscillation, affected by the slow dynamics of ice sheets and the ocean, contribute to this non-stationarity^{##REF##18239110##24##}. This introduces the challenge such that the behaviour of the signal outside the training data period (in our case, the GRACE and GRACE-FO period) may differ from that within the training data time span, even if the seasonal amplitudes remain relatively consistent^{##UREF##22##25##}. Like all data-driven approaches, deep learning (DL)-based methods adjust their parameters through optimization algorithms to find the best fit to available output data based on the corresponding input data. This optimization aims to establish a statistically optimal mapping from input to output, limited to the training data period. Consequently, deep learning models typically perform well when producing or predicting outputs for new input values falling within the range of input data used for training. However, predicting outcomes outside the range of the training data, referred to as ‘out of sample prediction,’ can be challenging. In other words, a deep learning model can provide reasonable results within the hyperspace defined by the boundary of the training data set, which can be seen as a high-dimensional interpolation, as long as the number of input data variables is fewer than 100^{##UREF##23##26##}.</p>", "<p id=\"Par13\">In the case of mass anomaly, an efficient extrapolation requires a priori knowledge about the signal behaviour in the extrapolation regime, i.e., in the time span out of the training data period. Unfortunately, this kind of information is usually not available, at least globally and at grid cell scale. However, the main differences of the mass anomaly signal as well as of the input climate data signal in the extrapolation regime from those in the interpolation regime (i.e., training data span) are in the long-term trends while the seasonal amplitudes do not vary much (see e.g., Supplementary Fig. ##SUPPL##0##S2##). Therefore, the main errors of extrapolation are due to mismodeling of the trend component which is retrieved from the training data. Some studies^{##UREF##5##6##,##UREF##6##7##} estimate and remove a linear trend using the data in the GRACE and GRACE-FO mission spans before calibrating their models based on residual signal and then extend and restore this trend to the extrapolation regime by assuming that their estimated linear trends also hold out of the training data period. Such an assumption is too optimistic and may not be valid globally. A typical example is the surface mass balance estimates at polar regions, e.g., Greenland and Antarctica (see The IMBIE team^{##REF##31822019##27##,##REF##29899482##28##}) exhibit relatively lower mass change rates until late 1990s followed with an onset of dramatic increase in mass loss after 2000 due to the accelerated ocean-driven melting of the ice sheets. Similar extrapolation errors are also reported^{##UREF##24##29##} for the long-term static gravity field solutions with co-estimated (TVC) time-variable coefficients (secular and seasonal periodic components), e.g. GOCO06S^{##UREF##25##30##}, by evaluating the differences of mass anomaly from the monthly GRACE-FO solutions and those extrapolated from the static field with TVC computed from the data solely within the GRACE era, suggesting that the static gravity models with TVC cannot be used for long-term (&gt;2-3 years) extrapolation and at least should be frequently updated with the newly available GRACE-FO data, e.g. for mass change studies as well as for improved precise orbit determination of low earth orbiting satellites. Also, Mouginot <italic>et al</italic>.^{##REF##31010924##31##} and Rignot <italic>et al</italic>.^{##REF##30642972##32##} discuss these mass changes for the last 40 years. Mouginot <italic>et al</italic>.^{##REF##31010924##31##} reported variability in the mass balance of the Greenland Ice Sheet since the 1980s, along with a sixfold increase in mass loss. This has resulted in a significant 13.7 mm contribution to global sea level rise since 1972, with half of this effect occurring during the period from 2010 to 2018. According to Rignot <italic>et al</italic>.^{##REF##30642972##32##}, the primary cause of mass loss in Antarctica is the glacier flow near warm, saline circumpolar deep-water regions, particularly in East Antarctica, with significant implications for future sea-level rise. In addition to these, Caceres <italic>et al</italic>.^{##UREF##26##33##} studied the land water storage except for the glaciers mass change. They reveal that from 1948 to 2016, continents contributed to a sea-level rise of 34–41 mm, with glacier mass loss responsible for 81% of the cumulative loss and land water storage anomalies accounting for the remaining 19%. Climate-driven land water storage anomalies are notably influenced by precipitation and linked to El Niño Southern Oscillation, although uncertainties persist in modelling these anomalies, particularly in relation to irrigation water use and artificial reservoirs.</p>", "<p id=\"Par14\">The power of DL, besides the computational resources, is attributed to the number of training data, that is, higher the number of data higher the accuracy can be achieved by large neural networks whose parameters are updated through deep learning algorithms (Aggarwal^{##UREF##27##34##}, pp. 3).</p>", "<p id=\"Par15\">After training the deep learning model with initial training and test data (see section Data Architecture for details of the training and test data) within the GRACE/-FO era and simulating the a priori monthly mass anomaly of 3-years backwards in time (i.e. from April 1999 to March 2002), here we applied a step-by-step piecewise trend correction approach where at each step the number of training data is incrementally increased backwards in time. The overall approach can be summarized as follows. We start with retraining our deep neural network after removing the first three years of the initial training data within the GRACE/-FO era (i.e., 3 years of data starting from April 2002 to March 2005) without altering the network architecture or any of the hyperparameters or the learning algorithm. The retrained model was then used to simulate global 1° × 1° monthly mass anomaly grids for the period coinciding with that of the removed 3-years of initial training data. For each grid cell, the linear trends from both the simulated and the corresponding original monthly mass anomaly data in the initial training set in these 3-years were estimated by least squares fit, and the difference between the two trends was computed. The computed trend difference at each grid cell was then applied to correct the simulated mass anomaly of the earlier 3-years (i.e., the monthly mass anomaly from April 1999 to March 2002). These new corrected simulations were added to the initial training data set which now constitutes the extended training data set for the next iteration. With the extended training data, the procedure above was repeated with removing the first 3-years (i.e., this time April 1999 to March 2002) and computing trend-corrected mass anomaly simulations for the previous 3-years (i.e., April 1996 to March 1999). This step-by-step correction process was applied once again with the updated training data from previous iteration so that the trend-corrected mass anomaly simulations from January 1994 to the beginning of GRACE/-FO era (i.e., April 2002) were completely obtained. The efficiency of the above procedure and the adequacy of the chosen step-size, i.e., 3-years have been verified with results shown in Technical Validation section.</p>", "<title>Descriptions of the input and output data of our DL-based simulation model</title>", "<p id=\"Par16\">Our DL architecture possess the multichannel input consisting of seven variables and a single output variable. Input are monthly coarse resolution SLR-only mass anomaly and five different Hydroclimatic/meteorological Variables (HV) from ERA5 (European Centre for Medium-Range Weather Forecast-ECMWF Reanalysis-5) as well as normalized (Day-of-Year) time epoch of these monthly dataset, i.e., nDOY. The five HV from ERA5 are PPT, TEMP, SST, Cumulative Water Storages Changes (CWSC) and mass anomaly retrieved from ERA5 model data while the single output is the monthly CSR RL06 Mascon (CSRM) mass anomaly solutions. The details of both input and output are given in the following.</p>", "<title>GRACE mass anomaly data</title>", "<p id=\"Par17\">The monthly mass anomaly of GRACE/-FO is derived from CSR RL06 Mascon (CSRM) solutions^{##UREF##28##35##,##UREF##29##36##}. The time span of this dataset is fragmented into two main parts that are April 2002–June 2017 for GRACE and May 2018–to 2021 for GRACE-FO missions. There is an 11 successive months of so-called intermission data gap between GRACE and GRACE-FO, but there are also missing months during the operational time period of each mission. All the standard post-processing corrections have been applied to CSRM models, i.e., degree 1 correction^{##UREF##30##37##}, replacement of C₂₀ and C₃₀ coefficients^{##UREF##31##38##}, Glacial Isostatic Adjustment (GIA)^{##UREF##32##39##} and Ellipsoidal correction^{##REF##30930553##40##}. The monthly mass anomalies are calculated considering the mean baseline between 2004.0 and 2009.9999. Besides, while the temporal resolution is monthly for CSRM, the spatial coverage is global, with a spatial sampling resolution of 0.25° × 0.25°. We resampled the original CSRM to 1.0° × 1.0° grids considering the native resolution of CSRM which determines the spatial resolution of our target mass anomaly simulations.</p>", "<title>SLR mass anomaly data</title>", "<p id=\"Par18\">The monthly SLR-only spherical harmonic gravity field models^{##UREF##7##8##} up to d/o 10 are provided by Dr. Anno Löcher from the Astronomical, Physical, and Mathematical Geodesy Institute of Bonn University via personal communication. The coarse resolution SLR-only monthly gravity field solutions are available from November 1992 to January 2021. The monthly mass anomalies from SLR-only models are calculated by applying post-processing to the spherical harmonic coefficients of those models after removing the same mean baseline (2004.0–2009.9999) with CSRM for consistency. A 1500 km Gaussian smoothing filter is applied, i.e., the filter radius is decided after applying both 1000 and 2000 km, but the optimum results are obtained from using the 1500 km radius showing a compromise between signal loss and noise reduction. No de-striping filter was applied as suggested by the data provider (Dr. Anno Löcher, pers. comm.) Mass anomalies are directly calculated for each 1.0° × 1.0° grids on the globe including both ocean and land.</p>", "<title>ERA5 data</title>", "<p id=\"Par19\">The ERA5 dataset is released by the European Centre for Medium-Range Weather Forecast (ECMWF - <ext-link ext-link-type=\"uri\" xlink:href=\"https://cds.climate.copernicus.eu/\">https://cds.climate.copernicus.eu/</ext-link>) and consists of both monthly averaged and hourly sub datasets. Besides, ERA5 has two different parameter levels, i.e., single and pressure levels. We used the monthly averaged single level from 1979 to present;^{##UREF##33##41##} a subset of ERA5 considering the timespan of the available SLR-only input data. Therefore, ERA5 was downloaded and used from November 1992 to January 2021. The basic input HV data are chosen in order to be used in the DL algorithm which are PPT, TEMP, SST, RunOff (RO), evapotranspiration (ET), snow water storage (SnWS), soil moisture storage (SMS) and Canopy Water Storage (CnWS). The input ERA5 mass anomaly and cumulative water storage changes (CWSC) are calculated using these downloaded variables applying the equations given in e.g., Uz <italic>et al</italic>.^{##UREF##10##11##}, and Mo <italic>et al</italic>.^{##UREF##34##42##}. Similar to the CSRM and SLR-only mass anomaly, each input data from ERA5 is referenced to the mean baseline, i.e., 2004.0–2009.9999, by removing its mean within this baseline period. Finally, all HV data are resampled from 0.25° × 0.25° to 1.0° × 1.0° to ensure consistency between all input and output data.</p>", "<title>Data architecture</title>", "<p id=\"Par20\">There are two considerations while designing the DL architecture which are addressed with the questions; (i) how do the temporal patterns of input and output data for each grid vary throughout their own time-spans? and (ii) how do the temporal correlations change with respect to time lags? The answers to these questions could give information about how many additional layers should be considered for the input data variables. In other words, how many channels should be used for the input data. We answered these questions by inspecting the Partial AutoCorrelations (PAC) computed for varying time lags in Amazon River basin. The Amazon River basin was chosen because it is a good example of reserving climate change signatures^{##UREF##35##43##}. The grid closest to the centre of the basin, according to the basin boundaries from Total Runoff Integrating Pathway (TRIP) database^{##UREF##36##44##}, was selected, and the PACs for each input and output were computed and plot. The time series of the input and output data and their corresponding PACs are given in Supplementary Fig. ##SUPPL##0##S2## and, except for SLR-only mass anomaly, all other input and output data show more or less a certain annual signal pattern. The SLR-only mass anomaly between December 1992 and December 1993 has higher variations with respect to those during the rest of the time span. Similar results are also reported in Löcher and Kusche^{##UREF##7##8##}. The obvious reason for this is that the SLR-only models in the first couple years were computed without using the Stella satellite, which joined the constellation in September 1993 and is an important part of SLR-only temporal gravity field recovery due to its low polar orbit and hence providing more redundant sets of normal equations for gravity inversion. Thus, the time series of SLR-only mass anomaly includes even higher noise until September 1993 (Dr. Anna Löcher, pers. comm.). On the other hand, PACs are calculated from the time series of input and output, starting from zero lag up to a 12-month time lags. All correlations are computed throughout the GRACE/-FO time period and are illustrated in Supplementary Fig. ##SUPPL##0##S2e##. It is clearly seen that almost all correlations reduce to below zero at a two-month lag and are almost zero beyond the two-months. Thus, we decided to set the number of additional layers to 2, i.e., the successive two months of relevant time epoch of input, i.e., <italic>t</italic> and <italic>t-1</italic>.</p>", "<p id=\"Par21\">According to the pre-analyses above, DL architecture is constituted considering both the time-span of SLR-only models with lower noise level and the computed temporal correlations. In addition, our initial training and testing data sets are randomly selected from the GRACE/-FO time period. While the total number of initial training months is 135, the number of testing months which is kept unchanged when applying the trend correction procedure is 57. The entire time series of predicted/simulated monthly mass anomaly cover both the pre-GRACE era as well as the existing data gaps within the GRACE/-FO time period after the step-by-step trend correction was applied. The number of predictions where no GRACE/-FO data is available is 136, starting from January 1994 due to a higher noise issue in SLR-only solutions between 1992 and 1994.</p>" ]

[]

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[]

[ "<p id=\"Par1\">Since April 2002, Gravity Recovery and Climate Experiment (GRACE) and GRACE-FO (FollowOn) satellite gravimetry missions have provided precious data for monitoring mass variations within the hydrosphere, cryosphere, and oceans with unprecedented accuracy and resolution. However, the long-term products of mass variations prior to GRACE-era may allow for a better understanding of spatio-temporal changes in climate-induced geophysical phenomena, e.g., terrestrial water cycle, ice sheet and glacier mass balance, sea level change and ocean bottom pressure (OBP). Here, climate-driven mass anomalies are simulated globally at 1.0° × 1.0° spatial and monthly temporal resolutions from January 1994 to January 2021 using an in-house developed hybrid Deep Learning architecture considering GRACE/-FO mascon and SLR-inferred gravimetry, ECMWF Reanalysis-5 data, and normalized time tag information as training datasets. Internally, we consider mathematical metrics such as RMSE, NSE and comparisons to previous studies, and externally, we compare our simulations to GRACE-independent datasets such as El-Nino and La-Nina indexes, Global Mean Sea Level, Earth Orientation Parameters-derived low-degree spherical harmonic coefficients, and <italic>in-situ</italic> OBP measurements for validation.</p>", "<title>Subject terms</title>" ]

[ "<title>Data Records</title>", "<p id=\"Par22\">The simulated monthly dataset is released both in the form of gridded mass anomalies with and spherical harmonic coefficients in accordance with the ICGEM (International Center for Global Earth Models) format. These datasets are available in figshare repository^{##UREF##37##45##}. The datasets cover the time span from January 1994 to December 2021, hence provides simulations for 324 months in total. The gridded dataset is available to users with data format identical to official CSR mascon data products in figshare as netcdf file with four variables: lat, lon, time and mass anomaly. Lat and lon are latitude and longitude vectors of dimension 180 × 1 and 360 × 1, representing the positions of the centre of 1.0° × 1.0° grid cells on the surface of the Earth. The mass anomaly represents the relative change in the water mass with respect to the mean baseline (2004.0-2009.9999) in terms of cm Equivalent Water Height (EWH); thus, the associated netcdf file has a dimension of 324 × 180 × 360 while the time is a column vector of dimension 324 × 1 with days since 1994 01 01 00 00 00.0 UTC. The monthly mass anomaly in the form of spherical harmonic coefficients from degree 2 up to degree/order 200 were released as ASCII files. The file naming convention similar to the official GRACE data processing centers was adopted, i.e., determined by the year and the day of the year corresponding to the first and last day of the respective month. Each file has a header that contains information regarding constant values.</p>", "<title>Technical Validation</title>", "<p id=\"Par23\">This section evaluates the simulation performance of our models and summarizes the findings in two categories: internal and external validation. Internal validation is performed by comparing our simulations to CSRM mass anomaly solutions and to those from previous studies in terms of common mathematical goodness-of-fit metrics such as Root Mean Square Error (RMSE), Nash - Sutcliffe efficiency (NSE^{##UREF##38##46##}), and Pearson Correlation Coefficient (PCC). On the contrary to internal validation, the simulated mass anomalies (or spherical harmonic coefficients derived from them) are validated externally by comparisons to GRACE-independent datasets, such as the long-term surface mass balance estimates of Greenland, the El Niño/La Niña SST index, global barystatic mean sea level changes, degree 2 order 1 spherical harmonic coefficients (a.k.a. C₂₁, S₂₁) retrieved from daily Earth Orientation Parameters (EOP) series, degree 2 order zero spherical harmonic coefficient (i.e., C₂₀) from SLR and <italic>in situ</italic> Ocean Bottom Pressure observations.</p>", "<title>Internal validation</title>", "<title>Comparison of different input scenarios</title>", "<p id=\"Par24\">The first step in our internal validation process is to assess the GRACE-like mass anomaly simulations generated by our deep learning algorithm, namely the ResDCAE model. Our objective is to determine how the inclusion of SLR and nDOY inputs in the model affects these simulations. To do so, we have devised four distinct simulation scenarios, which we refer to as DL models. These scenarios are grouped based on whether they incorporate SLR and/or nDOY inputs, while all DL models consistently include all four ERA5 layers.</p>", "<p id=\"Par25\">To make it easier to distinguish between these various combinations, we have assigned specific names to them: Sol1 (which includes both SLR and nDOY), Sol2 (which includes only SLR), Sol3 (which includes only nDOY), and Sol4 (which excludes both SLR and nDOY). We then compare the mass anomaly simulations from these four solutions with the reference mass anomaly data from original CSR Mascon solutions. In each of these comparisons, we calculate two commonly used metrics, namely RMSE (Root Mean Square Error) and NSE (Nash-Sutcliffe Efficiency), for all test months. These metrics allow us to effectively assess the performance of the models and gain insights into the impact of different input configurations.</p>", "<p id=\"Par26\">From April 2002 to August 2020, RMSE and NSE metrics for each of the randomly chosen 57 test months and their overall mean are computed and shown in Fig. ##FIG##0##1##. The overall mean values of the metrics are also listed in Table ##TAB##0##1##. Furthermore, the metrics are separately computed over (i) entire globe (land + ocean), (ii) land-only, and (iii) ocean-only areas. All three scenarios (i-iii) exhibit a level of accuracy in retrieving the missing test months, with RMSE of 4 cm, 5 cm, and 3 cm, respectively. The corresponding NSE metrics are computed as 0.86, 0.91, and 0.68. A similar comparison was made by Uz <italic>et al</italic>.^{##UREF##10##11##} between April 2014 and September 2020 for the thirteen test months but only over land areas. The simulative performance of test months that either encompass the ACC (accelerometer) transplanted^{##UREF##39##47##} time period or those which uses the piled data from two successive months to solve for the corresponding CSRM mass anomaly is worse than those within the other time spans (see Fig. 1 of Uz <italic>et al</italic>.^{##UREF##10##11##}). The same is also reported over oceans (see Fig. 2 of Chen <italic>et al</italic>.^{##REF##35535258##48##}). From this point of view, while the CSRM products of GRACE after November 2016 are calculated using transplanted ACC data to mitigate the effect of the GRACE-B battery issue^{##UREF##40##49##}, the ACC transplantation has been carried out since the start of the GRACE-FO mission because the standard ACC data derivation procedure from Level-1A (L1A) to Level-1B (L1B) does not ensure sufficient accuracy for gravity field recovery^{##UREF##41##50##,##UREF##42##51##}. Thus, the Science Data System (SDS) produces and distributes the transplanted and calibrated ACC data product on a regular basis. Additionally, the specifics of Level-2 (L2) data products, metadata including whether the ACC transplantation is applied or piled data from consecutive months used for gravity inversion, are listed in Table ##TAB##1##2## of the SDS Newsletter(<ext-link ext-link-type=\"uri\" xlink:href=\"https://isdc.gfz-potsdam.de/grace-isdc/grace-gravity-data-and-documentation/\">https://isdc.gfz-potsdam.de/grace-isdc/grace-gravity-data-and-documentation/</ext-link>). As expected, the RMSE of all solutions over land are greater than those over the oceans. This is because the mass anomaly signal over land is stronger while those over ocean has lower magnitude which is dominated by noise. Therefore, the computed NSE over ocean are lower due to the low signal-to-noise ratio of CSRM over ocean at grid cell scale.</p>", "<p id=\"Par27\">The lowest RMSE are found in time series between 2004 and 2010, which is attributed to the better orbit configuration and availability of telemetry data with minor gaps during this time span (Fig. ##FIG##0##1##). Additionally, the NSE over oceans is at its lowest level within this time period while the RMSE is still minimum (~2 cm), implying that the DL algorithm successfully mitigated the high frequency spatiotemporal ocean mass change errors of CSRM at grid cell scale (Fig. ##FIG##0##1c##). There is a significant jump in RMSE of all simulations, which is clearly seen in the comparison over land-only areas (Fig. ##FIG##0##1b##), around August 2014. According to the August 2014 SDS Newsletter, the swap manoeuvre, during when the satellite twins exchange positions^{##UREF##43##52##,##UREF##44##53##}, was carried out in July 2014. It may have impacted the satellite observations in August 2014 and in the following few months.</p>", "<p id=\"Par28\">The overall metrics in Table ##TAB##0##1## reveal the following. Over land, the RMSE and NSE of Sol1 and Sol2 are higher and lower, respectively, than those of Sol3 and Sol4. However, the metrics of Sol3 and Sol4 are nearly identical. The primary distinction between Sol1-2 and Sol3-4 is the presence of SLR-only mass anomaly as a training input. It is expected that SLR-only mass anomaly would be noisier and will propagate to the Sol1 and Sol2 simulated models. Thus, mass anomalies that incorporate the spatiotemporal variations of the SLR-only mass anomaly through input are noisier than those that do not. On the other hand, the relationships of the metrics with the simulation models over oceans are different from those over land. While Sol2 metrics continue to have the greatest RMSE and the lowest NSE values over the ocean, Sol1 metrics reveal the opposite. Differently from Sol2, Sol1 uses normalized time (nDOY) as an additional training input. Although Sol1 exhibits propagated noise from input SLR-only mass anomaly, the simulated model is also associated with the temporal changes of nDOY input. On Earth, the oceanic regions have more complicated dynamics, and gravitational signals are influenced by the higher-frequency mass redistributions at oceans. Thus, the temporal variations retrieved by the aid of nDOY parameter may guarantee to provide more accurate description of the signal throughout the ocean. The metrics computed considering both land and ocean (global) grids are used to assess which simulation model has the best mathematical fit.</p>", "<p id=\"Par29\">The simulation models are further evaluated based on the spatial distribution of RMSE, NSE, and PCC by the illustration given in Fig. ##FIG##1##2##. These metrics are calculated for each 1.0° × 1.0° grid cell from monthly differences between DL-based simulations and corresponding CSRM mass anomaly in the 57 testing months. While the rows of Fig. ##FIG##1##2## correspond to RMSE (a, b, c, and d), NSE (d, e, f, and g), and PCC (h, i, j, and k), the columns correspond to simulations Sol1 to Sol4, from left to right. The highest RMSE is seen in the same regions for all simulations, i.e., the Amazon, Ganges, Greenland, and Gulf of Alaska. This result is also observed by previous studies (e.g., see Fig. 2a of Humphrey and Gudmundsson^{##UREF##5##6##}, Fig. 2d of Li <italic>et al</italic>.^{##UREF##6##7##}, Fig. 4j-1 of Mo <italic>et al</italic>.^{##UREF##34##42##} and Fig. 3 of Uz <italic>et al</italic>.^{##UREF##10##11##}). These basins are hydrologically active in terms of signal variations and are the main contributors to ice sheet melting areas on Earth. For example, the Amazon is the largest drainage basin and is subject to the largest seasonal changes that can be surpassed by variations as much as 1 m of EWH in Total Water Storage (TWS) globally^{##UREF##45##54##}. The Ganges basin is under the coupled effects of groundwater depletion due to human intervention for irrigation^{##REF##29358394##55##} and the melting of ice sheets in High Mountain Asia glaciers. The ice sheet mass loss in Greenland and the Gulf of Alaska, moreover, contributes to global sea-level rise^{##REF##31822019##27##,##UREF##46##56##}. The mass anomaly signals and variations in these regions are higher than those in other basins on Earth. Thus, the discrepancy between CSRM and simulated mass anomaly is sourced from this outcome, and systematically worse or higher RMSE are calculated at these regions. In addition, the spatial distribution of RMSE for Sol1 and Sol2 is more intense than for Sol3 and Sol4 due to the propagation of the noise of the SLR-only input into the simulations. The Empirical Cumulative Distribution Functions (ECDF) of RMSE over land are illustrated in Fig. ##FIG##1##2m##. While 80% of RMSE are below 5 cm for all simulations, there is a significant difference between SLR-only mass anomaly included simulations, i.e., Sol1, Sol2, and not included ones, i.e., Sol3, Sol4. Similarly, NSE and PCC values over land can be evaluated by considering the spatial distributions of these metrics as shown in Fig. ##FIG##1##2##. Both NSE and PCC of all simulations are almost zero throughout arid regions on Earth, e.g., North Africa. These results are similar to those of Uz <italic>et al</italic>.^{##UREF##10##11##}. Although the RMSE at arid regions are almost the same and having the lowest values, there is also very little correlation between CSRM and simulated mass anomaly which can be explained by the low signal-to-noise ratio of CSRM at these regions.</p>", "<p id=\"Par30\">The simulative performances of the DL models over oceans are also demonstrated spatially with the same metrics. The geophysical dynamics are more complex in oceans. This complexity is sourced from the oceanographic variables and their temporal variations. Thus, the variations of these variables are more difficult to model when compared to land and seem to degrade the simulation performances over oceans in all four simulations. As shown in Fig. ##FIG##1##2a–d##, the RMSE at high latitude regions are even higher compared to the other parts. These regions may be influenced more by polar climatic characteristics. On the other hand, NSE and PCC over the ocean possess the lowest values not only in areas that are close to polar regions but also in different parts of the ocean, i.e., the Atlantic Ocean. There is a significant difference between Sol1 and other simulations, based on the ECDF of RMSE and NSE scores computed over the oceans (see Fig. ##FIG##1##2m,n## and Table ##TAB##0##1##). The simulations of Sol1 clearly exceeds other simulations in terms of metrics meaning that the oceans are better modelled using the input combination adopted for Sol1, i.e., when both SLR-only mass anomaly and nDOY are included as additional input data. For example, among all only the NSE of Sol1 is above zero, which indicates that only Sol1 modelled the mass change over oceans realistically. In general, if the NSE of a simulation model is below zero it means that the mean of observation is better than the simulation results^{##UREF##47##57##}. According to our analyses so far, Sol1 and Sol2 suffer from propagation of noise in SLR-only input mass anomaly over land, but Sol1 provides better simulations over oceans.</p>", "<title>Comparison with previous studies</title>", "<p id=\"Par31\">Based on the comparisons among the four DL simulation models in the previous section we pick the simulation results of Sol1 as our final reconstructed data products. Thus, the simulations of Sol1 are compared to the previous similar studies of Humphrey and Gudmundsson^{##UREF##5##6##}, Li <italic>et al</italic>.^{##UREF##6##7##}, and Löcher and Kusche^{##UREF##7##8##} (which are called Humphrey, Li, and Löcher in the rest of the paper, respectively) regarding the performance metrics. The chosen reconstruction of Humphrey uses both JPL RL06 mascon and ERA5 HV, and the spatial resolution is 0.5° × 0.5° covering the time span from January 1979 to July 2019. The simulated mass anomaly of Li is calculated based on the CSR RL06 mascon with a spatial resolution of 0.5° × 0.5° and covers the time span between July 1979 and June 2020. On the other hand, hybrid models of Löcher were released as spherical harmonic coefficients up to degree and order 60 from November 1992 to January 2021. Thus, mass anomaly is calculated from the model coefficients with spatial resolution of 1.0° × 1.0°, removing the mean-field between 2004.0 and 2009.9999. To ensure consistency, Humphrey and Li’s mass anomalies are also calculated by removing this mean-field and up-sampled to 1.0° × 1.0° grids. The time period of comparison of all studies was chosen to be within the GRACE/-FO time period. In total, comparisons based on overlapping 175 months are made, and the illustration of the metrics is given in Fig. ##FIG##2##3##. The columns of Fig. ##FIG##2##3## are represented by Sol1, Li, Humphrey, and Löcher from left to right, respectively. Similar to the results of the four DL simulations as shown in the previous section, the RMSE of all models are higher in hydrologically dominant regions on Earth. The order of RMSE performances of the models from best to worst with respect to CSRM mass anomaly are Sol1, Li, Humphrey, and Löcher. A similar performance order is also clearly seen for NSE and PCC. The illustrated ECDF in Fig. ##FIG##2##3m,n##,##FIG##2##o## also represent the model accuracies prominently. Throughout the GRACE/-FO era, the Sol1 simulation has the lowest spatial RMSE and the highest NSE and PCC with CSRM. In contrast to our DL-based simulation, the Li, Humphrey, and Löcher used different approaches or estimation strategies, therefore the long-term trend of each study may be different from the other. For consistency, the RMSE, NSE and PCC metrics are recalculated using detrended and detrended-deseasoned mass anomalies and are given in Supplementary Figs. ##SUPPL##0##S3## and ##SUPPL##0##S4##, respectively. These results reveal that the Sol1 provides significantly the best simulation and outperforms the previous studies when compared to CSRM within the GRACE/-FO period. For simplicity, we will use the name ‘ResDCAE’ to represent the DL model ‘Sol1’ in the rest of the paper.</p>", "<title>External Validation</title>", "<title>Comparison with Greenland long-term surface mass balance estimates</title>", "<p id=\"Par32\">In order to validate our simulation results, we performed a comparison with independent surface mass balance estimates of Greenland. Greenland is particularly chosen as a test bed because of the following three main reasons, among others. First, Greenland surface mass balance data is a unique independent data set which has a long temporal coverage, e.g., the IMBIE (Ice sheet Mass Balance Inter-comparison Exercise) surface mass balance data record starts ten years before the GRACE era. Second, the beginning of a dramatic increase in the ice mass loss trend was observed in ~2002^{##UREF##48##58##}, almost right before the launch of GRACE mission. Thus, Greenland is the most challenging region on the Earth with a perfect data set to test the performance of the strategy adopted to mitigate the trend error in backwards extrapolation in this study. Third, as a consequence of the exacerbated global climate change, the melting of the Greenland ice sheets, and its peripheral glaciers and ice caps is the major contributor to contemporary sea level rise^{##REF##28361871##59##}.</p>", "<p id=\"Par33\">Here, we use the most recent Greenland surface mass balance data from the IMBIE as the reference for comparison. The IMBIE data has been produced using 26 estimates of ice sheet mass balance derived from satellite altimetry (9 datasets), satellite gravimetry (14 datasets) and the input–output method (3 datasets) to assess changes in the Greenland Ice Sheet mass balance^{##REF##31822019##27##}. Prior to 2003, the estimates are solely from input-output method consistency of which with the estimates from satellite altimetry and gravimetry has been shown for common spatial and temporal domains after 2003 (see Fig. 2 of The IMBIE team^{##REF##31822019##27##} for the number of individual mass balance estimates and the temporal coverage of the ach measurement type used). The final IMBIE surface mass balance data is available as a reconciled time series of cumulative total mass changes between 1992 and 2018 along with the uncertainty estimates and can be downloaded from <ext-link ext-link-type=\"uri\" xlink:href=\"http://imbie.org/data-downloads/\">http://imbie.org/data-downloads/</ext-link>.</p>", "<p id=\"Par34\">The cumulative total mass change has been produced by integrating the rates of mass changes computed at annual intervals from time series of relative mass change using a 3-yr window. Therefore, for a fair comparison, we applied a 13-months moving average filter to our monthly mass change simulations from ResDCAE. The cumulative mass changes from the IMBIE surface mass balance and from ResDCAE (ResDCAE^Sm°°^thed) starting from 1994 are shown in Fig. ##FIG##3##4##. Note that the IMBIE data is referred to the mean baseline between 2004.000–2009.9999 to be consistent with the ResDCAE in the plot. The time series of original monthly ResDCAE mass change simulations as well as the estimated 1-σ uncertainties of the IMBIE are also presented in Fig. ##FIG##3##4##. Figure ##FIG##3##4## shows an almost excellent agreement between IMBIE and the ResDCAE simulations throughout the entire time span. The slight differences between ResDCAE and IMBIE cumulative mass change time series are all within the 1-σ envelope of the IMBIE. The standard deviation is computed as ± 90 Gt based on these differences. The results indicate that our simulations are not only accurate within the GRACE/-FO era in which the training of the DL model was performed, but also provide good predictions of mass anomaly for the pre-GRACE era; the trend error mitigation strategy seems to work reasonably well even if not perfect.</p>", "<title>Validation with ENSO events</title>", "<p id=\"Par35\">The most active climate variability on the interannual timescale affecting long-term mass anomaly values is the El Niño–Southern Oscillation (ENSO), which results from large-scale ocean–atmosphere interactions over the equatorial Pacific^{##UREF##49##60##–##UREF##52##63##}. Positive Sea Surface Temperature Anomalies (SSTA) in the eastern or central equatorial Pacific Ocean, as well as a weakening of equatorial trade, define the first phase of ENSO events, which occur every 2–7 years on average. El Niño events produce several severe droughts^{##UREF##53##64##,##REF##21292971##65##} in the western Pacific and floods^{##UREF##54##66##,##UREF##55##67##} in the eastern Pacific, affecting climate globally. Besides that, the negative phase of ENSO is La Niña, which is a phenomenon in the tropical Pacific causing exceptional cooling of SSTs. The Southern Oscillation is the other part of ENSO, and it is a large-scale see-saw trend in the sea level pressure between the eastern and western tropical Pacific. El Niño results in low sea-level pressure in the eastern Pacific and higher pressure in the western Pacific, whereas La Niña has the reverse effect. Statistical models are frequently employed to determine ENSO evolution, with the SSTA index of Nino3.4 (120°W–170°W, 5°N–5°S, as given in Supplementary Fig. ##SUPPL##0##S5##). These anomalies are the deviation of monthly SSTs from their long-term mean. When the Nino3.4 index surpasses + 0.5 °C and −0.5 °C for at least five consecutive months, it is considered an El Niño and a La Niña event, respectively. The 2015/16 El Niño is the most powerful event in recorded El Niño history. It exceeded the previous two extreme occurrences in 1997/98 and 1982/83^{##UREF##56##68##,##UREF##57##69##}.</p>", "<p id=\"Par36\">The ENSO has been demonstrated to have a significant impact on precipitation and air temperature in a variety of regions^{##UREF##58##70##–##UREF##60##72##}. Mass anomaly is highly dependent on the integrated water mass changes due to precipitation, evapotranspiration, and runoff. It is also heavily influenced by regional meteorological circumstances such as droughts, flooding, and extended periods of high temperatures. All these components, particularly precipitation, are linked to ENSO. As a result, it is possible to conclude that ENSO and mass anomaly are related, as well. Several studies^{##UREF##54##66##,##UREF##58##70##,##UREF##61##73##–##UREF##63##75##} have demonstrated the impact of ENSO on mass anomaly in different basins of the Earth. For instance, Chen <italic>et al</italic>.^{##UREF##54##66##} explored the relationship between interannual mass anomaly variations and ENSO occurrences in the Amazon basin. Furthermore, Ni <italic>et al</italic>.^{##UREF##63##75##} analysed this phenomenon globally and discovered that ENSO occurrences have a significant impact on local Precipitation Anomalies (PPTA) and interannual TWS variations.</p>", "<p id=\"Par37\">As the first external validation of our reconstruction, ResDCAE, a thorough assessment focused on the relationship between interannual mass anomaly and ENSO was carried out. It was also compared to previous studies which utilized the TRIP database for major river basin boundaries^{##UREF##36##44##}. The mass anomaly reconstruction models are validated both with GRACE/-FO mass anomaly data and through examining their relationship with precipitation anomalies from ERA5, the National Oceanic and Atmospheric Administration (NOAA) Climate Prediction Center (CPC)^{##UREF##64##76##} and the Global Precipitation Climatology Center (GPCC)^{##UREF##65##77##} global precipitation dataset. The three different precipitation data above is chosen in order to ensure fair comparison among simulated mass anomaly from different studies. This is because ERA5 precipitation is an input dataset in both our ResDCAE and Humphrey while Li used CPC as an input for reconstruction. Furthermore, GPCC was also chosen due to its independence, regardless of the relationship between the input and simulations. Before the dataset used in validation, both CPC and GPCC were resampled to monthly temporal and 1.0° × 1.0° spatial resolutions. First, the average signals of reconstructions and precipitation datasets over river basins are calculated considering TRIP basin boundary data, which are also illustrated in Supplementary Fig. ##SUPPL##0##S5##. In order to ensure consistency between these time series, 5-month moving average filters are applied as in the study by Ni <italic>et al</italic>.^{##UREF##63##75##}, and then these smoothed time series are temporally matched to obtain the same time coverage for all datasets. According to Ni <italic>et al</italic>.^{##UREF##63##75##}, the Amazon basin exhibits the highest correlation between mass anomaly and ENSO events, according to their global analyses, and it takes 5 months for the influence to appear in the basin. The illustration of average basin signals for the Amazon basin is given in Fig. ##FIG##4##5##. In Fig. ##FIG##4##5a##,the Nino3.4 index (<ext-link ext-link-type=\"uri\" xlink:href=\"http://www.cpc.ncep.noaa.gov/data/indices/\">http://www.cpc.ncep.noaa.gov/data/indices/</ext-link>) is employed as a measure of ENSO activity in a comparison to simulated mass anomaly in the Amazon basin. A similar comparison of simulated mass anomaly versus precipitation anomalies from three different precipitation data set is also given in Fig. ##FIG##4##5b##. Figure ##FIG##4##5a,b## show that interannual mass anomaly variations are strongly linked to ENSO and precipitation with several months of time lags over the Amazon basin, particularly during massive El Niño events in 1997/98 and 2015/16. Furthermore, when the time series of the models are compared to CSRM, it is found that the ResDCAE model distinguishes rapid changes more easily and matches well with the CSRM mass anomaly. The GRACE and pre-GRACE correlations between the time series of all investigated mass anomaly solutions and the SSTA time series are calculated separately. The calculated correlations are all maximal values with specific phase lags. The correlations for the comparison between ResDCAE and SSTA in the GRACE-period were −0.57 (6-month lags) and −0.85 (4-month lags) in the pre-GRACE period. Similarly, the correlations for the Li, Humphrey, Löcher, and CSRM time series were calculated as −0.65 (5), −0.73 (6), −0.60 (7), and −0.57 (5) for the GRACE period, whereas they (excluding CSRM) were calculated as −0.87 (6), −0.84 (7), and −0.63 (5), respectively for the pre-GRACE period. During the GRACE period, our ResDCAE solution has nearly the same correlation as the CSRM time series. This results in a significant convergence to the simulated CSRM data. On the other hand, our simulation is also consistent with other studies in both time periods.</p>", "<p id=\"Par38\">The time series of all three precipitation anomaly data are in good agreement with the mass anomaly simulations. In order to quantify the Cross-Correlations (CCR) between these precipitation anomalies and mass anomaly simulations at all river basins in the TRIP database, the CCR coherence spectrum is calculated by determining and considering time lags between all signals separately. Totally 176 river basins are considered in this comparison. Each comparison, e.g., ResDCAE vs ERA5 PPTA for all basins, has its own defined time lag, which is determined by the maximum correlation computed between the compared time series. Thus, both cross-correlations and time lags are calculated for all compared time series pairs at each river basin. CCR metrics are given in Fig. ##FIG##5##6a–l## for the comparison between mass anomaly simulations and ERA5, CPC, and GPCC datasets. The lags between the time series of mass anomaly simulations and precipitation are also given in Supplementary Fig. ##SUPPL##0##S6##. According to Fig. ##FIG##5##6##, almost similar CCR are calculated between all mass anomaly simulations and precipitation datasets, except for Löcher’s. The discrepancies between precipitation and mass anomaly simulations of Löcher is most likely due to the fact that, contrary to ResDCAE, Humprey and Li, no precipitation data was considered when Löcher’s spherical harmonic coefficient models are calculated. On the other hand, as expected, Humphrey’s simulations have slightly higher correlations with PPTA in all basins, because the precipitation data has been directly used as the dominant input in a linear water store model (see Eq. 1 of Humprey and Gudmundsson^{##UREF##5##6##}). Nevertheless, all simulations have similar CCR in almost all river basins. As it can be seen in Fig. ##FIG##5##6a–l##, the Amazon River basin has the highest CCR (&gt;0.75) in all comparisons.</p>", "<title>Validation with independent degree-2 spherical harmonic coefficient estimates</title>", "<p id=\"Par39\">The long-wavelength components of gravity change due to variations of mass redistribution on Earth can also be recovered from SLR tracking measurements or EOP, independently^{##REF##35535258##48##,##UREF##66##78##}. Thanks to advancements in satellite geodetic techniques, EOP- and SLR-derived low-degree spherical harmonic coefficients, i.e., ∆C₂₀, ∆C₂₁ and ∆S₂₁, can be determined with higher accuracy than GRACE/-FO observations^{##REF##35535258##48##}. These degree-2 coefficients are related to the different geophysical dynamics of mass redistribution on Earth^{##UREF##0##1##,##REF##35535258##48##}. These relationships could be exemplified by the fact that while SLR-derived ∆C₂₀ provides information about mass variations due to the oblateness of the Earth, ∆C₂₁ and ∆S₂₁ are related to the variations of the Earth’s rotational axis^{##UREF##67##79##,##UREF##68##80##}. Therefore, degree-2 spherical harmonic coefficients recovered from GRACE/-FO could be validated independently using EOP- and SLR-derived counterparts.</p>", "<p id=\"Par40\">In order to validate our simulations, global 1.0° × 1.0° gridded mass anomaly from ResDCAE were first converted to spherical harmonic coefficients using Eq. 35 of Wahr <italic>et al</italic>.^{##UREF##69##81##}. The maximum degree and order of the spherical harmonic expansion were chosen as 96. Note that these spherical harmonic coefficients represent the relative anomalies w.r.t the 2004.0–2009.9999 mean baseline, i.e., the coefficients are ∆C_nm and ∆S_nm. Similarly, CSRM mass anomaly were also converted to spherical harmonic coefficients (CSRM spherical harmonic coefficients). On the other hand, monthly C₂₀ coefficients estimated from SLR were taken from <ext-link ext-link-type=\"uri\" xlink:href=\"https://grace.jpl.nasa.gov/data/get-data/oblateness/\">https://grace.jpl.nasa.gov/data/get-data/oblateness/</ext-link> and the 2004.0–2009.9999 mean was removed from the entire time series to calculate the SLR ∆C₂₀ series consistent with those of ResDCAE and CSRM. Note that the background models adopted within CSRM and CSR GRACE/-FO RL06 processing chain have also been used for the recovery of SLR-only gravity field models for comparison in this study^{##UREF##66##78##,##UREF##67##79##}. Further, EOP-derived, ∆C₂₁, ∆S₂₁, and ∆C₂₀ coefficients were calculated from the mass term of the Earth Rotation excitations using Eq. 2 of Chen <italic>et al</italic>.^{##UREF##70##82##}. Mass excitations are obtained by subtracting motion terms from observed excitations of polar motion components (X, Y), and Length of Day (LOD), respectively. While the mass excitations are due to the mass load variations, the motion excitations arise from the angular momentum exchange between the Earth’s crust and the atmosphere, i.e., due to the frictions of atmospheric wind and ocean current fields on the Earth^{##UREF##70##82##,##UREF##71##83##}.</p>", "<p id=\"Par41\">Daily Polar motion (X, Y) and LOD observations are taken from the International Earth Rotation and Reference Systems (IERS) EOP 14 C04 series^{##UREF##72##84##}. In order to calculate mass terms, the daily observed and motion excitations were computed using interactive tools of the IERS (<ext-link ext-link-type=\"uri\" xlink:href=\"https://hpiers.obspm.fr/eop-pc/analysis/excitactive.html\">https://hpiers.obspm.fr/eop-pc/analysis/excitactive.html</ext-link>) with the Chandler period set at 433 days and quality factor at 100. The motion terms are also computed considering the angular momentum series, namely ECWMF and Max-Planck-Institute for Meteorology Ocean Model (MPIOM), that are provided by GFZ^{##UREF##73##85##} which also serve as a basis for atmosphere and ocean de-alising (AOD1B RL06) model in GRACE/-FO data processing^{##UREF##71##83##}. Thus, the consistency between ResDCAE and EOP-derived also ensured using these models as angular momentum components of motion terms. The daily mass excitations are obtained by removing motion terms from observed excitations and these datasets comprise different periodic signals, such as the 5.8-yr oscillation in the observed LOD series, which is sourced from core-mantle interaction and is not related to gravity change^{##UREF##74##86##}. Thus, first a zero phase-shift Butterworth high-pass filter with a cut-off frequency of 1/4 cpy is applied to remove this oscillation and other long-period signals from the LOD series. Besides, the linear trends were removed from all mass excitations using unweighted least squares trend estimation because the long-term variation was in good agreement with the EOP- or SLR-derived series at seasonal time scales. After that, the daily degree-2 coefficients were computed using these detrended mass excitations, and a low-pass filter with a cut-off frequency of 6 cpy was applied to remove from the every signal shorter than 2 months. Finally, the monthly ∆C₂₁, ∆S₂₁, and ∆C₂₀ were computed by averaging the daily ones in each month. In order to make a fair comparison, the corresponding time series of ResDCAE, CSRM and SLR were also detrended using unweighted least squares. All the detrended series are presented in Fig. ##FIG##6##7##.</p>", "<p id=\"Par42\">All degree-2 coefficient time series are in a good agreement of ± 2e-10 for both GRACE and GRACE FO period (Fig. ##FIG##6##7##). In contrast to ∆C₂₁ and ∆S₂₁, EOP-derived ∆C₂₀ has a phase-shift of about 4-months that may be sourced from high-pass filtering applied to the remove the long-period signals from ∆LOD, and therefore not shown in Fig. ##FIG##6##7a##. On the other hand, the seasonal amplitude of ∆S₂₁ series is higher than that of ∆C₂₁. This is because ∆S₂₁ is more sensitive to mass changes over land while ∆C₂₁ is more sensitive to mass changes over the oceans. Similar to the results in e.g., Meyrath <italic>et al</italic>.^{##UREF##74##86##} and Chen <italic>et al</italic>.^{##UREF##70##82##}, while GRACE/-FO-derived (ResDCAE and CSRM) ∆C₂₀ coefficients are more consistent to those estimated from SLR ones, the ∆C₂₁ and ∆S₂₁ from ResDCAE and CSRM are both closer to each other and the EOP-derived estimates. Zero phase-lag correlations, after removing annual/semiannual variations and linear trends using unweighted least squares fit, between every pair of estimates of ∆C₂₀, ΔC₂₁ and ΔS₂₁ shown in Fig. ##FIG##6##7## are listed in Table ##TAB##1##2##. As expected, for all three coefficients, the highest correlations (0.94-0.96) are observed between ResDCAE and CSRM estimates when compared to others within the GRACE/-FO era. The correlations between the ∆C₂₀ from ResDCAE and from SLR within and pre-GRACE era are still as high as 0.80 and 0.70, respectively. The correlations for ∆C₂₁ are slightly smaller than those for ∆S₂₁ in all comparisons, which may be due to the lower signal-to-noise ratio of ∆C₂₁ as it is more relevant to the mass change signal over the oceans. The correlations given in Table ##TAB##1##2## within GRACE/-FO era are all comparable to the results from earlier studies, e.g., Chen <italic>et al</italic>.^{##UREF##71##83##}, Meyrath <italic>et al</italic>.^{##UREF##74##86##} although they used the coefficient estimates from GRACE/-FO Level-2 GSM data rather than mascon solutions. This, as well as the different time spans used to compute correlations explain the slight differences between their results and the results in this study. It is worth noting that the degree-2 coefficients from our ResDCAE model and from GRACE/-FO are converted from mascon type mass grids on Earth. The coefficients from ResDCAE model simulations also reveal reasonably high correlations (0.58 for ∆C₂₁ and 0.65 for ∆S₂₁) with EOP-derived ones before the GRACE era which shows the efficiency of the methodology (DL + backwards trend error mitigation strategy) used in this study.</p>", "<title>Validation with global barystatic mean sea level change data</title>", "<p id=\"Par43\">Barystatic mean sea level change data is another independent data source used to validate GRACE/-FO temporal gravity solutions^{##REF##35535258##48##} on global scale. Satellite altimetry has been a well-established space geodetic technique for accurately measuring global sea level change for about three decades. Thus, we compared our simulation results over the oceans with the altimeter-observed Global Mean Sea Level (GMSL) change time series after appropriate preprocessing steps were applied. To this end, the time series of GMSL anomalies (from altimetry) and associated steric components are calculated and compared to mean ocean mass change retrieved from ResDCAE simulations and CSRM. GMSL records consist of both ocean mass and steric components and can be expressed with the sea level-budget equation as GMSL_altimetry = GMSL_steric + GMSL_oceanmass. While GMSL_steric represents the contributions of oceans’ thermal expansion and salinity to sea-level variations, GMSL_oceanmass represents the change in ocean mass^{##UREF##75##87##,##UREF##76##88##}. Before comparing the time series of GMSL to ocean mass anomalies from ResDCAE and CSRM, GMSL_steric must be removed from the GMSL_altimetry anomalies to obtain GMSL_oceanmass in order to make physically consistent comparison.</p>", "<p id=\"Par44\">The GMSL_altimetry dataset is provided by Copernicus Marine Environment Monitoring Service (CMEMS) and the Copernicus Climate Change Service (C3S) and derived from ECWMF database^{##UREF##77##89##,##UREF##78##90##}, which includes the gridded (with daily and 0.25° × 0.25° spatial resolutions) merged satellite altimetry sea level anomaly data products combining different satellite altimetry observations and covering the time span from January 1993 to present. We downloaded the gridded GMSL_altimetry dataset from ECWMF from January 1994 to December 2017. We confined our comparison between this time span as there is no reliable data available for computation of steric contribution beyond December 2017^{##UREF##79##91##}. Some pre-processing steps such as upsampling and some corrections, were applied before the calculating GMSL_altimetry time series from gridded dataset. First, GMSL dataset were upsampled to monthly 1.0° × 1.0° grids by averaging from its own native resolutions to ensure consistency with our simulations. After that, the so-called TOPEX-A instrumental drift corrections, which is sourced from the instrumental problems of satellite and spanning the period from January 1993 to December 1998^{##UREF##75##87##}, were added to each grid using provided correction values along with the dataset. On the other hand, GMSL_steric component was calculated from gridded steric-height anomalies that are retrieved from Camargo <italic>et al</italic>.^{##UREF##80##92##}, which is the ensemble mean derived from the 10 different temperature and salinity data sets and has monthly sampling with 1.0° × 1.0° spatial resolution covering oceans between 66° N–66° S latitudes from January 1993 to December 2017.</p>", "<p id=\"Par45\">GMSL_altimetry and GMSL_steric anomalies are calculated considering the mean baseline between 2004.0 and 2009.9999 to ensure consistency with ocean mass change from our simulations as well as from CSRM. Then the time series are obtained by averaging grids over the oceans, excluding a 300-km buffer zone along the coasts to avoid any signal leakage from land hydrology. The average monthly sampling of time series was obtained from the weighted ocean mass change grids. The weights were determined considering the surface area of each grid cell over the oceans within 65° N–65° S latitudes. In addition, GIA correction was applied to GMSL time series by adding a constant value of −0.23 mm/year derived from the ICE-6G_D VM5a model^{##UREF##32##39##}, which also was used as GIA correction CSRM. GMSL_oceanmass time series was then calculated by removing GMSL_steric from GMSL_altimetry time series to compare with ocean mass change from ResDCAE simulations and CSRM. The same averaging procedure was also applied to the time series of ResDCAE simulations (from January 1994 to December 2017) and CSRM (from April 2002 to December 2017). The GIA and GAD corrections are readily included in our simulation as it uses the corrected version of CSRM as output data for training the DL model. Finally, the seasonal (annual/semi-annual) signals were removed using unweighted least-squares from all time series and a moving average filter with a window length of 400 days was applied to each of the time series before comparison.</p>", "<p id=\"Par46\">Ocean mass change time series both from our ResDCAE simulation and CSRM as well as the altimetry derived GMSL_oceanmass are given in Fig. ##FIG##7##8##. While trend values are calculated for GRACE period as 2.11, 2.15, and 2.21 mm/yr, they are calculated for pre-GRACE period only for ResDCAE and GMSL_oceanmass as 0.13 and 0.77 mm/yr shown in Fig. ##FIG##7##8##. The long-term linear trends estimated from ResDCAE, CSRM, GMSL_altimetry, GMSL_steric and GMSL_oceanmass time series are 1.47, 2.15, 2.82, 1.47 and 1.34 mm/yr, respectively. The deseasoned time series of ocean mass change are consistent to each other especially after 2004. This improvement can be attributed to the accurate Argo-based steric height models developed in early 2005^{##UREF##75##87##,##UREF##81##93##,##UREF##82##94##}. Dieng <italic>et al</italic>.^{##UREF##75##87##} has shown that the ensemble members of steric heights used in various studies show significant differences between 1993 and 2004. However, Argo data from January 2005 to end of 2015 significantly reduce the uncertainties of the steric sea level change data products^{##UREF##75##87##,##UREF##82##94##}. We re-estimated the linear trends from all time series but for the time period from January 2005 to December 2017 and obtained 2.26, 2.26, 3.58, 0.88 and 2.70 mm/yr, respectively for ResDCAE, CSRM, GMSL_altimetry, GMSL_steric and GMSL_oceanmass. The GRACE-based (ResDCAE simulations and CSRM) ocean mass change time series and altimetry-steric (GMSL_oceanmass) are all in excellent agreement throughout the Argo data time-span. We also computed the correlations of the original (non-detrended/non-deseasoned) ResDCAE simulations as well as of the CSRM with the Altimetry-Steric (barystatic) sea level change time series. Within the GRACE era (April 2002 – December 2017) a correlation coefficient of 0.86 is obtained both with ResDCAE and CSRM. The correlation coefficient computed between ResDCAE and the Altimeter-derived barystatic sea level for the pre-GRACE era (January 1994 – March 2002) is still as high as 0.79, indicating reasonably well simulation performance and the effectiveness of the trend error mitigation strategy (c.f. section Methods) adopted in this study.</p>", "<title>Validation with <italic>in situ</italic> ocean bottom pressure data</title>", "<p id=\"Par47\">The simulated mass changes over oceans can be compared to <italic>in situ</italic> ocean bottom pressure (OBP) observations for qualitative validation. The detection of spatiotemporal mass variations over oceans from GRACE/-FO is more challenging than those of the continental hydrology signal since the detectable variations of gravity signal over the ocean are weaker^{##UREF##83##95##}. Moreover, the comparison of these variations to independent point-wise OBP variations is much more challenging due to the differences between spatial and temporal resolutions, the irregular distribution of OBP stations over the globe, or the necessity of isolating signals that are related to ocean circulation and sea level changes^{##UREF##84##96##}. Considering these aspects, <italic>in situ</italic> OBP observations that are publicly available in the Permanent Service for Mean Sea Level (PSMSL) database (<ext-link ext-link-type=\"uri\" xlink:href=\"https://www.psmsl.org/data/bottom_pressure/\">https://www.psmsl.org/data/bottom_pressure/</ext-link>) were used to compare to the gridded mass change time series from our simulation and CSRM. To this end, we chose two stations from PSMSL database considering the temporal coverage of data records (stations with data records available for longer time) and taking into account the proximity and distance from the land to test any possible leakage effect from the land. Thus, the stations Dark Passage South – DPS (60.9° S–54.7° W) and NDBC 51406 – Central South Pacific (8.5° S–125.0° W) were chosen the start and end deployments of which are November 1992 – June 2011 and September 2001 – February 2013, respectively.</p>", "<p id=\"Par48\">The daily sampled mean OBP data constructed from hourly observations at each deployment are readily available after removal of diurnal and shorter period tides by averaging 24 hourly values and sensor drift corrections. The long-term trends as well as the remaining drifts were first removed with a quadratic fit from daily OBP time series at each deployment as suggested by Poropat <italic>et al</italic>.^{##UREF##84##96##}\n<italic>in situ</italic>. In order to generate monthly sampled OBP time series, first the low-pass Butterworth filter was applied with 12 cpy cut-off frequency to remove any remaining signal with sub-monthly periods from the daily time series. The monthly time series were then computed by averaging these filtered daily samples. On the other hand, the monthly mass change time series from ResDCAE and CSRM at the two 1.0° × 1.0° grids which contain the selected OBP stations were first detrended. The monthly time series of <italic>in situ</italic> OBP and mass change usually need to be compared for long-wavelength signal content due to the complexity between observed mass change by GRACE and OBP induced by their differences in spatial and/or temporal resolutions. Therefore, six-months moving average filter was further applied to monthly sampled ResDCAE and CSRM time series. The resulting time series are given with a dual axis plot in Fig. ##FIG##8##9##.</p>", "<p id=\"Par49\">The agreement between <italic>in situ</italic> OBP and mass change time series is better at NDBC station than at DPS station as shown in Fig. ##FIG##8##9a,b##. This is most likely due to the signal leakage from land and sea ice at northern Antarctica to the mass change signal at DPS station. The NDBC station, on the other hand, is in the open ocean and thus almost no signal leakage from land hydrology exists. The comparison of GRACE-based mass change solutions to <italic>in situ</italic> OBP observations is highly dependent on both oceanographic priors and applied post-processing as well as the basin size adopted for averaging while generating the mass change time series^{##UREF##84##96##,##UREF##85##97##}. Despite the fact that the general signal preprocessing tools such as filtering and smoothing were applied to obtain time series of ResDCAE, CSRM, and OBP observations, the agreements with <italic>in situ</italic> OBP records at these two different locations over the oceans are comparable with those in previous studies^{##UREF##86##98##,##UREF##87##99##}.</p>", "<title>Summary and Future Perspectives</title>", "<p id=\"Par50\">In this study we employed a hybrid deep learning architecture called resDCAE to simulate mass anomalies at a spatial resolution of 1.0 degree by 1.0 degree and a monthly temporal resolution from January 1994 to January 2021. We proposed and successfully performed a strategy to reduce the error of the trend component in the simulations during the pre-GRACE period (1994 to 2002). The primary objective was to achieve a better understanding and characterization of various climate-induced geophysical phenomena, including the terrestrial water cycle, ice sheet and glacier mass balance, sea level changes, and variations in ocean bottom pressure by providing longer time series of global water storage changes both over continents and oceans. The research demonstrated that the use of a combination of ERA5 and SLR datasets, along with time channel information, provided better, if not the best, solution for simulations. This study contributes to the monitoring and comprehension of long-term global gravity field changes, offering valuable insights into climate change and other significant geophysical events. Such research is advancing our understanding of climate changes and their impacts on Earth’s water cycle. With the new data sets as well as advanced satellite gravity missions and developments in deep learning era and algorithms, improved simulations of the water mass change with enhanced resolutions will be possible in the future.</p>", "<title>Usage Notes</title>", "<p id=\"Par51\">The simulated data is available with no gaps from January 1994 to end of December 2020 both in the form of monthly 1.0° × 1.0° mass anomaly grids and spherical harmonic coefficients similar to official GSM data products but with a much higher resolution up to degree and order 200. The user should note that the data set may not include seismic signal and thus is not proper for e.g. earthquake signal detection. For conversion of mass anomaly grids to spherical harmonic coefficients, Equations 6–8, and load Love numbers in Table ##TAB##0##1## of Wahr <italic>et al</italic>.^{##UREF##69##81##} were used with ρ_ave =5517 kg/m³ as the average density of the solid Earth. Both data sets represent the anomalies relative to 2004.0–2009.9999 mean baseline similar to CSR mascon solutions. When using spherical harmonic coefficients data, no further destriping or smoothing filter is required.</p>", "<title>Supplementary information</title>", "<p>\n\n</p>" ]

[ "<title>Supplementary information</title>", "<p>The online version contains supplementary material available at 10.1038/s41597-023-02887-5.</p>", "<title>Acknowledgements</title>", "<p>This work is partially supported by Scientific and Technological Research Council of Turkey - TÜBİTAK (Contract number 119Y176), by the United States Agency for International Development (USAID)/Indian Partnerships Program (Cooperative Agreement: 72038621CA00002), by National Geospatial-Intelligence Agency (NGA) GEO-ESCON Program (No. HM157522D0009, Task 8.8), by NSF Partnerships for Innovation Program (2044704) and by NASA Earth Surface Interior Focused Area Program (80NSSC20K0494). Dr. Anno Löcher from University of Bonn is gratefully acknowledged for providing monthly gravity field solutions from SLR data. Special thanks to the editors and the two anonymous reviewers who significantly improved the quality of the manuscript with constructive comments. </p>", "<title>Author contributions</title>", "<p>This work is a part of M.U.’s PhD dissertation supervised by O.A. at Istanbul Technical University. M.U. conceived the idea with O.A. and C.K.S., and developed the initial framework. K.G.A. and M.U. designed and numerically implemented the deep learning models adopted in the study. S.O. and Ö.G. contributed in the data curation and preprocessing. M.U., O.A., C.K.S. and K.G.A. wrote the manuscript. All authors commented on the manuscript.</p>", "<title>Code availability</title>", "<p>There is no customized code in generation or processing of datasets. For setting up and training the Deep Learning Models, the publicly available codes in Python language from TensorFlow^{##UREF##8##9##} and Keras^{##UREF##9##10##} libraries were used. The trend error mitigation and all the figure plots in the paper were implemented using the existing routines/functions in MATLAB software.</p>", "<title>Competing interests</title>", "<p id=\"Par52\">The authors declare no competing interests.</p>" ]

[ "<fig id=\"Fig1\"><label>Fig. 1</label><caption><p>Comparison of overall RMSE and NSE values computed over (<bold>a</bold>) land + ocean (global), (<bold>b</bold>) land-only and (<bold>c</bold>) ocean-only areas on Earth from 57 test months.</p></caption></fig>", "<fig id=\"Fig2\"><label>Fig. 2</label><caption><p>Spatial distributions of RMSE (<bold>a</bold>–<bold>d</bold>), NSE (<bold>e</bold>–<bold>h</bold>) and PCC (<bold>i</bold>–<bold>l</bold>) metrics of different solutions (Sol1, Sol2, Sol3 and Sol4) calculated from 57 test months and the ECDF illustrations of (<bold>m</bold>) RMSE, (<bold>n</bold>) NSE and (<bold>o</bold>) PCC for all solutions over land and over oceans on Earth.</p></caption></fig>", "<fig id=\"Fig3\"><label>Fig. 3</label><caption><p>Spatial distributions of RMSE, NSE, and PCC computed over land areas (excluding Antarctica), with each measure represented in separate lines. Specifically, (<bold>a,</bold><bold>e,</bold><bold>i</bold>) correspond to Sol1, (<bold>b,</bold><bold>f,</bold><bold>j</bold>) to Li <italic>et al</italic>.^{##UREF##6##7##}, (<bold>c,</bold><bold>g,</bold><bold>k</bold>) to Humphrey and Gudmundsson^{##UREF##5##6##}, and (<bold>d,</bold><bold>h,</bold><bold>l</bold>) to Löcher and Kusche^{##UREF##7##8##}. These calculations are based on 175 common months of mass anomaly solutions from all four studies. Additionally, the figure includes Empirical Cumulative Distribution Function (ECDF) plots for (<bold>m</bold>) RMSE, (<bold>n</bold>) NSE, and (<bold>o</bold>) PCC.</p></caption></fig>", "<fig id=\"Fig4\"><label>Fig. 4</label><caption><p>Cumulative mass change time series in Greenland from smoothed ResDCAE simulations (<italic>dashed blue</italic>) and from the IMBIE surface mass balance estimates (<italic>dashed black</italic>). Monthly mass anomaly from ResDCAE including seasonal mass change signal (<italic>solid red</italic>) is also shown. The shaded envelope represents the estimated 1-σ uncertainties of the cumulative changes of IMBIE.</p></caption></fig>", "<fig id=\"Fig5\"><label>Fig. 5</label><caption><p>Time series of Nino3.4’s SSTA index (derived from the region 120°W–170°W, 5°N–5°S) and mass anomaly time series from previous studies, CSRM, and our simulation (ResDCAE) for the Amazon River basin calculated by averaging all grids in TRIP basin boundaries. (<bold>a</bold>) Comparison of the Amazon mass anomaly signal of all compared models to precipitation time series from ERA5, CPC, and GPCC datasets (<bold>b</bold>).</p></caption></fig>", "<fig id=\"Fig6\"><label>Fig. 6</label><caption><p>Cross-correlations between precipitation anomalies, from (<italic>top to bottom</italic>) ERA5, CPC, GPCC and mass anomaly simulations from (<italic>left to right</italic>) ResDCAE (<bold>a,</bold><bold>e</bold>, and <bold>i</bold>), Li (<bold>b,</bold><bold>f</bold>, and <bold>j</bold>), Humphrey (<bold>c,</bold><bold>g</bold>,and <bold>k</bold>) and Löcher (<bold>d,</bold><bold>h</bold> and <bold>l</bold>) at 176 river basins defined in the TRIP database, respectively.</p></caption></fig>", "<fig id=\"Fig7\"><label>Fig. 7</label><caption><p>Comparison of ResDCAE-, CSRM-, EOP- and SLR-derived low-degree spherical harmonic coefficients, (<bold>a</bold>) ∆C₂₀, (<bold>b</bold>) ∆C₂₁ and (<bold>c</bold>) ∆S₂₁.</p></caption></fig>", "<fig id=\"Fig8\"><label>Fig. 8</label><caption><p>Deseasoned global mean ocean mass change time series (in mm equivalent water height) from our ResDCAE simulation (<italic>solid red</italic>), from steric corrected altimetry (<italic>solid green</italic>) and from original CSRM computed over the ocean grids between 65° N and 65° S latitudes. The altimetry derived GMSL time series (<italic>dashed blue</italic>) is from Horwath <italic>et al</italic>.^{##UREF##82##94##} and the steric component (<italic>dashed yellow</italic>) is from Camargo <italic>et al</italic>.^{##UREF##80##92##} both are presented for completeness.</p></caption></fig>", "<fig id=\"Fig9\"><label>Fig. 9</label><caption><p>Time series comparisons of <italic>in situ</italic> OBP measurements (<italic>green</italic>) with mass anomaly data from ResDCAE (<italic>red</italic>) and CSRM (<italic>blue</italic>) at two selected stations: (<bold>a</bold>) Dark Passage South – DPS (60.9° S – 54.7° W) and (<bold>b</bold>) NDBC 51406 – Central South Pacific (8.5° S – 125.0° W). The mass anomaly plots represent the monthly values at the 1° × 1° grid which contains the location of the corresponding OBP station.</p></caption></fig>" ]

[ "<table-wrap id=\"Tab1\"><label>Table 1</label><caption><p>Overall monthly RMSE (cm) and NSE metrics for 57 test months.</p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th/><th>Region</th><th>Sol1</th><th>Sol2</th><th>Sol3</th><th>Sol4</th></tr></thead><tbody><tr><td rowspan=\"3\"><bold>RMSE</bold></td><td><bold>Global - RMSE</bold></td><td>3.9</td><td>4.4</td><td>3.8</td><td>3.7</td></tr><tr><td><bold>Land - RMSE</bold></td><td>5.6</td><td>6.0</td><td>4.6</td><td>4.6</td></tr><tr><td><bold>Ocean - RMSE</bold></td><td>2.8</td><td>3.4</td><td>3.3</td><td>3.3</td></tr><tr><td rowspan=\"3\"><bold>NSE</bold></td><td><bold>Global - NSE</bold></td><td>0.87</td><td>0.85</td><td>0.88</td><td>0.88</td></tr><tr><td><bold>Land - NSE</bold></td><td>0.90</td><td>0.89</td><td>0.93</td><td>0.93</td></tr><tr><td><bold>Ocean – NSE</bold></td><td>0.73</td><td>0.62</td><td>0.64</td><td>0.65</td></tr></tbody></table></table-wrap>", "<table-wrap id=\"Tab2\"><label>Table 2</label><caption><p>Correlations between ResDCAE-, CSRM-, EOP- and SLR-derived degree-2 spherical harmonic coefficients ∆C₂₀, ∆C₂₁ and ∆S₂₁ within GRACE/-FO and pre-GRACE era.</p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th/><th/><th>Pre-GRACE</th><th>GRACE/-FO</th></tr></thead><tbody><tr><td rowspan=\"3\">Δ<italic>C</italic>₂₀</td><td>ResDCAE - SLR</td><td>0.70</td><td>0.80</td></tr><tr><td>ResDCAE - CSRM</td><td>—</td><td>0.96</td></tr><tr><td>CSRM - SLR</td><td>—</td><td>0.80</td></tr><tr><td rowspan=\"3\">Δ<italic>C</italic>₂₁</td><td>ResDCAE - EOP</td><td>0.58</td><td>0.65</td></tr><tr><td>ResDCAE - CSRM</td><td>—</td><td>0.94</td></tr><tr><td>CSRM - EOP</td><td>—</td><td>0.68</td></tr><tr><td rowspan=\"3\">Δ<italic>S</italic>₂₁</td><td>ResDCAE - EOP</td><td>0.65</td><td>0.85</td></tr><tr><td>ResDCAE - CSRM</td><td>—</td><td>0.96</td></tr><tr><td>CSRM - EOP</td><td>—</td><td>0.86</td></tr></tbody></table></table-wrap>" ]

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[ "<table-wrap-foot><p>(Note that the pre-GRACE era is not shown for comparisons to CSRM because there are no CSRM solutions available for that period.).</p></table-wrap-foot>", "<fn-group><fn><p><bold>Publisher’s note</bold> Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.</p></fn></fn-group>" ]

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[ "<title>Introduction</title>", "<p id=\"Par3\">Neural networks have made significant contributions to the field of Artificial Intelligence, serving as both a tool for mathematical modeling and a means to understand brain function. The Hopfield paradigm^{##REF##6953413##1##,##REF##10031847##2##} has played a crucial role in this domain, utilizing a synaptic matrix to represent the interconnections between neurons. This matrix possesses the remarkable ability to store and recognize patterns, and serve as a fundamental framework for the realization of future content-addressable memory (CAM)^{##UREF##0##3##,##UREF##1##4##}.</p>", "<p id=\"Par4\">To store a memory consisting of <italic>N</italic> elements, the widely adopted approach is to employ Hebb’s rule^{##UREF##2##5##}. This rule entails constructing a synaptic matrix, denoted as <bold><italic>T</italic></bold>, by taking the tensorial product of the vector <bold><italic>ϕ</italic></bold>^* (representing the pattern to be stored) and its conjugate transpose (<bold><italic>ϕ</italic></bold>^*†):However, there is a fundamental limit to the number of memories that can be reliably stored using Hebbian-based approaches. As the network becomes more densely populated, the interactions between different memory elements can lead to the emergence of unintended and uncontrolled memory states^{##REF##10031847##2##}. To address this limitation, recent research has explored various methods to enhance the capacity of neural networks: dilution^{##REF##27065365##6##–##UREF##3##9##}, autapses^{##REF##28119595##10##,##REF##33267440##11##}, and convex probability flow^{##UREF##4##12##,##UREF##5##13##}.</p>", "<p id=\"Par5\">Recently, it was proposed to leverage the interaction among stored patterns in a constructive way: an emergent archetype may be stored by proposing to the network multiple prototypes that closely resemble the target pattern but are intentionally corrupted or filled with errors. The interaction between these prototypes serves to strengthen the emergence of the desired memory^{##REF##35158159##14##}. This paradigm is connected to the prototype concept developed in hierarchical clustering, in which prototypes are elements of the dataset representative of each cluster^{##UREF##6##15##}.</p>", "<p id=\"Par6\">In this study, we introduce a novel learning strategy called <italic>Stochastic Emergent Storage</italic> (SES). SES taps into the abundance of natural randomness to construct an emergent representation of the desired memory. Capitalizing a vast database of fully random patterns freely produced by a disordered, self-assembled structure, we select a set of prototypes that bear resemblance to the target memory through a similarity-based criterion. Subsequently, by performing a weighted sum of the synaptic matrices corresponding to these selected prototypes, we are able to effectively generate the desired pattern in an emergent fashion.</p>", "<p id=\"Par7\">Given the inherent advantages of photonic computation, such as ultra-fast wavefront transformation and parallel operation, it results that optics is the ideal domain to explore the SES paradigm. The convergence of photonics, artificial intelligence, and machine learning represents a highly active and promising area of research^{##UREF##0##3##,##REF##36604423##16##}, leading to novel interdisciplinary paradigms such as Diffractive Deep neural networks^{##REF##30049787##17##,##UREF##7##18##} photonic Ising machines^{##REF##31283311##19##} and photonic Boltzmann computing Machines^{##UREF##8##20##}. However, these approaches typically rely on direct control over optical properties of millions of scattering elements, which can be challenging and costly both with microfabrication or adaptive optical elements.</p>", "<p id=\"Par8\">In a departure from traditional approaches, disordered scattering structures have been proposed as a radically different avenue for optical computation in various applications: classification^{##UREF##9##21##}, vector-matrix multiplication^{##UREF##10##22##}, computation of statistical mechanics ensembles dynamics^{##REF##34021081##23##}, and others^{##UREF##11##24##}.</p>", "<p id=\"Par9\">Here, we propose to employ the scattering intrinsic patterns, the optical transmission matrices, to realize a SES-based optical hardware, the disordered classifier. This device is capable of efficiently performing pattern storage, and subsequent pattern retrieval. It is able to simultaneously compare an input pattern with thousands of stored elements, and it enables a two-layer architecture, providing categorical (deep) classification, which allows for more complex tasks.</p>" ]

[ "<title>Methods</title>", "<title>Background</title>", "<p id=\"Par36\">In our experiment (similarly to a typical wavefront shaping experiment), light from a coherent source is controlled by a spatial light modulator and transmitted after propagation through a disordered medium into the mode <italic>ν</italic>. The field transmitted at <italic>ν</italic> is described aswhere the index <italic>n</italic> runs on the controlled segments at the input of the disordered medium, is the field resulting from laser field from the <italic>n</italic><italic>t</italic><italic>h</italic> segment transformed by the transmission matrix element on the sensor <italic>ν</italic> and is the phase value from the wavefront modulator. In our experiment, we consider the simplified configuration in which .</p>", "<p id=\"Par37\">The field at <italic>ν</italic> can be separated in its two components: the field-at-the-segment <italic>A</italic>_{<italic>n</italic>} and transmission matrix elment Indeed the are Gaussianly distributed complex numbers:In the case in which just two segments <italic>n</italic> and <italic>m</italic> are active and in the + 1 configuration, we can ignore the <italic>ϕ</italic>_{<italic>n</italic>}:In absence of modulation, intensity is written as the modulus square of <italic>E</italic>^{<italic>ν</italic>}we recognize thatthusorIn general for <italic>N</italic> segments in an arbitrary configuration of the modulatorthe argument of the sum can be written in matrix form defining the matrix <bold>V</bold>^{<italic>ν</italic>} also named optical coupling matrix:Matrix <bold>V</bold>^{<italic>ν</italic>} is a bi-dyadic matrix and it can be rewritten in matricial notation:where the <bold><italic>notation</italic></bold> indicates a vector on lowercase Greek letters and a matrix on uppercase, while ^† is the conjugate transpose operator. Being bi-dyadic the matrix possesses the eigenvectors <bold><italic>ξ</italic></bold>^{<bold><italic>ν</italic></bold>} and <bold><italic>η</italic></bold>^{<bold><italic>ν</italic></bold>} by construction. Note that , <bold>V</bold> and is Hermitian.</p>", "<p id=\"Par38\">When modulation is present with an input modulation pattern <bold><italic>ϕ</italic></bold>Note that even if <bold>V</bold>^{<italic>ν</italic>} is a complex matrix, being Hermitian, the double scalar product produces a real scalar because inverted sign imaginary contributions from above and below the diagonal result eliminated reciprocally thus producing a positive real intensity. The optical operator <bold>V</bold>^{<italic>ν</italic>}, associates thus the pattern/array <bold><italic>ϕ</italic></bold> to the scalar <italic>I</italic>^{<italic>ν</italic>} which is a measure of the degree of similitude of <bold><italic>ϕ</italic></bold> to the first eigenvector of <bold>V</bold>^{<italic>ν</italic>}, EIG(<bold>V</bold>^{<italic>ν</italic>}) = <bold><italic>ξ</italic></bold>^{<bold><italic>ν</italic></bold>}.</p>", "<p id=\"Par39\">Note that to simplify the realization of the experiment, we operate in the configuration in which each mode <italic>ν</italic> corresponds to a single sensor. As we employ a camera to measure <italic>I</italic>^{<italic>ν</italic>}, e the one-mode-per-pixel configuration is obtained by properly tuning the optical magnification.</p>", "<title>Stochastic Hebb’s storage protocols details</title>", "<p id=\"Par40\">By summing intensity measured at two modes <italic>ν</italic>₁ and <italic>ν</italic>₂, and considering linearity of the process:Generalizing, i.e. summing intensity at an arbitrary number <italic>M</italic> of modes pertaining to the set , we retrievewithThus, the optical operator textbfJ associates a pattern/array <bold><italic>ϕ</italic></bold> to the scalar , the <italic>transformed intensity</italic>, which is a proxy of the degree of similitude of <bold><italic>ϕ</italic></bold> to the first eigenvector of <bold>J</bold>: EIG(<bold>J</bold>)= . To deliver an user-designed arbitrary optical operator, we introduce the tailored attenuation coefficients <italic>λ</italic>^{<italic>ν</italic>} ∈ [0, 1]. These can be both obtained in “software version” (multiplying each <italic>I</italic>^{<italic>ν</italic>} by an attenuation coefficient <italic>λ</italic>^{<italic>ν</italic>}) or by realizing a mode-specific hardware optical attenuator (such as proposed in the sketch in Fig. ##FIG##0##1##, fuchsia windows, see below).</p>", "<p id=\"Par41\"><italic>Transformed intensity</italic> with the addition of the attenuation coefficients reads as:In SHS, the absorption coefficients <bold><italic>λ</italic></bold> are the free parameters which enable to design the arbitrary optical operator <bold>J</bold>. For example, to replicate the dyadic matrix constructed with he Hebb’s rule <bold>T</bold> and capable to store the pattern <bold><italic>ϕ</italic></bold> (see Fig ##FIG##0##1##c of the main paper) one has to select <bold><italic>λ</italic></bold> so that the functionis minimized. We name the artificial optical synaptic matrix in which <bold><italic>λ</italic></bold> have been optimized to deliver the optical operator <bold>T</bold>, andthe relative <italic>transformed intensity</italic>.</p>", "<p id=\"Par42\">This approach employs the random, naturally-occurring optical synaptic matrices from the set as a random basis on which to build the target optical operator. Its effectiveness is thus dependent on the number of free parameters with respect to the constraints. The constraints are the number of independent elements that have to be tailored on <bold>T</bold>. These are as <bold>T</bold> is symmetric. Indeed as shown in Fig. ##FIG##1##2## of the main paper (inset of panel b) for the <italic>N</italic> = 81 case, it is possible to replicate almost identically <bold>T</bold> when <italic>M</italic> &gt; Π, that is when the number of free parameters (the <bold><italic>λ</italic></bold>) is comparable with the constraints.</p>", "<title>Storage error probability</title>", "<p id=\"Par43\">In our storage paradigm, the stored pattern corresponds to the eigenvector of the <bold>T</bold>. As we are employing binary patterns, the sign operation is needed. The stored pattern is thus SEIG()= <bold><italic>ξ</italic></bold>_{<bold>Σ</bold>}, where the <italic>S</italic><italic>E</italic><italic>I</italic><italic>G</italic>() operator retrieves the first eigenvalue of a matrix and applies the sign operation to it. The <italic>Storage Error Probability</italic> reported in Figs. ##FIG##2##3## and ##FIG##1##2## the storage process effectiveness. First, we calculate the number of elements of <bold><italic>ξ</italic></bold>_{<bold>Σ</bold>} which differ from the target memory <bold><italic>ϕ</italic></bold>^*, <italic>S</italic>_<italic>E</italic><italic>R</italic><italic>R</italic>. The value of <italic>S</italic>_<italic>E</italic><italic>R</italic><italic>R</italic> can be seen as the number of error pixels in the stored pattern image.</p>", "<p id=\"Par44\">Then we computeFor storage purposes, obviously the lower, the <italic>Storage Error Probability</italic> the better.</p>", "<title>Recognition error probability</title>", "<p id=\"Par45\">The optical operator <bold>J</bold> associates the <italic>transformed intensity</italic> scalar to each input pattern <bold><italic>ϕ</italic></bold>:we can thus employ the experimentally measured <italic>transformed intensity</italic> to recognize patterns. We employed a repository of <italic>P</italic> = 5000 patterns containing digits with random orientation (<ext-link ext-link-type=\"uri\" xlink:href=\"https://it.mathworks.com/help/deeplearning/ug/data-sets-for-deep-learning.html\">https://it.mathworks.com/help/deeplearning/ug/data-sets-for-deep-learning.html</ext-link>), labeling as recognized patterns, the ones producing a transformed intensity above 10 standard deviations from the values obtained probing randomly generated binary patterns. The value <italic>R</italic>_<italic>E</italic><italic>R</italic><italic>R</italic> is the number of wrongly identified patterns experimentally.</p>", "<p id=\"Par46\">Indeed, the <italic>transformed intensity</italic> is obtained experimentally optically presenting the pattern to our disordered classifier. The step-by-step presentation procedure is the following: <italic>i)</italic> the probe pattern <italic>ϕ</italic> is printed onto a propagating laser beam employing a DMD in binary phase modulation mode (see experimental section), <italic>ii)</italic> light scattered by the disordered medium is retrieved for the relevant mode/pixel set , <italic>iii)</italic> the transformed intensity measured by the selected sensors/camera pixels is obtained with Eq. (##FORMU##71##30##), <italic>iv)</italic> a pattern is defined as recognized if the <italic>trained transformed intensity</italic> results higher than the threshold. The <italic>Recognition Error Probability</italic> is then obtained asThe Supplementary Fig. ##SUPPL##0##4## visualizes for the classification/recognition process.</p>", "<p id=\"Par47\">Note that Storage Error Probability (<italic>S</italic>_<italic>E</italic><italic>R</italic><italic>R</italic>) and Recognition error probability (<italic>R</italic>_<italic>E</italic><italic>R</italic><italic>R</italic>) provide insights on two very different aspects of our storage platform performance. <italic>S</italic>_<italic>E</italic><italic>R</italic><italic>R</italic> is essentially a storage fidelity observable, counting the ratio of wrong/correct pixels in the pattern to be stored which differ from the target memory to be stored <bold><italic>ϕ</italic></bold>^*, and accounts for the efficiency of our approach (the emergent storage) to instantiate a target memory in a memory repository. <italic>R</italic>_<italic>E</italic><italic>R</italic><italic>R</italic> retrieves recognition efficiency, thus reports on the ratio of memory retrieval tests providing wrong memory addresses, when different input patterns from a repository are proposed as stimuli. The <italic>S</italic>_<italic>E</italic><italic>R</italic><italic>R</italic> influences <italic>R</italic>_<italic>E</italic><italic>R</italic><italic>R</italic>: i.e. if many error are present in the pattern injected in a repository the recognition fails. However <italic>R</italic>_<italic>E</italic><italic>R</italic><italic>R</italic> is also affected by other features such as for example the order of nolinearity (we use intensity do appreciate differences in the field thus we employ a second order nonlinearity) which influences the capability to differentiate similar patterns and also the structure of the repository (if the repository contains very similar patterns then the recognition task is more difficult). Thus the relation is not a simple proportionality, while the two observable look at two very different aspects of the memory process i.e. storage fidelity and recognition efficiency.</p>", "<p id=\"Par48\">In Deep-SES instead, a single probe pattern is compared with many memories. We performed this task with digital data analysis but all the processes can be realized analogically, by performing pixel selection and weighting with DMDs. In such a case the probe pattern is directly tested against many memories: all the ones composing the training set. For the 9 class digit classification reported in the Fig, 3969 individual memories (441 per class) have been used. Employing a DMD with 33 kHz frame rate would mean essentially performing optical classification in 0.1 seconds.</p>", "<title>Stochastic emergent storage protocol details</title>", "<p id=\"Par49\">In SES (see code and data at^{##UREF##21##35##}) we exploit the fact that any optical coupling matrix <bold><italic>V</italic></bold>^{<bold><italic>ν</italic></bold>} is a bi-dyadic thus hosting two intrinsic but random patterns:thus the optical coupling matrix at location <italic>ν</italic>, <bold>V</bold>^{<italic>ν</italic>}, hosts the two random memory vectors <bold><italic>ξ</italic></bold>^{<bold><italic>ν</italic></bold>} and <bold><italic>η</italic></bold>^{<bold><italic>ν</italic></bold>}.</p>", "<p id=\"Par50\">To employ these disorder-embedded structures as memories we resorted to the following multi step strategy.</p>", "<p id=\"Par51\">\n<list list-type=\"simple\"><list-item><label>i.</label><p id=\"Par52\">We start measuring the transmission matrices from a large set of modes employing the Complete Couplings Mapping Method (CCMM, see below). We monitor <italic>M</italic>^{<italic>L</italic>} = 65536 modes employing a region of interest for the camera of 256 × 256 pixels in the one-mode-per-pixel configuration. The retrieved transmission matrices are saved into a computer memory and compose our starting random structures repository .</p></list-item><list-item><label>ii.</label><p id=\"Par53\">We computationally find the first eigenvector <bold><italic>ξ</italic></bold>^{<italic>ν</italic>} for each measured matrix <bold>V</bold>^{<italic>ν</italic>}</p></list-item><list-item><label>iii.</label><p id=\"Par54\">The user, designs a target memory pattern to be stored <bold><italic>ϕ</italic></bold>^* and a number <italic>M</italic>^* of modes to be employed.</p></list-item><list-item><label>iv.</label><p id=\"Par55\">The target pattern <bold><italic>ϕ</italic></bold>^* is compared with all the eigenvectors in by computing the similitude degree :with the symbol indicating vector normalization: .</p></list-item><list-item><label>v.</label><p id=\"Par56\">The set of modes is similarity-decimated to the set , i.e. we select the <italic>M</italic>^* modes with the higher values to be part of the new, reduced repository .</p></list-item></list>\n</p>", "<p id=\"Par57\">Once is realized, we need to “train” the attenuation coefficients <bold><italic>λ</italic></bold>. The attenuation values are selected between 16 values degrees of absorption in the ∈ [0, 1] range, so that they are identified with a 4 bits number.</p>", "<p id=\"Par58\">After initializing the lambda and computing the initial configuration optical operatorthe <bold><italic>λ</italic></bold> are optimized with a Monte Carlo algorithm. At each optimization step a single <italic>λ</italic>^{<italic>ν</italic>} is modified and the change is accepted if the eigenvector similarity functiondecreases. Note that in Eq. (##FORMU##93##37##), <bold><italic>ξ</italic></bold>_Σ is the first eigenvector of <bold>J</bold>.</p>", "<p id=\"Par59\">After a sufficiently large number of steps is minimized and form the final configuration of <bold><italic>λ</italic></bold> we obtain the final version of the optical operator: .</p>", "<p id=\"Par60\">Note that the previous procedure can be cast in a computationally lighter version replacing some digital operations with optical measurements. The similarity selection can be substituted with intensity measurement. Indeed intensity itself is a direct measure (see Eq. (##FORMU##57##24##)) of the degree of similarity of the probe pattern with the correspondent <bold><italic>t</italic></bold>^{<italic>ν</italic>} vector, thus similarity selection can be substituted by an optical operation with cost <italic>M</italic>^{<italic>L</italic>}.</p>", "<title>Experimental setup and CCMM</title>", "<p id=\"Par61\">The same experimental setup is employed for two tasks. The first is the measurement of the optical synaptic matrices <bold>V</bold>^{<italic>ν</italic>}, the second is to perform classification, presenting to the disordered classifier a test pattern <bold><italic>ϕ</italic></bold> and retrieving the <italic>transformed intensities</italic> for each trained memory. A sketch of the experimental setup is provided in Supplementary Fig. ##SUPPL##0##1## in supplementary information file.</p>", "<p id=\"Par62\">We employ a single mode laser (AzurLight 532, 0,5W) with beam to about 1 cm. Then it is fragmented into <italic>N</italic> individually modulated light rays controlled by a Digital Micromirror Device (DMD)^{##REF##33414495##36##} composed by 1024 × 768 (Vialux, V-7000, pixel Pitch 13.68 μm, 22 kHz max frame rate) flipping mirrors which can be tuned into two configurations (on or off). Phase modulation is obtained employing the super-pixel method (see refs. ^{##REF##34021081##23##,##REF##25089419##37##}) which require a spatial filtering to isolate the selected diffraction orders. DMD pixels are organized into <italic>N</italic> 4-elements super-pixels (segments) capable to produce a 0 or <italic>π</italic> phase pre-factors equivalent to field multiplication by <italic>ϕ</italic>_{<italic>n</italic>} = ∈ { − 1, 1}. The bundle of light rays is then scrambled by a diffusive, multiple scattering medium (60 μ layer of ZnO obtained from ZnO powder from Sigma Aldrich item 544906-50g, transport mean free path 8 μm^{##REF##33397941##38##}). The <italic>N</italic> super-pixels are organized on the DMD in a square array, which is illuminated by an expanded laser Gaussian beam (diameter of about 1 cm). Indeed, the DMD surface is imaged onto the Diffusive medium (0.3 × de-magnification). This de-magnification is required to ensure the diffused image to fit into the selected detection camera ROI. Then, the back layer of the disordered structure is imaged on the detection camera (11 × magnification). This magnification has been chosen to minimize the speckle grain size in order to work in the one-mode-per-pixel configuration (one-pixel-per-speckle-grain). The optical collection apparatus, does not require a particular performance, indeed we employed a commercial, low-cost 25.45 mm focal bi-convex lens for the light collection from the far side of the sample. Several constraints have to be considered in the experimental design. When light from a DMD super-pixel emerges from the disordered medium, it is diffused into a larger disk-shaped area. For this reason, we have to ensure that each these light disks are interfering with all the disks generated by other super-ixels in the detection camera ROI, and this introduces a constraint on the maximum ROI size (<italic>M</italic>^{<italic>L</italic>}). The size of these diffusion disks is regulated by the thickness of the disordered scattering medium. Nevertheless, note that increasing the scattered thickness also decreases the light intensity on the camera and the stability of the speckle pattern thus a trade-off between thickness and signal-stability should be found at the experimental design step.</p>", "<p id=\"Par63\">Superpixel method is obtained thanks to 2.66 mm aperture iris in front of the DMD. As shown in Eq. (##FORMU##50##19##) when two DMD mirrors are activated:while if a single segment is activatedThus putting together Eq. (##FORMU##97##38##) and Eq. (##FORMU##98##39##) one obtains</p>", "<p id=\"Par64\">Thus to determine one single element of the optical synaptic matrix, one has to perform three intensity measurements. The total number of measurement to reconstruct the full synaptic matrix is Π = <italic>N</italic>(<italic>N</italic> − 1)/2 (as i.e. the optical synaptic matrix is symmetric then just the above-the-diagonal elements need to be measured). For <italic>N</italic> = 256 this means that 32896 measurements are required, which can be obtained in maximum 5 minutes employing our DMD-Camera experimental setup (speed bottleneck from the camera sensor which works at ~ 150 frames per second). At each measurement we take a image from a Region Of Interest (ROI) of 256 × 256 pixels, thus collecting info for <italic>M</italic>^{<italic>L</italic>} = 65536 modes. This measurement realizes modes, random memories and optical synaptic matrix for the database . Experimental data are organized into a 32896 × 256 × 256 matrix. In our case increasing the size of <italic>N</italic> or <italic>M</italic>^{<italic>L</italic>} is limited by the size of the Random Access Memory size of the computing workstation.</p>", "<title>Satistics and reproducibility</title>", "<p id=\"Par65\">In error bars in Figs. ##FIG##1##2##–##FIG##3##4## represent standard error, obtained realizing 10 different target matrices <bold>T</bold> for each <italic>M/N</italic> value, and measuring standard deviation <italic>σ</italic> for each dataset and calculating standard error as ERR. <bold>T</bold> matrices where blindly and randomly extracted at each measure from a 5000 elements pattern repository. No statistical method was used to predetermine sample size. No data were excluded from the analyses.</p>" ]

[ "<title>Results</title>", "<p id=\"Par10\">The idea stems from the fact that intensity scattered by a disordered medium into a mode <italic>ν</italic> resulting from an input pattern <bold><italic>ϕ</italic></bold>) may be written as:with the scattering process driven by the matrix <bold>V</bold> :generated from the tensorial product of the transmission matrix row (transmission vector) <bold><italic>ξ</italic></bold>^{<italic>ν</italic>} ()with its conjugate transpose <bold><italic>ξ</italic></bold>^{<italic>ν</italic>†}.</p>", "<p id=\"Par11\">Indeed <italic>I</italic>^{<italic>ν</italic>}(<bold><italic>ϕ</italic></bold>) is maximized if <bold><italic>ϕ</italic></bold>∥<bold><italic>ξ</italic></bold>^{<italic>ν</italic>}: this paradigm is at the basis of the wavefront shaping techniques^{##UREF##12##25##,##UREF##13##26##}, in which the input pattern is adapted to the transmission matrix elements. Thus, scattering into a mode (corresponding to one of our camera pixels, see methods) is described by the same mathematics of the Hopfield Hamiltonian and a pattern is “recognized” (produces maximal intensity) if it matches the <bold><italic>ξ</italic></bold>^{<italic>ν</italic>} vector. Given this mapping, <bold>V</bold>^{<italic>ν</italic>} may be named an optical synaptic matrix relative to the <bold><italic>ξ</italic></bold>^{<italic>ν</italic>} memory.</p>", "<p id=\"Par12\">In naturally occurring scattering, one has no control over the pattern <bold><italic>ξ</italic></bold>^{<italic>ν</italic>} and the relative optical synaptic matrix <bold>V</bold>^{<italic>ν</italic>} because it results from a multitude of subsequent scattering events with micro-nano particles of unknown shape, optical properties, and location. Here, we propose to store an arbitrary, user-defined, memory (or pattern) in naturally occurring scattering media, by exploiting the fact that a scattering process generated billions of output modes, each with a unique and random embedded memory pattern <bold><italic>ξ</italic></bold>^{<italic>ν</italic>} and the relative <bold><italic>V</italic></bold>^{<italic>ν</italic>}. Thus we propose a new method to realize a photonic linear combination of <bold>V</bold>^{<italic>ν</italic>} to generate an artificial, (user-designed) optical synaptic matrix. This method is based on the realization of a sensor collecting the <italic>transformed intensity</italic>resulting from the incoherent sum of <italic>M</italic> intensities realized from that many transmitted optical modes from which is a subset of all the modes monitored . Coefficient <italic>λ</italic>^{<italic>ν</italic>} ( ∈ {0 − 1} and identified by a 4 bit positive real number) represent attenuation coefficients realized by mode-specific neutral density filters. Then employing the Eq. (##FORMU##1##2##) in Eq. (##FORMU##5##4##) we obtainThen, we propose two techniques to design the optical operator <bold>J</bold>: 1) the Stochastic Hebb’s Storage (SHS) which enables to realize an arbitrary optical operator, 2) the Stochastic Emergent Storage (SES) which instead aimed to the realization of optical memories.</p>", "<title>Stochastic Hebb’s storage</title>", "<p id=\"Par13\">First, we will employ this to realize an optical equivalent of the Hebbs rule: the <italic>stochastic Hebbs storage</italic> (SHS). Then we will show how the storage and recognition performance is greatly improved if SES is exploited.</p>", "<p id=\"Par14\">With the SHS we want to generate a synaptic optical matrix <bold>J</bold> equivalent to an Hebb’s matrix <bold>T</bold> with the aim to store the pattern <bold><italic>ϕ</italic></bold>^*. To do this, we rely on a linear combination of a set of random optical synaptic matrices resulting from uncontrolled scattering:Thus given Eq. (##FORMU##8##5##), the <italic>transformed intensity</italic> with the optical operator emulates the Hamiltonian function associated to Hebb’s synaptic matrix <bold>T</bold>. Indeed the matrix is connected to the intensities of the modes pertaining to the set with the following equation:The values for coefficients <italic>λ</italic>^{<italic>ν</italic>} are obtained by a Monte Carlo algorithm, (see methods) minimizing the difference between the target matrix and <bold>J</bold>. Each coefficient may be then realized in hardware (mode-specific neutral density filters) or software fashion.</p>", "<p id=\"Par15\">Employing SHS we can design any arbitrary optical operator if the two following ingredients are available: i) the access to the intensity <italic>I</italic>^{<italic>ν</italic>}(<bold><italic>ϕ</italic></bold>) produced by a sufficiently large number of modes and ii) the correspondent optical synaptic matrix <bold>V</bold>^{<italic>ν</italic>} for each mode. This is now possible with the Complete Couplings Mapping Method (CCMM, see methods), which enables the measurement of the intrinsic (no interference with a reference) <bold>V</bold>^{<italic>ν</italic>} with a Digital Micromirror Device (DMD).</p>", "<p id=\"Par16\">With the CCMM, and the experimental apparatus shown in Fig. ##FIG##0##1## see methods we are able to gather a repository of tens of thousands (<italic>M</italic>^{<italic>L</italic>} = 65536) of optical synaptic matrices in minutes from which we sample a random subset (with <italic>M</italic> random samples) which we use as bases to construct our target artificial synaptic matrix.</p>", "<p id=\"Par17\">The performance of this optical learning approach is shown in Fig. ##FIG##1##2##, in which we realized an Hebb’s dyadic-like optical synaptic matrix (see insets of Fig. ##FIG##1##2##) from a ZnO scattering layer.</p>", "<p id=\"Par18\">The memory pattern stored in our system is , with SEIG(<bold>H</bold>) the operator that finds the eigenvector correspondent to the largest eigenvalue of <bold>H</bold> and then produces a binary vector with its elements’ sign. Performance in storage and recognition for SHS are reported respectively in Fig. ##FIG##1##2##a, b (see methods). There we report the <italic>Storage Error Probability</italic> (the lower the better, indicates the average number of pixels differing between the stored and the target pattern, full definition in the, methods) and <italic>Recognition Error</italic> (the lower the better, the percentage of wrongly recognized memory elements out of a repository of 5000 presented patterns, full definition in the, methods).</p>", "<p id=\"Par19\">SHS is basis hungry, requiring a large number of random optical synaptic matrices (which means modes/sensors/pixels) to successfully construct a memory element. his is connected to the fact that the target matrix T is constructed on <italic>N</italic> × <italic>N</italic>/2 parameters (is symmetrical) acting as constraints, while we have <italic>M</italic> free parameters to emulate it. A full emulation of <italic>T</italic> is expected thus to be successful for <italic>M</italic> &gt; <italic>N</italic> × <italic>N</italic>/2 which is consistent with what we retrieve in Fig. ##FIG##1##2## (Data for <italic>M</italic> = 4096 are out of scale as storage and recognition error is negligible).</p>", "<title>Stochastic emergent storage</title>", "<p id=\"Par20\">For the remainder of the paper, we will discuss how the performance drastically improves with SES. We recognize that each optical synaptic matrix contains the strongest of two memories <bold><italic>ξ</italic></bold>^{<italic>ν</italic>} = SEIG(<bold>V</bold>^{<italic>ν</italic>}) then (instead of randomly extracting modes) we perform a similarity selection (see Fig. ##FIG##0##1##a and Supplementary Figs. ##SUPPL##0##1## and ##SUPPL##0##2## in the supplementary information file) in which we extract a set <italic>M</italic>^* whose intrinsic memories are the closest possible to the target pattern <bold><italic>ϕ</italic></bold>^* (see insets of Fig. ##FIG##0##1##). The fact that in a mesoscopic laser scattering process, billions of independent modes can be produced and millions of them can be measured at once with modern cameras, is strategically employed in SES to boost the performance.</p>", "<p id=\"Par21\">To perform the similarity selection with the optical modes the target pattern <bold><italic>ϕ</italic></bold>^* is compared with the eigenvectors of all the modes in the repository of characterized modes . The comparison is driven by the parameter that quantifies the degree of similarity between the first eigenvector of mode <italic>ν</italic>, <bold><italic>ξ</italic></bold>^{<italic>ν</italic>}, and <bold><italic>ϕ</italic></bold>^*.</p>", "<p id=\"Par22\">The modes <italic>ν</italic> providing the higher are selected to feed a restricted repository of modes . The correspondent eigenvectors <bold><italic>ξ</italic></bold>^{<italic>ν</italic>} can be seen as prototypes of the target archetype, i.e. imperfect representations of the pattern to be stored (such as the one in Fig. ##FIG##0##1##c).</p>", "<p id=\"Par23\">In SES, these prototypes interact constructively, generating a representation of the memory <bold><italic>ϕ</italic></bold>^* in an emergent fashion^{##REF##35158159##14##}. The interaction is obtained by the incoherent sum of the intensity of several pixels/modes with proper attenuation coefficients/weights <italic>λ</italic>.</p>", "<p id=\"Par24\">The attenuation coefficients <bold><italic>λ</italic></bold> are found by minimizing the distance between the archetype pattern to be stored <bold><italic>ϕ</italic></bold>^* and the matrix first eigenvector SEIG (see methods). Thus substituting in Eq. (##FORMU##8##5##) the transformed intensity in SES reads :The potential of SES is clarified in Fig. ##FIG##2##3##: the panels on the top left represent the stored pattern (target pattern is reported in Fig. ##FIG##0##1##c) for various sizes <italic>M</italic>^* of the restricted repository. Note that SES greatly outperforms the random selection approach where emergent storage is absent (panel on the right).</p>", "<p id=\"Par25\">Figure ##FIG##2##3##a shows the storage capability of the system. Blue triangles are relative to patterns with <italic>N</italic> = 81 elements, while for golden diamonds <italic>N</italic> = 256. The <italic>Storage Error Probability</italic> (Fig. ##FIG##2##3##c) improves more than an order of magnitude with respect to random selection (red circles). <italic>Recognition Error Probability</italic> (Fig. ##FIG##2##3##b) is three to four orders of magnitude better with respect to the randomly selected database. Note that the SES enormously outperforms SHS, indeed it is possible to perform recognition in the <italic>M</italic> &lt; &lt; <italic>N</italic> configuration, i.e. employing a number of camera pixels(<italic>M</italic>) much smaller than the elements composing the pattern <italic>N</italic>.</p>", "<p id=\"Par26\">Thus, in summary, SHS enables to create an optical operator of arbitrary nature, which can effectively execute diverse tasks. This versatility arises from its capability to construct an artificial optical synaptic matrix designed by the user, effectively emulating a matricial operator T. Conversely, SES focuses its functionality on generating an operator designed primarily for memory storage, excelling in this singular aspect. Consequently, it demands significantly less computational power and a smaller optical hardware setup (with a smaller <italic>M</italic>^*, see below), and enables lossy data compression (see supplementary information file and Supplementary Fig. ##SUPPL##0##3)##.</p>", "<p id=\"Par27\">This distinction influences the optimization procedure: SHS optimization relies on distances between matrices (measuring such distance computational cost scales as <italic>N</italic> × <italic>N</italic>), while SES optimization is driven by distances between vectors (measuring such distance computational cost scales as <italic>N</italic>). Secondly, SES leverages preliminary similarity selection to identify the most relevant pixels/modes, a feature absent in SHS. As a result, the modes chosen for SES provide higher contrast in the classification task, especially in the <italic>M</italic> &lt; <italic>N</italic> regime. In contrast, in the <italic>M</italic> &gt; <italic>N</italic> regime (more degrees of freedom than constraints), both approaches achieve essentially the same level of efficiency.</p>", "<p id=\"Par28\">Figure ##FIG##2##3##c, d shows a recognition process example. The emergent learning process has been employed to store the pattern <bold><italic>ϕ</italic></bold> with index <italic>j</italic> = 241 from a repository of 5000 patterns. Figure ##FIG##2##3##c reports <italic>transformed intensity</italic> for the first 600 repository elements: a clear peak is distinguishable at <italic>j</italic> = 241 this implies that the pattern is recognized). The same graph is shown for the case of the random basis case (no similarity selection), in which recognition is more noisy.</p>", "<title>Photonic disordered classifier</title>", "<p id=\"Par29\">Our <italic>disordered classifier</italic> can work in parallel, simultaneously comparing an input with all memories stored, effectively working as a content addressable memory^{##UREF##1##4##}.</p>", "<p id=\"Par30\">The experimentally retrieved <italic>transformed intensity</italic> for 4096 different memory elements <bold><italic>ϕ</italic></bold>^* is reported in Fig. ##FIG##3##4##a (organized in a camera-like 64 × 64 pixels diagram) for the proposed pattern <bold><italic>ϕ</italic></bold>. Each value of represent the degree of similitude of <bold><italic>ϕ</italic></bold> to <bold><italic>ϕ</italic></bold>^*. The patterns to the right side of the panel report the proposed pattern <bold><italic>ϕ</italic></bold> and the stored patterns relative to each arrow-indicated pixel. The pixel indicated with a red circle contains the pattern most similar to <bold><italic>ϕ</italic></bold> thus as expected produces the highest intensity. The system effectively works as a CAM in which an input query <bold><italic>ϕ</italic></bold> is tested in parallel against a list of stored patterns (the <bold><italic>ϕ</italic></bold>^*) identifying the matching memory as the most intense <italic>transformed intensity</italic> pixel.</p>", "<p id=\"Par31\">The interplay between Hopfield networks and Deep learning has been recently proposed and investigated^{##UREF##14##27##,##UREF##15##28##}. In this framework here we demonstrate a new approach to perform higher rank categorical classification employing the cashed memories as features^{##UREF##16##29##,##UREF##17##30##}: the deep-SES. We tested it on a 4500 randomly tilted digits images repository which is organized into 9 categories (digits from 1 to 9). We stored 3969 patterns/features in the disordered classifier (<italic>m</italic> = 441 per each digit), leaving 59 patterns per category for validation. In the camera-like diagrams (Fig. ##FIG##3##4##b, d) each category is found in the correspondent quadrant of the image. The two panels show the response of the disordered classifier to the inputs on the left for which the correspondent quadrants show a high number of intense pixels. Figure ##FIG##3##4##c shows integrated intensity after threshold. Figure ##FIG##3##4##e reports the confusion matrix for all labels, demonstrating categorical recognition efficiency above 90% which eventually may be enhanced employing error correction algorithms^{##UREF##18##31##}. This result demonstrates the possibility to generate deeper optical machine learning achitectures and perform training by simply grouping memories. The potential of Deep-SES is further demonstrated by Fig. ##FIG##3##4##f, where we report a figure of merit comparing the efficiency of Deep-SES with Ridge Regression with Speckles (RRS)^{##UREF##9##21##} (simulated). Note, while Deep-SES reaches an efficiency 90% for <italic>M</italic>^* = 40, the RRS suprasses this threshold for <italic>M</italic>^* = 1600. As <italic>M</italic>^* represent the number of physical camera pixel employed in the classification, SES is capable of delivering a classifier with a much smaller hardware and computational complexity. The origin of this advantage, emerges form the fact that our memory writing process, selects pixels/modes which are the most correlated with the pattern to be recognized thus outperform with respect to randomly chosen ones. Moreover deep-SES enables thus to reorganize memories into new classes (reshuffling of classes) with almost no computational cost, a task which typically requires a new training in standard digital or optical architectures (see methods and supplementary information file, “Comparison with other platforms” section and Supplemenatary Table ##SUPPL##0##1)##.</p>" ]

[ "<title>Discussion</title>", "<p id=\"Par32\">In summary, the Stochastic Emergent Storage (SES) paradigm enables classification with a significantly smaller number of sensors/pixels/modes compared to the elements composing the pattern. This opens up the possibility of fabricating complex pattern classifiers with only a few detecting elements, eliminating the fabrication processes. Deep-SES offers a new paradigm for network training, enabling to generate classes just by grouping memories, and it opens the way to a computation-free rearrangement of classes.</p>", "<p id=\"Par33\">The paradigms presented here can be potentially exported to other disordered systems, such as biological neural networks or neuromorphic computer architectures while exploring the emergent learning process in these systems can also provide valuable insights into the memory formation process in the brain.</p>", "<p id=\"Par34\">The results presented in this study contributes to the ongoing challenge of understanding the biological memory formation process. There are currently two major hypotheses that are the subject of debate, the connectionist hypothesis^{##UREF##2##5##}, which suggests that neural networks form new links or adjust existing ones when storing new patterns, and the innate hypothesis^{##REF##21383177##32##}, which posits that patterns are stored using pre-existing neural assemblies with fixed connectivity. One central aspect in this ongoing debate pertains to the ’efficiency’ of the network, a facet that, in both artificial and natural networks, immediately invokes considerations related to energy consumption. On one side, it has long been established that in Hebbian networks, the number of memories (W) scales linearly with the number of nodes (N), expressed as <italic>W</italic> = <italic>α</italic><italic>N</italic>. For this reason, many research efforts are dedicated to optimizing the proportionality constant <italic>α</italic>, which however appears to be upper bounded to two. On the other side, it has been recently demonstrated, both numerically^{##REF##29705670##7##,##UREF##3##9##} and theoretically^{##UREF##19##33##,##UREF##20##34##}, that in the stochastic innate approach, the number of memory increases exponentially with the number of nodes: <italic>W</italic> ∝ <italic>e</italic>^{<italic>a</italic><italic>N</italic>}. In other words, for larger system sizes, the innate approach predicts a significantly greater number of memories compared to the connectionist perspective. The “complexity” of the system (artificial neural network or brain), denoted as , tends to zero for the connectionists, whereas it remains non-zero for the innatists.</p>", "<p id=\"Par35\">SES introduces a fresh perspective to the problem by leveraging the Hebbian structure of the synaptic matrix, with a foundation of the connectionist hypothesis. However, SES goes beyond by exploring the potential of a stochastic innate network in which, pre-existent random synaptic structures are combined to generate memory elements in an emergent manner. Whether the SES could bring a new point of view, lumping together the innatism and connectivism, is a fascinating hypothesis, that must be explored in the future.</p>" ]

[]

[ "<p id=\"Par1\">Disorder is a pervasive characteristic of natural systems, offering a wealth of non-repeating patterns. In this study, we present a novel storage method that harnesses naturally-occurring random structures to store an arbitrary pattern in a memory device. This method, the Stochastic Emergent Storage (SES), builds upon the concept of emergent archetypes, where a training set of imperfect examples (prototypes) is employed to instantiate an archetype in a Hopfield-like network through emergent processes. We demonstrate this non-Hebbian paradigm in the photonic domain by utilizing random transmission matrices, which govern light scattering in a white-paint turbid medium, as prototypes. Through the implementation of programmable hardware, we successfully realize and experimentally validate the capability to store an arbitrary archetype and perform classification at the speed of light. Leveraging the vast number of modes excited by mesoscopic diffusion, our approach enables the simultaneous storage of thousands of memories without requiring any additional fabrication efforts. Similar to a content addressable memory, all stored memories can be collectively assessed against a given pattern to identify the matching element. Furthermore, by organizing memories spatially into distinct classes, they become features within a higher-level categorical (deeper) optical classification layer.</p>", "<p id=\"Par2\">Photonic Stochastic Emergent Storage is a neuromorphic photonic device for image storage and classification based on scattering-intrinsic patterns. Here, the authors show emergent storage employs stochastic prototype scattering-induced light patterns to generate categories corresponding to emergent archetypes.</p>", "<title>Subject terms</title>" ]

[ "<title>Supplementary information</title>", "<p>\n\n\n</p>" ]

[ "<title>Supplementary information</title>", "<p>The online version contains supplementary material available at 10.1038/s41467-023-44498-z.</p>", "<title>Acknowledgements</title>", "<p>This research was funded by: Regione Lazio, Project LOCALSCENT, Grant PROT. A0375-2020-36549, Call POR-FESR “Gruppi di Ricerca 2020” (to M.L.); ERC-2019-Synergy Grant (ASTRA, n. 855923; to GR); EIC-2022-PathfinderOpen (ivBM-4PAP, n. 101098989; to G.R.); Project “National Center for Gene Therapy and Drugs based on RNA Technology” (CN00000041) financed by NextGeneration EU PNRR MUR - M4C2 - Action 1.4 - Call “Potenziamento strutture di ricerca e creazione di campioni nazionali di R&amp;S” (CUP J33C22001130001) (to G.R.). The authors Acknowledge Enrico Ventura, and Luigi Loreti for fruitful discussions.</p>", "<title>Author contributions</title>", "<p>M.L. Conceived the Idea, Designed and Realized the experiments, Analyzed the data. G.G. analyzed the data, developed the geometrical interpretation and performed numerical simulations of RRS. G.R. conceived the mapping with the innate and connectionist conjectures. All authors contributed to data interpretation and writing the manuscript.</p>", "<title>Peer review</title>", "<title>Peer review information</title>", "<p id=\"Par66\"><italic>Nature Communications</italic> thanks the anonymous reviewer(s) for their contribution to the peer review of this work. A ##SUPPL##1##peer review file## is available.</p>", "<title>Data availability</title>", "<p>Experimental and generated data related to the generated in this study are deposited in the GitHub repository at the address 10.5281/zenodo.10222344^{##UREF##21##35##}.</p>", "<title>Code availability</title>", "<p>Code realized in this study are deposited in the GitHub repository at the address 10.5281/zenodo.10222344^{##UREF##21##35##}.</p>", "<title>Competing interests</title>", "<p id=\"Par67\">The authors declare no competing interests.</p>" ]

[ "<fig id=\"Fig1\"><label>Fig. 1</label><caption><title>Emergent memory storage scheme.</title><p>The sketch describes both the input query <bold><italic>ϕ</italic></bold> presentation and the measurement of the optical synaptic matrix <bold>V</bold>^{<italic>ν</italic>} (details in the Methods section). The first set of lenses (A-B) demagnifies (by a factor of 0.3) the DMD image, accommodating the scrambled input pattern of the scattering medium within Lens1’s field of view. The second set of lenses (1-2) images the opaque medium’s backplane onto the camera plane, with a magnification (factor of 11) that ensures the 1 speckle grain/mode per pixel imaging regime. <bold>a</bold>–<bold>c</bold> Illustrate the process of instantiating a memory in our architecture. <bold>a</bold> Represents the similarity selection stage, wherein optical synaptic matrix (<italic>V</italic>^{<italic>ν</italic>}) are chosen based on their similarity with the target memory (<bold><italic>ϕ</italic></bold>^*). <bold>b</bold> Illustrates the construction of an emergent memory through the summation of relative optical synaptic matrices (), resulting in the memory element <bold><italic>ξ</italic></bold>_Σ, obtained getting the largest eigenvector and computing the sign function (SEIG function). <bold>c</bold> Shows the pattern to be instantiated in memory <bold><italic>ϕ</italic></bold>^*, its vectorization, and the corresponding coupling matrix constructed using Hebb’s Rule (as employed in SHS).</p></caption></fig>", "<fig id=\"Fig2\"><label>Fig. 2</label><caption><title>Photonic Stochastic Hebb’s Storage.</title><p>SHS realizes arbitrary optical operators emulating the target matrix <bold>T</bold> that stores the pattern <bold><italic>ϕ</italic></bold>^* (as exemplified in Fig ##FIG##0##1##c). Our experimental setup utilizes 9 × 9 binary patterns (<italic>N</italic> = 81, <italic>M</italic>^{<italic>L</italic>} = 65536) and <italic>M</italic> modes or camera pixels. <bold>a</bold> Presents the Mean Squared Difference (MSD) between the target and probe artificial optical synaptic matrices plotted against <italic>M</italic>/<italic>N</italic>. In <bold>b</bold>, we show the Storage Error Probability versus <italic>M</italic>/<italic>N</italic>. Finally, in <bold>c</bold>, we illustrate the Recognition Error Probability as a function of <italic>M</italic>/<italic>N</italic>. The insets within the figure display images of the reconstructed <bold>J</bold> matrix for various values of <italic>M</italic>. Note that the point at <italic>M</italic>/<italic>N</italic> ~ 50 is out of scale because of a very low error rate. In all panels error bar represent standard error, obtained realizing 10 different target matrices <bold>T</bold> for each <italic>M/N</italic> value, and measuring standard deviation <italic>σ</italic> for each dataset and calculating standard error as ERR.</p></caption></fig>", "<fig id=\"Fig3\"><label>Fig. 3</label><caption><title>Photonic Stocastic Emergent storage.</title><p>Top panels show the stored patterns obtained with SES for different values of <italic>M</italic>^* with Similarity selection (three patterns on the left), and with random selection (pattern on the right). <bold>a</bold> Shows <italic>Storage Error Probability</italic> while <bold>b</bold> the <italic>Recognition error probability</italic>. Both with respect to <italic>M</italic>^*/<italic>N</italic>. <bold>c</bold> Transformed intensity for 600 patterns in the repository (pattern index <italic>j</italic> on the ordinate axis) for the SES with the similarity selection (Sel.) stage and SES without the similarity selection. The mode <italic>j</italic> = 241, (with red circled intensity), correspond to the stored (recognized) pattern. <bold>d</bold> same as <bold>c</bold> but without similarity selection. The insets between <bold>c</bold> and <bold>d</bold> report the obtained <bold>J</bold>. Error bars in <bold>a</bold> and <bold>b</bold> are constructed as in Fig. ##FIG##1##2##.</p></caption></fig>", "<fig id=\"Fig4\"><label>Fig. 4</label><caption><title>Parallel/Categorical photonic classification.</title><p><bold>a</bold> Shows the <italic>transformed intensities</italic> relative to 4096 stored memories (<italic>N</italic> = 256, <italic>M</italic> = 120) organized in 64 × 64 camera-like diagram. The <italic>transformed intensities</italic> are all generated as the pattern <bold><italic>ϕ</italic></bold> (shown on the right, highlighted by a red frame) is presented to the disordered classifier. For the circled pixels, the correspondent stored memory patterns are shown on the right (indicated by the arrow). The red circled pixel is associated to the memory which is the most similar to the pattern presented with the DMD. The diagram in <bold>b</bold> is similar to the previous but memories associated with 9 numerical categories, are organized in quadrants. The presented pattern <bold><italic>ϕ</italic></bold> is the “three” on the left). <bold>c</bold> Shows the thresholded and integrated intensity corresponding to the measure in <bold>b</bold>. <bold>d</bold> Is the same as <bold>b</bold> but with a “nine” pattern presented as input. <bold>e</bold> Reports the confusion matrix for the categorical (number class) recognition. <bold>f</bold> Reports the classification efficiency on the same digit database for Deep-SES and the Ridge Regression with Speckles (RRS) (for <italic>M</italic> = 120 recognition efficiency = 91.71% ± 0.8 (95% confidence interval, size of the statistics = 59))^{##UREF##9##21##}, versus <italic>M</italic>^* (see Methods). Error bars in <bold>f</bold> are constructed as in Fig. ##FIG##1##2##.</p></caption></fig>" ]

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mathvariant=\"bold-italic\">*</mml:mi><mml:mo>†</mml:mo></mml:mrow></mml:msup></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ2\"><label>2</label><alternatives><tex-math id=\"M3\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${I}^{\\nu }({{{{{{{\\boldsymbol{\\phi }}}}}}}})={{{{{{{\\boldsymbol{\\phi }}}}}}}}\\cdot {{{{{{{\\bf{{V}}}}}}}^{\\nu }}}\\cdot {{{{{{{{\\boldsymbol{\\phi }}}}}}}}}^{{{{\\dagger}}} }$$\\end{document}</tex-math><mml:math id=\"M4\"><mml:msup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi 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id=\"M6\"><mml:msup><mml:mrow/><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msup><mml:mo>∈</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant=\"double-struck\">C</mml:mi></mml:mrow><mml:mrow><mml:mi>N</mml:mi><mml:mo>×</mml:mo><mml:mi>N</mml:mi></mml:mrow></mml:msup></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ3\"><label>3</label><alternatives><tex-math id=\"M7\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${{{{{{{{\\bf{V}}}}}}}}}^{\\nu } \\sim {{{{{{{{\\boldsymbol{\\xi }}}}}}}}}^{\\nu }\\otimes {{{{{{{{\\boldsymbol{\\xi }}}}}}}}}^{\\nu {{{\\dagger}}} }$$\\end{document}</tex-math><mml:math id=\"M8\"><mml:msup><mml:mrow><mml:mi mathvariant=\"bold\">V</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msup><mml:mo>~</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant=\"bold-italic\">ξ</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msup><mml:mo>⊗</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant=\"bold-italic\">ξ</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi><mml:mo>†</mml:mo></mml:mrow></mml:msup></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq2\"><alternatives><tex-math id=\"M9\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\in {{\\mathbb{C}}}^{N}$$\\end{document}</tex-math><mml:math id=\"M10\"><mml:mo>∈</mml:mo><mml:msup><mml:mrow><mml:mi 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\n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${{{{{{{\\boldsymbol{{{{{{{{\\mathcal{M}}}}}}}}}}}}}}}}$$\\end{document}</tex-math><mml:math id=\"M14\"><mml:mi mathvariant=\"bold-script\">M</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq4\"><alternatives><tex-math id=\"M15\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${{{{{{{{\\boldsymbol{{{{{{{{\\mathcal{M}}}}}}}}}}}}}}}}}^{L}$$\\end{document}</tex-math><mml:math id=\"M16\"><mml:msup><mml:mrow><mml:mi mathvariant=\"bold-script\">M</mml:mi></mml:mrow><mml:mrow><mml:mi>L</mml:mi></mml:mrow></mml:msup></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ5\"><label>5</label><alternatives><tex-math id=\"M17\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${I}^{{{{{{{{\\boldsymbol{{{{{{{{\\mathcal{M}}}}}}}}}}}}}},\\lambda }}}({{{{{{{\\boldsymbol{\\phi }}}}}}}})={{{{{{{\\boldsymbol{\\phi }}}}}}}}\\cdot \\left(\\mathop{\\sum }\\limits_{\\nu }^{M}{\\lambda }^{\\nu }{{{{{{{{\\bf{V}}}}}}}}}^{\\nu }\\right)\\cdot {{{{{{{{\\boldsymbol{\\phi }}}}}}}}}^{{{{\\dagger}}} }={{{{{{{\\boldsymbol{\\phi }}}}}}}}\\cdot {{{{{{{{\\bf{J}}}}}}}}}^{{{{{{{{\\boldsymbol{{{{{{{{\\mathcal{M}}}}}}}}}}}}}},\\lambda }}}\\cdot {{{{{{{{\\boldsymbol{\\phi }}}}}}}}}^{{{{\\dagger}}} }.$$\\end{document}</tex-math><mml:math id=\"M18\"><mml:msup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-script\">M</mml:mi><mml:mo>,</mml:mo><mml:mi>λ</mml:mi></mml:mrow></mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi mathvariant=\"bold-italic\">ϕ</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mi mathvariant=\"bold-italic\">ϕ</mml:mi><mml:mo>⋅</mml:mo><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:munderover accent=\"false\" accentunder=\"false\"><mml:mrow><mml:mo>∑</mml:mo></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow><mml:mrow><mml:mi>M</mml:mi></mml:mrow></mml:munderover><mml:msup><mml:mrow><mml:mi>λ</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mi mathvariant=\"bold\">V</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:mfenced><mml:mo>⋅</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant=\"bold-italic\">ϕ</mml:mi></mml:mrow><mml:mrow><mml:mo>†</mml:mo></mml:mrow></mml:msup><mml:mo>=</mml:mo><mml:mi mathvariant=\"bold-italic\">ϕ</mml:mi><mml:mo>⋅</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant=\"bold\">J</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-script\">M</mml:mi><mml:mo>,</mml:mo><mml:mi>λ</mml:mi></mml:mrow></mml:msup><mml:mo>⋅</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant=\"bold-italic\">ϕ</mml:mi></mml:mrow><mml:mrow><mml:mo>†</mml:mo></mml:mrow></mml:msup><mml:mo>.</mml:mo></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq5\"><alternatives><tex-math id=\"M19\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${}^{{{{{{{{\\boldsymbol{{{{{{{{\\mathcal{M}}}}}}}}}}}}}},\\lambda }}}$$\\end{document}</tex-math><mml:math id=\"M20\"><mml:msup><mml:mrow/><mml:mrow><mml:mi mathvariant=\"bold-script\">M</mml:mi><mml:mo>,</mml:mo><mml:mi>λ</mml:mi></mml:mrow></mml:msup></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq6\"><alternatives><tex-math id=\"M21\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${}_{{{{{{{{\\bf{T}}}}}}}}}^{{{{{{{{\\boldsymbol{{{{{{{{\\mathcal{M}}}}}}}}}}}}}}}},{{{{{{{\\boldsymbol{\\lambda }}}}}}}}}$$\\end{document}</tex-math><mml:math id=\"M22\"><mml:msubsup><mml:mrow/><mml:mrow><mml:mi mathvariant=\"bold\">T</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-script\">M</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant=\"bold-italic\">λ</mml:mi></mml:mrow></mml:msubsup></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq7\"><alternatives><tex-math id=\"M23\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${{{{{{{\\boldsymbol{{{{{{{{\\mathcal{M}}}}}}}}}}}}}}}}=\\{{{{{{{{{\\bf{V}}}}}}}}}^{1},{{{{{{{{\\bf{V}}}}}}}}}^{2}\\ldots {{{{{{{{\\bf{V}}}}}}}}}^{M}\\}$$\\end{document}</tex-math><mml:math id=\"M24\"><mml:mi mathvariant=\"bold-script\">M</mml:mi><mml:mo>=</mml:mo><mml:mrow><mml:mo>{</mml:mo><mml:mrow><mml:msup><mml:mrow><mml:mi mathvariant=\"bold\">V</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mo>,</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant=\"bold\">V</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo>…</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant=\"bold\">V</mml:mi></mml:mrow><mml:mrow><mml:mi>M</mml:mi></mml:mrow></mml:msup></mml:mrow><mml:mo>}</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ6\"><label>6</label><alternatives><tex-math id=\"M25\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${{{{{{{{\\bf{J}}}}}}}}}_{{{{{{{{\\bf{T}}}}}}}}}^{{{{{{{{\\boldsymbol{{{{{{{{\\mathcal{M}}}}}}}}}}}}}}}},{{{{{{{\\boldsymbol{\\lambda }}}}}}}}}=\\mathop{\\sum }\\limits_{\\nu }^{M}{\\lambda }^{\\nu }{{{{{{{{\\bf{V}}}}}}}}}^{\\nu }$$\\end{document}</tex-math><mml:math id=\"M26\"><mml:msubsup><mml:mrow><mml:mi mathvariant=\"bold\">J</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold\">T</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-script\">M</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant=\"bold-italic\">λ</mml:mi></mml:mrow></mml:msubsup><mml:mo>=</mml:mo><mml:munderover accent=\"false\" accentunder=\"false\"><mml:mrow><mml:mo>∑</mml:mo></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow><mml:mrow><mml:mi>M</mml:mi></mml:mrow></mml:munderover><mml:msup><mml:mrow><mml:mi>λ</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mi mathvariant=\"bold\">V</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msup></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq8\"><alternatives><tex-math id=\"M27\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${I}_{{{{{{{{\\bf{T}}}}}}}}}^{{{{{{{{\\boldsymbol{{{{{{{{\\mathcal{M}}}}}}}}}}}}}}}},{{{{{{{\\boldsymbol{\\lambda }}}}}}}}}({{{{{{{\\boldsymbol{\\phi }}}}}}}})$$\\end{document}</tex-math><mml:math id=\"M28\"><mml:msubsup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold\">T</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-script\">M</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant=\"bold-italic\">λ</mml:mi></mml:mrow></mml:msubsup><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi mathvariant=\"bold-italic\">ϕ</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq9\"><alternatives><tex-math id=\"M29\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${{{{{{{{\\bf{J}}}}}}}}}_{{{{{{{{\\bf{T}}}}}}}}}^{{{{{{{{\\boldsymbol{{{{{{{{\\mathcal{M}}}}}}}}}}}}}}}},{{{{{{{\\boldsymbol{\\lambda }}}}}}}}}$$\\end{document}</tex-math><mml:math id=\"M30\"><mml:msubsup><mml:mrow><mml:mi mathvariant=\"bold\">J</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold\">T</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-script\">M</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant=\"bold-italic\">λ</mml:mi></mml:mrow></mml:msubsup></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq10\"><alternatives><tex-math id=\"M31\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${{{{{{{{\\bf{J}}}}}}}}}_{{{{{{{{\\bf{T}}}}}}}}}^{{{{{{{{\\boldsymbol{{{{{{{{\\mathcal{M}}}}}}}}}}}}}}}},{{{{{{{\\boldsymbol{\\lambda }}}}}}}}}$$\\end{document}</tex-math><mml:math id=\"M32\"><mml:msubsup><mml:mrow><mml:mi mathvariant=\"bold\">J</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold\">T</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-script\">M</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant=\"bold-italic\">λ</mml:mi></mml:mrow></mml:msubsup></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq11\"><alternatives><tex-math id=\"M33\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${{{{{{{\\boldsymbol{{{{{{{{\\mathcal{M}}}}}}}}}}}}}}}}$$\\end{document}</tex-math><mml:math id=\"M34\"><mml:mi mathvariant=\"bold-script\">M</mml:mi></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ7\"><label>7</label><alternatives><tex-math id=\"M35\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${{{{{{{\\boldsymbol{\\phi }}}}}}}}\\cdot {{{{{{{{\\bf{J}}}}}}}}}_{{{{{{{{\\bf{T}}}}}}}}}^{{{{{{{{\\boldsymbol{{{{{{{{\\mathcal{M}}}}}}}}}}}}}}}},{{{{{{{\\boldsymbol{\\lambda }}}}}}}}}\\cdot {{{{{{{{\\boldsymbol{\\phi }}}}}}}}}^{{{{\\dagger}}} }=\\mathop{\\sum }\\limits_{\\nu }^{M}{\\lambda }^{\\nu }{I}^{\\nu }({{{{{{{\\boldsymbol{\\phi }}}}}}}})={I}_{{{{{{{{\\bf{T}}}}}}}}}^{{{{{{{{\\boldsymbol{{{{{{{{\\mathcal{M}}}}}}}}}}}}}}}},{{{{{{{\\boldsymbol{\\lambda }}}}}}}}}({{{{{{{\\boldsymbol{\\phi }}}}}}}}).$$\\end{document}</tex-math><mml:math id=\"M36\"><mml:mi mathvariant=\"bold-italic\">ϕ</mml:mi><mml:mo>⋅</mml:mo><mml:msubsup><mml:mrow><mml:mi mathvariant=\"bold\">J</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold\">T</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-script\">M</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant=\"bold-italic\">λ</mml:mi></mml:mrow></mml:msubsup><mml:mo>⋅</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant=\"bold-italic\">ϕ</mml:mi></mml:mrow><mml:mrow><mml:mo>†</mml:mo></mml:mrow></mml:msup><mml:mo>=</mml:mo><mml:munderover accent=\"false\" accentunder=\"false\"><mml:mrow><mml:mo>∑</mml:mo></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow><mml:mrow><mml:mi>M</mml:mi></mml:mrow></mml:munderover><mml:msup><mml:mrow><mml:mi>λ</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi mathvariant=\"bold-italic\">ϕ</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:msubsup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold\">T</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-script\">M</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant=\"bold-italic\">λ</mml:mi></mml:mrow></mml:msubsup><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi mathvariant=\"bold-italic\">ϕ</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>.</mml:mo></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq12\"><alternatives><tex-math id=\"M37\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${}_{{{{{{{{\\bf{T}}}}}}}}}^{{{{{{{{\\boldsymbol{{{{{{{{\\mathcal{M}}}}}}}}}}}}}}}},{{{{{{{\\boldsymbol{\\lambda }}}}}}}}}$$\\end{document}</tex-math><mml:math id=\"M38\"><mml:msubsup><mml:mrow/><mml:mrow><mml:mi mathvariant=\"bold\">T</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-script\">M</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant=\"bold-italic\">λ</mml:mi></mml:mrow></mml:msubsup></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq13\"><alternatives><tex-math id=\"M39\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${{{{{{{{\\boldsymbol{{{{{{{{\\mathcal{M}}}}}}}}}}}}}}}}}^{{{{{{{{\\rm{L}}}}}}}}}$$\\end{document}</tex-math><mml:math id=\"M40\"><mml:msup><mml:mrow><mml:mi mathvariant=\"bold-script\">M</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">L</mml:mi></mml:mrow></mml:msup></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq14\"><alternatives><tex-math id=\"M41\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${{{{{{{\\boldsymbol{{{{{{{{\\mathcal{M}}}}}}}}}}}}}}}}$$\\end{document}</tex-math><mml:math id=\"M42\"><mml:mi mathvariant=\"bold-script\">M</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq15\"><alternatives><tex-math id=\"M43\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\sum }^{M}{{{{{{{{\\bf{V}}}}}}}}}^{\\nu }={{{{{{{{\\bf{J}}}}}}}}}_{{{{{{{{\\boldsymbol{{\\phi }}}}}}}^{*}}}}^{{{{{{{{{\\boldsymbol{{{{{{{{\\mathcal{M}}}}}}}}}}}}}}}}}^{*},{{{{{{{\\boldsymbol{\\lambda }}}}}}}}}$$\\end{document}</tex-math><mml:math id=\"M44\"><mml:msup><mml:mrow><mml:mo>∑</mml:mo></mml:mrow><mml:mrow><mml:mi>M</mml:mi></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mi mathvariant=\"bold\">V</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msup><mml:mo>=</mml:mo><mml:msubsup><mml:mrow><mml:mi mathvariant=\"bold\">J</mml:mi></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mi mathvariant=\"bold-italic\">ϕ</mml:mi></mml:mrow><mml:mrow><mml:mo>*</mml:mo></mml:mrow></mml:msup></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mi mathvariant=\"bold-script\">M</mml:mi></mml:mrow><mml:mrow><mml:mo>*</mml:mo></mml:mrow></mml:msup><mml:mo>,</mml:mo><mml:mi mathvariant=\"bold-italic\">λ</mml:mi></mml:mrow></mml:msubsup></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq16\"><alternatives><tex-math id=\"M45\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${}_{{{{{{{{\\boldsymbol{T}}}}}}}}}^{{{{{{{{\\boldsymbol{{{{{{{{\\mathcal{M}}}}}}}}}}}}}}}},{{{{{{{\\boldsymbol{\\lambda }}}}}}}}}$$\\end{document}</tex-math><mml:math id=\"M46\"><mml:msubsup><mml:mrow/><mml:mrow><mml:mi mathvariant=\"bold-italic\">T</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-script\">M</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant=\"bold-italic\">λ</mml:mi></mml:mrow></mml:msubsup></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq17\"><alternatives><tex-math id=\"M47\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$=\\sigma /\\sqrt{(10-1)}$$\\end{document}</tex-math><mml:math id=\"M48\"><mml:mo>=</mml:mo><mml:mi>σ</mml:mi><mml:mo>/</mml:mo><mml:msqrt><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mn>10</mml:mn><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:msqrt></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq18\"><alternatives><tex-math id=\"M49\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${{{{{{{\\boldsymbol{{\\xi }}}}}}}_{\\Sigma }}}={{{{{{{\\rm{SEIG}}}}}}}}({{{{{{{{\\boldsymbol{J}}}}}}}}}_{{{{{{{{\\boldsymbol{T}}}}}}}}}^{{{{{{{{\\boldsymbol{{{{{{{{\\mathcal{M}}}}}}}}}}}}}}}},{{{{{{{\\boldsymbol{\\lambda }}}}}}}}})$$\\end{document}</tex-math><mml:math id=\"M50\"><mml:msub><mml:mrow><mml:mi mathvariant=\"bold-italic\">ξ</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">Σ</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mi mathvariant=\"normal\">SEIG</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msubsup><mml:mrow><mml:mi mathvariant=\"bold-italic\">J</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">T</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-script\">M</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant=\"bold-italic\">λ</mml:mi></mml:mrow></mml:msubsup></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq19\"><alternatives><tex-math id=\"M51\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${{{{{{{{\\boldsymbol{{{{{{{{\\mathcal{M}}}}}}}}}}}}}}}}}^{{{{{{{{\\rm{L}}}}}}}}}$$\\end{document}</tex-math><mml:math id=\"M52\"><mml:msup><mml:mrow><mml:mi mathvariant=\"bold-script\">M</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">L</mml:mi></mml:mrow></mml:msup></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq20\"><alternatives><tex-math id=\"M53\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${{{{{{{{\\mathcal{S}}}}}}}}}^{\\nu }$$\\end{document}</tex-math><mml:math id=\"M54\"><mml:msup><mml:mrow><mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">S</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msup></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ8\"><label>8</label><alternatives><tex-math id=\"M55\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${{{{{{{{\\mathcal{S}}}}}}}}}^{\\nu }={\\hat{{{{{{{{\\boldsymbol{\\phi }}}}}}}}}}^{*}\\cdot {\\hat{{{{{{{{\\boldsymbol{\\xi }}}}}}}}}}^{\\nu }$$\\end{document}</tex-math><mml:math id=\"M56\"><mml:msup><mml:mrow><mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">S</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msup><mml:mo>=</mml:mo><mml:msup><mml:mrow><mml:mover accent=\"true\"><mml:mrow><mml:mi mathvariant=\"bold-italic\">ϕ</mml:mi></mml:mrow><mml:mrow><mml:mo>^</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mo>*</mml:mo></mml:mrow></mml:msup><mml:mo>⋅</mml:mo><mml:msup><mml:mrow><mml:mover accent=\"true\"><mml:mrow><mml:mi mathvariant=\"bold-italic\">ξ</mml:mi></mml:mrow><mml:mrow><mml:mo>^</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msup></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq21\"><alternatives><tex-math id=\"M57\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${{{{{{{{\\mathcal{S}}}}}}}}}^{\\nu }$$\\end{document}</tex-math><mml:math id=\"M58\"><mml:msup><mml:mrow><mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">S</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msup></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq22\"><alternatives><tex-math id=\"M59\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${{{{{{{{\\mathcal{M}}}}}}}}}^{*}$$\\end{document}</tex-math><mml:math id=\"M60\"><mml:msup><mml:mrow><mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">M</mml:mi></mml:mrow><mml:mrow><mml:mo>*</mml:mo></mml:mrow></mml:msup></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq23\"><alternatives><tex-math id=\"M61\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$({{{{{{{{\\bf{J}}}}}}}}}_{{{{{{{{{\\boldsymbol{\\phi }}}}}}}}}^{*}}^{{{{{{{{{\\boldsymbol{{{{{{{{\\mathcal{M}}}}}}}}}}}}}}}}}^{*},{{{{{{{\\boldsymbol{\\lambda }}}}}}}}})={{{{{{{{\\boldsymbol{\\xi }}}}}}}}}_{{{{{{{{\\boldsymbol{\\Sigma }}}}}}}}}$$\\end{document}</tex-math><mml:math id=\"M62\"><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msubsup><mml:mrow><mml:mi mathvariant=\"bold\">J</mml:mi></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mi mathvariant=\"bold-italic\">ϕ</mml:mi></mml:mrow><mml:mrow><mml:mo>*</mml:mo></mml:mrow></mml:msup></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mi mathvariant=\"bold-script\">M</mml:mi></mml:mrow><mml:mrow><mml:mo>*</mml:mo></mml:mrow></mml:msup><mml:mo>,</mml:mo><mml:mi mathvariant=\"bold-italic\">λ</mml:mi></mml:mrow></mml:msubsup></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant=\"bold-italic\">ξ</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">Σ</mml:mi></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq24\"><alternatives><tex-math id=\"M63\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${{{{{{{{\\bf{J}}}}}}}}}_{{{{{{{{{\\boldsymbol{\\phi }}}}}}}}}^{*}}^{{{{{{{{{\\boldsymbol{{{{{{{{\\mathcal{M}}}}}}}}}}}}}}}}}^{*},{{{{{{{\\boldsymbol{\\lambda }}}}}}}}}$$\\end{document}</tex-math><mml:math id=\"M64\"><mml:msubsup><mml:mrow><mml:mi mathvariant=\"bold\">J</mml:mi></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mi mathvariant=\"bold-italic\">ϕ</mml:mi></mml:mrow><mml:mrow><mml:mo>*</mml:mo></mml:mrow></mml:msup></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mi mathvariant=\"bold-script\">M</mml:mi></mml:mrow><mml:mrow><mml:mo>*</mml:mo></mml:mrow></mml:msup><mml:mo>,</mml:mo><mml:mi mathvariant=\"bold-italic\">λ</mml:mi></mml:mrow></mml:msubsup></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ9\"><label>9</label><alternatives><tex-math id=\"M65\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${I}_{{{{{{{{\\boldsymbol{{\\phi }}}}}}}^{*}}}}^{{{{{{{{{\\boldsymbol{{{{{{{{\\mathcal{M}}}}}}}}}}}}}}}}}^{*},{{{{{{{\\boldsymbol{\\lambda }}}}}}}}}({{{{{{{\\boldsymbol{\\phi }}}}}}}})={{{{{{{\\boldsymbol{\\phi }}}}}}}}\\cdot {{{{{{{{\\bf{J}}}}}}}}}_{{{{{{{{\\boldsymbol{{\\phi }}}}}}}^{*}}}}^{{{{{{{{{\\boldsymbol{{{{{{{{\\mathcal{M}}}}}}}}}}}}}}}}}^{*},{{{{{{{\\boldsymbol{\\lambda }}}}}}}}}\\cdot {{{{{{{{\\boldsymbol{\\phi }}}}}}}}}^{{{{\\dagger}}} }=\\mathop{\\sum }\\limits_{\\nu }^{{M}^{*}}{\\lambda }^{\\nu }{I}^{\\nu }({{{{{{{\\boldsymbol{\\phi }}}}}}}}).$$\\end{document}</tex-math><mml:math id=\"M66\"><mml:msubsup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mi mathvariant=\"bold-italic\">ϕ</mml:mi></mml:mrow><mml:mrow><mml:mo>*</mml:mo></mml:mrow></mml:msup></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mi mathvariant=\"bold-script\">M</mml:mi></mml:mrow><mml:mrow><mml:mo>*</mml:mo></mml:mrow></mml:msup><mml:mo>,</mml:mo><mml:mi mathvariant=\"bold-italic\">λ</mml:mi></mml:mrow></mml:msubsup><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi mathvariant=\"bold-italic\">ϕ</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mi mathvariant=\"bold-italic\">ϕ</mml:mi><mml:mo>⋅</mml:mo><mml:msubsup><mml:mrow><mml:mi mathvariant=\"bold\">J</mml:mi></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mi mathvariant=\"bold-italic\">ϕ</mml:mi></mml:mrow><mml:mrow><mml:mo>*</mml:mo></mml:mrow></mml:msup></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mi mathvariant=\"bold-script\">M</mml:mi></mml:mrow><mml:mrow><mml:mo>*</mml:mo></mml:mrow></mml:msup><mml:mo>,</mml:mo><mml:mi mathvariant=\"bold-italic\">λ</mml:mi></mml:mrow></mml:msubsup><mml:mo>⋅</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant=\"bold-italic\">ϕ</mml:mi></mml:mrow><mml:mrow><mml:mo>†</mml:mo></mml:mrow></mml:msup><mml:mo>=</mml:mo><mml:munderover accent=\"false\" accentunder=\"false\"><mml:mrow><mml:mo>∑</mml:mo></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mo>*</mml:mo></mml:mrow></mml:msup></mml:mrow></mml:munderover><mml:msup><mml:mrow><mml:mi>λ</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi mathvariant=\"bold-italic\">ϕ</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>.</mml:mo></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq25\"><alternatives><tex-math id=\"M67\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${}_{{{{{{{{{\\boldsymbol{\\phi }}}}}}}}}^{*}}^{{{{{{{{{\\boldsymbol{{{{{{{{\\mathcal{M}}}}}}}}}}}}}}}}}^{*},{{{{{{{\\boldsymbol{\\lambda }}}}}}}}}$$\\end{document}</tex-math><mml:math id=\"M68\"><mml:msubsup><mml:mrow/><mml:mrow><mml:msup><mml:mrow><mml:mi mathvariant=\"bold-italic\">ϕ</mml:mi></mml:mrow><mml:mrow><mml:mo>*</mml:mo></mml:mrow></mml:msup></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mi mathvariant=\"bold-script\">M</mml:mi></mml:mrow><mml:mrow><mml:mo>*</mml:mo></mml:mrow></mml:msup><mml:mo>,</mml:mo><mml:mi mathvariant=\"bold-italic\">λ</mml:mi></mml:mrow></mml:msubsup></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq26\"><alternatives><tex-math id=\"M69\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${I}_{{{{{{{{\\boldsymbol{{\\phi }}}}}}}^{*}}}}^{{{{{{{{{\\boldsymbol{{{{{{{{\\mathcal{M}}}}}}}}}}}}}}}}}^{*},{{{{{{{\\boldsymbol{\\lambda }}}}}}}}}({{{{{{{\\boldsymbol{\\phi }}}}}}}})$$\\end{document}</tex-math><mml:math id=\"M70\"><mml:msubsup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mi mathvariant=\"bold-italic\">ϕ</mml:mi></mml:mrow><mml:mrow><mml:mo>*</mml:mo></mml:mrow></mml:msup></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mi mathvariant=\"bold-script\">M</mml:mi></mml:mrow><mml:mrow><mml:mo>*</mml:mo></mml:mrow></mml:msup><mml:mo>,</mml:mo><mml:mi mathvariant=\"bold-italic\">λ</mml:mi></mml:mrow></mml:msubsup><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi mathvariant=\"bold-italic\">ϕ</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq27\"><alternatives><tex-math id=\"M71\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$S=\\mathop{\\lim }\\nolimits_{N\\to \\infty }\\log (W)/N$$\\end{document}</tex-math><mml:math id=\"M72\"><mml:mi>S</mml:mi><mml:mo>=</mml:mo><mml:msub><mml:mrow><mml:mi>lim</mml:mi></mml:mrow><mml:mrow><mml:mi>N</mml:mi><mml:mo>→</mml:mo><mml:mi>∞</mml:mi></mml:mrow></mml:msub><mml:mi>log</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>W</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>/</mml:mo><mml:mi>N</mml:mi></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ10\"><label>10</label><alternatives><tex-math id=\"M73\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${E}^{\\nu }=\\mathop{\\sum }\\limits_{n=1}^{N}{E}_{n}^{\\nu }{\\phi }_{n}$$\\end{document}</tex-math><mml:math id=\"M74\"><mml:msup><mml:mrow><mml:mi>E</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msup><mml:mo>=</mml:mo><mml:munderover accent=\"false\" accentunder=\"false\"><mml:mrow><mml:mo>∑</mml:mo></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mi>N</mml:mi></mml:mrow></mml:munderover><mml:msubsup><mml:mrow><mml:mi>E</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msubsup><mml:msub><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msub></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq28\"><alternatives><tex-math id=\"M75\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${E}_{n}^{\\nu }$$\\end{document}</tex-math><mml:math id=\"M76\"><mml:msubsup><mml:mrow><mml:mi>E</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msubsup></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq29\"><alternatives><tex-math id=\"M77\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\phi }_{n}^{\\nu }$$\\end{document}</tex-math><mml:math id=\"M78\"><mml:msubsup><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msubsup></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq30\"><alternatives><tex-math id=\"M79\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\phi }_{n}^{\\nu } \\in \\{-1,+1\\}$$\\end{document}</tex-math><mml:math id=\"M80\"><mml:msubsup><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msubsup><mml:mo>∈</mml:mo><mml:mrow><mml:mo>{</mml:mo><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mo>}</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq31\"><alternatives><tex-math id=\"M81\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${t}_{n}^{\\nu }$$\\end{document}</tex-math><mml:math id=\"M82\"><mml:msubsup><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msubsup></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ11\"><label>11</label><alternatives><tex-math id=\"M83\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${E}_{n}^{\\nu }={A}_{n}{t}_{n}^{\\nu }$$\\end{document}</tex-math><mml:math id=\"M84\"><mml:msubsup><mml:mrow><mml:mi>E</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msubsup><mml:mo>=</mml:mo><mml:msub><mml:mrow><mml:mi>A</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msubsup></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq32\"><alternatives><tex-math id=\"M85\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${E}_{n}^{\\nu }$$\\end{document}</tex-math><mml:math id=\"M86\"><mml:msubsup><mml:mrow><mml:mi>E</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msubsup></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ12\"><label>12</label><alternatives><tex-math id=\"M87\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${E}_{n}^{\\nu }={\\xi }_{n}^{\\nu }+i{\\eta }_{n}^{\\nu }.$$\\end{document}</tex-math><mml:math id=\"M88\"><mml:msubsup><mml:mrow><mml:mi>E</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msubsup><mml:mo>=</mml:mo><mml:msubsup><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msubsup><mml:mo>+</mml:mo><mml:mi>i</mml:mi><mml:msubsup><mml:mrow><mml:mi>η</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msubsup><mml:mo>.</mml:mo></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ13\"><label>13</label><alternatives><tex-math id=\"M89\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${E}^{\\nu }={E}_{n}^{\\nu }+{E}_{m}^{\\nu }={\\xi }_{n}^{\\nu }+i{\\eta }_{n}^{\\nu }+{\\xi }_{m}^{\\nu }+i{\\eta }_{m}^{\\nu }.$$\\end{document}</tex-math><mml:math id=\"M90\"><mml:msup><mml:mrow><mml:mi>E</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msup><mml:mo>=</mml:mo><mml:msubsup><mml:mrow><mml:mi>E</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msubsup><mml:mo>+</mml:mo><mml:msubsup><mml:mrow><mml:mi>E</mml:mi></mml:mrow><mml:mrow><mml:mi>m</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msubsup><mml:mo>=</mml:mo><mml:msubsup><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msubsup><mml:mo>+</mml:mo><mml:mi>i</mml:mi><mml:msubsup><mml:mrow><mml:mi>η</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msubsup><mml:mo>+</mml:mo><mml:msubsup><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mrow><mml:mi>m</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msubsup><mml:mo>+</mml:mo><mml:mi>i</mml:mi><mml:msubsup><mml:mrow><mml:mi>η</mml:mi></mml:mrow><mml:mrow><mml:mi>m</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msubsup><mml:mo>.</mml:mo></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ14\"><label>14</label><alternatives><tex-math id=\"M91\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${I}^{\\nu }=\t| {E}_{n}^{\\nu }+{E}_{m}^{\\nu }| | {E}_{n}^{\\nu {{{\\dagger}}} }+{E}_{m}^{\\nu {{{\\dagger}}} }| \\\\=\t| {E}_{n}^{\\nu }{| }^{2}+| {E}_{m}^{\\nu }{| }^{2}+| {E}_{n}^{\\nu }| | {E}_{m}^{\\nu {{{\\dagger}}} }|+| {E}_{n}^{\\nu {{{\\dagger}}} }| | {E}_{m}^{\\nu }| .$$\\end{document}</tex-math><mml:math id=\"M92\"><mml:msup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msup><mml:mo>=</mml:mo><mml:mo>∣</mml:mo><mml:msubsup><mml:mrow><mml:mi>E</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msubsup><mml:mo>+</mml:mo><mml:msubsup><mml:mrow><mml:mi>E</mml:mi></mml:mrow><mml:mrow><mml:mi>m</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msubsup><mml:mo>∣</mml:mo><mml:mo>∣</mml:mo><mml:msubsup><mml:mrow><mml:mi>E</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi><mml:mo>†</mml:mo></mml:mrow></mml:msubsup><mml:mo>+</mml:mo><mml:msubsup><mml:mrow><mml:mi>E</mml:mi></mml:mrow><mml:mrow><mml:mi>m</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi><mml:mo>†</mml:mo></mml:mrow></mml:msubsup><mml:mo>∣</mml:mo><mml:mo>=</mml:mo><mml:mo>∣</mml:mo><mml:msubsup><mml:mrow><mml:mi>E</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msubsup><mml:msup><mml:mrow><mml:mo>∣</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo>+</mml:mo><mml:mo>∣</mml:mo><mml:msubsup><mml:mrow><mml:mi>E</mml:mi></mml:mrow><mml:mrow><mml:mi>m</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msubsup><mml:msup><mml:mrow><mml:mo>∣</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo>+</mml:mo><mml:mo>∣</mml:mo><mml:msubsup><mml:mrow><mml:mi>E</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msubsup><mml:mo>∣</mml:mo><mml:mo>∣</mml:mo><mml:msubsup><mml:mrow><mml:mi>E</mml:mi></mml:mrow><mml:mrow><mml:mi>m</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi><mml:mo>†</mml:mo></mml:mrow></mml:msubsup><mml:mo>∣</mml:mo><mml:mo>+</mml:mo><mml:mo>∣</mml:mo><mml:msubsup><mml:mrow><mml:mi>E</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi><mml:mo>†</mml:mo></mml:mrow></mml:msubsup><mml:mo>∣</mml:mo><mml:mo>∣</mml:mo><mml:msubsup><mml:mrow><mml:mi>E</mml:mi></mml:mrow><mml:mrow><mml:mi>m</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msubsup><mml:mo>∣</mml:mo><mml:mo>.</mml:mo></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ15\"><label>15</label><alternatives><tex-math id=\"M93\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$| {E}_{n}^{\\nu }| | {E}_{m}^{\\nu {{{\\dagger}}} }|={\\xi }_{n}^{\\nu }{\\xi }_{m}^{\\nu }-i{\\xi }_{n}^{\\nu }{\\eta }_{m}^{\\nu }+i{\\eta }_{n}^{\\nu }{\\xi }_{m}^{\\nu }+{\\eta }_{n}^{\\nu }{\\eta }_{m}^{\\nu }$$\\end{document}</tex-math><mml:math id=\"M94\"><mml:mo>∣</mml:mo><mml:msubsup><mml:mrow><mml:mi>E</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msubsup><mml:mo>∣</mml:mo><mml:mo>∣</mml:mo><mml:msubsup><mml:mrow><mml:mi>E</mml:mi></mml:mrow><mml:mrow><mml:mi>m</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi><mml:mo>†</mml:mo></mml:mrow></mml:msubsup><mml:mo>∣</mml:mo><mml:mo>=</mml:mo><mml:msubsup><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mrow><mml:mi>m</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msubsup><mml:mo>−</mml:mo><mml:mi>i</mml:mi><mml:msubsup><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mi>η</mml:mi></mml:mrow><mml:mrow><mml:mi>m</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msubsup><mml:mo>+</mml:mo><mml:mi>i</mml:mi><mml:msubsup><mml:mrow><mml:mi>η</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mrow><mml:mi>m</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msubsup><mml:mo>+</mml:mo><mml:msubsup><mml:mrow><mml:mi>η</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mi>η</mml:mi></mml:mrow><mml:mrow><mml:mi>m</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msubsup></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ16\"><label>16</label><alternatives><tex-math id=\"M95\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$| {E}_{n}^{\\nu {{{\\dagger}}} }| | {E}_{m}^{\\nu }|={\\xi }_{n}^{\\nu }{\\xi }_{m}^{\\nu }+i{\\xi }_{n}^{\\nu }{\\eta }_{m}^{\\nu }-i{\\eta }_{n}^{\\nu }{\\xi }_{m}^{\\nu }+{\\eta }_{n}^{\\nu }{\\eta }_{m}^{\\nu }$$\\end{document}</tex-math><mml:math id=\"M96\"><mml:mo>∣</mml:mo><mml:msubsup><mml:mrow><mml:mi>E</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi><mml:mo>†</mml:mo></mml:mrow></mml:msubsup><mml:mo>∣</mml:mo><mml:mo>∣</mml:mo><mml:msubsup><mml:mrow><mml:mi>E</mml:mi></mml:mrow><mml:mrow><mml:mi>m</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msubsup><mml:mo>∣</mml:mo><mml:mo>=</mml:mo><mml:msubsup><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mrow><mml:mi>m</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msubsup><mml:mo>+</mml:mo><mml:mi>i</mml:mi><mml:msubsup><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mi>η</mml:mi></mml:mrow><mml:mrow><mml:mi>m</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msubsup><mml:mo>−</mml:mo><mml:mi>i</mml:mi><mml:msubsup><mml:mrow><mml:mi>η</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mrow><mml:mi>m</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msubsup><mml:mo>+</mml:mo><mml:msubsup><mml:mrow><mml:mi>η</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mi>η</mml:mi></mml:mrow><mml:mrow><mml:mi>m</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msubsup></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ17\"><label>17</label><alternatives><tex-math id=\"M97\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$| {E}_{n}^{\\nu }| | {E}_{m}^{\\nu {{{\\dagger}}} }|+| {E}_{n}^{\\nu {{{\\dagger}}} }| | {E}_{m}^{\\nu }|=2{\\xi }_{n}^{\\nu }{\\xi }_{m}^{\\nu }+2{\\eta }_{n}^{\\nu }{\\eta }_{m}^{\\nu }.$$\\end{document}</tex-math><mml:math id=\"M98\"><mml:mo>∣</mml:mo><mml:msubsup><mml:mrow><mml:mi>E</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msubsup><mml:mo>∣</mml:mo><mml:mo>∣</mml:mo><mml:msubsup><mml:mrow><mml:mi>E</mml:mi></mml:mrow><mml:mrow><mml:mi>m</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi><mml:mo>†</mml:mo></mml:mrow></mml:msubsup><mml:mo>∣</mml:mo><mml:mo>+</mml:mo><mml:mo>∣</mml:mo><mml:msubsup><mml:mrow><mml:mi>E</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi><mml:mo>†</mml:mo></mml:mrow></mml:msubsup><mml:mo>∣</mml:mo><mml:mo>∣</mml:mo><mml:msubsup><mml:mrow><mml:mi>E</mml:mi></mml:mrow><mml:mrow><mml:mi>m</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msubsup><mml:mo>∣</mml:mo><mml:mo>=</mml:mo><mml:mn>2</mml:mn><mml:msubsup><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mrow><mml:mi>m</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msubsup><mml:mo>+</mml:mo><mml:mn>2</mml:mn><mml:msubsup><mml:mrow><mml:mi>η</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mi>η</mml:mi></mml:mrow><mml:mrow><mml:mi>m</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msubsup><mml:mo>.</mml:mo></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ18\"><label>18</label><alternatives><tex-math id=\"M99\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${I}^{\\nu }={\\xi }_{n}^{\\nu 2}+{\\eta }_{n}^{\\nu 2}+{\\xi }_{m}^{\\nu 2}+{\\eta }_{m}^{\\nu 2}+2{\\xi }_{n}^{\\nu }{\\xi }_{m}^{\\nu }+2{\\eta }_{n}^{\\nu }{\\eta }_{m}^{\\nu }.$$\\end{document}</tex-math><mml:math id=\"M100\"><mml:msup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msup><mml:mo>=</mml:mo><mml:msubsup><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo>+</mml:mo><mml:msubsup><mml:mrow><mml:mi>η</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo>+</mml:mo><mml:msubsup><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mrow><mml:mi>m</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo>+</mml:mo><mml:msubsup><mml:mrow><mml:mi>η</mml:mi></mml:mrow><mml:mrow><mml:mi>m</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo>+</mml:mo><mml:mn>2</mml:mn><mml:msubsup><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mrow><mml:mi>m</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msubsup><mml:mo>+</mml:mo><mml:mn>2</mml:mn><mml:msubsup><mml:mrow><mml:mi>η</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mi>η</mml:mi></mml:mrow><mml:mrow><mml:mi>m</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msubsup><mml:mo>.</mml:mo></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ19\"><label>19</label><alternatives><tex-math id=\"M101\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${I}^{\\nu }={E}_{n}^{\\nu 2}+{E}_{m}^{\\nu 2}+2{\\xi }_{n}^{\\nu }{\\xi }_{m}^{\\nu }+2{\\eta }_{n}^{\\nu }{\\eta }_{m}^{\\nu }.$$\\end{document}</tex-math><mml:math id=\"M102\"><mml:msup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msup><mml:mo>=</mml:mo><mml:msubsup><mml:mrow><mml:mi>E</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo>+</mml:mo><mml:msubsup><mml:mrow><mml:mi>E</mml:mi></mml:mrow><mml:mrow><mml:mi>m</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo>+</mml:mo><mml:mn>2</mml:mn><mml:msubsup><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mrow><mml:mi>m</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msubsup><mml:mo>+</mml:mo><mml:mn>2</mml:mn><mml:msubsup><mml:mrow><mml:mi>η</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mi>η</mml:mi></mml:mrow><mml:mrow><mml:mi>m</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msubsup><mml:mo>.</mml:mo></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ20\"><label>20</label><alternatives><tex-math id=\"M103\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${I}^{\\nu }=\\mathop{\\sum }\\limits_{n,m}^{N}{E}_{n}^{\\nu 2}+{E}_{m}^{\\nu 2}+2({\\xi }_{n}^{\\nu }{\\xi }_{m}^{\\nu }+{\\eta }_{n}^{\\nu }{\\eta }_{m}^{\\nu }){\\phi }_{n}{\\phi }_{m}.$$\\end{document}</tex-math><mml:math id=\"M104\"><mml:msup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msup><mml:mo>=</mml:mo><mml:munderover accent=\"false\" accentunder=\"false\"><mml:mrow><mml:mo>∑</mml:mo></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mo>,</mml:mo><mml:mi>m</mml:mi></mml:mrow><mml:mrow><mml:mi>N</mml:mi></mml:mrow></mml:munderover><mml:msubsup><mml:mrow><mml:mi>E</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo>+</mml:mo><mml:msubsup><mml:mrow><mml:mi>E</mml:mi></mml:mrow><mml:mrow><mml:mi>m</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo>+</mml:mo><mml:mn>2</mml:mn><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msubsup><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mrow><mml:mi>m</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msubsup><mml:mo>+</mml:mo><mml:msubsup><mml:mrow><mml:mi>η</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mi>η</mml:mi></mml:mrow><mml:mrow><mml:mi>m</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msubsup></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:msub><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mi>m</mml:mi></mml:mrow></mml:msub><mml:mo>.</mml:mo></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ21\"><label>21</label><alternatives><tex-math id=\"M105\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${V}_{nn}^{\\nu }={E}_{n}^{\\nu 2}={\\xi }_{n}^{\\nu 2}+{\\eta }_{n}^{\\nu 2}$$\\end{document}</tex-math><mml:math id=\"M106\"><mml:msubsup><mml:mrow><mml:mi>V</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mi>n</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msubsup><mml:mo>=</mml:mo><mml:msubsup><mml:mrow><mml:mi>E</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo>=</mml:mo><mml:msubsup><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo>+</mml:mo><mml:msubsup><mml:mrow><mml:mi>η</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi><mml:mn>2</mml:mn></mml:mrow></mml:msubsup></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ22\"><label>22</label><alternatives><tex-math id=\"M107\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${V}_{nm}^{\\nu }={\\xi }_{n}^{\\nu }{\\xi }_{m}^{\\nu }+{\\eta }_{n}^{\\nu }{\\eta }_{m}^{\\nu }$$\\end{document}</tex-math><mml:math id=\"M108\"><mml:msubsup><mml:mrow><mml:mi>V</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mi>m</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msubsup><mml:mo>=</mml:mo><mml:msubsup><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow><mml:mrow><mml:mi>m</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msubsup><mml:mo>+</mml:mo><mml:msubsup><mml:mrow><mml:mi>η</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msubsup><mml:msubsup><mml:mrow><mml:mi>η</mml:mi></mml:mrow><mml:mrow><mml:mi>m</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msubsup></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ23\"><label>23</label><alternatives><tex-math id=\"M109\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${{{{{{{\\bf{{V}}}}}}}^{\\nu }}}={{{{{{{{\\boldsymbol{\\xi }}}}}}}}}^{\\nu }\\otimes {{{{{{{{\\boldsymbol{\\xi }}}}}}}}}^{\\nu {{{\\dagger}}} }+{{{{{{{{\\boldsymbol{\\eta }}}}}}}}}^{\\nu }\\otimes {{{{{{{{\\boldsymbol{\\eta }}}}}}}}}^{\\nu {{{\\dagger}}} }$$\\end{document}</tex-math><mml:math id=\"M110\"><mml:msup><mml:mrow><mml:mi mathvariant=\"bold\">V</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msup><mml:mo>=</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant=\"bold-italic\">ξ</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msup><mml:mo>⊗</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant=\"bold-italic\">ξ</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi><mml:mo>†</mml:mo></mml:mrow></mml:msup><mml:mo>+</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant=\"bold-italic\">η</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msup><mml:mo>⊗</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant=\"bold-italic\">η</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi><mml:mo>†</mml:mo></mml:mrow></mml:msup></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq33\"><alternatives><tex-math id=\"M111\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${{{{{{{\\boldsymbol{{\\xi }}}}}}}^{\\nu }}},{{{{{{{{\\boldsymbol{\\eta }}}}}}}}}^{\\nu }\\in {{\\mathbb{C}}}^{N}$$\\end{document}</tex-math><mml:math id=\"M112\"><mml:msup><mml:mrow><mml:mi mathvariant=\"bold-italic\">ξ</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msup><mml:mo>,</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant=\"bold-italic\">η</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msup><mml:mo>∈</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant=\"double-struck\">C</mml:mi></mml:mrow><mml:mrow><mml:mi>N</mml:mi></mml:mrow></mml:msup></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq34\"><alternatives><tex-math id=\"M113\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${}^{\\nu }\\in {{\\mathbb{C}}}^{N\\times N}$$\\end{document}</tex-math><mml:math id=\"M114\"><mml:msup><mml:mrow/><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msup><mml:mo>∈</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant=\"double-struck\">C</mml:mi></mml:mrow><mml:mrow><mml:mi>N</mml:mi><mml:mo>×</mml:mo><mml:mi>N</mml:mi></mml:mrow></mml:msup></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ24\"><label>24</label><alternatives><tex-math id=\"M115\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${I}^{\\nu }({{{{{{{\\boldsymbol{\\phi }}}}}}}})=\\mathop{\\sum }\\limits_{n,m}^{N}{V}_{nm}^{\\nu }{\\phi }_{n}{\\phi }_{m}={{{{{{{\\boldsymbol{\\phi }}}}}}}}\\cdot {{{{{{{{\\bf{V}}}}}}}}}^{\\nu }\\cdot {{{{{{{{\\boldsymbol{\\phi }}}}}}}}}^{{{{\\dagger}}} }$$\\end{document}</tex-math><mml:math id=\"M116\"><mml:msup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi mathvariant=\"bold-italic\">ϕ</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:munderover accent=\"false\" accentunder=\"false\"><mml:mrow><mml:mo>∑</mml:mo></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mo>,</mml:mo><mml:mi>m</mml:mi></mml:mrow><mml:mrow><mml:mi>N</mml:mi></mml:mrow></mml:munderover><mml:msubsup><mml:mrow><mml:mi>V</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mi>m</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msubsup><mml:msub><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mi>m</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mi mathvariant=\"bold-italic\">ϕ</mml:mi><mml:mo>⋅</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant=\"bold\">V</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msup><mml:mo>⋅</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant=\"bold-italic\">ϕ</mml:mi></mml:mrow><mml:mrow><mml:mo>†</mml:mo></mml:mrow></mml:msup></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ25\"><label>25</label><alternatives><tex-math id=\"M117\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${I}^{{\\nu }_{1}}+{I}^{{\\nu }_{2}}=\\,\t{{{{{{{\\boldsymbol{\\phi }}}}}}}}\\cdot {{{{{{{{\\bf{V}}}}}}}}}^{{\\nu }_{1}}\\cdot {{{{{{{{\\boldsymbol{\\phi }}}}}}}}}^{{{{\\dagger}}} }+{{{{{{{\\boldsymbol{\\phi }}}}}}}}\\cdot {{{{{{{{\\bf{V}}}}}}}}}^{{\\nu }_{2}}\\cdot {{{{{{{{\\boldsymbol{\\phi }}}}}}}}}^{{{{\\dagger}}} }=\\\\=\\,\t{{{{{{{\\boldsymbol{\\phi }}}}}}}}\\cdot {{{{{{{{\\bf{J}}}}}}}}}^{{\\nu }_{1},{\\nu }_{2}}\\cdot {{{{{{{{\\boldsymbol{\\phi }}}}}}}}}^{{{{\\dagger}}} }.$$\\end{document}</tex-math><mml:math id=\"M118\"><mml:msup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>ν</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:msup><mml:mo>+</mml:mo><mml:msup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>ν</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:msup><mml:mo>=</mml:mo><mml:mspace width=\"0.25em\"/><mml:mi mathvariant=\"bold-italic\">ϕ</mml:mi><mml:mo>⋅</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant=\"bold\">V</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>ν</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:msup><mml:mo>⋅</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant=\"bold-italic\">ϕ</mml:mi></mml:mrow><mml:mrow><mml:mo>†</mml:mo></mml:mrow></mml:msup><mml:mo>+</mml:mo><mml:mi mathvariant=\"bold-italic\">ϕ</mml:mi><mml:mo>⋅</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant=\"bold\">V</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>ν</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:msup><mml:mo>⋅</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant=\"bold-italic\">ϕ</mml:mi></mml:mrow><mml:mrow><mml:mo>†</mml:mo></mml:mrow></mml:msup><mml:mo>=</mml:mo><mml:mo>=</mml:mo><mml:mspace width=\"0.25em\"/><mml:mi mathvariant=\"bold-italic\">ϕ</mml:mi><mml:mo>⋅</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant=\"bold\">J</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>ν</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mi>ν</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:msup><mml:mo>⋅</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant=\"bold-italic\">ϕ</mml:mi></mml:mrow><mml:mrow><mml:mo>†</mml:mo></mml:mrow></mml:msup><mml:mo>.</mml:mo></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq35\"><alternatives><tex-math id=\"M119\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${{{{{{{\\boldsymbol{{{{{{{{\\mathcal{M}}}}}}}}}}}}}}}}=\\{{\\nu }_{1},{\\nu }_{2},...{\\nu }_{M}\\}$$\\end{document}</tex-math><mml:math id=\"M120\"><mml:mi mathvariant=\"bold-script\">M</mml:mi><mml:mo>=</mml:mo><mml:mrow><mml:mo>{</mml:mo><mml:mrow><mml:msub><mml:mrow><mml:mi>ν</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mi>ν</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:mo>.</mml:mo><mml:mo>.</mml:mo><mml:mo>.</mml:mo><mml:msub><mml:mrow><mml:mi>ν</mml:mi></mml:mrow><mml:mrow><mml:mi>M</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo>}</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ26\"><label>26</label><alternatives><tex-math id=\"M121\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${I}^{{{{{{{{\\boldsymbol{{{{{{{{\\mathcal{M}}}}}}}}}}}}}}}}}=\\mathop{\\sum }\\limits_{\\nu }^{M}{{{{{{{\\boldsymbol{\\phi }}}}}}}}\\cdot {{{{{{{{\\bf{V}}}}}}}}}^{\\nu }\\cdot {{{{{{{{\\boldsymbol{\\phi }}}}}}}}}^{{{{\\dagger}}} }={{{{{{{\\boldsymbol{\\phi }}}}}}}}\\cdot {{{{{{{{\\bf{J}}}}}}}}}^{{{{{{{{\\boldsymbol{{{{{{{{\\mathcal{M}}}}}}}}}}}}}}}}}\\cdot {{{{{{{{\\boldsymbol{\\phi }}}}}}}}}^{{{{\\dagger}}} }$$\\end{document}</tex-math><mml:math id=\"M122\"><mml:msup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-script\">M</mml:mi></mml:mrow></mml:msup><mml:mo>=</mml:mo><mml:munderover accent=\"false\" accentunder=\"false\"><mml:mrow><mml:mo>∑</mml:mo></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow><mml:mrow><mml:mi>M</mml:mi></mml:mrow></mml:munderover><mml:mi mathvariant=\"bold-italic\">ϕ</mml:mi><mml:mo>⋅</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant=\"bold\">V</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msup><mml:mo>⋅</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant=\"bold-italic\">ϕ</mml:mi></mml:mrow><mml:mrow><mml:mo>†</mml:mo></mml:mrow></mml:msup><mml:mo>=</mml:mo><mml:mi mathvariant=\"bold-italic\">ϕ</mml:mi><mml:mo>⋅</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant=\"bold\">J</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-script\">M</mml:mi></mml:mrow></mml:msup><mml:mo>⋅</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant=\"bold-italic\">ϕ</mml:mi></mml:mrow><mml:mrow><mml:mo>†</mml:mo></mml:mrow></mml:msup></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ27\"><label>27</label><alternatives><tex-math id=\"M123\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${J}_{nm}^{{{{{{{{\\boldsymbol{{{{{{{{\\mathcal{M}}}}}}}}}}}}}}}}}=\\mathop{\\sum }\\limits_{\\nu }^{M}{V}_{nm}^{\\nu }$$\\end{document}</tex-math><mml:math id=\"M124\"><mml:msubsup><mml:mrow><mml:mi>J</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mi>m</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-script\">M</mml:mi></mml:mrow></mml:msubsup><mml:mo>=</mml:mo><mml:munderover accent=\"false\" accentunder=\"false\"><mml:mrow><mml:mo>∑</mml:mo></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow><mml:mrow><mml:mi>M</mml:mi></mml:mrow></mml:munderover><mml:msubsup><mml:mrow><mml:mi>V</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mi>m</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msubsup></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq36\"><alternatives><tex-math id=\"M125\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${}^{{{{{{{{\\boldsymbol{{{{{{{{\\mathcal{M}}}}}}}}}}}}}}}}}$$\\end{document}</tex-math><mml:math id=\"M126\"><mml:msup><mml:mrow/><mml:mrow><mml:mi mathvariant=\"bold-script\">M</mml:mi></mml:mrow></mml:msup></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq37\"><alternatives><tex-math id=\"M127\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${I}^{{{{{{{{\\boldsymbol{{{{{{{{\\mathcal{M}}}}}}}}}}}}}}}}}$$\\end{document}</tex-math><mml:math id=\"M128\"><mml:msup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-script\">M</mml:mi></mml:mrow></mml:msup></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq38\"><alternatives><tex-math id=\"M129\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${}^{{{{{{{{\\boldsymbol{{{{{{{{\\mathcal{M}}}}}}}}}}}}}}}}}$$\\end{document}</tex-math><mml:math id=\"M130\"><mml:msup><mml:mrow/><mml:mrow><mml:mi mathvariant=\"bold-script\">M</mml:mi></mml:mrow></mml:msup></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq39\"><alternatives><tex-math id=\"M131\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${}^{{{{{{{{\\boldsymbol{{{{{{{{\\mathcal{M}}}}}}}}}}}}}}}}}$$\\end{document}</tex-math><mml:math id=\"M132\"><mml:msup><mml:mrow/><mml:mrow><mml:mi mathvariant=\"bold-script\">M</mml:mi></mml:mrow></mml:msup></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq40\"><alternatives><tex-math id=\"M133\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${{{{{{{{\\boldsymbol{\\xi }}}}}}}}}_{{{{{{{{\\boldsymbol{{{{{{{{\\mathcal{M}}}}}}}}}}}}}}}}}$$\\end{document}</tex-math><mml:math id=\"M134\"><mml:msub><mml:mrow><mml:mi mathvariant=\"bold-italic\">ξ</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-script\">M</mml:mi></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ28\"><label>28</label><alternatives><tex-math id=\"M135\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${I}^{{{{{{{{\\boldsymbol{{{{{{{{\\mathcal{M}}}}}}}}}}}}}}}},{{{{{{{\\boldsymbol{\\lambda }}}}}}}}}=\\mathop{\\sum }\\limits_{\\nu }^{M}{{{{{{{\\boldsymbol{\\phi }}}}}}}}\\cdot {\\lambda }^{\\nu }{{{{{{{{\\bf{V}}}}}}}}}^{\\nu }\\cdot {{{{{{{{\\boldsymbol{\\phi }}}}}}}}}^{{{{\\dagger}}} }={{{{{{{\\boldsymbol{\\phi }}}}}}}}\\cdot {{{{{{{{\\bf{J}}}}}}}}}^{{{{{{{{\\boldsymbol{{{{{{{{\\mathcal{M}}}}}}}}}}}}}}}},{{{{{{{\\boldsymbol{\\lambda }}}}}}}}}\\cdot {{{{{{{{\\boldsymbol{\\phi }}}}}}}}}^{{{{\\dagger}}} }$$\\end{document}</tex-math><mml:math id=\"M136\"><mml:msup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-script\">M</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant=\"bold-italic\">λ</mml:mi></mml:mrow></mml:msup><mml:mo>=</mml:mo><mml:munderover accent=\"false\" accentunder=\"false\"><mml:mrow><mml:mo>∑</mml:mo></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow><mml:mrow><mml:mi>M</mml:mi></mml:mrow></mml:munderover><mml:mi mathvariant=\"bold-italic\">ϕ</mml:mi><mml:mo>⋅</mml:mo><mml:msup><mml:mrow><mml:mi>λ</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mi mathvariant=\"bold\">V</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msup><mml:mo>⋅</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant=\"bold-italic\">ϕ</mml:mi></mml:mrow><mml:mrow><mml:mo>†</mml:mo></mml:mrow></mml:msup><mml:mo>=</mml:mo><mml:mi mathvariant=\"bold-italic\">ϕ</mml:mi><mml:mo>⋅</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant=\"bold\">J</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-script\">M</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant=\"bold-italic\">λ</mml:mi></mml:mrow></mml:msup><mml:mo>⋅</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant=\"bold-italic\">ϕ</mml:mi></mml:mrow><mml:mrow><mml:mo>†</mml:mo></mml:mrow></mml:msup></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq41\"><alternatives><tex-math id=\"M137\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${}^{{{{{{{{\\boldsymbol{{{{{{{{\\mathcal{M}}}}}}}}}}}}}}}},{{{{{{{\\boldsymbol{\\lambda }}}}}}}}}$$\\end{document}</tex-math><mml:math id=\"M138\"><mml:msup><mml:mrow/><mml:mrow><mml:mi mathvariant=\"bold-script\">M</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant=\"bold-italic\">λ</mml:mi></mml:mrow></mml:msup></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ29\"><label>29</label><alternatives><tex-math id=\"M139\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{array}{r}{{{{{{{\\mathcal{F}}}}}}}}({{{{{{{\\boldsymbol{{{{{{{{\\mathcal{M}}}}}}}}}}}}}}}},{{{{{{{\\boldsymbol{\\lambda }}}}}}}})=\\mathop{\\sum }\\limits_{n,m}^{N}{\\left | \\mathop{\\sum }\\limits_{\\nu }^{M}{\\lambda }^{\\nu }{V}_{nm}^{\\nu }-{T}_{nm}\\right | }^{2}=\\\\={{{{{{{\\rm{DIST}}}}}}}}\\left({{{{{{{{\\bf{J}}}}}}}}}^{M,{{{{{{{\\boldsymbol{\\lambda }}}}}}}}},{{{{{{{\\bf{T}}}}}}}}\\right)\\end{array}$$\\end{document}</tex-math><mml:math id=\"M140\"><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">F</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi mathvariant=\"bold-script\">M</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant=\"bold-italic\">λ</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:munderover accent=\"false\" accentunder=\"false\"><mml:mrow><mml:mo>∑</mml:mo></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mo>,</mml:mo><mml:mi>m</mml:mi></mml:mrow><mml:mrow><mml:mi>N</mml:mi></mml:mrow></mml:munderover><mml:msup><mml:mrow><mml:mfenced close=\"∣\" open=\"∣\"><mml:mrow><mml:munderover accent=\"false\" accentunder=\"false\"><mml:mrow><mml:mo>∑</mml:mo></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow><mml:mrow><mml:mi>M</mml:mi></mml:mrow></mml:munderover><mml:msup><mml:mrow><mml:mi>λ</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msup><mml:msubsup><mml:mrow><mml:mi>V</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mi>m</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msubsup><mml:mo>−</mml:mo><mml:msub><mml:mrow><mml:mi>T</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mi>m</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:mfenced></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo>=</mml:mo></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign=\"right\"><mml:mo>=</mml:mo><mml:mi mathvariant=\"normal\">DIST</mml:mi><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:msup><mml:mrow><mml:mi mathvariant=\"bold\">J</mml:mi></mml:mrow><mml:mrow><mml:mi>M</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant=\"bold-italic\">λ</mml:mi></mml:mrow></mml:msup><mml:mo>,</mml:mo><mml:mi mathvariant=\"bold\">T</mml:mi></mml:mrow></mml:mfenced></mml:mtd></mml:mtr></mml:mtable></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq42\"><alternatives><tex-math id=\"M141\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${{{{{{{{\\bf{J}}}}}}}}}_{{{{{{{{\\bf{T}}}}}}}}}^{{{{{{{{\\boldsymbol{M,\\lambda }}}}}}}}}$$\\end{document}</tex-math><mml:math id=\"M142\"><mml:msubsup><mml:mrow><mml:mi mathvariant=\"bold\">J</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold\">T</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">M,λ</mml:mi></mml:mrow></mml:msubsup></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ30\"><label>30</label><alternatives><tex-math id=\"M143\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${I}_{{{{{{{{\\bf{T}}}}}}}}}^{{{{{{{{\\boldsymbol{M,\\lambda }}}}}}}}}=\\mathop{\\sum }\\limits_{\\nu }^{M}{\\lambda }^{\\nu }{I}^{\\nu }$$\\end{document}</tex-math><mml:math id=\"M144\"><mml:msubsup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold\">T</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">M,λ</mml:mi></mml:mrow></mml:msubsup><mml:mo>=</mml:mo><mml:munderover accent=\"false\" accentunder=\"false\"><mml:mrow><mml:mo>∑</mml:mo></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow><mml:mrow><mml:mi>M</mml:mi></mml:mrow></mml:munderover><mml:msup><mml:mrow><mml:mi>λ</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msup></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq43\"><alternatives><tex-math id=\"M145\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${{{{{{{\\boldsymbol{{{{{{{{\\mathcal{M}}}}}}}}}}}}}}}}$$\\end{document}</tex-math><mml:math id=\"M146\"><mml:mi mathvariant=\"bold-script\">M</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq44\"><alternatives><tex-math id=\"M147\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\Pi=\\left.\\right(N(N-1)/2$$\\end{document}</tex-math><mml:math id=\"M148\"><mml:mi mathvariant=\"normal\">Π</mml:mi><mml:mo>=</mml:mo><mml:mfenced close=\"(\"><mml:mrow/></mml:mfenced><mml:mi>N</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>N</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>/</mml:mo><mml:mn>2</mml:mn></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq45\"><alternatives><tex-math id=\"M149\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${{{{{{{{\\bf{J}}}}}}}}}_{{{{{{{{{\\boldsymbol{\\phi }}}}}}}}}^{*}}^{{{{{{{{{\\boldsymbol{{{{{{{{\\mathcal{M}}}}}}}}}}}}}}}}}^{*},{{{{{{{\\boldsymbol{\\lambda }}}}}}}}}$$\\end{document}</tex-math><mml:math id=\"M150\"><mml:msubsup><mml:mrow><mml:mi mathvariant=\"bold\">J</mml:mi></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mi mathvariant=\"bold-italic\">ϕ</mml:mi></mml:mrow><mml:mrow><mml:mo>*</mml:mo></mml:mrow></mml:msup></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mi mathvariant=\"bold-script\">M</mml:mi></mml:mrow><mml:mrow><mml:mo>*</mml:mo></mml:mrow></mml:msup><mml:mo>,</mml:mo><mml:mi mathvariant=\"bold-italic\">λ</mml:mi></mml:mrow></mml:msubsup></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ31\"><label>31</label><alternatives><tex-math id=\"M151\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${Storage \\, Error \\, Probability}=S\\_ERR/N.$$\\end{document}</tex-math><mml:math id=\"M152\"><mml:mi>S</mml:mi><mml:mi>t</mml:mi><mml:mi>o</mml:mi><mml:mi>r</mml:mi><mml:mi>a</mml:mi><mml:mi>g</mml:mi><mml:mi>e</mml:mi><mml:mspace width=\"0.25em\"/><mml:mi>E</mml:mi><mml:mi>r</mml:mi><mml:mi>r</mml:mi><mml:mi>o</mml:mi><mml:mi>r</mml:mi><mml:mspace width=\"0.25em\"/><mml:mi>P</mml:mi><mml:mi>r</mml:mi><mml:mi>o</mml:mi><mml:mi>b</mml:mi><mml:mi>a</mml:mi><mml:mi>b</mml:mi><mml:mi>i</mml:mi><mml:mi>l</mml:mi><mml:mi>i</mml:mi><mml:mi>t</mml:mi><mml:mi>y</mml:mi><mml:mo>=</mml:mo><mml:mi>S</mml:mi><mml:mo>_</mml:mo><mml:mi>E</mml:mi><mml:mi>R</mml:mi><mml:mi>R</mml:mi><mml:mo>/</mml:mo><mml:mi>N</mml:mi><mml:mo>.</mml:mo></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq46\"><alternatives><tex-math id=\"M153\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${}_{{{{{{{{\\boldsymbol{T}}}}}}}}}^{{{{{{{{\\boldsymbol{M,\\lambda }}}}}}}}}$$\\end{document}</tex-math><mml:math id=\"M154\"><mml:msubsup><mml:mrow/><mml:mrow><mml:mi mathvariant=\"bold-italic\">T</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">M,λ</mml:mi></mml:mrow></mml:msubsup></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ32\"><label>32</label><alternatives><tex-math id=\"M155\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${I}_{{{{{{{{\\bf{T}}}}}}}}}^{{{{{{{{\\boldsymbol{{{{{{{{\\mathcal{M}}}}}}}}}}}}}}}},{{{{{{{\\boldsymbol{\\lambda }}}}}}}}}={{{{{{{\\boldsymbol{\\phi }}}}}}}}\\cdot {{{{{{{{\\boldsymbol{J}}}}}}}}}_{{{{{{{{\\bf{T}}}}}}}}}^{{{{{{{{\\boldsymbol{{{{{{{{\\mathcal{M}}}}}}}}}}}}}}}}},{{{{{{{\\boldsymbol{\\lambda }}}}}}}}\\cdot {{{{{{{{\\boldsymbol{\\phi }}}}}}}}}^{{{{\\dagger}}} }.$$\\end{document}</tex-math><mml:math id=\"M156\"><mml:msubsup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold\">T</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-script\">M</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant=\"bold-italic\">λ</mml:mi></mml:mrow></mml:msubsup><mml:mo>=</mml:mo><mml:mi mathvariant=\"bold-italic\">ϕ</mml:mi><mml:mo>⋅</mml:mo><mml:msubsup><mml:mrow><mml:mi mathvariant=\"bold-italic\">J</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold\">T</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-script\">M</mml:mi></mml:mrow></mml:msubsup><mml:mo>,</mml:mo><mml:mi mathvariant=\"bold-italic\">λ</mml:mi><mml:mo>⋅</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant=\"bold-italic\">ϕ</mml:mi></mml:mrow><mml:mrow><mml:mo>†</mml:mo></mml:mrow></mml:msup><mml:mo>.</mml:mo></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq47\"><alternatives><tex-math id=\"M157\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${{{{{{{\\boldsymbol{{{{{{{{\\mathcal{M}}}}}}}}}}}}}}}}$$\\end{document}</tex-math><mml:math id=\"M158\"><mml:mi mathvariant=\"bold-script\">M</mml:mi></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ33\"><label>33</label><alternatives><tex-math id=\"M159\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${Recognition \\, Error \\, Probability}=R\\_ERR/P.$$\\end{document}</tex-math><mml:math id=\"M160\"><mml:mi>R</mml:mi><mml:mi>e</mml:mi><mml:mi>c</mml:mi><mml:mi>o</mml:mi><mml:mi>g</mml:mi><mml:mi>n</mml:mi><mml:mi>i</mml:mi><mml:mi>t</mml:mi><mml:mi>i</mml:mi><mml:mi>o</mml:mi><mml:mi>n</mml:mi><mml:mspace width=\"0.25em\"/><mml:mi>E</mml:mi><mml:mi>r</mml:mi><mml:mi>r</mml:mi><mml:mi>o</mml:mi><mml:mi>r</mml:mi><mml:mspace width=\"0.25em\"/><mml:mi>P</mml:mi><mml:mi>r</mml:mi><mml:mi>o</mml:mi><mml:mi>b</mml:mi><mml:mi>a</mml:mi><mml:mi>b</mml:mi><mml:mi>i</mml:mi><mml:mi>l</mml:mi><mml:mi>i</mml:mi><mml:mi>t</mml:mi><mml:mi>y</mml:mi><mml:mo>=</mml:mo><mml:mi>R</mml:mi><mml:mo>_</mml:mo><mml:mi>E</mml:mi><mml:mi>R</mml:mi><mml:mi>R</mml:mi><mml:mo>/</mml:mo><mml:mi>P</mml:mi><mml:mo>.</mml:mo></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ34\"><label>34</label><alternatives><tex-math id=\"M161\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${{{{{{{{\\bf{V}}}}}}}}}^{\\nu }={{{{{{{{\\boldsymbol{\\xi }}}}}}}}}^{\\nu }\\otimes {{{{{{{{\\boldsymbol{\\xi }}}}}}}}}^{\\nu {{{\\dagger}}} }+{{{{{{{{\\boldsymbol{\\eta }}}}}}}}}^{\\nu }\\otimes {{{{{{{{\\boldsymbol{\\eta }}}}}}}}}^{\\nu {{{\\dagger}}} }$$\\end{document}</tex-math><mml:math id=\"M162\"><mml:msup><mml:mrow><mml:mi mathvariant=\"bold\">V</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msup><mml:mo>=</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant=\"bold-italic\">ξ</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msup><mml:mo>⊗</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant=\"bold-italic\">ξ</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi><mml:mo>†</mml:mo></mml:mrow></mml:msup><mml:mo>+</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant=\"bold-italic\">η</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msup><mml:mo>⊗</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant=\"bold-italic\">η</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi><mml:mo>†</mml:mo></mml:mrow></mml:msup></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq48\"><alternatives><tex-math id=\"M163\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${{{{{{{{\\boldsymbol{{{{{{{{\\mathcal{M}}}}}}}}}}}}}}}}}^{L}$$\\end{document}</tex-math><mml:math id=\"M164\"><mml:msup><mml:mrow><mml:mi mathvariant=\"bold-script\">M</mml:mi></mml:mrow><mml:mrow><mml:mi>L</mml:mi></mml:mrow></mml:msup></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq49\"><alternatives><tex-math id=\"M165\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${{{{{{{{\\boldsymbol{{{{{{{{\\mathcal{M}}}}}}}}}}}}}}}}}^{L}$$\\end{document}</tex-math><mml:math id=\"M166\"><mml:msup><mml:mrow><mml:mi mathvariant=\"bold-script\">M</mml:mi></mml:mrow><mml:mrow><mml:mi>L</mml:mi></mml:mrow></mml:msup></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq50\"><alternatives><tex-math id=\"M167\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${{{{{{{\\mathcal{S}}}}}}}}$$\\end{document}</tex-math><mml:math id=\"M168\"><mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">S</mml:mi></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ35\"><label>35</label><alternatives><tex-math id=\"M169\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${{{{{{{{\\mathcal{S}}}}}}}}}^{\\nu }={\\hat{{{{{{{{\\boldsymbol{\\phi }}}}}}}}}}^{*}\\cdot {\\hat{{{{{{{{\\boldsymbol{\\xi }}}}}}}}}}^{\\nu }$$\\end{document}</tex-math><mml:math id=\"M170\"><mml:msup><mml:mrow><mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">S</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msup><mml:mo>=</mml:mo><mml:msup><mml:mrow><mml:mover accent=\"true\"><mml:mrow><mml:mi mathvariant=\"bold-italic\">ϕ</mml:mi></mml:mrow><mml:mrow><mml:mo>^</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mo>*</mml:mo></mml:mrow></mml:msup><mml:mo>⋅</mml:mo><mml:msup><mml:mrow><mml:mover accent=\"true\"><mml:mrow><mml:mi mathvariant=\"bold-italic\">ξ</mml:mi></mml:mrow><mml:mrow><mml:mo>^</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msup></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq51\"><alternatives><tex-math id=\"M171\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\hat{{{{{{{{\\boldsymbol{i}}}}}}}}}$$\\end{document}</tex-math><mml:math id=\"M172\"><mml:mover accent=\"true\"><mml:mrow><mml:mi mathvariant=\"bold-italic\">i</mml:mi></mml:mrow><mml:mrow><mml:mo>^</mml:mo></mml:mrow></mml:mover></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq52\"><alternatives><tex-math id=\"M173\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\hat{i}\\cdot \\hat{i}=1$$\\end{document}</tex-math><mml:math id=\"M174\"><mml:mover accent=\"true\"><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mo>^</mml:mo></mml:mrow></mml:mover><mml:mo>⋅</mml:mo><mml:mover accent=\"true\"><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mo>^</mml:mo></mml:mrow></mml:mover><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq53\"><alternatives><tex-math id=\"M175\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${{{{{{{{\\boldsymbol{{{{{{{{\\mathcal{M}}}}}}}}}}}}}}}}}^{L}$$\\end{document}</tex-math><mml:math id=\"M176\"><mml:msup><mml:mrow><mml:mi mathvariant=\"bold-script\">M</mml:mi></mml:mrow><mml:mrow><mml:mi>L</mml:mi></mml:mrow></mml:msup></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq54\"><alternatives><tex-math id=\"M177\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${{{{{{{{\\boldsymbol{{{{{{{{\\mathcal{M}}}}}}}}}}}}}}}}}^{*}$$\\end{document}</tex-math><mml:math id=\"M178\"><mml:msup><mml:mrow><mml:mi mathvariant=\"bold-script\">M</mml:mi></mml:mrow><mml:mrow><mml:mo>*</mml:mo></mml:mrow></mml:msup></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq55\"><alternatives><tex-math id=\"M179\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${{{{{{{{\\mathcal{S}}}}}}}}}^{\\nu }$$\\end{document}</tex-math><mml:math id=\"M180\"><mml:msup><mml:mrow><mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">S</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msup></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq56\"><alternatives><tex-math id=\"M181\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${{{{{{{{\\boldsymbol{{{{{{{{\\mathcal{M}}}}}}}}}}}}}}}}}^{*}$$\\end{document}</tex-math><mml:math id=\"M182\"><mml:msup><mml:mrow><mml:mi mathvariant=\"bold-script\">M</mml:mi></mml:mrow><mml:mrow><mml:mo>*</mml:mo></mml:mrow></mml:msup></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq57\"><alternatives><tex-math id=\"M183\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${{{{{{{{\\boldsymbol{{{{{{{{\\mathcal{M}}}}}}}}}}}}}}}}}^{*}$$\\end{document}</tex-math><mml:math id=\"M184\"><mml:msup><mml:mrow><mml:mi mathvariant=\"bold-script\">M</mml:mi></mml:mrow><mml:mrow><mml:mo>*</mml:mo></mml:mrow></mml:msup></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ36\"><label>36</label><alternatives><tex-math id=\"M185\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${{{{{{{{\\bf{J}}}}}}}}}^{{{{{{{{{\\boldsymbol{{{{{{{{\\mathcal{M}}}}}}}}}}}}}}}}}^{*},{{{{{{{\\boldsymbol{\\lambda }}}}}}}}}=\\mathop{\\sum }\\limits_{\\nu }^{{M}^{*}}{\\lambda }^{\\nu }{{{{{{{{\\bf{J}}}}}}}}}^{\\nu }$$\\end{document}</tex-math><mml:math id=\"M186\"><mml:msup><mml:mrow><mml:mi mathvariant=\"bold\">J</mml:mi></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mi mathvariant=\"bold-script\">M</mml:mi></mml:mrow><mml:mrow><mml:mo>*</mml:mo></mml:mrow></mml:msup><mml:mo>,</mml:mo><mml:mi mathvariant=\"bold-italic\">λ</mml:mi></mml:mrow></mml:msup><mml:mo>=</mml:mo><mml:munderover accent=\"false\" accentunder=\"false\"><mml:mrow><mml:mo>∑</mml:mo></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mo>*</mml:mo></mml:mrow></mml:msup></mml:mrow></mml:munderover><mml:msup><mml:mrow><mml:mi>λ</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mi mathvariant=\"bold\">J</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msup></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ37\"><label>37</label><alternatives><tex-math id=\"M187\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${{{{{{{{\\mathcal{F}}}}}}}}}^{*}({{{{{{{\\boldsymbol{\\lambda }}}}}}}},{{{{{{{{\\boldsymbol{\\phi }}}}}}}}}^{*},{{{{{{{{\\boldsymbol{{{{{{{{\\mathcal{M}}}}}}}}}}}}}}}}}^{*})={\\hat{{{{{{{{\\boldsymbol{\\xi }}}}}}}}}}_{\\Sigma }\\cdot {\\hat{{{{{{{{\\boldsymbol{\\phi }}}}}}}}}}^{*}$$\\end{document}</tex-math><mml:math id=\"M188\"><mml:msup><mml:mrow><mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">F</mml:mi></mml:mrow><mml:mrow><mml:mo>*</mml:mo></mml:mrow></mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi mathvariant=\"bold-italic\">λ</mml:mi><mml:mo>,</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant=\"bold-italic\">ϕ</mml:mi></mml:mrow><mml:mrow><mml:mo>*</mml:mo></mml:mrow></mml:msup><mml:mo>,</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant=\"bold-script\">M</mml:mi></mml:mrow><mml:mrow><mml:mo>*</mml:mo></mml:mrow></mml:msup></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:msub><mml:mrow><mml:mover accent=\"true\"><mml:mrow><mml:mi mathvariant=\"bold-italic\">ξ</mml:mi></mml:mrow><mml:mrow><mml:mo>^</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">Σ</mml:mi></mml:mrow></mml:msub><mml:mo>⋅</mml:mo><mml:msup><mml:mrow><mml:mover accent=\"true\"><mml:mrow><mml:mi mathvariant=\"bold-italic\">ϕ</mml:mi></mml:mrow><mml:mrow><mml:mo>^</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mo>*</mml:mo></mml:mrow></mml:msup></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq58\"><alternatives><tex-math id=\"M189\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${}^{{{{{{{{{\\boldsymbol{{{{{{{{\\mathcal{M}}}}}}}}}}}}}}}}}^{*},{{{{{{{\\boldsymbol{\\lambda }}}}}}}}}$$\\end{document}</tex-math><mml:math id=\"M190\"><mml:msup><mml:mrow/><mml:mrow><mml:msup><mml:mrow><mml:mi mathvariant=\"bold-script\">M</mml:mi></mml:mrow><mml:mrow><mml:mo>*</mml:mo></mml:mrow></mml:msup><mml:mo>,</mml:mo><mml:mi mathvariant=\"bold-italic\">λ</mml:mi></mml:mrow></mml:msup></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq59\"><alternatives><tex-math id=\"M191\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${{{{{{{{\\mathcal{F}}}}}}}}}^{*}({{{{{{{\\boldsymbol{\\lambda }}}}}}}},{{{{{{{\\boldsymbol{{\\phi }}}}}}}^{*}}},{{{{{{{{\\boldsymbol{{{{{{{{\\mathcal{M}}}}}}}}}}}}}}}}}^{*})$$\\end{document}</tex-math><mml:math id=\"M192\"><mml:msup><mml:mrow><mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">F</mml:mi></mml:mrow><mml:mrow><mml:mo>*</mml:mo></mml:mrow></mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi mathvariant=\"bold-italic\">λ</mml:mi><mml:mo>,</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant=\"bold-italic\">ϕ</mml:mi></mml:mrow><mml:mrow><mml:mo>*</mml:mo></mml:mrow></mml:msup><mml:mo>,</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant=\"bold-script\">M</mml:mi></mml:mrow><mml:mrow><mml:mo>*</mml:mo></mml:mrow></mml:msup></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq60\"><alternatives><tex-math id=\"M193\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${{{{{{{{\\bf{J}}}}}}}}}_{{{{{{{{{\\boldsymbol{\\phi }}}}}}}}}^{*}}^{{{{{{{{{\\boldsymbol{{{{{{{{\\mathcal{M}}}}}}}}}}}}}}}}}^{*},{{{{{{{\\boldsymbol{\\lambda }}}}}}}}}$$\\end{document}</tex-math><mml:math id=\"M194\"><mml:msubsup><mml:mrow><mml:mi mathvariant=\"bold\">J</mml:mi></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mi mathvariant=\"bold-italic\">ϕ</mml:mi></mml:mrow><mml:mrow><mml:mo>*</mml:mo></mml:mrow></mml:msup></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mi mathvariant=\"bold-script\">M</mml:mi></mml:mrow><mml:mrow><mml:mo>*</mml:mo></mml:mrow></mml:msup><mml:mo>,</mml:mo><mml:mi mathvariant=\"bold-italic\">λ</mml:mi></mml:mrow></mml:msubsup></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ38\"><label>38</label><alternatives><tex-math id=\"M195\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${I}_{n,m}^{\\nu }={E}_{n}^{\\nu 2}+{E}_{m}^{\\nu 2}+2{\\xi }_{n}^{\\nu }{\\xi }_{m}^{\\nu }+2{\\eta }_{n}^{\\nu }{\\eta }_{m}^{\\nu }.$$\\end{document}</tex-math><mml:math 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"<disp-formula id=\"Equ39\"><label>39</label><alternatives><tex-math id=\"M197\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${I}_{n}^{\\nu }={E}_{n}^{\\nu 2}.$$\\end{document}</tex-math><mml:math id=\"M198\"><mml:msubsup><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msubsup><mml:mo>=</mml:mo><mml:msubsup><mml:mrow><mml:mi>E</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo>.</mml:mo></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ40\"><label>40</label><alternatives><tex-math id=\"M199\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${V}_{nm}^{\\nu }={\\xi }_{n}^{\\nu }{\\xi }_{m}^{\\nu }+{\\eta }_{n}^{\\nu }{\\eta }_{m}^{\\nu }=\\frac{{I}_{nm}^{\\nu }-{I}_{n}^{\\nu }-{I}_{m}^{\\nu }}{2}.$$\\end{document}</tex-math><mml:math 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"<inline-formula id=\"IEq61\"><alternatives><tex-math id=\"M201\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${V}_{nm}^{\\nu }={V}_{mn}^{\\nu }$$\\end{document}</tex-math><mml:math id=\"M202\"><mml:msubsup><mml:mrow><mml:mi>V</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mi>m</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msubsup><mml:mo>=</mml:mo><mml:msubsup><mml:mrow><mml:mi>V</mml:mi></mml:mrow><mml:mrow><mml:mi>m</mml:mi><mml:mi>n</mml:mi></mml:mrow><mml:mrow><mml:mi>ν</mml:mi></mml:mrow></mml:msubsup></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq62\"><alternatives><tex-math id=\"M203\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${{{{{{{{\\mathcal{M}}}}}}}}}^{L}$$\\end{document}</tex-math><mml:math id=\"M204\"><mml:msup><mml:mrow><mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">M</mml:mi></mml:mrow><mml:mrow><mml:mi>L</mml:mi></mml:mrow></mml:msup></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq63\"><alternatives><tex-math id=\"M205\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$=\\sigma /\\sqrt{(10-1)}$$\\end{document}</tex-math><mml:math id=\"M206\"><mml:mo>=</mml:mo><mml:mi>σ</mml:mi><mml:mo>/</mml:mo><mml:msqrt><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mn>10</mml:mn><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:msqrt></mml:math></alternatives></inline-formula>" ]

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[ "<supplementary-material content-type=\"local-data\" id=\"MOESM1\"></supplementary-material>", "<supplementary-material content-type=\"local-data\" id=\"MOESM2\"></supplementary-material>" ]

[ "<fn-group><fn><p><bold>Publisher’s note</bold> Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.</p></fn></fn-group>" ]

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[ "<media xlink:href=\"41467_2023_44498_MOESM1_ESM.pdf\"><caption><p>Supplementary informations</p></caption></media>", "<media xlink:href=\"41467_2023_44498_MOESM2_ESM.pdf\"><caption><p>Peer Review File</p></caption></media>" ]

{ "acronym": [], "definition": [] }

[ "<title>Background &amp; Summary</title>", "<p id=\"Par2\"><italic>Decapterus maruadsi</italic>, a pelagic fish in the family Carangidae, lives widely in distributed warm offshore waters of East and Southeast Asia^{##REF##32744461##1##}. And it is especially abundant along the coasts of the South China Sea^{##UREF##0##2##}. From last century, <italic>D. maruadsi</italic> has become one of the most commercially valuable marine fishery species in Chinese aquaculture. It is also one of the main species captured by pelagic trawls and light-luring fishing vessels^{##UREF##1##3##}. A short lifespan and fast growth and reproduction rates are the most notable features of <italic>D. maruadsi</italic>^{##UREF##2##4##}. Meanwhile, as an r-selection strategy species, it is vulnerable to the environmental deterioration and fishing intensity including those unregulated fishing methods and advanced technologies^{##UREF##3##5##,##UREF##4##6##}. In recent decades, under the multiple stresses of continuous high-intensity fishing, increasing temperature and feed structure changes caused by global climate change, the population of <italic>D. maruadsi</italic> has been subjected to strong selection pressure, which gradually showing adaptive evolution phenomena such as miniaturization, sex precocity, and the population size has also been decreasing year by year^{##REF##25667602##7##,##UREF##5##8##}. To deal with this dilemma, artificial cultivation of juvenile fish of <italic>D. maruadsi</italic> has been gradually realized in the offshore area of Dongshan island at present.</p>", "<p id=\"Par3\">To ensure the preservation of economically significant species, it is crucial to safeguard their germplasm resources and prevent any potential decline^{##UREF##6##9##}. Genomic data are essential tools for investigating species germplasm resources and assessing population genetic structure and diversity. These resources are of great significance for managing fishery resources and promoting their sustainable use. High-quality reference genomes are essential basic genetic data, and their application value to the aquaculture is also very important. Furthermore, the value of long Nanopore reads which includes low cost, high-throughput sequencing, and high-quality assembly of genomes has been reported by many researches^{##REF##27153285##10##,##REF##18846088##11##}. By combining third-generation sequencing and high-through chromosome conformation capture (Hi-C)^{##REF##22652625##12##} technologies, we can assemble the chromosome-level genome rapidly, efficiently and accurately. On this basis, the annotation of <italic>D. maruadsi</italic> genome can be completed. In this report, we provided a high-quality genome assembly of <italic>D. maruadsi</italic> using Illumina short-reads sequencing, Nanopore sequencing, and Hi-C technologies. We obtained a total of 47.61 Gb clean reads by Illumina platform, and through K-mer frequency distribution analysis, the genome size of <italic>D. maruadsi</italic> was about 720.70 Mb. For Nanopore genome sequencing, we assembled a total genome length of 723.69 Mb, which includes a total of 169 contigs. In addition, N50 and N90 lengths of filtered reads were respectively 24.67 Mb and 2.78 Mb, and contigs with a length of 2Kb accounted for 100%. We generated 76.37 Gb of Hi-C filtered data, after chromosome-level scaffolding, there are 23 chromosomes with a total length of 713.58 Mb, resulting in a scaffold N50 of 32.35 Mb. The reference genome of <italic>D. maruadsi</italic> can assist in subsequent population genomics and adaptive genome microevolution studies^{##REF##19627488##13##}.</p>" ]

[ "<title>Methods</title>", "<title>Ethics statement</title>", "<p id=\"Par4\">The <italic>D. maruadsi</italic> in our experiments were collected from Dongshan, Zhangzhou City, Fujian Province, China. Furthermore, the methods used in this work are strictly in accordance with the Guidelines for The Care and Use of Laboratory Animals and followed by the Laboratory Animal Laboratory Committee School of Ocean and Earth Sciences, Xiamen University.</p>", "<title>Sample collection and nucleic acid preparation</title>", "<p id=\"Par5\">A healthy alive female <italic>D. maruadsi</italic> was collected from the Dongshan Pacific Ocean Observation and Experiment Station, Xiamen University. Ten fresh tissue samples, including muscle, eye, skin, gill, kidney, liver, intestine, spleen, heart and stomach, were frozen in liquid nitrogen immediately and then stored in −80 °C. Following the standard protocol of QIAGEN DNeasy Blood &amp; Tissue Kit (Qiagen, Shanghai, China), genomic DNA (gDNA) of muscle was extracted. Total RNA was extracted from ten tissues by a TRIzoL kit (Invitrogen, Shanghai, China) and mixed equally for RNA-seq. The quality of nucleic acid was detected by 1.0% agarose gel electrophoresis and quantified by a Qubit 4.0 fluorometer (Thermo Fisher Scientific, Waltham, MA).</p>", "<title>Library construction and sequencing</title>", "<p id=\"Par6\">For Illumina data, a pair-end sequencing library with 350 bp insert size was constructed using the Illumina TruSeq Nano DNA Library Prep Kit (Illumina, San Diego, CA, USA) and sequenced on the Illumina HiSeq X Ten platform with the 2 × 150 bp read strategy at Novogene company (Tianjin, China). A total of 47.85 Gb raw data were obtained and 47.61 Gb clean data were retained after quality filtering by fastp (V.0.23.1)^{##REF##30423086##14##} software (Table ##TAB##0##1##). For the Nanopore sequencing, the frozen muscle sample was lysed in SDS digestion buffer, and then the lysate was purified with AMPure XP microbeads (Beckman Coulter, High Wycombe, UK) to obtain High-Molecular-Weight(HMW) gDNA. DNA fragment sizes were selected with the BluePippin system (Sage Science, Beverly, MA, USA) and fragments larger than 20 kb were used for subsequent Nanopore sequencing. The Nanopore libraries were prepared using the Ligation Sequencing Kit (SQK-LSK109, Oxford Nanopore Technologies, Oxford, UK) according to the manufacturer’s instructions and sequenced on the flow cells of the PromethION sequencer at Novogene company (Tianjin, China). Finally, we obtained 86.39 Gb Nanopore data, which average and N50 read length were 22.07Kb and 26.23Kb, respectively. The Nanopore data were further screened before assembly to remove reads less than 1500 bp in length. For Hi-C sequencing, DNA fixed with formaldehyde was digested with the restriction enzyme (<italic>DpnII</italic>), and after being repaired by 5’overhangs biotinylated and blunt-end ligation, these fragments are connected <italic>in situ</italic>, the DNA is cross-linked and purified^{##REF##25497547##15##}. In the end, the Hi-C sequencing library was performed on the Illumina HiSeq X Ten platform with a strategy of 2 × 150 bp and generated 76.37 Gb raw reads overall. The RNA-seq library was constructed using Illumina standard protocol (San Diego, CA, USA) and sequenced on the Illumina HiSeq X Ten platform. Finally, we obtained 33.65 Gb paired-end raw reads and 32.61 Gb paired-end clean reads for the following gene prediction (Table ##TAB##0##1##).</p>", "<title>De novo genome assembly</title>", "<p id=\"Par7\">The Illumina clean reads were used for further assembly and estimation of genome size using 17-kmer analysis. With K-mer numbers of 434,759,917,857 and a dominant peak depth of 47.24, the genome size was approximately 720.70 Mb, which was similar to the species in Genus <italic>Decapterus</italic> and the heterozygosity and repetitive sequence content were about 0.69 and 32.6%, respectively^{##REF##14824511##16##} (Supplementary Table ##SUPPL##1##1## &amp; Supplementary Fig. ##SUPPL##0##1##).</p>", "<p id=\"Par8\">NextDenovo was used for genome assembly based on the overlap layout-consensus algorithm with default parameters. To obtain the contig-level genome, we utilized Racon^{##REF##28100585##17##} for three iterations of polish using the three-generation Nanopore data. Nextpolish^{##REF##31778144##18##} was then employed to correct the genome based on the Illumina data. Lastly, we utilized Purge_Dups^{##REF##31971576##19##} (v.1.25) to de-redundant the genome, resulting in the final contig-level genome. The assembled genome size was 723.69 Mb, including 169 contigs in total, with a contig N50 of 24.67 Mb (Table ##TAB##1##2##). The assembled genome size almost matched the estimated results of genome survey, which reflected the high assembly integrity.</p>", "<p id=\"Par9\">Hi-C sequencing data was used for chromosome assembly of <italic>D. maruadsi</italic>. Firstly, we filtered out Hi-C raw reads(low-quality and duplicated reads) using HiC-Pro^{##REF##25583448##20##}. Juicer^{##REF##27467249##21##} was used to map Hi-C clean reads to the reference genome. Subsequently, we used the genomic proximity signal in the Hi-C data sets to get chromosome-level scaffolds. Then, the 3D-DNA pipeline^{##REF##28336562##22##} was used to scaffold the <italic>D. maruadsi</italic> genome. Afterwards, scaffolds were fine-tuned to correct the misassemblies by Juicebox^{##REF##27467250##23##} assembly tools. Finally, we generated a chromosome-level genome assembly of 724.05 Mb and scaffold N50 is up to 32.35 Mb (Table ##TAB##1##2##). The genome assembly contained 23 chromosomes, with a total length of 713.58 Mb (98.6% of the total length of all contigs). Chromosome sizes ranged from 21.36 to 45.1 Mb, with an average chromosome length of 31.03 Mb (Fig. ##FIG##0##1A##,##FIG##0##B## &amp; Table ##TAB##2##3##).</p>", "<title>Anotation of repeat sequences</title>", "<p id=\"Par10\">Both homology-based and <italic>de novo</italic> methods were used to annotate repeat sequences in the <italic>D. maruadsi</italic> genome. RepeatModeler^{##REF##32300014##24##} (v.2.0.1) and LTR_Finder^{##REF##17485477##25##} (v.1.07) were utilized to detect repetitive sequences in the <italic>D. maruadsi</italic> genome and generate a <italic>de novo</italic> repeat library. Combined with Repbase^{##UREF##7##26##}, the final repeat library was constructed. RepeatMasker^{##UREF##8##27##} (v.4.1.0) was used to search and classify repeats based on this library. Unclassified repeats were further annotated using TEclass^{##REF##19349283##28##} (v.2.1.3). Transposable Elements (TEs) annotation results were summarized by adopting the buildSummary.pl of RepeatMasker. Moreover, calcDivergenceFromalign.pl was used to calculate the Kimura divergence value of TEs and createRepeatLandscape.pl was used to draw TEs landscapes^{##REF##7463489##29##}. To estimate the insertion age, we compared the nucleotide distances between all copies of each TE using the Kimura two-parameter method^{##REF##7463489##29##}. We identified tandem repeats using the Tandem Repeats Finder^{##REF##9862982##30##} (v.4.0.9) and soft-masked all repetitive regions except for tandem repeats in the process of protein-coding gene annotation. Finally, a total of 199.49 Mb (27.57% in genome) of consistent and non-redundant repeat sequences were obtained by combining novel, known and tandem repeats. The most abundant repetitive elements were DNA transposons, which spanned more than 102.57 Mb, accounting for 14.17% of the genome of <italic>D. maruadsi</italic>. Besides, the repetitive sequences were also composed of long interspersed elements (LINE) in 37.62 Mb (5.20% in genome), short interspersed nuclear elements (SINEs) in 2.82 Mb (0.39% in genome) and long terminal repeats (LTRs) in 40.99 Mb (5.66% in genome) (Fig. ##FIG##1##2A## &amp; Table ##TAB##3##4##).</p>", "<title>Prediction and functional annotation of protein-coding genes</title>", "<p id=\"Par11\">For non-coding RNA (ncRNA) annotation, the programs tRNAScan-SE^{##REF##9023104##31##} (v.1.3.1) and RNAmmer^{##REF##17452365##32##} (v.1.2) were used to predict tRNA and rRNA respectively. The other ncRNAs were predicted by searching the Rfam database^{##REF##33211869##33##} (<ext-link ext-link-type=\"uri\" xlink:href=\"http://eggnogdb.embl.de/\">http://eggnogdb.embl.de/</ext-link>). As a result, we annotated four types of non-coding RNAs, including 1,285 miRNAs, 3,820 tRNAs, 1,592 rRNAs and 762 snRNAs (Table ##TAB##3##4##).</p>", "<p id=\"Par12\">For gene structure prediction, ab-initio strategies, homologous searching and transcriptome-assisted approaches were used to predict protein-coding genes in the <italic>D. maruadsi</italic> genome after soft-masking all repeat sequences. In homology-based prediction, the genetically proximal coding sequences of related species, containing <italic>Oryzias latipes</italic>, <italic>Seriola lalandi</italic>, <italic>Seriola dumerili</italic>, <italic>Oreochromis niloticus</italic>, and <italic>Trachinotus ovatus</italic> were downloaded from European Nucleotide Archive and provided to GenomeTreader^{##UREF##9##34##} (v.1.7.0) (Supplementary Table ##SUPPL##1##4##). Additionally, the RNA-seq data was subjected to the assembly using Trinity^{##REF##21572440##35##} (v.2.10.0). The ab-initio gene prediction was performed using the transcripts assembled from RNA-seq and known genes of <italic>O. latipes</italic>, <italic>S. lalandi</italic>, <italic>S. dumerili</italic>, <italic>O. niloticus</italic>, and <italic>T. ovatus</italic> by Braker2^{##REF##33575650##36##}. After two rounds of model training, the optimal parameters are determined. Another gene prediction method involved aligning RNA-seq data to the <italic>D. maruadsi</italic> genome to assemble the transcriptome using Hisat2^{##REF##31375807##37##} and StringTie^{##REF##25690850##38##} (v.2.1.4). Then, the open reading frame (ORF) regions were predicted using TransDecoder (v.5.5.0). Ultimately, EvidenceModeler was utilized to create a thorough gene set, which was then further annotated for protein-coding gene structure via PASA^{##UREF##10##39##} (v.2.4.1). For functional annotation of predicted gene, Diamond^{##REF##25402007##40##} (v.2.0.6) was applied to align protein-coding genes to the Swiss-Prot (<ext-link ext-link-type=\"uri\" xlink:href=\"http://www.uniprot.org/\">http://www.uniprot.org/</ext-link>), InterPro(<ext-link ext-link-type=\"uri\" xlink:href=\"https://www.ebi.ac.uk/interpro/\">https://www.ebi.ac.uk/interpro/</ext-link>) and NR protein databases with E-values &lt; 1*10⁻⁵. Additionally, GO and KEGG pathway annotations were performed by InterProScan^{##REF##11590104##41##} (v.4.8) (<ext-link ext-link-type=\"uri\" xlink:href=\"https://www.ebi.ac.uk/interpro/\">https://www.ebi.ac.uk/interpro/</ext-link>) and KEGG Automatic Annotation Server (KAAS^{##REF##17526522##42##}, <ext-link ext-link-type=\"uri\" xlink:href=\"https://www.genome.jp/tools/kaas/\">https://www.genome.jp/tools/kaas/</ext-link>) (Table ##TAB##4##5##).</p>", "<p id=\"Par13\">In this study, a high-quality reference genome of <italic>D. maruadsi</italic> was generated, which could provide a solid foundation for species diversity and population genetic studies in the future. Nowadays, genomics is gradually being applied in every stage of large-scale aquaculture production and domestication. As an important aquatic economic fish, <italic>D. maruadsi</italic> is necessary to identify genetic diversity under phenotypic traits by a high-precision chromosome-scale genome to improve the economic benefits of aquaculture species. In addition, high-quality genome-wide maps are important as essential basic genetic data for industrial and scientific research applications, providing a genetic basis and more accurate genetic evaluation tools for the management and sustainable use of <italic>D. maruadsi</italic> fisheries resources. Finally, as a potential cultured fish, the genome of <italic>D. maruadsi</italic> will help to the breeding program for selecting excellent growth-related traits.</p>" ]

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[ "<p id=\"Par1\"><italic>Decapterus maruadsi</italic> is one of the representative offshore fish in the Western Pacific. Since the last century, it has become a commercially valuable marine fishery species in the Western Pacific region. Despite its high economic value, there is still a lack of high-quality reference genome of <italic>D. maruadsi</italic> in germplasm resource evaluation research. Here we report a chromosome-level reference genome of <italic>D. maruadsi</italic> based on Nanopore sequencing and Hi-C technologies. The whole genome was assembled through 169 contigs with a total length of 723.69 Mb and a contig N50 length of 24.67 Mb. By chromosome scaffolding, 23 chromosomes with a total length of 713.58 Mb were constructed. In addition, a total of 199.49 Mb repetitive elements, 33,515 protein-coding genes, and 6,431 ncRNAs were annotated in the reference genome. This reference genome of <italic>D. maruadsi</italic> will provide a solid theoretical basis not only for the subsequent development of genomic resources of <italic>D. maruadsi</italic> but also for the formulation of policies related to the protection of <italic>D. maruadsi</italic>.</p>", "<title>Subject terms</title>" ]

[ "<title>Data Records</title>", "<p id=\"Par14\">The raw sequencing reads of all libraries have been deposited into NCBI SRA database via the accession number of SRP408505^{##UREF##11##43##}. The assembled genome has been deposited at Genbank under the accession number GCA_030347415.2^{##UREF##12##44##}. Moreover, data of the assembled genome and sequence annotations are available at Figshare^{##UREF##13##45##}.</p>", "<title>Technical Validation</title>", "<title>Genome assembly and annotation completeness evaluation</title>", "<p id=\"Par15\">To ensure the accuracy and integrity of the assembly, we assessed the completeness of the final genome assembly using Benchmarking Universal Single-Copy Orthologues (BUSCO)^{##REF##26059717##46##} with the Actinopterygii_odb10 lineage database. Out of 3,640 single-copy orthologues, approximately 97.8% were completely identified in the <italic>D. maruadsi</italic> genome (Supplementary Table ##SUPPL##1##3##). Besides, the Illumina short reads were aligned to the genome using the BWA MEM algorithm. Subsequently, employing samtools on the generated BAM files, we calculated the sequencing depth across the genome. The non-zero sequencing depth positions were tallied and summed, then compared to the total base positions for the final coverage percentage. This yielded a mapping ratio of 99.76% and a genome coverage of 98.80% (Supplementary Table ##SUPPL##1##2##). Moreover, a total of 33,515 protein-coding genes were successfully obtained by combining ab-initio strategies, homologous searching and transcriptome-assisted approaches. A total of 25,933 genes were successfully functionally annotated in at least one of these databases (Fig. ##FIG##1##2B## &amp; Table ##TAB##4##5##). The high integration efficiency, mapping ratio, recognition rate of single-copy orthologues and gene number collectively suggest that the assembled <italic>D. maruadsi</italic> genome was of superior quality.</p>", "<title>Genome assembly accuracy evaluation</title>", "<p id=\"Par16\">To validate the precise arrangement of the <italic>D. maruadsi</italic> genome, we aligned the assembly to the <italic>O. latipes</italic> genome using minimap2^{##REF##29750242##47##} with a unit of 1 Kbp (Fig. ##FIG##0##1C##). Additionally, we performed the same alignment method with <italic>Trachurus trachurus</italic>, a closely related species in the Carangidae family. The 23 chromosomes identified in the <italic>D. maruadsi</italic> genome showed a significant level of collinearity with the other two species, indicating the high genomic continuity of our assembly (Supplementary Fig. ##SUPPL##0##2## &amp; Supplementary Fig. ##SUPPL##0##3##).</p>", "<p id=\"Par17\">Notably, Chromosome 2 of <italic>D. maruadsi</italic> aligns with both chromosome 2 and chromosome 4 of <italic>O. latipes</italic> and <italic>T. trachurus</italic>. To confirm the accuracy of the chromosome number, we performed a nucmer^{##REF##29373581##48##} alignment of chromosome 2 of <italic>D. maruadsi</italic> with chromosome 2 and chromosome 4 of <italic>O. latipes</italic> and <italic>T. trachurus</italic>. The results revealed that chromosomes 4 and 2 of <italic>O. latipes</italic> aligned to the regions 0.34 M - 31.19 M and 31.82 M - 44.96 M, respectively, on chromosome 2 of <italic>D. maruadsi</italic>. Similarly, chromosomes 4 and 2 of <italic>T. trachurus</italic> aligned to the regions 0.09 M - 31.62 M and 31.66 M - 45.06 M, respectively, on chromosome 2 of <italic>D. maruadsi</italic> (Supplementary Fig. ##SUPPL##0##4B##). These comparative analyses collectively indicate the presence of a distinct alignment gap within the 31 M - 32 M region of the reported chromosome 2 of <italic>D. maruadsi</italic>. This alignment gap suggests the possibility of a structural alteration and connection region between the two chromosomes in this location.</p>", "<p id=\"Par18\">Moreover, based on the Hi-C assisted assembly data, we have identified that this connection region is completely covered by the precisely assembled contig (Supplementary Fig. ##SUPPL##0##4##). Additionally, we selected the genomic range from 31 M to 32 M and utilized the minimap2 tool to align Illumina and Nanopore reads to this region. Subsequently, we implemented a sliding window approach with a window size of 50 kb to calculate the average depth at each genomic position. In this important region, both Illumina and Nanopore data consistently exhibited stable depth profiles, with no significant decrease in depth observed (Supplementary Fig. ##SUPPL##0##5##). These observations further emphasize the integrity and continuity of our assembly results.</p>", "<title>Supplementary information</title>", "<p>\n\n\n</p>" ]

[ "<title>Supplementary information</title>", "<p>The online version contains supplementary material available at 10.1038/s41597-024-02912-1.</p>", "<title>Acknowledgements</title>", "<p>We acknowledge financial support from the Fundamental Research Funds for the Central Universities (No.20720200119).</p>", "<title>Author contributions</title>", "<p>P.X. conceived and supervised the study. Z.X.Z., Y.C.D. and P.X. colledcted the sample. L.Y.C., Z.X.Z. and Z.Y.Z. performed bioinformatics analysis. L.Y.C., Z.X.Z. and J.Y.Y. drafted the manuscript. F.P. helped on manuscript preparation. P.X., T.Z. and Y.L.B. provided review and modification of the manuscript. All authors read and approved the final manuscript.</p>", "<title>Code availability</title>", "<p>Genome annotation:</p>", "<p>(1) RepeatMasker: parameters: -e ncbi -a -nolow -no_is -norna.</p>", "<p>(2) TE-class: parameters: all parameters were set as default.</p>", "<p>(3) Braker2: parameters: all parameters were set as default.</p>", "<p>(4) PASA: --ALIGNERS blat.</p>", "<p>(5) EvidenceModeler: parameters: all parameters were set as default.</p>", "<p>Genome assembly:</p>", "<p>(1) NextDenovo: parameters: all parameters were set as default.</p>", "<p>Gene family identifcation and phylogenetic analysis:</p>", "<p>(1) RAxML: parameters: -f a -m PROTGAMMAAUTO.</p>", "<p>(2) MCMCTREE: parameters: all parameters were set as default. Other analysis modules that were not mentioned parameters were used default parameters. The other custom codes used in this analysis were mentioned in methods sections.</p>", "<title>Competing interests</title>", "<p id=\"Par19\">The authors declare no competing interests.</p>" ]

[ "<fig id=\"Fig1\"><label>Fig. 1</label><caption><p>Characteristics of <italic>Decapterus maruadsi</italic> genome assembly. (<bold>A</bold>) Contact map of chromosomal interactions in the <italic>D. maruadsi</italic> genome using Hi-C data. (<bold>B</bold>) A circos plot of 23 chromosomes in <italic>D. maruadsi</italic> genome, the tracks from outside to inside are: a. Lines represent <italic>D. maruadsi</italic> chromosomes; b. GC content; c. Gene density; d. Repeat element density. (<bold>C</bold>) Circos diagram showing synteny relations between <italic>D. maruadsi</italic> and <italic>O. latipes</italic>. Each coloured line represents a 1 Kb fragment match between two species. We reordered the chromosome numbers of <italic>D. maruadsi</italic> for better illustration.</p></caption></fig>", "<fig id=\"Fig2\"><label>Fig. 2</label><caption><p>Gene and repeat annotations of the <italic>D. maruadsi</italic> genome. (<bold>A</bold>) Distribution of divergence rate for each type of TEs in the <italic>D. maruadsi</italic> genome. (<bold>B</bold>) Venn diagram of the functionally annotated protein-coding genes based on diferent databases.</p></caption></fig>" ]

[ "<table-wrap id=\"Tab1\"><label>Table 1</label><caption><p>Statistics for the sequencing data of the <italic>D. maruadsi</italic> genome.</p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th>Platform</th><th>Insert size (bp)</th><th>Raw data (Gb)</th><th>Clean data (Gb)</th><th>Average Read Length (bp)</th><th>N50 Read Length (bp)</th><th>Coverage (X)</th></tr></thead><tbody><tr><td>Illumina</td><td>350</td><td>47.85</td><td>47.61</td><td>150</td><td>150</td><td>66.06</td></tr><tr><td>Nanopore</td><td>20,000</td><td>86.39</td><td>—</td><td>220,70.2</td><td>26,227</td><td>119.87</td></tr><tr><td>Hi-C</td><td>—</td><td>76.37</td><td>75.77</td><td>150</td><td>150</td><td>105.13</td></tr><tr><td>RNA-seq</td><td>—</td><td>33.65</td><td>32.61</td><td>150</td><td>150</td><td>45.25</td></tr><tr><td>Total</td><td>—</td><td>244.26</td><td>—</td><td>—</td><td>—</td><td>336.31</td></tr></tbody></table></table-wrap>", "<table-wrap id=\"Tab2\"><label>Table 2</label><caption><p>Statistics of the <italic>D. maruadsi</italic> genome assembly.</p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th>Statistic type</th><th>Contig assembly</th><th>Chromosome assembly</th></tr></thead><tbody><tr><td>N50</td><td>24,671,527</td><td>32,348,610</td></tr><tr><td>N90</td><td>2,783,646</td><td>23,005,116</td></tr><tr><td>Contig/Scaffold number(&gt;100 bp)</td><td>169</td><td>139</td></tr><tr><td>Total length</td><td>723,689,783</td><td>724,050,554</td></tr><tr><td>Maximum length</td><td>42,935,365</td><td>45,095,783</td></tr><tr><td>Mean length</td><td>4,282,188</td><td>31,025,152</td></tr></tbody></table></table-wrap>", "<table-wrap id=\"Tab3\"><label>Table 3</label><caption><p>Statistics of 23 chromosomes of <italic>D. maruadsi</italic> genome.</p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th>Chromosome</th><th>Length (bp)</th><th>Number of Scaffolds</th></tr></thead><tbody><tr><td>Chr1</td><td>41,317,494</td><td>12</td></tr><tr><td>Chr2</td><td>45,095,783</td><td>2</td></tr><tr><td>Chr3</td><td>26,889,330</td><td>5</td></tr><tr><td>Chr4</td><td>23,735,607</td><td>4</td></tr><tr><td>Chr5</td><td>21,359,908</td><td>4</td></tr><tr><td>Chr6</td><td>22,930,063</td><td>4</td></tr><tr><td>Chr7</td><td>26,730,606</td><td>8</td></tr><tr><td>Chr8</td><td>28,300,000</td><td>5</td></tr><tr><td>Chr9</td><td>31,512,500</td><td>4</td></tr><tr><td>Chr10</td><td>31,747,162</td><td>8</td></tr><tr><td>Chr11</td><td>33,111,663</td><td>4</td></tr><tr><td>Chr12</td><td>33,226,575</td><td>9</td></tr><tr><td>Chr13</td><td>27,185,605</td><td>8</td></tr><tr><td>Chr14</td><td>23,005,116</td><td>4</td></tr><tr><td>Chr15</td><td>33,635,488</td><td>5</td></tr><tr><td>Chr16</td><td>36,885,046</td><td>5</td></tr><tr><td>Chr17</td><td>34,731,214</td><td>2</td></tr><tr><td>Chr18</td><td>29,852,700</td><td>6</td></tr><tr><td>Chr19</td><td>37,058,507</td><td>5</td></tr><tr><td>Chr20</td><td>27,911,733</td><td>14</td></tr><tr><td>Chr21</td><td>29,349,286</td><td>6</td></tr><tr><td>Chr22</td><td>32,348,910</td><td>5</td></tr><tr><td>Chr23</td><td>35,658,500</td><td>10</td></tr><tr><td>Mean</td><td>31,025,152</td><td>6</td></tr><tr><td>Total</td><td>713,578,496</td><td>139</td></tr></tbody></table></table-wrap>", "<table-wrap id=\"Tab4\"><label>Table 4</label><caption><p>Classification of repetitive sequences and ncRNAs of the <italic>D. maruadsi</italic> genome.</p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th colspan=\"2\">Repeat type</th><th>Denovo + Repbase Length (bp)</th><th>Proportion in Genome (%)</th></tr></thead><tbody><tr><td colspan=\"2\">DNA</td><td>102,573,344</td><td>14.17</td></tr><tr><td colspan=\"2\">LINE</td><td>37,618,924</td><td>5.2</td></tr><tr><td colspan=\"2\">SINE</td><td>2,824,647</td><td>0.39</td></tr><tr><td colspan=\"2\">LTR</td><td>40,988,688</td><td>5.66</td></tr><tr><td colspan=\"2\">Simple Repeat</td><td>865,852</td><td>0.12</td></tr><tr><td colspan=\"2\">Unkown</td><td>398,722</td><td>0.06</td></tr><tr><td colspan=\"2\">Total</td><td>199,488,111</td><td>27.57</td></tr><tr><td colspan=\"2\"><bold>ncRNA type</bold></td><td><bold>Copy</bold></td><td><bold>Proportion in Genome (%)</bold></td></tr><tr><td>miRNA</td><td/><td>1,285</td><td>0.025</td></tr><tr><td>tRNA</td><td/><td>3,820</td><td>0.039</td></tr><tr><td>rRNA</td><td>18 S</td><td>510</td><td>0.014</td></tr><tr><td/><td>28 S</td><td>150</td><td>0.01</td></tr><tr><td/><td>5.8 S</td><td>17</td><td>0</td></tr><tr><td/><td>5 S</td><td>915</td><td>0.015</td></tr><tr><td/><td>Total</td><td>1,592</td><td>0.038</td></tr><tr><td>sRNA</td><td>CD-box</td><td>165</td><td>0.003</td></tr><tr><td/><td>HACA-box</td><td>85</td><td>0.002</td></tr><tr><td/><td>Splicing</td><td>501</td><td>0.01</td></tr><tr><td/><td>Total</td><td>762</td><td>0.015</td></tr></tbody></table></table-wrap>", "<table-wrap id=\"Tab5\"><label>Table 5</label><caption><p>Statistics of gene structure and fuctional annotation of the <italic>D. maruadsi</italic> genome.</p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th>Gene structure Annotation</th><th/></tr></thead><tbody><tr><td>Number of protein-coding gene</td><td>33,515</td></tr><tr><td>Average transcript length (bp)</td><td>8,571.59</td></tr><tr><td>Average exons per gene</td><td>7.82</td></tr><tr><td>Average exon length (bp)</td><td>166.1</td></tr><tr><td>Average CDS length (bp)</td><td>166.1</td></tr><tr><td>Gene fuction Annotation</td><td>Number (Percent)</td></tr><tr><td>Swissport</td><td>21,345(63.69%)</td></tr><tr><td>NR</td><td>25,902(77.28%)</td></tr><tr><td>Interpro</td><td>20,813(62.10%)</td></tr><tr><td>GO</td><td>13,123(39.16%)</td></tr><tr><td>KEGG</td><td>16,081(47.98%)</td></tr><tr><td>Annotated</td><td>25,933(77.38%)</td></tr><tr><td>Unanotated</td><td>7,582(22.62%)</td></tr></tbody></table></table-wrap>" ]

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[ "<supplementary-material content-type=\"local-data\" id=\"MOESM1\"></supplementary-material>", "<supplementary-material content-type=\"local-data\" id=\"MOESM2\"></supplementary-material>" ]

[ "<table-wrap-foot><p>Note: Genome size estimated by genome survey (720.70 Mb) were used for sequencing coverage calculation</p></table-wrap-foot>", "<fn-group><fn><p><bold>Publisher’s note</bold> Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.</p></fn><fn><p>These authors contributed equally: Longyu Chen, Zhixiong Zhou.</p></fn></fn-group>" ]

[ "<graphic xlink:href=\"41597_2024_2912_Fig1_HTML\" id=\"d32e621\"/>", "<graphic xlink:href=\"41597_2024_2912_Fig2_HTML\" id=\"d32e900\"/>" ]

[ "<media xlink:href=\"41597_2024_2912_MOESM1_ESM.pdf\"><caption><p>Supplement Figure</p></caption></media>", "<media xlink:href=\"41597_2024_2912_MOESM2_ESM.xlsx\"><caption><p>Supplement Table</p></caption></media>" ]

{ "acronym": [], "definition": [] }

[ "<title>Introduction</title>", "<p id=\"Par3\">All-carbon quaternary stereocenters are an important synthetic motif found in natural products and bioactive molecules (Fig. ##FIG##0##1##) that are especially difficult to synthesize enantioselectively^{##REF##31515549##1##–##UREF##2##4##}. Successful strategies have recently been developed for cyclic systems (Fig. ##FIG##1##2A##)^{##UREF##0##2##,##REF##25715056##5##–##REF##34014613##10##}; however, constructing quaternary centers in acyclic molecules remains a significant synthetic challenge due to the combination of high levels of steric congestion and greater conformational freedom^{##REF##28910092##11##–##REF##19728707##21##}. Despite its known benefits, organocatalysis^{##UREF##7##22##–##REF##33496291##24##} has been used for asymmetric quaternary center construction only a few classes of acyclic systems^{##UREF##9##25##–##REF##34795297##28##}, though asymmetric 1,4-conjugate addition for tertiary carbon synthesis is well documented^{##REF##18072803##29##–##REF##12762713##33##}. This report details the successful organocatalytic synthesis of valuable acyclic 1,4-dicarbonyl products with vinylated and arylated quaternary centers^{##REF##33432109##34##}.</p>", "<p id=\"Par4\">Organocatalyzed Michael additions to acyclic proquaternary substrates have been reported for nitromethane or cyanide but have otherwise been rare (Fig. ##FIG##1##2B##)^{##UREF##14##35##–##REF##25390507##42##}. A newer class of easily synthesized BINOL–derived enantioselective 1,4–addition organocatalysts have proven to be useful, recyclable, and functional group tolerant in many transformations^{##REF##17402741##43##–##REF##35749308##53##}; however, these reactions have only produced chiral tertiary carbon centers to date. In fact, β,β–disubstituted enones were investigated for quaternary carbon formation but completely lacked reactivity^{##REF##31247785##54##}.</p>" ]

[]

[ "<title>Results and discussion</title>", "<p id=\"Par5\">To overcome steric deactivation via increased electrophilic activation, we looked to enones bearing additional electron-withdrawing groups. In particular, the use of 2–ene–1,4–diones (1) could allow an approach to often difficult-to-access chiral 1,4-diketones with beta quaternary carbons (Fig. ##FIG##1##2C##) and functionality for the total synthesis of natural products^{##UREF##21##55##–##REF##30115803##57##}. However, few precedents related to such a β-vinylation or arylation of ketones to construct quaternary centers exist^{##REF##18331035##58##–##UREF##24##63##}.</p>", "<p id=\"Par6\">To test the diketone activation hypothesis, enediketone <bold>1a</bold> was synthesized as a mixture of cis and trans isomers, which was then purified, and potassium styrenyl trifluoroborate was chosen as an exploratory nucleophile (Fig. ##FIG##2##3##). Unsurprisingly, the <italic>E</italic>–isomer was almost completely unreactive (&lt;2% yield, ~77:23 er), so further experimentation was conducted with pure (<italic>Z</italic>)–enediketone. While we fully expected competing regioselectivity with addition at both alkene positions, substrate <bold>(</bold><bold><italic>Z</italic></bold><bold>)–1a</bold> reacted with superb regioselectivity with the use of typical conditions for BINOL 4–catalyzed conjugate additions^{##UREF##3##9##,##REF##34014613##10##} to give a single product. In fact, <bold>(</bold><bold><italic>Z</italic></bold><bold>)−1a</bold> defied all initial expectations of addition at the less hindered carbon and provided quaternary carbon-bearing 1,4-diketone <bold>3a</bold> in 99% yield and 96:4 er. A control experiment without any catalyst gave <bold>3a</bold> from <bold>(</bold><bold><italic>Z</italic></bold><bold>)–1a</bold> in 69% yield after 18 h, indicating that a significant racemic background reaction was operative. These results suggested that two cis-disposed ketone carbonyls must be present for reactivity and may provide <italic>Z</italic>-dependent cooperative activation. We were then able to demonstrate a one-pot reaction using <bold>(</bold><bold><italic>E</italic></bold><bold>)−1a</bold>, which was first converted to the <italic>Z</italic>-form via photo-isomerization^{##REF##23590409##64##} and then could undergo conjugate addition to provide the quaternary carbon product in good yield (79%) and the same er obtained from pure (<italic>Z</italic>)-enediketone. Such an isomerization may even have been the source of activity seen in Fig. ##FIG##2##3A##. The presence or absence of light had no effect on the reaction using pure <italic>Z</italic>-enediketone. Various BINOL derivatives were tested for increased reactivity and stereoselectivity, and the effects of solvent and temperature were also investigated, but with no improvement (See “Synthesis of BINOL-based catalysts” in the Supplementary Information).</p>", "<p id=\"Par7\">Various enediketones showed productive reactivity with the catalytic conditions identified above. Both electron–donating and electron–withdrawing groups on the ketone’s aromatic ring provided effective reaction (<bold>3a</bold>–<bold>3e</bold>, Fig. ##FIG##3##4##). A heteroaromatic enone substituent was likewise accommodated (see <bold>3f</bold>). However, changing the phenylketone to a methylketone resulted in the formation of product <bold>3</bold><bold>g</bold> with slightly reduced stereoselectivity and competing regioselectivity to form a tertiary stereocenter in 11% yield (see <bold>5g</bold>). Despite both carbonyls being equally Lewis basic, quaternary carbon formation was still favored in a 4:1 ratio. Interestingly, moving the branching vinyl methyl from the alkyl ketone side of the alkene to the phenyl ketone side reversed the regioselectivity so that the major diastereomer of dione <bold>5h</bold> was formed with low enantioselectivity in 63% yield. The minor diastereomer of <bold>5h</bold> was produced along with the quaternary product in 17% yield as an inseparable 2:1 mixture. An initial regioselectivity hypothesis was that the relative locations of the phenyl and aliphatic ketones have a directing effect on the addition, where C–C bond formation is more strongly favored at the β-carbon of an aryl ketone than that of an alkyl ketone^{##REF##17402741##43##–##REF##35749308##53##}. Based on work by Goodman and Pellegrinet^{##REF##18543992##44##}, we believed that the phenyl ketone directed an intramolecular 5-exo-trig 1,4-addition that reinforced the favored quaternary carbon formation.</p>", "<p id=\"Par8\">Replacing the methyl groups of <bold>1a</bold> with ethyl groups gave improved stereoselectivity (<bold>3i</bold>) but slightly lowered the yield, likely due to an increase in steric repulsion. Changing the methyl ketone of <bold>1a</bold> to a phenyl ketone afforded the quaternary center in <bold>3j</bold> with moderate er; however, replacing both methyl groups with phenyls precluded reaction so triphenyl product <bold>3k</bold> was not observed. The series <bold>3a,</bold>\n<bold>3j</bold>, and <bold>3k</bold> shows the negative impact of increasing the size of substituents on yield. It is noteworthy that a cyclic diketone system gave the alpha quaternary center in <bold>3l</bold> in high yield but with little enantioselectivity. Apparently, without rotation of the carbonyl-olefin bond where C–C bond formation occurs reactivity is retained, but the stereoselectivity is almost completely lost.</p>", "<p id=\"Par9\">Various vinyl, alkynyl, and heteroaromatic nucleophiles were also examined. A few substrates gave a lower er, but use of trifluorotoluene and/or an increased loading of the catalyst improved the enantioselectivity (Fig. ##FIG##4##5##). For example, the electron rich styrenyl nucleophiles that afforded <bold>3m</bold> and <bold>3n</bold> originally showed a significant racemic background reaction, but increasing the catalyst loading improved the er to 80:20 and 92:8, respectively. An electron-withdrawing group on the styrene system in <bold>3o</bold> produced a high yield and er without adjustment. Nucleophiles with alkyl chains gave the dienyl adducts <bold>3p,</bold>\n<bold>3q</bold>, and trans alkenyl <bold>3r</bold>–<bold>3x</bold> in high yield and enantioselectivity. Having two vinyl substituents resulted in diminished enantioselectivity (see <bold>3y</bold> and <bold>3z</bold>), but a synthetically useful bromo vinyl borate synthesized <bold>3aa</bold> in moderate yield and improved er. Alternatives to the vinyl nucleophilic system were also tested. Alkynyl reagents provided useful reactivity, but decreased stereoselectivity (see <bold>3ab</bold> and <bold>3ac</bold>). It is worth noting that the isomerization from <bold>(</bold><bold><italic>Z</italic></bold><bold>)−1a</bold> to <bold>(</bold><bold><italic>E</italic></bold><bold>)−1a</bold> occurred competitively during the formation of <bold>3z</bold> and <bold>3ab</bold>, which may have reduced both the yield and stereoselectivity for those reactions. The use of other strong nucleophiles, like furanyl borate, similarly formed quaternary carbons in high yield but with low enantioselectivity due to the competitiveness of the background reaction (see <bold>3ad</bold>)^{##REF##26576776##65##}. On the other hand, a thienyl borate produced <bold>3ae</bold> in good yield and high er.</p>", "<p id=\"Par10\">To explain (A) why only the <italic>Z</italic>-isomer was reactive and (B) why quaternary regioselectivity was favored over tertiary carbon formation, we pursued a computational investigation using substrates <bold>(</bold><bold><italic>Z</italic></bold><bold>)−1a</bold> and <bold>(</bold><bold><italic>E</italic></bold><bold>)−1a</bold> with styrenyl boronate. Based on prior mechanistic investigations relevant to tertiary carbon formation via similar catalysis^{##UREF##19##47##}, it is likely that the potassium trifluoroborate salt dissociates fluoride and condenses with the BINOL <bold>4</bold> to form an activated chiral boronate ester that then coordinates to the enone carbonyl. To simplify the calculations, BINOL <bold>4</bold> was modelled as 3,3´-difluorobisphenol. We initially modeled the formation of the Lewis acid/Lewis base complexed boronate-ketone adduct <bold>6</bold>, and our calculations supported Goodman’s finding^{##REF##18543992##44##} that this complex formed as a discrete mechanistic intermediate prior to the transition state (Fig. ##FIG##5##6##). Note that this stable intermediate was taken as the zero-point reference for all other calculated geometries. Conjugate addition transition states derived from ketone-coordinated boronates with both endo and exo modes of addition^{##UREF##25##66##,##UREF##26##67##} were next examined (Figs. ##FIG##5##6## and ##FIG##6##7##). Where Goodman’s work showed 6-endo cyclization (see <bold>8c</bold> and <bold>8e</bold>), we found that 5-exo modes^{##UREF##27##68##} were lower in energy for both quaternary and tertiary carbon formation (compare <bold>8a</bold> to <bold>8c</bold> and <bold>8d</bold> to <bold>8e</bold>). This new mode of reactivity is enabled by the additional ketone. Close examination of the exo transition states revealed a fascinating stabilizing effect; the ketone distal to the Lewis acid coordination not only enabled the 5-exo addition but also participated in an n→π* donation to the bound carbonyl (<bold>8a</bold>)^{##REF##30774885##69##}. This ouroboros-like activation^{##UREF##28##70##,##UREF##29##71##} is evidenced by the short C = O→C = O bond (1.53 Å in <bold>8a</bold>) and the tetrahedral geometry of the carbon of the bound C = O (C16). Such interactions have been described for static protein structure^{##REF##28735540##72##} and utilized for the synthesis of Lewis acid/base heteroaromatics^{##REF##30774885##69##}, but to our knowledge has not been proposed as a stabilizing factor in reaction catalysis^{##REF##31607125##73##}. The LUMO of the C16 carbonyl thus acts as a Lewis acid activating the planar enone for 5-exo-trig conjugate addition, lowering the LUMO energy, and the electron donation of the planar ketone to the C16 carbonyl simultaneously increases the electron density in the nucleophilic system (C28), raising the HOMO energy. Additionally, this stabilizing interaction was not accessible in the 6-endo geometries (see longer O to C distances in <bold>8c</bold> and <bold>8e</bold>), and we believe this to be a reason for their relatively higher energy pathways. In investigating the generation of the ouroboros stabilization, we could identify that the formation of iso-furan <bold>7</bold> occurred prior to C–C bond formation. Some pathways, such as that shown in Fig. ##FIG##5##6##, have <bold>7</bold> formed as a meta stable intermediate as a local minimum. In others, it is a shoulder or part of a continuous slope to the transition state. These calculations also showed that the (<italic>R</italic>)-biaryl introduces torsion in the coordinated system that favors <bold>8a</bold> over <bold>8b</bold>, giving the major observed enantiomer. In considering why reactivity is unfavorable for (<italic>E</italic>)-enediketones, several potential transitions states derived from methyl or phenyl ketone-coordinated isomers of <bold>6</bold> were examined, but only the transition states <bold>8f</bold> and <bold>8g</bold> converged reliably. Notably, ouroboros activation was not observable for any reasonable geometries corresponding to transition states derived from (<italic>E</italic>)-substrates, the 6-endo-trig transition state was thus lower in energy, and the resulting higher overall barrier explains the lack of reactivity of β-disubstituted enones in all previous studies^{##UREF##3##9##}. Given other recent efforts that also observe such a dependence on (<italic>Z</italic>)-enone geometry^{##UREF##9##25##–##REF##34795297##28##}, ouroboros stabilization may be operative in many catalytic reactions. The poor er arising from the lack of rotation about the alkene-ketone bond in forming <bold>3l</bold> also aligns with this hypothesis, as the carbonyl could not fully rotate out of plane to provide 5-exo reactivity, forcing it through a 6-endo transition state like <bold>8e</bold>, which also lacks ouroboros activation and therefore has reduced stereocontrol. The decreased enantioselectivity and altered regioselectivity seen for <bold>5g</bold> and <bold>5h</bold> could also be due to competitive 6-endo reactivity for those substrates rather than due to Lewis basicity.</p>", "<p id=\"Par11\">An examination of the calculated LUMOs in intermediate <bold>6</bold> and its phenylketone-coordinated isomer showed localization on the planar enone (HOMO/LUMO illustrated in the Source Data file), but the carbons undergoing nucleophilic attack (C13 or C14) bear different proportions of the LUMO^{##UREF##30##74##}. For both isomers, C14 has significant LUMO character, leading to better HOMO/LUMO overlap, but less of the LUMO is located on C13. The relative localization of the LUMO in these structures and the relative energies of the subsequent transition states correspond to previously characterized experimental rate dependencies on the stabilization of developing cationic charge at the β-carbon of the enone in this class of conjugate additions^{##UREF##31##75##,##UREF##32##76##}. An additional insight into the regioselectivity was obtained by examining two possible n→π* interactions in the Z-enediketone. That arising from the Ph-ketone donating into a twisted Me-ketone resulted in a 1.7 kcal/mol more stable conformation than that arising from the Me-ketone donating into a twisted Ph-ketone. The stereoelectronic and steric interactions in these conformations are also likely to be present in <bold>8a</bold> and <bold>8d</bold>, and the relative stability of their geometries contributing to the difference in regioisomeric transition state energies.</p>", "<p id=\"Par12\">To demonstrate the utility of these quaternary 1,4-diketone products^{##REF##18331035##58##–##UREF##24##63##,##REF##17914838##77##}, examples were transformed into key synthetic building blocks (Fig. ##FIG##7##8##). Chiral cyclopentenones (see 8), which exist widely in bioactive compounds^{##UREF##33##78##–##UREF##35##80##}, could be formed in high yield and er via aldol condensation. The absolute stereochemistry of <bold>9</bold> was confirmed by X-ray crystallography^{##UREF##36##81##}. Chemoselective hydrogenation reduced the benzoyl carbonyl and the alkene of <bold>3a</bold> or <bold>3q</bold> to form the aliphatic quaternary carbon centers in <bold>10a</bold> and <bold>10q</bold>, respectively, with the latter containing an otherwise difficult to access alkylated quaternary center with high er. The oxidative cleavage of the styrenyl olefin gave ketoaldehyde <bold>11</bold>, which would be useful in recently reported pyrrolidine syntheses^{##REF##17914838##77##,##UREF##37##82##}. Non-planar heterocycle dihydropyridazine <bold>12</bold>, a bioactive pharmacophore^{##UREF##38##83##,##REF##19827778##84##}, could be generated in good yield. Using the bromo substrate <bold>3e</bold> to incorporate an intramolecular Heck coupling reaction gave the quinone-like derivative <bold>12</bold>^{##REF##17935323##85##}.</p>", "<p id=\"Par13\">In conclusion, we successfully synthesized challenging quaternary centers enantioselectively from (<italic>Z</italic>)-1,4-enediketones via organocatalyzed conjugate addition. Control experiments showed that the cis-relationship of the ketones was vital to reactivity, and keto-ene bond rotation at the location of C–C bond formation was important for enantioselectivity. DFT calculations showed that the additional ketone provided 5-exo-trig reactivity and a stabilizing interaction through an n→π*cyclic ouroboros activation. The regioselectivity for quaternary carbon formation appeared to be based primarily on the greater HOMO/LUMO overlap in the formation of the quaternary carbon relative to the tertiary carbon. A broad substrate scope of chiral α-quaternary 1,4-diketones were synthesized. Further transformations to quaternary carbon-containing enantio-enriched cyclopentenones, linear hydrocarbons, dihydropyridazines, and quinone methides were demonstrated in good yield and er. These building blocks will enable synthetic endeavors in many areas.</p>" ]

[ "<title>Results and discussion</title>", "<p id=\"Par5\">To overcome steric deactivation via increased electrophilic activation, we looked to enones bearing additional electron-withdrawing groups. In particular, the use of 2–ene–1,4–diones (1) could allow an approach to often difficult-to-access chiral 1,4-diketones with beta quaternary carbons (Fig. ##FIG##1##2C##) and functionality for the total synthesis of natural products^{##UREF##21##55##–##REF##30115803##57##}. However, few precedents related to such a β-vinylation or arylation of ketones to construct quaternary centers exist^{##REF##18331035##58##–##UREF##24##63##}.</p>", "<p id=\"Par6\">To test the diketone activation hypothesis, enediketone <bold>1a</bold> was synthesized as a mixture of cis and trans isomers, which was then purified, and potassium styrenyl trifluoroborate was chosen as an exploratory nucleophile (Fig. ##FIG##2##3##). Unsurprisingly, the <italic>E</italic>–isomer was almost completely unreactive (&lt;2% yield, ~77:23 er), so further experimentation was conducted with pure (<italic>Z</italic>)–enediketone. While we fully expected competing regioselectivity with addition at both alkene positions, substrate <bold>(</bold><bold><italic>Z</italic></bold><bold>)–1a</bold> reacted with superb regioselectivity with the use of typical conditions for BINOL 4–catalyzed conjugate additions^{##UREF##3##9##,##REF##34014613##10##} to give a single product. In fact, <bold>(</bold><bold><italic>Z</italic></bold><bold>)−1a</bold> defied all initial expectations of addition at the less hindered carbon and provided quaternary carbon-bearing 1,4-diketone <bold>3a</bold> in 99% yield and 96:4 er. A control experiment without any catalyst gave <bold>3a</bold> from <bold>(</bold><bold><italic>Z</italic></bold><bold>)–1a</bold> in 69% yield after 18 h, indicating that a significant racemic background reaction was operative. These results suggested that two cis-disposed ketone carbonyls must be present for reactivity and may provide <italic>Z</italic>-dependent cooperative activation. We were then able to demonstrate a one-pot reaction using <bold>(</bold><bold><italic>E</italic></bold><bold>)−1a</bold>, which was first converted to the <italic>Z</italic>-form via photo-isomerization^{##REF##23590409##64##} and then could undergo conjugate addition to provide the quaternary carbon product in good yield (79%) and the same er obtained from pure (<italic>Z</italic>)-enediketone. Such an isomerization may even have been the source of activity seen in Fig. ##FIG##2##3A##. The presence or absence of light had no effect on the reaction using pure <italic>Z</italic>-enediketone. Various BINOL derivatives were tested for increased reactivity and stereoselectivity, and the effects of solvent and temperature were also investigated, but with no improvement (See “Synthesis of BINOL-based catalysts” in the Supplementary Information).</p>", "<p id=\"Par7\">Various enediketones showed productive reactivity with the catalytic conditions identified above. Both electron–donating and electron–withdrawing groups on the ketone’s aromatic ring provided effective reaction (<bold>3a</bold>–<bold>3e</bold>, Fig. ##FIG##3##4##). A heteroaromatic enone substituent was likewise accommodated (see <bold>3f</bold>). However, changing the phenylketone to a methylketone resulted in the formation of product <bold>3</bold><bold>g</bold> with slightly reduced stereoselectivity and competing regioselectivity to form a tertiary stereocenter in 11% yield (see <bold>5g</bold>). Despite both carbonyls being equally Lewis basic, quaternary carbon formation was still favored in a 4:1 ratio. Interestingly, moving the branching vinyl methyl from the alkyl ketone side of the alkene to the phenyl ketone side reversed the regioselectivity so that the major diastereomer of dione <bold>5h</bold> was formed with low enantioselectivity in 63% yield. The minor diastereomer of <bold>5h</bold> was produced along with the quaternary product in 17% yield as an inseparable 2:1 mixture. An initial regioselectivity hypothesis was that the relative locations of the phenyl and aliphatic ketones have a directing effect on the addition, where C–C bond formation is more strongly favored at the β-carbon of an aryl ketone than that of an alkyl ketone^{##REF##17402741##43##–##REF##35749308##53##}. Based on work by Goodman and Pellegrinet^{##REF##18543992##44##}, we believed that the phenyl ketone directed an intramolecular 5-exo-trig 1,4-addition that reinforced the favored quaternary carbon formation.</p>", "<p id=\"Par8\">Replacing the methyl groups of <bold>1a</bold> with ethyl groups gave improved stereoselectivity (<bold>3i</bold>) but slightly lowered the yield, likely due to an increase in steric repulsion. Changing the methyl ketone of <bold>1a</bold> to a phenyl ketone afforded the quaternary center in <bold>3j</bold> with moderate er; however, replacing both methyl groups with phenyls precluded reaction so triphenyl product <bold>3k</bold> was not observed. The series <bold>3a,</bold>\n<bold>3j</bold>, and <bold>3k</bold> shows the negative impact of increasing the size of substituents on yield. It is noteworthy that a cyclic diketone system gave the alpha quaternary center in <bold>3l</bold> in high yield but with little enantioselectivity. Apparently, without rotation of the carbonyl-olefin bond where C–C bond formation occurs reactivity is retained, but the stereoselectivity is almost completely lost.</p>", "<p id=\"Par9\">Various vinyl, alkynyl, and heteroaromatic nucleophiles were also examined. A few substrates gave a lower er, but use of trifluorotoluene and/or an increased loading of the catalyst improved the enantioselectivity (Fig. ##FIG##4##5##). For example, the electron rich styrenyl nucleophiles that afforded <bold>3m</bold> and <bold>3n</bold> originally showed a significant racemic background reaction, but increasing the catalyst loading improved the er to 80:20 and 92:8, respectively. An electron-withdrawing group on the styrene system in <bold>3o</bold> produced a high yield and er without adjustment. Nucleophiles with alkyl chains gave the dienyl adducts <bold>3p,</bold>\n<bold>3q</bold>, and trans alkenyl <bold>3r</bold>–<bold>3x</bold> in high yield and enantioselectivity. Having two vinyl substituents resulted in diminished enantioselectivity (see <bold>3y</bold> and <bold>3z</bold>), but a synthetically useful bromo vinyl borate synthesized <bold>3aa</bold> in moderate yield and improved er. Alternatives to the vinyl nucleophilic system were also tested. Alkynyl reagents provided useful reactivity, but decreased stereoselectivity (see <bold>3ab</bold> and <bold>3ac</bold>). It is worth noting that the isomerization from <bold>(</bold><bold><italic>Z</italic></bold><bold>)−1a</bold> to <bold>(</bold><bold><italic>E</italic></bold><bold>)−1a</bold> occurred competitively during the formation of <bold>3z</bold> and <bold>3ab</bold>, which may have reduced both the yield and stereoselectivity for those reactions. The use of other strong nucleophiles, like furanyl borate, similarly formed quaternary carbons in high yield but with low enantioselectivity due to the competitiveness of the background reaction (see <bold>3ad</bold>)^{##REF##26576776##65##}. On the other hand, a thienyl borate produced <bold>3ae</bold> in good yield and high er.</p>", "<p id=\"Par10\">To explain (A) why only the <italic>Z</italic>-isomer was reactive and (B) why quaternary regioselectivity was favored over tertiary carbon formation, we pursued a computational investigation using substrates <bold>(</bold><bold><italic>Z</italic></bold><bold>)−1a</bold> and <bold>(</bold><bold><italic>E</italic></bold><bold>)−1a</bold> with styrenyl boronate. Based on prior mechanistic investigations relevant to tertiary carbon formation via similar catalysis^{##UREF##19##47##}, it is likely that the potassium trifluoroborate salt dissociates fluoride and condenses with the BINOL <bold>4</bold> to form an activated chiral boronate ester that then coordinates to the enone carbonyl. To simplify the calculations, BINOL <bold>4</bold> was modelled as 3,3´-difluorobisphenol. We initially modeled the formation of the Lewis acid/Lewis base complexed boronate-ketone adduct <bold>6</bold>, and our calculations supported Goodman’s finding^{##REF##18543992##44##} that this complex formed as a discrete mechanistic intermediate prior to the transition state (Fig. ##FIG##5##6##). Note that this stable intermediate was taken as the zero-point reference for all other calculated geometries. Conjugate addition transition states derived from ketone-coordinated boronates with both endo and exo modes of addition^{##UREF##25##66##,##UREF##26##67##} were next examined (Figs. ##FIG##5##6## and ##FIG##6##7##). Where Goodman’s work showed 6-endo cyclization (see <bold>8c</bold> and <bold>8e</bold>), we found that 5-exo modes^{##UREF##27##68##} were lower in energy for both quaternary and tertiary carbon formation (compare <bold>8a</bold> to <bold>8c</bold> and <bold>8d</bold> to <bold>8e</bold>). This new mode of reactivity is enabled by the additional ketone. Close examination of the exo transition states revealed a fascinating stabilizing effect; the ketone distal to the Lewis acid coordination not only enabled the 5-exo addition but also participated in an n→π* donation to the bound carbonyl (<bold>8a</bold>)^{##REF##30774885##69##}. This ouroboros-like activation^{##UREF##28##70##,##UREF##29##71##} is evidenced by the short C = O→C = O bond (1.53 Å in <bold>8a</bold>) and the tetrahedral geometry of the carbon of the bound C = O (C16). Such interactions have been described for static protein structure^{##REF##28735540##72##} and utilized for the synthesis of Lewis acid/base heteroaromatics^{##REF##30774885##69##}, but to our knowledge has not been proposed as a stabilizing factor in reaction catalysis^{##REF##31607125##73##}. The LUMO of the C16 carbonyl thus acts as a Lewis acid activating the planar enone for 5-exo-trig conjugate addition, lowering the LUMO energy, and the electron donation of the planar ketone to the C16 carbonyl simultaneously increases the electron density in the nucleophilic system (C28), raising the HOMO energy. Additionally, this stabilizing interaction was not accessible in the 6-endo geometries (see longer O to C distances in <bold>8c</bold> and <bold>8e</bold>), and we believe this to be a reason for their relatively higher energy pathways. In investigating the generation of the ouroboros stabilization, we could identify that the formation of iso-furan <bold>7</bold> occurred prior to C–C bond formation. Some pathways, such as that shown in Fig. ##FIG##5##6##, have <bold>7</bold> formed as a meta stable intermediate as a local minimum. In others, it is a shoulder or part of a continuous slope to the transition state. These calculations also showed that the (<italic>R</italic>)-biaryl introduces torsion in the coordinated system that favors <bold>8a</bold> over <bold>8b</bold>, giving the major observed enantiomer. In considering why reactivity is unfavorable for (<italic>E</italic>)-enediketones, several potential transitions states derived from methyl or phenyl ketone-coordinated isomers of <bold>6</bold> were examined, but only the transition states <bold>8f</bold> and <bold>8g</bold> converged reliably. Notably, ouroboros activation was not observable for any reasonable geometries corresponding to transition states derived from (<italic>E</italic>)-substrates, the 6-endo-trig transition state was thus lower in energy, and the resulting higher overall barrier explains the lack of reactivity of β-disubstituted enones in all previous studies^{##UREF##3##9##}. Given other recent efforts that also observe such a dependence on (<italic>Z</italic>)-enone geometry^{##UREF##9##25##–##REF##34795297##28##}, ouroboros stabilization may be operative in many catalytic reactions. The poor er arising from the lack of rotation about the alkene-ketone bond in forming <bold>3l</bold> also aligns with this hypothesis, as the carbonyl could not fully rotate out of plane to provide 5-exo reactivity, forcing it through a 6-endo transition state like <bold>8e</bold>, which also lacks ouroboros activation and therefore has reduced stereocontrol. The decreased enantioselectivity and altered regioselectivity seen for <bold>5g</bold> and <bold>5h</bold> could also be due to competitive 6-endo reactivity for those substrates rather than due to Lewis basicity.</p>", "<p id=\"Par11\">An examination of the calculated LUMOs in intermediate <bold>6</bold> and its phenylketone-coordinated isomer showed localization on the planar enone (HOMO/LUMO illustrated in the Source Data file), but the carbons undergoing nucleophilic attack (C13 or C14) bear different proportions of the LUMO^{##UREF##30##74##}. For both isomers, C14 has significant LUMO character, leading to better HOMO/LUMO overlap, but less of the LUMO is located on C13. The relative localization of the LUMO in these structures and the relative energies of the subsequent transition states correspond to previously characterized experimental rate dependencies on the stabilization of developing cationic charge at the β-carbon of the enone in this class of conjugate additions^{##UREF##31##75##,##UREF##32##76##}. An additional insight into the regioselectivity was obtained by examining two possible n→π* interactions in the Z-enediketone. That arising from the Ph-ketone donating into a twisted Me-ketone resulted in a 1.7 kcal/mol more stable conformation than that arising from the Me-ketone donating into a twisted Ph-ketone. The stereoelectronic and steric interactions in these conformations are also likely to be present in <bold>8a</bold> and <bold>8d</bold>, and the relative stability of their geometries contributing to the difference in regioisomeric transition state energies.</p>", "<p id=\"Par12\">To demonstrate the utility of these quaternary 1,4-diketone products^{##REF##18331035##58##–##UREF##24##63##,##REF##17914838##77##}, examples were transformed into key synthetic building blocks (Fig. ##FIG##7##8##). Chiral cyclopentenones (see 8), which exist widely in bioactive compounds^{##UREF##33##78##–##UREF##35##80##}, could be formed in high yield and er via aldol condensation. The absolute stereochemistry of <bold>9</bold> was confirmed by X-ray crystallography^{##UREF##36##81##}. Chemoselective hydrogenation reduced the benzoyl carbonyl and the alkene of <bold>3a</bold> or <bold>3q</bold> to form the aliphatic quaternary carbon centers in <bold>10a</bold> and <bold>10q</bold>, respectively, with the latter containing an otherwise difficult to access alkylated quaternary center with high er. The oxidative cleavage of the styrenyl olefin gave ketoaldehyde <bold>11</bold>, which would be useful in recently reported pyrrolidine syntheses^{##REF##17914838##77##,##UREF##37##82##}. Non-planar heterocycle dihydropyridazine <bold>12</bold>, a bioactive pharmacophore^{##UREF##38##83##,##REF##19827778##84##}, could be generated in good yield. Using the bromo substrate <bold>3e</bold> to incorporate an intramolecular Heck coupling reaction gave the quinone-like derivative <bold>12</bold>^{##REF##17935323##85##}.</p>", "<p id=\"Par13\">In conclusion, we successfully synthesized challenging quaternary centers enantioselectively from (<italic>Z</italic>)-1,4-enediketones via organocatalyzed conjugate addition. Control experiments showed that the cis-relationship of the ketones was vital to reactivity, and keto-ene bond rotation at the location of C–C bond formation was important for enantioselectivity. DFT calculations showed that the additional ketone provided 5-exo-trig reactivity and a stabilizing interaction through an n→π*cyclic ouroboros activation. The regioselectivity for quaternary carbon formation appeared to be based primarily on the greater HOMO/LUMO overlap in the formation of the quaternary carbon relative to the tertiary carbon. A broad substrate scope of chiral α-quaternary 1,4-diketones were synthesized. Further transformations to quaternary carbon-containing enantio-enriched cyclopentenones, linear hydrocarbons, dihydropyridazines, and quinone methides were demonstrated in good yield and er. These building blocks will enable synthetic endeavors in many areas.</p>" ]

[]

[ "<p id=\"Par1\">The chemical synthesis of molecules with closely packed atoms having their bond coordination saturated is a challenge to synthetic chemists, especially when three-dimensional control is required. The organocatalyzed asymmetric synthesis of acyclic alkenylated, alkynylated and heteroarylated quaternary carbon stereocenters via 1,4-conjugate addition is here catalyzed by 3,3´-bisperfluorotoluyl-BINOL. The highly useful products (31 examples) are produced in up to 99% yield and 97:3 er using enediketone substrates and potassium trifluoroorganoborate nucleophiles. In addition, mechanistic experiments show that the (<italic>Z</italic>)–isomer is the reactive form, ketone rotation at the site of bond formation is needed for enantioselectivity, and quaternary carbon formation is favored over tertiary. Density functional theory-based calculations show that reactivity and selectivity depend on a key n→π* donation by the unbound ketone’s oxygen lone pair to the boronate-coordinated ketone in a 5-exo-trig cyclic ouroboros transition state. Transformations of the conjugate addition products to key quaternary carbon-bearing synthetic building blocks proceed in good yield.</p>", "<p id=\"Par2\">All-carbon quaternary stereocenters are an important synthetic motif but are especially difficult to synthesize enantioselectively. Here, the authors demonstrate the organocatalytic regio- and enantioselective synthesis of valuable acyclic 1,4-dicarbonyl products with vinylated and arylated quaternary centers.</p>", "<title>Subject terms</title>" ]

[ "<title>Supplementary information</title>", "<p>\n\n\n</p>", "<title>Source data</title>", "<p>\n\n</p>" ]

[ "<title>Supplementary information</title>", "<p>The online version contains supplementary material available at 10.1038/s41467-024-44744-y.</p>", "<title>Acknowledgements</title>", "<p>The authors are grateful for generous financial support from the NSF (grant CHE-2102282, JAM) and the Welch Foundation (grant E−1744, JAM). Professor Judy Wu at the University of Houston is thanked for aid in access to computational facilities. The computational work was completed in part with resources provided by the Research Computing Data Core at the University of Houston. We are grateful to Dr. Sasha Sundstrom and AbbVie North Chicago for aiding in the acquisition of specific rotation data.</p>", "<title>Author contributions</title>", "<p>P.-K.P.: Conceptualization, experimental methodology development and investigation, validation of structures, writing of first draft. A.I.: computational model development and investigation, validation of transitiona states and intermediates, and electronic data curation. J.A.M.: conceptualization of project, writing and editing, project supervision, and acquisition of funding.</p>", "<title>Peer review</title>", "<title>Peer review information</title>", "<p id=\"Par14\"><italic>Nature Communications</italic> thanks Giovanni Ghigo, Silvina Pellegrinet and the other, anonymous, reviewer for their contribution to the peer review of this work. A peer review file is available.</p>", "<title>Data availability</title>", "<p>The experimental data generated in this study and computational procedures with optimized structures are provided in the Supplementary Information file. The molecular coordinate data generated in this study are provided in the Source Data file. All primary data files, such as.fid files for NMR spectra or coordinate files for molecular structures, are available from the corresponding author for free upon request. The X-ray crystallographic coordinates for structures reported in this study have been deposited at the Cambridge Crystallographic Data Centre (CCDC), under deposition number 2121715 (<bold>9</bold>). These data can be obtained free of charge from The Cambridge Crystallographic Data Centre via <ext-link ext-link-type=\"uri\" xlink:href=\"http://www.ccdc.cam.ac.uk/data_request/cif\">www.ccdc.cam.ac.uk/data_request/cif</ext-link>. <xref ref-type=\"sec\" rid=\"Sec4\">Source data</xref> are provided with this paper.</p>", "<title>Competing interests</title>", "<p id=\"Par15\">The authors declare no competing interests.</p>" ]

[ "<fig id=\"Fig1\"><label>Fig. 1</label><caption><title>Representative natural products bearing alkenylated quaternary carbons.</title><p>Inset shows how quaternary carbon formation applies to a synthesis^{##UREF##39##86##}.</p></caption></fig>", "<fig id=\"Fig2\"><label>Fig. 2</label><caption><title>Enantioselective synthesis of quaternary carbons.</title><p><bold>A</bold> Organometallic catalysis for cyclic quaternary carbon synthesis. <bold>B</bold> Organocatalytic Michael addition of nitromethane to chalcones. <bold>C</bold> The synthesis of acyclic quaternary carbons with alkenyl, alkynyl, and aryl substituents (orange “R´” substituent) via an ouroboros transition state discussed herein.</p></caption></fig>", "<fig id=\"Fig3\"><label>Fig. 3</label><caption><title>Preliminary results and control experiments.</title><p><bold>A</bold> Nearly no reactivity was observed with the <italic>trans</italic> diketone. <bold>B</bold> The <italic>cis</italic> substrate afforded catalytic activity that produced the acyclic quaternary carbon product in high yield and with high stereoselectivity despite a significant background reaction also being operative. <bold>C</bold> Pure <italic>trans</italic> diketone or a mixture of <italic>cis</italic> and <italic>trans</italic> could be converted to <italic>cis</italic>-enriched substrate that then reacted in a similar manner to pure <italic>cis</italic> diketone. 4Å-MS 4Å molecular sieves, PhMe toluene, er enantiomeric ratio, CFL Compact Fluorescent Lightbulb.</p></caption></fig>", "<fig id=\"Fig4\"><label>Fig. 4</label><caption><title>Scope of products from various enediketones.</title><p>Reaction yields are of purified isolated products. Enantiomeric ratios were determined by HPLC with chiral stationary phase. For <bold>5g</bold> and <bold>5h</bold> the stereochemistry was not determined. For <bold>3j</bold> the use of both 20 mol % (30 h) and 40 mol % (18 h) of <bold>4</bold> is illustrated.</p></caption></fig>", "<fig id=\"Fig5\"><label>Fig. 5</label><caption><title>Substrate scope of products from various nucleophiles.</title><p>Reaction yields are of purified isolated products, with the average of at least 2 trials presented. Enantiomeric ratios were determined by HPLC with chiral stationary phase. For <bold>3m–n</bold> and <bold>3s–3x</bold>, the use of 40 mol % of <bold>4</bold> is presented. For <bold>3s–x</bold> and <bold>3ae</bold>, the solvent was PhCF₃. For <bold>3aa</bold> the use 30 mol % of <bold>4</bold> allowed reduction of organoborate nucleophile to 1.5 equiv.</p></caption></fig>", "<fig id=\"Fig6\"><label>Fig. 6</label><caption><title>Proposed mechanistic intermediates for lowest energy reaction pathway.</title><p>The BINOL catalyst was modelled as 3,3´-difluorobisphenol. The pre-transition state Lewis acid/Lewis base complex was defined as 0 kcal/mol and other optimized structures are reported relative that energy.</p></caption></fig>", "<fig id=\"Fig7\"><label>Fig. 7</label><caption><title>Alternative Transition States with and without ouroboros activation.</title><p>Energies are defined relative to <bold>B(Me)-Z-1a (6)</bold> in Fig. ##FIG##5##6##. <bold>A</bold> Transition state structures derived from <bold>(</bold><bold><italic>Z</italic></bold><bold>)−1a</bold>. <bold>B</bold> Transition state structures derived from <bold>(</bold><bold><italic>E</italic></bold><bold>)−1a</bold>. (Me) and (Ph) define whether the methyl ketone or phenyl ketone are bound by the Lewis acidic boron, respectively. 5-exo and 6-endo define the geometry of C–C bond formation according to Baldwin’s rules. <bold>8b</bold> and <bold>8d</bold> show significant ouroboros stabilization.</p></caption></fig>", "<fig id=\"Fig8\"><label>Fig. 8</label><caption><title>Utility of quaternary diketone products.</title><p>The chemical synthesis of several important molecular motifs is illustrated. Examples include α-quaternary cyclopentenes (<bold>9</bold>), quaternary alkanes that are achiral (<bold>10a</bold>) or chiral and enantioenriched (<bold>10q</bold>), tricarbonyls (<bold>11</bold>), dihydropyridazines (<bold>12</bold>), and methide-quinones (<bold>13</bold>).</p></caption></fig>" ]

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[ "<media xlink:href=\"41467_2024_44744_MOESM1_ESM.pdf\"><caption><p>Supplementary Information</p></caption></media>", "<media xlink:href=\"41467_2024_44744_MOESM2_ESM.pdf\"><caption><p>Peer Review File</p></caption></media>", "<media xlink:href=\"41467_2024_44744_MOESM3_ESM.xlsx\"><caption><p>Source Data</p></caption></media>" ]

{ "acronym": [], "definition": [] }

[ "<title>Introduction</title>", "<p id=\"Par2\">The axillary nerve is a branch of the posterior cord of the brachial plexus and contains nerve components from the fifth and sixth cervical spinal segments^{##UREF##0##1##}. It passes through the posterior wall of the axilla via the quadrangular space with the posterior circumflex humeral vessels^{##UREF##1##2##}. The axillary nerve can be injured by blunt trauma, traction injury, penetrating trauma, and nerve compression in the quadrangular space^{##REF##11575912##3##}. The most common cause of axillary nerve injury is trauma during orthopedic surgery such as shoulder arthroscopy, thermal shrinkage of the shoulder capsule, and plate fixation on the proximal humerus^{##REF##18766295##4##}.</p>", "<p id=\"Par3\">Axillary nerve injury can result in partial or total inactivation of the deltoid and teres minor muscles^{##REF##18766295##4##,##REF##28671874##5##}. Because damage to the axillary nerve can result in shoulder instability or dysfunction, it is important to protect the axillary nerve and its branches and to restore the damaged nerves to maintain the functions of shoulder.</p>", "<p id=\"Par4\">Nerve transfer is one of the therapeutic options for functional restoration of denervated muscles^{##REF##28671874##5##,##REF##18751774##6##}. To restore the deltoid muscle, nerve transfer using triceps motor branches of the radial nerve to the axillary nerve are widely used^{##REF##17695392##7##–##REF##12877852##10##}. The most appropriate choice of branch of the radial nerve to use as the donor and which branch of the axillary nerve to use as the recipient can improve the results of nerve transfer.</p>", "<p id=\"Par5\">The purpose of this study was to understand the anatomy of axillary and radial nerves to prevent nerve damage and restore muscle action. Additionally, this study aimed to identify potential donor branches of the radial nerve that are good matches for potential recipient branches of the axillary nerve in nerve transfer.</p>" ]

[ "<title>Materials and methods</title>", "<p id=\"Par23\">Fifty upper extremities, 24 right and 26 left, from 29 formalin-embalmed cadavers (mean age, 75.8 years; range 52–95 years) with no pathologies or surgical history in the upper extremity were used in this study. All study procedures approved by the Surgical Anatomy Education Centre, Yonsei University College of Medicine (approval number: YSAEC: 23-006). The participants have provided informed consent to donate their bodies for research purposes. The authors state that every effort was made to follow all local and international ethical guidelines and law that pertain to the use of human cadaveric donors in anatomical research^{##REF##35218594##27##}.</p>", "<p id=\"Par24\">After removal of the skin and subcutaneous tissue of the shoulder, the deltoid and teres minor muscles were identified. The deltoid was detached from its origin on the clavicle, acromion, and scapular spine and divided into three parts according to origin. The axillary nerve was traced from the level of the quadrangular space to the points of entry into the perimysium of the deltoid and teres minor muscles. Number and length of axillary nerve branches from the posterior and anterior divisions were compared. We measured the location where the axillary nerve entered the deltoid muscle, with the acromial end as 0% and the deltoid tuberosity as 100%.</p>", "<p id=\"Par25\">To expose the medial branch of the radial nerve, the lateral head of the triceps brachii was divided. Other branches of the radial nerve were traced proximally to their origins from the radial nerve. The number, diameter, and length of branches were recorded.</p>", "<p id=\"Par26\">From ten sides of fresh cadavers, 5-mm-long segments were harvested from the main trunk of the axillary nerve, its anterior and posterior divisions, and radial branches, to estimate the number of axonal fibers in these nerve branches. Branches were fixed overnight in 4% paraformaldehyde, dehydrated, and then cleared in toluene. They were then embedded in paraffin and cut into transverse sections of 2 µm thickness. Sections were stained with toluidine blue and photographed with a microscope-mounted camera. Morphometric measurements were performed at 100-fold magnification. The number of axons in randomly selected 0.04 mm² cross sectional areas was measured by ImageJ software version 1.53t (NIH, Bethesda, MD, USA).</p>", "<p id=\"Par27\">Independent t-tests and chi-square tests were used to compare the significance of differences between body sides or genders. Statistical analyses were performed using the software SPSS version 21 (IBM SPSS Software, Armonk, NY, USA). One-way ANOVA was used to verify the significance of differences in measurements between the types of nerves.</p>" ]

[ "<title>Results</title>", "<title>Branching patterns of the axillary nerve</title>", "<p id=\"Par6\">The clavicular and acromial parts of the deltoid muscle were constantly innervated by the anterior division of the axillary nerve. The teres minor muscle was constantly innervated by the posterior division of the axillary nerve. Variations were observed in the innervation of the spinous part of the deltoid muscle, which was innervated by the posterior division, anterior division, or both.</p>", "<p id=\"Par7\">According to the origin of the nerve branches to the spinous part of the deltoid muscle, variations in the distribution of the axillary nerve to the deltoid and teres minor muscles were grouped into three types (Fig. ##FIG##0##1##A–F). In 48.0% of upper limbs, the axillary nerve was “mixed type,” as the anterior division of the axillary nerve innervated all parts of deltoid muscle and the posterior division innervated the spinous part of the deltoid muscle as well as the teres minor muscle (Fig. ##FIG##0##1##A, D). In these upper limbs, the spinous part of the deltoid muscle was innervated by both the anterior and posterior divisions. In 38.0%, the axillary nerve was “anterior dominant type,” as the anterior division innervated all three parts of the deltoid muscle and the posterior division innervated the teres minor muscle only (Fig. ##FIG##0##1##B, E). In these upper limbs, the spinous part of the deltoid muscle was innervated by only the anterior division. In 14.0%, the axillary nerve was “posterior dominant type” with the anterior division innervating the clavicular and acromial parts of the deltoid muscle, while the posterior division innervated the teres minor and spinous part of the deltoid muscle (Fig. ##FIG##0##1##C, ##FIG##0##F##).</p>", "<title>Size of the axillary nerve</title>", "<p id=\"Par8\">The axillary nerve bifurcated into anterior and posterior divisions before passing through the quadrangular space in every upper extremity (Fig. ##FIG##1##2##). The length of the axillary nerve from its origin from the posterior cord to its bifurcation was 44.0 ± 10.5 mm on average. This length was 6.5 mm longer in males than in females, which was a statistically significant difference (p &lt; 0.05).</p>", "<p id=\"Par9\">Ramification of the muscular branches from the anterior division was distal to that of the muscular branches from the posterior division. The location of the first ramification of a muscular branch from the anterior division was an average of 23.6 ± 10.0 mm from the bifurcation. The first ramification of a muscular branch from the posterior division was an average of 16.9 ± 9.4 mm from the bifurcation. The first ramification of a muscular branch from the anterior division was distal to the quadrangular space in 78.0% of upper limbs. In contrast, the first ramification of a muscular branch from the posterior division was proximal to the quadrangular space in 66.0% of upper limbs.</p>", "<p id=\"Par10\">In 82.0% of upper limbs, the diameter of the anterior division was larger than the diameter of the posterior division. The average diameters of the axillary nerve, the anterior division, and the posterior division were 3.0 ± 0.5 mm, 2.4 ± 0.5 mm, and 2.0 ± 0.5 mm, respectively.</p>", "<title>Divisions and muscular branches of the axillary nerve</title>", "<p id=\"Par11\">The average number of deltoid branches from the axillary nerve was 11.1 ± 2.4. The spinous part of the deltoid muscle was innervated by the smallest number of nerve branches among the parts (Table ##TAB##0##1##). Mixed type axillary nerves provided more branches to the spinous part than the other axillary nerve types (p &lt; 0.05). The number of branches to the spinous part was 2.5 for mixed type axillary nerves, and 57.7% of these were from the anterior division.</p>", "<p id=\"Par12\">The average lengths of the terminal branches from the main division were 9.4 ± 3.7 mm to the clavicular part, 22.4 ± 11.9 mm to the acromial part, and 46.3 ± 14.7 mm to the spinous part. Branch lengths to the spinous part were greater on the left than the right with statistical significance (p &lt; 0.05).</p>", "<title>Site of junction of the nerve branch to the deltoid muscle</title>", "<p id=\"Par13\">Nerve terminals on the deltoid muscle were concentrated in the second upper quarter of the deltoid muscle. The junction of nerve branches to the deltoid muscle was located between the level of 28.7 ± 7.6% and the level of 47.0 ± 6.7% from the acromial tip to the deltoid tuberosity (Fig. ##FIG##2##3##). In 80.0% of shoulders, the highest entry point of nerve branches into the deltoid muscle was the acromial part. The lowest entry point was also in the acromial part in 86.0% of shoulders, with a diamond-shaped area of entry.</p>", "<title>Teres minor branch of the axillary nerve</title>", "<p id=\"Par14\">In all upper limbs, the teres minor muscle was innervated only by the posterior division of the axillary nerve. The mean number of branches innervating the teres minor was 1.5 ± 0.6, and the mean length of the longest branch to the teres minor was 36.8 ± 11.2 mm. Entry points of nerve branches were on the deeper side of the teres minor muscle.</p>", "<title>Anatomy of the radial nerve to the triceps brachii muscle</title>", "<p id=\"Par15\">All muscular branches to the triceps brachii muscle branched from the radial nerve before passing through a triangular inlet. Each head of the triceps brachii muscle was innervated by branches that arose separately from the radial nerve or by common branches for multiple triceps heads. In 74.0% of upper limbs, there was a common branch to the medial and lateral heads. In contrast, in 26.0% of upper limbs, all triceps heads were innervated by muscular nervous branches that arose from the radial nerve separately.</p>", "<p id=\"Par16\">There were more branches to the long head than to the other heads. The average numbers of branches to the long head, lateral head, medial head, and common branch to the lateral and medial heads were 2.1 ± 0.8, 1.6 ± 0.9, 1.5 ± 1.0, and 0.7 ± 0.6, respectively. The number of branches was greater for the long and lateral heads than for the medial head, with statistical significance (p &lt; 0.05).</p>", "<p id=\"Par17\">The nerve branch to the medial head was longer than the branches to the other heads. Lengths of the medial head branch, lateral head branch, long head branch, and the common branch to the lateral and medial heads were 63.5 ± 36.4 mm, 38.1 ± 26.7 mm, 23.2 ± 19.5 mm, and 48.5 ± 25.8 mm, respectively.</p>", "<p id=\"Par18\">Branches to the long head had a larger diameter than branches to the other heads. Diameters of branches to the long head, lateral head, and medial head were 1.4 ± 0.3 mm, 1.3 ± 0.3 mm, and 1.1 ± 0.4 mm, respectively. The average diameter of common branches to the lateral and medial heads was 1.7 ± 0.4 mm, which was significantly larger than that of branches to each head.</p>", "<title>Cross-section between the axillary nerve and radial nerve</title>", "<p id=\"Par19\">To identify the most suitable branch for nerve transfer, the diameter and axon numbers per nit area of branches between the axillary and radial nerves were compared. The diameter of the muscular branch was larger in the axillary nerve and its divisions than in the radial nerve and the branches to the triceps brachii (Table ##TAB##1##2##). The mean diameter of the axillary nerve before division was 3.0 ± 0.5 mm. In the anterior and posterior divisions, this diameter was 2.5 ± 0.6 mm and 2.3 ± 0.6 mm, respectively. In the radial nerve, the diameters of branches to the triceps brachii long head, lateral head, medial head, and common branch of the medial and lateral head were 1.4 ± 0.3 mm, 1.3 ± 0.3 mm, 1.1 ± 0.4 mm, and 1.7 ± 0.4 mm, respectively. In cross-section, the mean number of axons was 214.8 ± 43 and the number of axons in a 0.04 mm² area was similar among all branches (Fig. ##FIG##3##4##).</p>" ]

[ "<title>Discussion</title>", "<p id=\"Par20\">The axillary nerve ran a mean distance of 44.0 mm in the axilla and divided into anterior and posterior divisions before passing through the quadrangular space in every specimen we examined. Early bifurcation of the axillary nerve before passing through the quadrangular space is an expected finding because of innervation of the teres minor muscle on the muscle’s deep surface, which is proximal to the quadrangular space. If the division was distal to the space, branches to the teres minor muscle would have to take a recurrent pathway to re-enter the axilla from the posterior side of the arm. The axillary nerve is one of the most commonly injured nerves during surgical procedures of the shoulder^{##REF##18766295##4##,##REF##19916067##11##–##REF##25954611##13##} and axillary nerve injuries account for 6–10% of all brachial plexus injuries^{##REF##19916067##11##,##REF##16683246##12##}.The entire axillary nerve or its divisions can be damaged by an injury near the quadrangular space. The axillary nerve is damaged most commonly at the posterior opening of the quadrangular space^{##REF##11575912##3##}. Owing to its early bifurcation, when the axillary nerve is injured near the quadrangular space, innervation of the teres minor muscle is protected by the muscle itself, as there is no superficial exposure to the quadrangular space. In contrast, denervation of the deltoid muscle can be caused by injury of the anterior division or injury of single muscular branches^{##REF##25954611##13##–##REF##11550872##15##}. Each division of the axillary nerve is related to different movements of the glenohumeral joint. Therefore, the function of individual branches in terms of their effect on the deltoid muscle should be considered when performing nerve replacement. In shoulders with a mixed type axillary nerve where the spinous part of the deltoid is innervated by both divisions, the anterior division is responsible for medial rotation, flexion, abduction, extension, and lateral rotation of the shoulder; while the posterior division is responsible for extension and lateral rotation of the shoulder by the deltoid and teres minor muscles. In the anterior dominant type, the anterior division contributes to medial rotation, flexion, abduction, extension, and lateral rotation of the shoulder; while the posterior division is responsible only for extension and lateral rotation of the shoulder by the teres minor muscle. In the posterior dominant type, the anterior division helps the shoulder rotate medially, flex, and abduct but is not involved in lateral rotation or extension of the glenohumeral joint, which is performed by only the posterior division. This type corresponds with the nerve distribution of the deltoid muscle as described by Frohse and Frenkel^{##UREF##2##16##}. In these types, the anterior division is important for abduction of the shoulder, for which other muscles such as the supraspinatus cannot compensate properly. In contrast, dysfunction of the teres minor and spinous part of the deltoid can be compensated by other muscles such as the infraspinatus, teres major, and latissimus dorsi^{##REF##17931906##17##–##REF##8175814##19##}. Therefore, the posterior division may not be as important as the anterior division. Damage of the anterior division is therefore more serious than damage of the posterior division because of the possibility of compensation by other muscles in the latter case. The axillary nerve is vulnerable to injury from intramuscular injection of the deltoid muscle for vaccination^{##UREF##3##20##}. In the present study, junctions of the axillary nerve branches to the deltoid muscle were confined to the second upper quarter of the muscle (Fig. ##FIG##2##3##). In most cases, the thickness of the deltoid muscle will protect against needle penetration^{##REF##25382907##21##}. However, especially in thin patients with small deltoid muscles, avoidance of this area during needle insertion is recommended. The distribution of the junctions is also important to consider when performing intraosseous infusion into the humeral head. During intraosseous access, the humerus is medially rotated, and the needle is inserted into the greater tubercle through its superolateral surface at a 45° angle to the horizontal plane^{##UREF##4##22##}. According to our findings, insertion of the intraosseous needle is superior to the area of nerve-muscle junctions and favorable. There are two important principles to maximize the outcomes of nerve transfers. The first is to reinnervate the recipient nerve as close to the target muscle as possible^{##REF##25657729##23##}. Although some studies recommend nerve transfer using the branch to the long head of triceps brachii using a posterior approach^{##REF##12877852##10##,##REF##14751116##24##}, the length of this branch was relatively short in our study. Because of the possibility of damage to the posterior branch of the axillary nerve and difficulty in dissecting the teres minor branch through a posterior approach^{##UREF##5##25##,##REF##22535814##26##}, a deltopectoral approach (anterior approach) is preferable for axillary nerve transfer ^{##REF##17695392##7##,##REF##22535814##26##}. The radial nerve branch to the lateral head is not easily accessible because it is covered by the triceps brachii muscle as it usually arises from the radial groove. Instead, use of the branch to the medial head of triceps brachii is recommended for restoration of the deltoid muscle. The medial branch of the radial nerve is easy to find because it runs across the medial head superficially. The second principle is to use combinations of similarly behaving neuromuscular units, maximized when agonistic donor and recipients are chosen, as cortical readaptation is the physiological basis for functional recovery^{##REF##25657729##23##}. In our study, the number of axons in a given cross-sectional area was similar between nerve branches, which means that the number of axons is proportional to the diameter of the nerve branch. It is reasonable to transfer a donor nerve with a similar diameter to the recipient nerve.</p>", "<p id=\"Par21\">This study has several limitations. The most significant limitation is that the measurement of lengths was conducted on formalin-fixed cadavers, which may result in deviations due to tissue shrinkage and damage from microdissection, potentially causing differences from values measured in living subjects. Additionally, the limited number of samples used for tissue staining means the findings may not fully represent the variation seen in a larger population, indicating the need for further studies with increased sample sizes to validate and extend our conclusions.</p>" ]

[ "<title>Conclusions</title>", "<p id=\"Par22\">The significant role of the axillary nerve in maintaining deltoid muscle functionality and in preventing palsy or malfunction highlights its vital importance in clinical and surgical procedures. Functional loss of the anterior division may be more critical because it can lead to difficulty in abduction by the acromial part of the deltoid^{##REF##28671874##5##}. Consequently, for axillary nerve transfer, the medial and long branches of the radial nerve, with their suitable length and axon density, are recommended as optimal donor choices for effective functional restoration.</p>" ]

[ "<p id=\"Par1\">This study investigated the anatomical details of the axillary and radial nerves in 50 upper limbs from 29 adult formalin-embalmed cadavers, and ten fresh upper limbs. The focus was on understanding the course, division, and ramifications of these nerves to improve treatment of shoulder dysfunction caused by axillary nerve damage. The axillary nerve divided anteriorly and posteriorly before passing the quadrangular space in all specimens, with specific distances to the first ramifications. It was found that the deltoid muscle's clavicular and acromial parts were always innervated by the anterior division of the axillary nerve, whereas the spinous part was variably innervated. The longest and thickest branches of the radial nerve to the triceps muscles were identified, with no statistically significant differences in fiber numbers among triceps branches. The study concludes that nerve transfer to the anterior division of the axillary nerve can restore the deltoid muscle in about 86% of shoulders, and the teres minor muscle can be restored by nerve transfer to the posterior division. The medial head branch and long head branch of radial nerve were identified as the best donor options.</p>", "<title>Subject terms</title>" ]

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[ "<title>Acknowledgements</title>", "<p>The authors sincerely thank those who donated their bodies to science so that anatomical research could be performed. Results from such research can potentially increase mankind's overall knowledge that can then improve patient care. Therefore, these donors and their families deserve our highest gratitude^{##REF##32808702##28##}.</p>", "<title>Author contributions</title>", "<p>S.-J.K., J.-H.B., H.-J.Y., S.-H.M., Y.-R.C., and H.-Y.L. contributed to the study’s conception and design. S.-J.K., and J.-H.B. proceeded with gross dissection, histological experiments, and photographic work. H.-J.Y. provided assistance in histologic analysis and gave technical advice. S.-J.K., and J.-H.B. performed formal analysis, and writing of the original manuscript. H.-J.Y., S.-H.M., and Y.-R.C. contributed to reviewing and editing. H.-Y.L. proceeded with project administration, manuscript editing, and critical revision of the manuscript for intellectual content. All authors read and provided final approval of the version to be published.</p>", "<title>Data availability</title>", "<p>The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.</p>", "<title>Competing interests</title>", "<p id=\"Par28\">The authors declare no competing interests.</p>" ]

[ "<fig id=\"Fig1\"><label>Figure 1</label><caption><p>The three types of branching patterns of the axillary nerve to the deltoid and teres minor muscles. (<bold>A,D</bold>) Mixed type: the anterior division innervates all parts of the deltoid and the posterior division innervates both the spinous part of the deltoid and the teres minor; (<bold>B,E</bold>) anterior dominant type: the anterior division innervates all parts of the deltoid, and the posterior division innervates the teres minor; (<bold>C,F</bold>) posterior dominant type: the anterior division innervates the clavicular and acromial parts of the deltoid, while the posterior division innerves the spinous part of the deltoid and the teres minor; <italic>C</italic> clavicular part of the deltoid, <italic>A</italic> acromial part of the deltoid, <italic>S</italic> spinous part of the deltoid, <italic>Tm</italic> teres minor muscle, <italic>QS</italic> quadrangular space.</p></caption></fig>", "<fig id=\"Fig2\"><label>Figure 2</label><caption><p>Anterior view of the branches of the axillary nerve (Ax) to the subscapularis and teres minor muscles. The posterior cord of the brachial plexus bifurcates to the radial and axillary nerves (arrow). The axillary nerve branches into lower subscapular nerves (arrowhead) and thoracodorsal nerves. The anterior cords of the brachial plexus and radial nerve are reflected for observation. <italic>MCN</italic> musculocutaneous nerve, <italic>MN</italic> median nerve, <italic>UN</italic> ulnar nerve, <italic>PC</italic> posterior cord, <italic>USN</italic> upper subscapular nerve, <italic>MBCN</italic> medial brachial cutaneous nerve, <italic>TD</italic> thoracodorsal nerve, <italic>LTN</italic> long thoracic nerve, <italic>RN</italic> radial nerve, <italic>Ax</italic> axillary nerve, <italic>SS</italic> subscapularis muscle, <italic>TM</italic> teres major muscle, <italic>Tm</italic> teres minor muscle, <italic>LoH</italic> long head of the triceps brachii muscle.</p></caption></fig>", "<fig id=\"Fig3\"><label>Figure 3</label><caption><p>Entry points of axillary nerve branches to the deltoid muscle. Nerve terminals onto the deltoid muscle are concentrated between the upper quarter and middle of the muscle. <italic>Tm</italic> teres minor muscle.</p></caption></fig>", "<fig id=\"Fig4\"><label>Figure 4</label><caption><p>Cross-sections of the axillary nerve and radial nerve. Obtained tissue sections were serially sectioned into 2-μm-thick slices and stained with Toluidine blue O. (<bold>A</bold>) Long branch of the radial nerve, (<bold>B</bold>) medial branch of the radial nerve, (<bold>C</bold>) lateral branch of the radial nerve, (<bold>D</bold>) common branch of the medial and lateral branch of the radial nerve, (<bold>E</bold>) axillary nerve, (<bold>F</bold>) anterior division of the axillary nerve, and (<bold>G</bold>) posterior division of the axillary nerve. <italic>AD</italic> anterior division of the axillary nerve, <italic>PD</italic> posterior division of the axillary nerve, <italic>CB</italic> common branch of the medial and lateral branch of the radial nerve, <italic>LaB</italic> lateral branch of the radial nerve, <italic>LoB</italic> long branch of the radial nerve, <italic>MB</italic> medial branch of the radial nerve, <italic>D</italic> deltoid muscle, <italic>LoH</italic> long head of the triceps brachii muscle, <italic>MH</italic> medial head of the triceps brachii muscle.</p></caption></fig>" ]

[ "<table-wrap id=\"Tab1\"><label>Table 1</label><caption><p>Number and lengths of each part of the axillary nerve.</p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th align=\"left\"/><th align=\"left\">Spinous part</th><th align=\"left\">Acromial part</th><th align=\"left\">Clavicular part</th></tr></thead><tbody><tr><td align=\"left\">Number of branches</td><td align=\"left\">2.0 ± 0.9 (1–4)</td><td align=\"left\">4.9 ± 2.1 (2–9)</td><td char=\".\" align=\"char\">4.2 ± 1.5 (1–8)</td></tr><tr><td align=\"left\">Length</td><td align=\"left\">46.3 ± 14.7</td><td align=\"left\">22.4 ± 11.9</td><td char=\".\" align=\"char\">9.4 ± 3.7</td></tr></tbody></table></table-wrap>", "<table-wrap id=\"Tab2\"><label>Table 2</label><caption><p>Diameter and axon number within axillary and radial nerve fascicles.</p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th align=\"left\"/><th align=\"left\">Diameter</th><th align=\"left\">Axon number per unit area</th></tr></thead><tbody><tr><td align=\"left\">Axillary nerve</td><td char=\".\" align=\"char\">3.0 ± 0.5 (2.36–3.96)</td><td char=\".\" align=\"char\">218.0 ± 28.6 (182–262)</td></tr><tr><td align=\"left\">Anterior division</td><td char=\".\" align=\"char\">2.5 ± 0.6 (1.23–3.08)</td><td char=\".\" align=\"char\">225.3 ± 31.0 (195–294)</td></tr><tr><td align=\"left\">Posterior division</td><td char=\".\" align=\"char\">2.3 ± 0.6 (1.70–3.55)</td><td char=\".\" align=\"char\">206.4 ± 37.6 (136–271)</td></tr><tr><td align=\"left\" colspan=\"3\">Radial nerve</td></tr><tr><td align=\"left\"> Long branch</td><td char=\".\" align=\"char\">1.4 ± 0.3 (1.08–1.79)</td><td char=\"–\" align=\"char\">222.1 ± 67.3 (164–352)</td></tr><tr><td align=\"left\"> Lateral branch</td><td char=\".\" align=\"char\">1.3 ± 0.3 (0.53–1.78)</td><td char=\"–\" align=\"char\">231.3 ± 61.7 (154–379)</td></tr><tr><td align=\"left\"> Medial branch</td><td char=\".\" align=\"char\">1.1 ± 0.4 (0.92–1.81)</td><td char=\"–\" align=\"char\">196.7 ± 45.7 (135–256)</td></tr><tr><td align=\"left\"> Common branch</td><td char=\".\" align=\"char\">1.7 ± 0.4 (1.18–2.33)</td><td char=\"–\" align=\"char\">207.3 ± 18.9 (169–238)</td></tr></tbody></table></table-wrap>" ]

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[ "<table-wrap-foot><p><italic>p</italic> &lt; 0.05.</p></table-wrap-foot>", "<table-wrap-foot><p>Data presented as mean ± SD, unit area: 0.04 mm².</p><p><italic>p</italic> &lt; 0.05.</p></table-wrap-foot>", "<fn-group><fn><p><bold>Publisher's note</bold></p><p>Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.</p></fn></fn-group>" ]

[ "<graphic xlink:href=\"41598_2024_51923_Fig1_HTML\" id=\"MO1\"/>", "<graphic xlink:href=\"41598_2024_51923_Fig2_HTML\" id=\"MO2\"/>", "<graphic xlink:href=\"41598_2024_51923_Fig3_HTML\" id=\"MO3\"/>", "<graphic xlink:href=\"41598_2024_51923_Fig4_HTML\" id=\"MO4\"/>" ]

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{ "acronym": [], "definition": [] }

[ "<title>Introduction</title>", "<p id=\"Par3\">Reducing fossil fuel use and global climate change requires a fast energy transition, and nations across the globe have successively set out their own targets and pathways to carbon neutrality^{##UREF##0##1##}. Since 2009, as the fastest-growing renewable power source, the generating capacity of solar photovoltaic (PV) energy has grown globally by 41% per year^{##UREF##1##2##}. It has put forward higher requirements for the conversion efficiency and capital cost reduction of PV energy generation^{##REF##34635590##3##}, which is always impacted by cloud cover, aerosol and panel soiling^{##UREF##2##4##–##UREF##6##9##}. Yet, in a stark contrast to aerosol and panel soiling, cloud cover or advection can dramatically and intermittently affect incident solar radiation, resulting in unbalance between the load demand and PV energy generation, which poses a considerable risk to the stability of power grids^{##UREF##7##10##–##UREF##9##12##}. Therefore, reliable and powerful PV energy generation or global tilted irradiance (GTI, the radiation captured by solar photovoltaic panels) forecast technique, particularly short-term forecasts of the intra-day GTI or PV power generation (at the leading time of 0–4 h), is also highly beneficial to power smoothing processes and other load-following applications^{##UREF##6##9##,##UREF##10##13##}. In addition, currently, in most European countries, short-term prices for the physical delivery of electricity are formed by spot markets, such as the European Power Exchange SPOT (<ext-link ext-link-type=\"uri\" xlink:href=\"https://www.europex.org/members/epex-spot/\">https://www.europex.org/members/epex-spot/</ext-link>). Although ~80% of trade volume is controlled by the day-ahead trading market, the intra-day auctions from hourly to 15-min intervals determine real-time electricity prices^{##UREF##11##14##}. Thus, sophisticated solar PV power generation nowcasting technique not only can improve the stability of power generation, but also facilitates the developments of more commercially viable PV systems, the current electricity market and price transactions, and increases the competitiveness of the solar PV energy source^{##UREF##12##15##,##REF##26658608##16##}.</p>", "<p id=\"Par4\">In recent years, rapid advances in artificial intelligence have promoted the application of data-driven machine learning-based approaches in Earth system science^{##REF##30760912##17##,##REF##34588668##18##}. Particularly, some recent studies^{##UREF##9##12##,##UREF##10##13##,##UREF##13##19##–##REF##36747538##21##} on the prediction of solar radiation also explicitly indicate that advanced prediction approaches based on machine learning perform better compared with empirical models, time series, and hybrid algorithms, such as artificial neural networks and support vector machines. Nevertheless, it is still a great challenge to predict cloud motion, formation, deformation and dissipation under complex atmospheric dynamics, geography, and climatic conditions^{##UREF##6##9##,##UREF##15##22##,##UREF##16##23##}. Thus, there is still no solar radiation forecast model that can work well in every region and at every time^{##REF##36747538##21##}.</p>", "<p id=\"Par5\">Cloud cover nowcasting remains a field of interest for forecasting the electricity production of PV plants^{##UREF##17##24##}. We are committed to developing a daytime hourly intra-day cloud fraction (CF) prediction algorithm for small areas over PV plants. Based on the recurrent-neural-networks-based (RNNs) long short-term memory (LSTM) algorithm framework, the newly developed PredRNN and PredRNN++ (an extended and latest version of PredRNN)^{##UREF##18##25##,##REF##35380958##26##} can well learn to predict long-term future imageries in various spatio-temporal tasks by modeling their spatial and temporal dependencies, including video frame prediction, human motion prediction, etc. Therefore, our primary objective is to develop an innovative and easy-to-promote algorithm or system based on the key framework of the PredRNN++ model. Through this algorithm, the 0–4 h CF at solar PV plants under all weather conditions can be predicted by using sequential Himawari-8/9 geostationary satellite images with high spatio-temporal resolutions^{##UREF##19##27##}. Compared with the previous study^{##UREF##20##28##}, it only used a single visible channel of geostationary satellite and a constant model to predict cloudiness. Some former studies directly used surface solar global horizontal irradiance (GHI) as model input to predict GHI values in the next few hours^{##UREF##21##29##}, achieving the purpose of estimating the power generation of PV plants. Nevertheless, the presence of clouds is still identified as the primary uncertainty in current surface solar GHI forecasts^{##UREF##22##30##}. In contrast, our investigation only predicts geostationary satellite Level 1B (L1B) radiance data. With the prediction results of satellite L1B radiance data and accurate cloud detection algorithm, this approach is expected to provide reliable and variable CF information for further improving the predictability of current GTI or PV power generation.</p>" ]

[ "<title>Methods</title>", "<p id=\"Par14\">The newly developed EPM and fast cloud mask algorithm in the NCP_CF system are applied to predict CF (or CSR) at the leading time of 0–4 h at two test PV plants. In this system, the −4–0 h geostationary satellite radiance data is used as input to dynamically provide 0–4 h satellite cloud images and fractions. To verify the reliability of the EPM, we first use the CF observations from twelve widely distributed ground-based manual stations and three all-sky imager stations for the period from 2019 to 2022 as true values to compare with the predictions in the corresponding period. Moreover, correlations of the predicted CSRs with the actual PV power generation and surface solar radiation at five test PV plants from October 2022 to March 2023 are analyzed. The benefits of the geostationary satellite data with high spatio-temporal resolutions and the advanced EPM to improve the PV power generation efficiency are investigated, as well as the wide applicability and generalizable value of the EPM system.</p>", "<title>Ground-based observation data</title>", "<p id=\"Par15\">The total cloud cover or CF (~20 km × 20 km square area) used in this study is obtained through manual observation at  twelve ground-based meteorological stations (Fig. ##FIG##0##1##) in January, April, July, and October of 2019. Note that due to the relatively large errors in low-visibility conditions, the CF data with the matched and automatically measured visibility &lt;2 km is removed. Besides, the view zenith angles of ground-based stations from H8/AHI field of view used here, as stated in Supplementary Table ##SUPPL##0##2##, are smaller than 60°. Therefore, the parallax effect is negligible (error &lt; 1 km) in the collocation between satellite pixels and ground-based stations for this study (For more explanations and details, please refer to ##SUPPL##0##Supplementary Note## and Supplementary Figs. ##SUPPL##0##7## and ##SUPPL##0##8##). Firstly, considering the daytime nowcasting applications and sunshine conditions at PV plants, only the manual observations at 11:00, 14:00 and 17:00 are collected for validation in this research. Secondly, three ground-based all-sky imager stations (equipped with a Japan EKO ASI-16 all-sky imager, <ext-link ext-link-type=\"uri\" xlink:href=\"https://www.eko-instruments.com/us/categories/products/all-sky-imagers/asi-16-all-sky-imager\">https://www.eko-instruments.com/us/categories/products/all-sky-imagers/asi-16-all-sky-imager</ext-link>) can provide the high temporal resolution (5 min) and valuable CF and cloud cover data during the daytime, which are retrieved by the standard EKO ASI-16 cloud detection algorithm (<ext-link ext-link-type=\"uri\" xlink:href=\"https://www.eko-instruments.com/media/z2aalysq/asi-16-software-manual-find-clouds.pdf\">https://www.eko-instruments.com/media/z2aalysq/asi-16-software-manual-find-clouds.pdf</ext-link>)^{##UREF##28##36##}. ASI, equipped with a digital camera coupled with an upward looking fisheye lens, could provide field of view (FOV) of ~180°, but pixels at a FOV &gt; 140 ° are excluded due to distortion. Digital images of the sky obtained by ASI are classified pixel by pixel into clear sky, optically thin and optically thick clouds, respectively. The cloud detection and opacity classification (CDOC) algorithm developed by Ghonima et al., 2012^{##UREF##28##36##} could provide 96%, 60%, and 96.3% accuracy in the validation for clear, thin, and thick cloud, respectively. Finer CF data allow more accurate validation of the NCP_CF system, thereby demonstrating the specific prediction effect at the forecast leading time of 0–4 h. The study periods of the local CF data from three all-sky imagers located in Zhuhai, Nanjing and Beijing (Fig. ##FIG##0##1##) are from September 2022 to February 2023, April 2021 to September 2021, and September 2022 to December 2022, respectively. In addition, the actual power generation (MW, temporal resolution of 15 min) and GTI (W·m⁻²) measured at Sangge and Leling PV plants from November 2022 to March 2023 and at Xiaochengzi, Lijiamen and Shiziyan PV plants in November 2022 (Fig. ##FIG##0##1##) are also used to analyze the agreement with the predicted CF. These real power generation data are obtained from the SCADA (Supervisory Control and Data Acquisition) system of China General Nuclear Power Group Wind Energy Co. Ltd.</p>", "<title>Geostationary satellite data and calculating the cloud fraction</title>", "<p id=\"Par16\">The NRT 16-band full-disk AHI level-1B radiance data from the Himawari-8/9 satellite (the new-generation Japanese geostationary meteorological satellite) with spatio-temporal resolutions of 1–4 km and 10 min are obtained from the Japan Meteorological Agency Himawari-Cast in China^{##UREF##24##32##}. Additionally, the offline Himawari-8/9 data at the original resolution (0.5–2 km) are also available for free download from the JAXA (Japan Aerospace Exploration Agency) Himawari satellite data FTP (File Transfer Protocol) site (ftp.ptree.jaxa.jp) from July 7, 2015 (<ext-link ext-link-type=\"uri\" xlink:href=\"http://www.jma-net.go.jp/msc/en/\">http://www.jma-net.go.jp/msc/en/</ext-link>). The nadir point of the Himawari-8/9 satellite is located at 140.7°E, and the coverage of this satellite includes the Japan island and the eastern parts of China.</p>", "<p id=\"Par17\">Based on the real-time Himawari-8/9 AHI full-disk observation data, each site would be centered around its precise location and matched with a 32 × 32 pixels box as the experimental area. The special fast cloud mask algorithm^{##UREF##24##32##} combines five inherited and improved cloudy/clear pixel tests in visible and infrared bands to determine the final confidence value (<italic>c</italic>) of every pixel of the satellite imager, i.e., <italic>c</italic> &gt; 0.99 = clear, 0.95 &lt; <italic>c</italic> ≤ 0.99 = probably clear, 0.66 &lt; <italic>c</italic> ≤ 0.95 = probably cloudy, and <italic>c</italic> ≤ 0.66 = cloudy. As the real viewing field at a ground-based station approximates a 20 km × 20 km square area, a 5 × 5 pixels box of cloud mask centered around a targeted PV plant is used in this study to calculate the CF predictions, which is expressed as Eq. (##FORMU##0##1##).where and indicate the total numbers of the cloudy and probably cloudy pixels in the 5 × 5 pixel box^{##UREF##29##37##}, respectively. The complementary CSR is equal to 1-CF.</p>", "<title>Model</title>", "<p id=\"Par18\">The PredRNN++ model^{##UREF##17##24##} (an improved prediction RNN), dedicated to short-term prediction and nowcasting, is adopted as a key model in this investigation for 0–4 h CF nowcasting. This advanced neural network successfully overcomes the spatio-temporal predictive learning dilemma between deep-in-time structure and vanishing gradient. Previous research has demonstrated that the PredRNN++ consistently outperforms the ConvLSTM, TrajGRU, Discrete Fracture Network, MCnet and PredRNN at every future time step for both peak signal-to-noise ratio and structural similarity index measure^{##UREF##17##24##}. To achieve greater spatio-temporal modeling capability, in this investigation, we re-design and develop the EPM with five convolutional layers, whose elaborated structure is shown in Fig. ##FIG##4##5##. The details of the casual LSTM and the Gradient Highway Unit (GHU)^{##UREF##17##24##} in the EPM structure are also illustrated in Supplementary Fig. ##SUPPL##0##2##.</p>", "<p id=\"Par19\">The causal LSTM, an upgraded version of the LSTM, increases the recurrence depth from one time step to the next and derives a more powerful modeling capability for stronger spatial correlations and short-term dynamics. As shown in Supplementary Fig. ##SUPPL##0##2a##, a causal LSTM unit contains two memories, namely a temporal memory and a spatial memory , where the superscripts and <italic>t</italic> denote the <italic>k</italic>th hidden layer in the stacked causal LSTM network and the <italic>t</italic>th time step, respectively. The temporal memory depends on its preceding state and is controlled by an input gate , a forget gate and an input modulation gate . The spatial memory relies on which is in the deep transition route. Notably, the topmost spatial memory is assigned to the bottom spatial memory . For the <italic>k</italic>th layer, the updated equations of the causal LSTM can be expressed as Eqs. (##FORMU##23##2##–##FORMU##28##7##).where “⊚\" denotes the convolution, \"⨂\" the element-wise multiplication, tan<italic>h</italic> the element-wise hyperbolic tangent function, σ the element-wise sigmoid function, and “[]” a concatenation of tensors. the convolutional filters. All the equations in the causal LSTM can be briefly expressed as Eq. (##FORMU##30##8##).where is replaced by and between the first and second layers.</p>", "<p id=\"Par20\">The gradient highway, a shorter route from future outputs back to distant inputs, can alleviate the vanishing gradient problem. As shown in Supplementary Fig. ##SUPPL##0##2b##, in a GHU, represents the convolutional filters, is a switch gate and enables adaptive learning between the transformed input and the hidden state . The equation of the GHU can be written as Eqs. (##FORMU##38##9##–##FORMU##40##11##).</p>", "<p id=\"Par21\">The equations of the GHU can be briefly expressed as:</p>", "<p id=\"Par22\">As presented in Fig. ##FIG##4##5a##, combined with Eqs. (##FORMU##30##8##) and (##FORMU##41##12##), the key equations of the entire EPM framework can be written as Eqs. (##FORMU##42##13##–##FORMU##47##18##).</p>", "<p id=\"Par23\">In the EPM framework, the GHU is injected between the first and second causal LSTMs. The causal LSTM and GUH respectively capture short-term and long-term data or image dependencies. The gradient highway (blue line in Fig. ##FIG##4##5a##) supplies a quick path from the first to the last time step by quickly updating hidden state . It is worth noting that, unlike temporal skip connections, the GHU controls the proportions of and the deep transition feature through , which allows the EPM to adaptively learn both short-term and long-term frame relations. In the causal LSTM of the EPM, the spatial memory is a function of the temporal memory through another set of gate structures. As the recurrence depth along the spatio-temporal transition paths grows considerably, each pixel in the final generated frame has a bigger receptive field of the input sequence at each time step, which is why the EPM has a better ability to model short-term video dynamics and sudden changes. Figure ##FIG##4##5b## displays the data flow process in the spatial memory of the EPM. The dimension of the input sequential satellite data (4 h and a time interval of 10 min) is 24 × 32 × 32. At the first convolutional layer, the input terms and are concatenated to form a larger tensor, and then is generated by the convolution calculation. The EPM performs a total of five convolution calculations, with dimensions of hidden state from 10 × 32 × 32 to 1 × 32 × 32.</p>", "<p id=\"Par24\">The EPM training process involves the use of the Adam optimizer with a learning rate of 0.001 and the mean square error as the loss metric. The input is a four-dimensional tensor of size [<italic>c, t, h, w</italic>] (6, 24, 32, 32), where <italic>c</italic> represents the number of input channels, <italic>t</italic> represents time (spanning 4 h), <italic>h</italic> represents height, and <italic>w</italic> represents width. Our research area focuses on PV plants, which provides us with images measuring 32 × 32 pixels (~128 km × 128 km). To train our model, we created a dataset consisting of sequences of 48 images for each channel, spanning 4 h. During the initial training process of the optimal network structure, data were obtained from resized AHI images at six bands over twelve manual observation stations and three all-sky imager stations. The training set utilized data from January to October in 2018, while the remaining months of 2018 were used as the validation set. In the application scenario of PV plants, the training method is described in the section of NRT and cyclically updated prediction system.</p>", "<p id=\"Par25\"><bold>Validation and assessment</bold>. The primary metrics used to evaluate the accuracy of the NCP_CF system for forecasting the CF are the RMSE, MBE and R, defined as shown in Eqs. (##FORMU##55##19##)-(##FORMU##57##21##).where <italic>P</italic>_{<italic>i</italic>} denotes the predicted CF, <italic>T</italic>_{<italic>i</italic>} denotes the actual CF obtained from ground-based observations mentioned above, and <italic>N</italic> is the total number of the matched samples.</p>", "<title>Reporting summary</title>", "<p id=\"Par26\">Further information on research design is available in the ##SUPPL##2##Nature Portfolio Reporting Summary## linked to this article.</p>" ]

[ "<title>Results</title>", "<title>Cloud fraction nowcasting and validations</title>", "<p id=\"Par6\">In order to better simulate real application scenarios, a quasi-operational and near real-time (NRT) and cyclically updated prediction system is newly developed for 0–4 h CF nowcasting at solar PV plants (hereafter referred to as the NCP_CF). The predicted CF from this NCP_CF nowcasting system is also compared with the real PV power generation and the GTI to verify its feasibility, reliability and adaptability. The results from the NCP_CF system are examined and validated by using the observed CF values from twelve manual meteorological observation stations of the China Meteorological Administration (only the observations at 14:00 and 17:00 are used) and three all-sky imager stations (Fig. ##FIG##0##1##). Figure ##FIG##1##2## shows the root mean square errors (RMSEs) and mean bias errors (MBEs) of the CF predictions at targeted stations with the forecast horizon, local time (diurnal cycle) and time series. In terms of forecast horizon (Figs. ##FIG##1##2##a–##FIG##1##2f##, Supplementary Table ##SUPPL##0##1##), the RMSE increases from 0.18 to 0.35 at all stations for 0–4 h forecast periods, and the MBE fluctuates around −0.1. Notably, the RMSE is less than 0.25 within the 2-h leading period, but the forecast accuracy decreases faster when the forecast leading period exceeds 2 h, indicating that the forecast performance threshold of this system is ~2-h leading time. Considering the continuity and coverage of observation time, Fig. ##FIG##1##2g## only shows the diurnal cycle of CF forecast accuracy at three all-sky imager stations. Within a one-day forecast window, the most and least accurate predictions occur during 12:00–17:00 and 08:00–09:30, respectively, whereas the relatively moderate decrease in the forecast accuracy before 09:30 is mainly attributed to the invalid satellite visible images before 08:00. Regarding the predicted performance for different months, the monthly mean RMSE and MBE values in time series at three all-sky imager stations do not vary considerably, indicating the weak monthly dependence and stability of this cyclically updated CF nowcasting algorithm and system.</p>", "<p id=\"Par7\">Furthermore, the correlation coefficients (R) between the 1–4 h predicted clear sky ratios (CSRs; CSR = 1−CF) from the NCP_CF system and the actual power generation (GTI) are calculated. Figure ##FIG##2##3## displays the comparisons among the 1–4 h predicted CSRs, the actual power generation and the GTI at Sangge, Leling, Xiaochengzi, Lijiamen, and Shiziyan PV plants from 09:00 to 17:00 in November 2022. The mean <italic>R</italic> values between the 1–4 h predicted CSRs and the actual PV power generation (the GTI) in November 2022 at five PV plants are 0.81, 0.73, 0.65, and 0.55 (0.81, 0.72, 0.64, and 0.55). This result highlights the good consistency of the predicted CSRs with the actual PV power generation and the GTI, especially for the first 2-h leading time. Overall, the EPM-model-based NCP_CF system developed in this research is applicable to provide high-quality CF estimations at PV plants in advance, which can be used to predict the GTI and power generation at the forecast leading time of 0–4 h. The results at Sangge and Leling PV plants from December 2022 to March 2023 are displayed in Supplementary Figs. ##SUPPL##0##3##–##SUPPL##0##6##.</p>", "<title>Near real-time and cyclically updated prediction system</title>", "<p id=\"Par8\">The satellite-based NRT and cyclically updated prediction system (Fig. ##FIG##3##4##) for 0–4 h CF nowcasting, operating at five real PV plants (Sangge, Leling, Xiaochengzi, Lijiamen and Shiziyan PV plants) belonged to China General Nuclear Power Group Wind Energy Co. Ltd., mainly consists of three subsystems, i.e., preprocessing, prediction and retrieval modules. Specifically, the preprocessing module regularly adjusts the real-time down-sampling Himawari-8/9 Advanced Himawari Imager (AHI) data received from the direct broadcast receiving system. The AHI is an advanced imager with 16 spectral bands ranging from 0.47 μm to 13.3 μm, which has spatio-temporal resolutions of 4 km and 10 min^{##UREF##23##31##}. The full-disk Level-1B radiance data at 0.65 μm, 0.86 μm, 3.9 μm, 7.0 μm, 11.2 μm and 12.3 μm with high-quality geolocation and radiometric calibration is grided into a 32 × 32 pixel box (~128 km × 128 km) centered around the targeted PV plant. Then, these sequential and resized AHI images at six bands are converted into a tensor [tile size = 6 (band number) × 24 (4-h time sequence) × 32 × 32] of the prediction model for forecasting the following 0–4 h satellite images (6 × 24 × 32 × 32).</p>", "<p id=\"Par9\">As a key function of the prediction module, an enhanced PredRNN + + model (EPM; more details in “Model” section) is developed in this study to predict 0–4 h sequential geostationary satellite images. In order to better track the fast and stochastic changes in cloud images, the neural network of the EPM is always cyclically generated by the scheduled training process of the model, which has a 1-h update frequency. The cumbersome cyclic process should take 40–50 min when using the latest sequential satellite images (6 × 24 × 32 × 32) between −5 h and −1 h as training samples and one graphics processing unit processor (NVIDIA Tesla-V100). For instance, a cyclic training process of the model starts at the scheduled local time of 12:08 and ends at ~12:53, the imported training samples are from 08:00 to 12:00, and the latest updated EPM is timely activated for CF nowcasting at ~13:08 (nowcasting from 13:00 to 17:00) and 13:38 (nowcasting from 13:30 to 17:30), respectively, with an update frequency of a frequency of half an hour. During the same period, the network of the next EPM is also trained simultaneously by using the sequential satellite images from 09:00 to 13:00. This cyclical procedure will continually update and replace the existing EPM every hour, ensuring that we always have the most up-to-date nowcasting model. Considering the use of satellite visible images, the NCP_CF system only operates from 07:20 to 17:20, which still meets the requirement of CF nowcasting at PV plants.</p>", "<p id=\"Par10\">In the retrieval module, the 0–4 h predicted and resized cloud images mentioned above are used by a fast cloud mask algorithm^{##UREF##24##32##}, which is able to calculate the number of cloudy pixels and the CF or cloud cover in the observation field of the targeted PV plant. Note that compared with the operational cloud mask algorithm, the fast cloud mask algorithm can fast retrieve the CF without any ancillary data, which is crucial for CF nowcasting. Supplementary Figure ##SUPPL##0##1## presents the comparisons between the predicted and actual satellite images and cloud mask results at Zhuhai station on 17 November 2022, illustrating the good agreement between them.</p>" ]

[ "<title>Discussion</title>", "<p id=\"Par11\">Our study demonstrates that the NCP_CF system can provide high-efficiency, high-quality and adaptable 0–4 h CF nowcasting data for PV plants. As shown in Figs. ##FIG##1##2##a–##FIG##1##f##, the mean RMSE (MBE) values are 0.21 (−0.09), 0.25 (−0.08), 0.3 (−0.07), and 0.35 (−0.03) for the forecast leading time of 1 h, 2 h, 3 h, and 4 h, respectively. Particularly, the CF nowcasting results from the NCP_CF system remain highly reliable within the forecast leading time of 2 h, with RMSE values staying almost at 0.2 and not increasing within the 1-h leading time. Conversely, the prediction performance of the NCP_CF system gradually deteriorates as the forecast leading time increases to more than 2 h, which may be due to the vanishing gradient problem^{##UREF##15##22##}. By the limited spatial domain, the rapid movement of clouds may cause a small bias between the predicted CF and the actual CF. Further analyses on the daily and seasonal scales are also conducted, as shown in Fig. ##FIG##1##2##g, a–f, respectively. On the daily scale, the NCP_CF system performs particularly well from 09:30 to 18:00, whereas it shows poor performance before dawn, mainly due to the poor quality of satellite data at the visible band during that time. Fortunately, this issue is mitigated due to the low power generation of PV plants before dawn, and thus cloud cover has a low impact on power generation in this period. For the seasonal scale, except for the forecast results at Nanjing station in May 2021, the NCP_CF system shows stable forecast performance and seasonal biases at different stations and in different seasons. Its salient adaptability thus is the largest advantage compared with other solar radiation nowcasting methods summarized in the previous review^{##REF##36747538##21##}. Overall, the CF nowcasting results of the NCP_CF system have good stability, strong generalizability and non-sensitivity to geographical locations and climatic characteristics.</p>", "<p id=\"Par12\">Given that the present electricity spot markets in Europe work within different time horizons, specific and professional forecast techniques are required for each leading time^{##UREF##10##13##}. The NCP_CF system with the 0–4 h forecast leading time within a 10-min interval shows more advantages than other existing nowcasting methods, such as the manners based on all-sky imager observation (the forecast leading time only ranging from 0 to 20 mins)^{##UREF##25##33##} and numerical weather prediction (from 6 h to day-ahead time frames)^{##UREF##26##34##} for solar PV power generation. Principally, this system is applicable to fast varying small-scale weather and environmental conditions and can accurately capture cloud motion over PV plants without relying on long-term historical in-situ meteorological data. Besides, the predictions of the NCP_CF system are not markedly affected by seasonal climate changes on long-term scales, which underscores the stable operation of this system. The system shows excellent forecast performance within the first 2-h leading time, with an average <italic>R</italic> value between the predicted CF and the actual power generation or GTI at PV plants close to or more than 0.80.</p>", "<p id=\"Par13\">Since one of the greatest challenges facing solar PV renewable energy production is its instability and intermittency, accurate CF nowcasting is still vital for the efficient operation of PV plants and their power systems. Improving the stability of PV power production can directly facilitate policy-making of feed-in tariffs and attract more investment in solar PV power generation^{##UREF##11##14##,##UREF##27##35##}. However, most importantly, the nowcasting technique developed in this research deserves attention in terms of promoting the overall penetration of solar power on the electric grid and having a non-negligible impact on electricity price trading in the intra-day spot market^{##UREF##11##14##}. Furthermore, it is evident that increasing the share of renewable energy in the global energy system can contribute to the reduction of global carbon emissions^{##REF##34635590##3##}. Therefore, our future mission is to further promote applications and improve the accuracy of this cloud cover nowcasting technique, especially for the forecast leading time of more than 2 h, by using higher spatial-resolution satellite data (i.e., 1–2 km) and combining the short-term forecast data from a rapidly updated regional high-resolution numerical weather prediction.</p>" ]

[]

[ "<p id=\"Par1\">Accurate nowcasting for cloud fraction is still intractable challenge for stable solar photovoltaic electricity generation. By combining continuous radiance images measured by geostationary satellite and an advanced recurrent neural network, we develop a nowcasting algorithm for predicting cloud fraction at the leading time of 0–4 h at photovoltaic plants. Based on this algorithm, a cyclically updated prediction system is also established and tested at five photovoltaic plants and several stations with cloud fraction observations in China. The results demonstrate that the cloud fraction nowcasting is efficient, high quality and adaptable. Particularly, it shows an excellent forecast performance within the first 2-hour leading time, with an average correlation coefficient close to 0.8 between the predicted clear sky ratio and actual power generation at photovoltaic plants. Our findings highlight the benefits and potential of this technique to improve the competitiveness of solar photovoltaic energy in electricity market.</p>", "<p id=\"Par2\">Accurate nowcasting of cloud cover or fraction and its movement remains a significant challenge for stable solar photovoltaic electricity generation. Here, the authors combine continuous radiance images with high spatio-temporal resolutions to develop a nowcasting algorithm for predicting cloud cover at a leading time of 0–4 h.</p>", "<title>Subject terms</title>" ]

[ "<title>Supplementary information</title>", "<p>\n\n\n\n</p>", "<title>Source data</title>", "<p>\n\n</p>" ]

[ "<title>Supplementary information</title>", "<p>The online version contains supplementary material available at 10.1038/s41467-023-44666-1.</p>", "<title>Acknowledgements</title>", "<p>We appreciate the Facebook Inc. for freely providing the Pytorch toolkit (<ext-link ext-link-type=\"uri\" xlink:href=\"https://pytorch.org\">https://pytorch.org</ext-link>). This study was supported partly by the Innovation Group Project of Southern Marine Science and Engineering Guangdong Laboratory (Zhuhai) (No.SML2023SP208), the National Natural Science Foundation of China (Grants 42175086 and U2142201), the FengYun Meteorological Satellite Innovation Foundation under Grant FY-APP-ZX-2022.0207, and the Science and Technology Planning Project of Guangdong Province (2023B1212060019).</p>", "<title>Author contributions</title>", "<p>P.X. and M.M. conceived and designed the study. P.X., M.M., L.Z. and J.L. collaborated in discussing the results and writing the paper. Y.W. provided the actual power generation and surface solar radiation data. Y.Y. and S.J. provided the ground-based observation data. All authors contributed edits.</p>", "<title>Peer review</title>", "<title>Peer review information</title>", "<p id=\"Par27\"><italic>Nature Communications</italic> thanks Sakshi Mishra and the other, anonymous, reviewer(s) for their contribution to the peer review of this work. A peer review file is available.</p>", "<title>Data availability</title>", "<p>The Himawari-8/9 data are available for free download from website [<ext-link ext-link-type=\"uri\" xlink:href=\"http://www.jma-net.go.jp/msc/en/\">http://www.jma-net.go.jp/msc/en/</ext-link>]. The photos above PV plants are available for free exported from Google Earth Pro. <xref ref-type=\"sec\" rid=\"Sec12\">Source data</xref> are provided with this paper.</p>", "<title>Code availability</title>", "<p>Data processing, drawing and FCMA were conducted using PYTHON. Those codes can be accessed at [<ext-link ext-link-type=\"uri\" xlink:href=\"https://zenodo.org/doi/10.5281/zenodo.10148796\">https://zenodo.org/doi/10.5281/zenodo.10148796</ext-link>]. The EPM code generated during the current study is available from the corresponding author upon request.</p>", "<title>Competing interests</title>", "<p id=\"Par28\">The authors declare no competing interests.</p>" ]

[ "<fig id=\"Fig1\"><label>Fig. 1</label><caption><title>Geographical distributions and photographs of ground-based sites.</title><p>PV plants (dark green small solid circles), 12 manual observation stations (red solid circles), 3 all-sky imager stations (blue solid circles), and 5 PV power test plants (yellow solid circles). The photos above the PV plants are exported from Google Earth Pro. Sangge Map Data: Google, mage ©2023 CNES/Airbus; Leling Map Data: Google, Image ©2023 Maxar Technologies; Xiaochengzi Map Data: Google, Image ©2023 Maxar Technologies; Lijiamen Map Data: Google, Image ©2023 CNES/Airbus; Shiziyan Map Data: Google, Image ©2023 CNES/Airbus. Source data are provided as a Source Data file.</p></caption></fig>", "<fig id=\"Fig2\"><label>Fig. 2</label><caption><title>RMSE and MBE between the predicted CF and the CF obtained by all-sky imagers.</title><p>Subfigures (<bold>a</bold>), (<bold>c</bold>), (<bold>e</bold>) and (<bold>b</bold>), (<bold>d</bold>), (<bold>f</bold>) are the RMSEs and MBEs for Beijing, Nanjing and Zhuhai all-sky imager stations, respectively. The different colored lines represent the results for different months, and the dashed black line represents the mean of all the lines. Subfigure (<bold>g</bold>) depicts the mean RMSE and MBE of three all-sky imager sites at different local time of one day. Source data are provided as a Source Data file.</p></caption></fig>", "<fig id=\"Fig3\"><label>Fig. 3</label><caption><title>Power and GTI of PV plants and corresponding 0–4 h CSR.</title><p>Time series of the power (MW), GTI (W·m⁻²) and predicted clear sky ratio (CSR) at (<bold>a</bold>) Sangge, (<bold>b</bold>) Leling, (<bold>c</bold>) Xiaochengzi, (<bold>d</bold>) Lijiamen, and (<bold>e</bold>) Shiziyan PV plants from 09:00 to 17:00 (local time or Beijing time) on each day in November of 2022. R1–R4 and R5–R8 indicate the correlation coefficients (R) of the predicted CSR with the power and the GTI for forecast leading time of 1–4 h, respectively. Note that the missing and invalid data are not shown. Source data are provided as a Source Data file.</p></caption></fig>", "<fig id=\"Fig4\"><label>Fig. 4</label><caption><title>CF prediction system schematic diagram.</title><p>The near real-time and dynamically updated prediction system for the cloud image and fraction nowcasting at the leading time of 0–4 h at the photovoltaic (PV) plants. For satellite images in this figure, red (white) color signifies high temperature (reflectance) and blue (black) color represents low temperature (reflectance). About Cloud Fraction pictures, The white, gray, light green, and dark green spots represent cloudy, probably cloudy, probably clear and clear pixel labels, respectively.</p></caption></fig>", "<fig id=\"Fig5\"><label>Fig. 5</label><caption><title>Enhanced PredRNN++.</title><p><bold>a</bold> Framework of the enhanced PredRNN++ model with five convolutional layers, and (<bold>b</bold>) visual illustration of the flow of input data in the spatial memory. In Fig. 5a, the Gradient Highway Unit (GHU; blue) is embedded between the first and the second convolutional layers, the horizontal red arrows denote the deep transition paths of the spatial memory, the vertical black arrows represent the updating direction of the temporal memory, and the blue parts indicate the gradient highway connecting the current time step directly with previous inputs. In Fig. 5b, “” denotes the convolution, the temporal memory, the spatial memory, and the hidden state. , and are concatenated to form a larger tensor, and then is generated by the convolution. The numbers below each memory indicate the dimensions of the corresponding tensors.</p></caption></fig>" ]

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id=\"IEq18\"><alternatives><tex-math id=\"M37\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${g}_{t}$$\\end{document}</tex-math><mml:math id=\"M38\"><mml:msub><mml:mrow><mml:mi>g</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq19\"><alternatives><tex-math id=\"M39\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${M}_{t}^{k}$$\\end{document}</tex-math><mml:math id=\"M40\"><mml:msubsup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mi>k</mml:mi></mml:mrow></mml:msubsup></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq20\"><alternatives><tex-math id=\"M41\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${M}_{t}^{k-1}$$\\end{document}</tex-math><mml:math id=\"M42\"><mml:msubsup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mi>k</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq21\"><alternatives><tex-math id=\"M43\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${M}_{t-1}^{5}$$\\end{document}</tex-math><mml:math id=\"M44\"><mml:msubsup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mn>5</mml:mn></mml:mrow></mml:msubsup></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq22\"><alternatives><tex-math id=\"M45\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${M}_{t}^{0}$$\\end{document}</tex-math><mml:math id=\"M46\"><mml:msubsup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msubsup></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ2\"><label>2</label><alternatives><tex-math id=\"M47\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\left(\\begin{array}{c}\\begin{array}{c}{g}_{t}\\\\ {i}_{t}\\end{array}\\\\ {f}_{t}\\end{array}\\right)=\\left(\\begin{array}{c}\\begin{array}{c}\\tanh \\\\ \\sigma \\end{array}\\\\ \\sigma \\end{array}\\right){W}_{1} \\circledcirc [{X}_{t},{H}_{t-1}^{k},{C}_{t-1}^{k}],$$\\end{document}</tex-math><mml:math id=\"M48\"><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"center\"><mml:mtable><mml:mtr><mml:mtd columnalign=\"center\"><mml:msub><mml:mrow><mml:mi>g</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign=\"center\"><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub></mml:mtd></mml:mtr></mml:mtable></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign=\"center\"><mml:msub><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced><mml:mo>=</mml:mo><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"center\"><mml:mtable><mml:mtr><mml:mtd columnalign=\"center\"><mml:mi>tanh</mml:mi></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign=\"center\"><mml:mi>σ</mml:mi></mml:mtd></mml:mtr></mml:mtable></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign=\"center\"><mml:mi>σ</mml:mi></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced><mml:msub><mml:mrow><mml:mi>W</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>⊚</mml:mo><mml:mrow><mml:mo>[</mml:mo><mml:mrow><mml:msub><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msubsup><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mi>k</mml:mi></mml:mrow></mml:msubsup><mml:mo>,</mml:mo><mml:msubsup><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mi>k</mml:mi></mml:mrow></mml:msubsup></mml:mrow><mml:mo>]</mml:mo></mml:mrow><mml:mo>,</mml:mo></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ3\"><label>3</label><alternatives><tex-math id=\"M49\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${C}_{t}^{k}={g}_{t}\\otimes {i}_{t}+{f}_{t}\\otimes {\\,C}_{t-1}^{k},$$\\end{document}</tex-math><mml:math id=\"M50\"><mml:msubsup><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mi>k</mml:mi></mml:mrow></mml:msubsup><mml:mo>=</mml:mo><mml:msub><mml:mrow><mml:mi>g</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo>⊗</mml:mo><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo>⊗</mml:mo><mml:msubsup><mml:mrow><mml:mspace width=\"0.25em\"/><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mi>k</mml:mi></mml:mrow></mml:msubsup><mml:mo>,</mml:mo></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ4\"><label>4</label><alternatives><tex-math id=\"M51\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\left(\\begin{array}{c}\\begin{array}{c}{g}_{t}^{,}\\\\ {i}_{t}^{,}\\end{array}\\\\ {f}_{t}^{,}\\end{array}\\right)=\\left(\\begin{array}{c}\\begin{array}{c}\\tanh \\\\ \\sigma \\end{array}\\\\ \\sigma \\end{array}\\right){W}_{2} \\circledcirc [{X}_{t},{C}_{t}^{k},{M}_{t}^{k-1}],$$\\end{document}</tex-math><mml:math id=\"M52\"><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"center\"><mml:mtable><mml:mtr><mml:mtd columnalign=\"center\"><mml:msubsup><mml:mrow><mml:mi>g</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo></mml:mrow></mml:msubsup></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign=\"center\"><mml:msubsup><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo></mml:mrow></mml:msubsup></mml:mtd></mml:mtr></mml:mtable></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign=\"center\"><mml:msubsup><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo></mml:mrow></mml:msubsup></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced><mml:mo>=</mml:mo><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"center\"><mml:mtable><mml:mtr><mml:mtd columnalign=\"center\"><mml:mi>tanh</mml:mi></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign=\"center\"><mml:mi>σ</mml:mi></mml:mtd></mml:mtr></mml:mtable></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign=\"center\"><mml:mi>σ</mml:mi></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced><mml:msub><mml:mrow><mml:mi>W</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mo>⊚</mml:mo><mml:mrow><mml:mo>[</mml:mo><mml:mrow><mml:msub><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msubsup><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mi>k</mml:mi></mml:mrow></mml:msubsup><mml:mo>,</mml:mo><mml:msubsup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mi>k</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup></mml:mrow><mml:mo>]</mml:mo></mml:mrow><mml:mo>,</mml:mo></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ5\"><label>5</label><alternatives><tex-math id=\"M53\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${M}_{t}^{k}={g}_{t}^{,}\\otimes {\\,i}_{t}^{,}+{f}_{t}^{,}\\otimes \\tanh ({W}_{3} \\circledcirc {\\,M}_{t}^{k-1}),$$\\end{document}</tex-math><mml:math id=\"M54\"><mml:msubsup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mi>k</mml:mi></mml:mrow></mml:msubsup><mml:mo>=</mml:mo><mml:msubsup><mml:mrow><mml:mi>g</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo></mml:mrow></mml:msubsup><mml:mo>⊗</mml:mo><mml:msubsup><mml:mrow><mml:mspace width=\"0.25em\"/><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo></mml:mrow></mml:msubsup><mml:mo>+</mml:mo><mml:msubsup><mml:mrow><mml:mi>f</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mo>,</mml:mo></mml:mrow></mml:msubsup><mml:mo>⊗</mml:mo><mml:mi>tanh</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mrow><mml:mi>W</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msub><mml:mo>⊚</mml:mo><mml:msubsup><mml:mrow><mml:mspace width=\"0.25em\"/><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mi>k</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>,</mml:mo></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ6\"><label>6</label><alternatives><tex-math id=\"M55\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${O}_{t}=\\tanh ({W}_{4} \\circledcirc [{X}_{t},{\\,C}_{t}^{k},{\\,M}_{t}^{k}]),$$\\end{document}</tex-math><mml:math id=\"M56\"><mml:msub><mml:mrow><mml:mi>O</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mi>tanh</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mrow><mml:mi>W</mml:mi></mml:mrow><mml:mrow><mml:mn>4</mml:mn></mml:mrow></mml:msub><mml:mo>⊚</mml:mo><mml:mrow><mml:mo>[</mml:mo><mml:mrow><mml:msub><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msubsup><mml:mrow><mml:mspace width=\"0.25em\"/><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mi>k</mml:mi></mml:mrow></mml:msubsup><mml:mo>,</mml:mo><mml:msubsup><mml:mrow><mml:mspace width=\"0.25em\"/><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mi>k</mml:mi></mml:mrow></mml:msubsup></mml:mrow><mml:mo>]</mml:mo></mml:mrow></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>,</mml:mo></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ7\"><label>7</label><alternatives><tex-math id=\"M57\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${H}_{t}^{k}={O}_{t}\\otimes \\tanh ({W}_{5} \\circledcirc [{\\,C}_{t}^{k},{\\,M}_{t}^{k}]),$$\\end{document}</tex-math><mml:math id=\"M58\"><mml:msubsup><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mi>k</mml:mi></mml:mrow></mml:msubsup><mml:mo>=</mml:mo><mml:msub><mml:mrow><mml:mi>O</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo>⊗</mml:mo><mml:mi>tanh</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mrow><mml:mi>W</mml:mi></mml:mrow><mml:mrow><mml:mn>5</mml:mn></mml:mrow></mml:msub><mml:mo>⊚</mml:mo><mml:mrow><mml:mo>[</mml:mo><mml:mrow><mml:msubsup><mml:mrow><mml:mspace width=\"0.25em\"/><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mi>k</mml:mi></mml:mrow></mml:msubsup><mml:mo>,</mml:mo><mml:msubsup><mml:mrow><mml:mspace width=\"0.25em\"/><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mi>k</mml:mi></mml:mrow></mml:msubsup></mml:mrow><mml:mo>]</mml:mo></mml:mrow></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>,</mml:mo></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq23\"><alternatives><tex-math id=\"M59\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${W}_{1{{{{{\\rm{\\hbox{--}}}}}}}5}$$\\end{document}</tex-math><mml:math id=\"M60\"><mml:msub><mml:mrow><mml:mi>W</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn><mml:mi mathvariant=\"normal\">–</mml:mi><mml:mn>5</mml:mn></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ8\"><label>8</label><alternatives><tex-math id=\"M61\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${H}_{t}^{k},{C}_{t}^{k},{M}_{t}^{k}={CauLST}{M}_{k}({H}_{t}^{k-1},{H}_{t-1}^{k},{C}_{t-1}^{k},{M}_{t}^{k-1}),$$\\end{document}</tex-math><mml:math id=\"M62\"><mml:msubsup><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mi>k</mml:mi></mml:mrow></mml:msubsup><mml:mo>,</mml:mo><mml:msubsup><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mi>k</mml:mi></mml:mrow></mml:msubsup><mml:mo>,</mml:mo><mml:msubsup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mi>k</mml:mi></mml:mrow></mml:msubsup><mml:mo>=</mml:mo><mml:mi>C</mml:mi><mml:mi>a</mml:mi><mml:mi>u</mml:mi><mml:mi>L</mml:mi><mml:mi>S</mml:mi><mml:mi>T</mml:mi><mml:msub><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi>k</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msubsup><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mi>k</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:mo>,</mml:mo><mml:msubsup><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mi>k</mml:mi></mml:mrow></mml:msubsup><mml:mo>,</mml:mo><mml:msubsup><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mi>k</mml:mi></mml:mrow></mml:msubsup><mml:mo>,</mml:mo><mml:msubsup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mi>k</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>,</mml:mo></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq24\"><alternatives><tex-math id=\"M63\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${H}_{t}^{k-1}$$\\end{document}</tex-math><mml:math id=\"M64\"><mml:msubsup><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mi>k</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq25\"><alternatives><tex-math id=\"M65\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${X}_{t}$$\\end{document}</tex-math><mml:math id=\"M66\"><mml:msub><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq26\"><alternatives><tex-math id=\"M67\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${Z}_{t}$$\\end{document}</tex-math><mml:math id=\"M68\"><mml:msub><mml:mrow><mml:mi>Z</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq27\"><alternatives><tex-math id=\"M69\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$W$$\\end{document}</tex-math><mml:math id=\"M70\"><mml:mi>W</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq28\"><alternatives><tex-math id=\"M71\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${S}_{t}$$\\end{document}</tex-math><mml:math id=\"M72\"><mml:msub><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq29\"><alternatives><tex-math id=\"M73\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${P}_{t}$$\\end{document}</tex-math><mml:math id=\"M74\"><mml:msub><mml:mrow><mml:mi>P</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq30\"><alternatives><tex-math id=\"M75\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${Z}_{t-1}$$\\end{document}</tex-math><mml:math id=\"M76\"><mml:msub><mml:mrow><mml:mi>Z</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ9\"><label>9</label><alternatives><tex-math id=\"M77\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${P}_{t}=\\tanh ({W}_{{pz}} \\circledcirc {Z}_{t-1}+{W}_{{ph}} \\circledcirc {H}_{t}^{1}),$$\\end{document}</tex-math><mml:math id=\"M78\"><mml:msub><mml:mrow><mml:mi>P</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mi>tanh</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mrow><mml:mi>W</mml:mi></mml:mrow><mml:mrow><mml:mi>p</mml:mi><mml:mi>z</mml:mi></mml:mrow></mml:msub><mml:mo>⊚</mml:mo><mml:msub><mml:mrow><mml:mi>Z</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mi>W</mml:mi></mml:mrow><mml:mrow><mml:mi>p</mml:mi><mml:mi>h</mml:mi></mml:mrow></mml:msub><mml:mo>⊚</mml:mo><mml:msubsup><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msubsup></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>,</mml:mo></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ10\"><label>10</label><alternatives><tex-math id=\"M79\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${S}_{t}=\\sigma ({W}_{{sz}} \\circledcirc {Z}_{t-1}+{W}_{{sh}} \\circledcirc {H}_{t}^{1}),$$\\end{document}</tex-math><mml:math id=\"M80\"><mml:msub><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mi>σ</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mrow><mml:mi>W</mml:mi></mml:mrow><mml:mrow><mml:mi>s</mml:mi><mml:mi>z</mml:mi></mml:mrow></mml:msub><mml:mo>⊚</mml:mo><mml:msub><mml:mrow><mml:mi>Z</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mi>W</mml:mi></mml:mrow><mml:mrow><mml:mi>s</mml:mi><mml:mi>h</mml:mi></mml:mrow></mml:msub><mml:mo>⊚</mml:mo><mml:msubsup><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msubsup></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>,</mml:mo></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ11\"><label>11</label><alternatives><tex-math id=\"M81\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${Z}_{t}={S}_{t}\\otimes {P}_{t}+(1-{S}_{t})\\otimes {Z}_{t-1}$$\\end{document}</tex-math><mml:math id=\"M82\"><mml:msub><mml:mrow><mml:mi>Z</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:msub><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo>⊗</mml:mo><mml:msub><mml:mrow><mml:mi>P</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mn>1</mml:mn><mml:mo>−</mml:mo><mml:msub><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>⊗</mml:mo><mml:msub><mml:mrow><mml:mi>Z</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ12\"><label>12</label><alternatives><tex-math id=\"M83\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${Z}_{t}={GHU}({H}_{t}^{1},{Z}_{t-1})$$\\end{document}</tex-math><mml:math id=\"M84\"><mml:msub><mml:mrow><mml:mi>Z</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mi>G</mml:mi><mml:mi>H</mml:mi><mml:mi>U</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msubsup><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mi>Z</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ13\"><label>13</label><alternatives><tex-math id=\"M85\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${H}_{t}^{1},{C}_{t}^{1},{M}_{t}^{1}={CauLST}{M}_{1}({X}_{t},{H}_{t-1}^{1},{C}_{t-1}^{1},{M}_{t-1}^{5})$$\\end{document}</tex-math><mml:math id=\"M86\"><mml:msubsup><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:mo>,</mml:mo><mml:msubsup><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:mo>,</mml:mo><mml:msubsup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:mo>=</mml:mo><mml:mi>C</mml:mi><mml:mi>a</mml:mi><mml:mi>u</mml:mi><mml:mi>L</mml:mi><mml:mi>S</mml:mi><mml:mi>T</mml:mi><mml:msub><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msubsup><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:mo>,</mml:mo><mml:msubsup><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:mo>,</mml:mo><mml:msubsup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mn>5</mml:mn></mml:mrow></mml:msubsup></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ14\"><label>14</label><alternatives><tex-math id=\"M87\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${Z}_{t}={GHU}({H}_{t}^{1},{Z}_{t-1})$$\\end{document}</tex-math><mml:math id=\"M88\"><mml:msub><mml:mrow><mml:mi>Z</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mi>G</mml:mi><mml:mi>H</mml:mi><mml:mi>U</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msubsup><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mi>Z</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ15\"><label>15</label><alternatives><tex-math id=\"M89\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${H}_{t}^{2},{C}_{t}^{2},{M}_{t}^{2}={CauLST}{M}_{2}({Z}_{t},{H}_{t-1}^{1},{C}_{t-1}^{1},{M}_{t}^{1}),$$\\end{document}</tex-math><mml:math id=\"M90\"><mml:msubsup><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo>,</mml:mo><mml:msubsup><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo>,</mml:mo><mml:msubsup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo>=</mml:mo><mml:mi>C</mml:mi><mml:mi>a</mml:mi><mml:mi>u</mml:mi><mml:mi>L</mml:mi><mml:mi>S</mml:mi><mml:mi>T</mml:mi><mml:msub><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mrow><mml:mi>Z</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msubsup><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:mo>,</mml:mo><mml:msubsup><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:mo>,</mml:mo><mml:msubsup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msubsup></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>,</mml:mo></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ16\"><label>16</label><alternatives><tex-math id=\"M91\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${H}_{t}^{3},{C}_{t}^{3},{M}_{t}^{3}={CauLST}{M}_{3}({H}_{t}^{2},{H}_{t-1}^{3},{C}_{t-1}^{3},{M}_{t}^{2}),$$\\end{document}</tex-math><mml:math id=\"M92\"><mml:msubsup><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msubsup><mml:mo>,</mml:mo><mml:msubsup><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msubsup><mml:mo>,</mml:mo><mml:msubsup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msubsup><mml:mo>=</mml:mo><mml:mi>C</mml:mi><mml:mi>a</mml:mi><mml:mi>u</mml:mi><mml:mi>L</mml:mi><mml:mi>S</mml:mi><mml:mi>T</mml:mi><mml:msub><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msubsup><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mo>,</mml:mo><mml:msubsup><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msubsup><mml:mo>,</mml:mo><mml:msubsup><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msubsup><mml:mo>,</mml:mo><mml:msubsup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msubsup></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>,</mml:mo></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ17\"><label>17</label><alternatives><tex-math id=\"M93\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${H}_{t}^{4},{C}_{t}^{4},{M}_{t}^{4}={CauLST}{M}_{4}({H}_{t}^{3},{H}_{t-1}^{4},{C}_{t-1}^{4},{M}_{t}^{3}),$$\\end{document}</tex-math><mml:math id=\"M94\"><mml:msubsup><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mn>4</mml:mn></mml:mrow></mml:msubsup><mml:mo>,</mml:mo><mml:msubsup><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mn>4</mml:mn></mml:mrow></mml:msubsup><mml:mo>,</mml:mo><mml:msubsup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mn>4</mml:mn></mml:mrow></mml:msubsup><mml:mo>=</mml:mo><mml:mi>C</mml:mi><mml:mi>a</mml:mi><mml:mi>u</mml:mi><mml:mi>L</mml:mi><mml:mi>S</mml:mi><mml:mi>T</mml:mi><mml:msub><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mn>4</mml:mn></mml:mrow></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msubsup><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msubsup><mml:mo>,</mml:mo><mml:msubsup><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mn>4</mml:mn></mml:mrow></mml:msubsup><mml:mo>,</mml:mo><mml:msubsup><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mn>4</mml:mn></mml:mrow></mml:msubsup><mml:mo>,</mml:mo><mml:msubsup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msubsup></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>,</mml:mo></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ18\"><label>18</label><alternatives><tex-math id=\"M95\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\,{H}_{t}^{5},{C}_{t}^{5},{M}_{t}^{5}={CauLST}{M}_{5}({H}_{t}^{4},{H}_{t-1}^{5},{C}_{t-1}^{5},{M}_{t}^{4}),$$\\end{document}</tex-math><mml:math id=\"M96\"><mml:mspace width=\"0.25em\"/><mml:msubsup><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mn>5</mml:mn></mml:mrow></mml:msubsup><mml:mo>,</mml:mo><mml:msubsup><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mn>5</mml:mn></mml:mrow></mml:msubsup><mml:mo>,</mml:mo><mml:msubsup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mn>5</mml:mn></mml:mrow></mml:msubsup><mml:mo>=</mml:mo><mml:mi>C</mml:mi><mml:mi>a</mml:mi><mml:mi>u</mml:mi><mml:mi>L</mml:mi><mml:mi>S</mml:mi><mml:mi>T</mml:mi><mml:msub><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mn>5</mml:mn></mml:mrow></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msubsup><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mn>4</mml:mn></mml:mrow></mml:msubsup><mml:mo>,</mml:mo><mml:msubsup><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mn>5</mml:mn></mml:mrow></mml:msubsup><mml:mo>,</mml:mo><mml:msubsup><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mn>5</mml:mn></mml:mrow></mml:msubsup><mml:mo>,</mml:mo><mml:msubsup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mn>4</mml:mn></mml:mrow></mml:msubsup></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>,</mml:mo></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq31\"><alternatives><tex-math id=\"M97\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${Z}_{t}$$\\end{document}</tex-math><mml:math id=\"M98\"><mml:msub><mml:mrow><mml:mi>Z</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq32\"><alternatives><tex-math id=\"M99\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${Z}_{t-1}$$\\end{document}</tex-math><mml:math id=\"M100\"><mml:msub><mml:mrow><mml:mi>Z</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq33\"><alternatives><tex-math id=\"M101\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${H}_{t}^{1}$$\\end{document}</tex-math><mml:math id=\"M102\"><mml:msubsup><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msubsup></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq34\"><alternatives><tex-math id=\"M103\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${S}_{t}$$\\end{document}</tex-math><mml:math id=\"M104\"><mml:msub><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq35\"><alternatives><tex-math id=\"M105\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${M}_{n-1}^{5}$$\\end{document}</tex-math><mml:math id=\"M106\"><mml:msubsup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mn>5</mml:mn></mml:mrow></mml:msubsup></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq36\"><alternatives><tex-math id=\"M107\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${C}_{n}^{1}$$\\end{document}</tex-math><mml:math id=\"M108\"><mml:msubsup><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msubsup></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq37\"><alternatives><tex-math id=\"M109\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${H}_{n}^{1}$$\\end{document}</tex-math><mml:math id=\"M110\"><mml:msubsup><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msubsup></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ19\"><label>19</label><alternatives><tex-math id=\"M111\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${{{{{\\rm{RMSE}}}}}}=\\sqrt{\\frac{1}{N}\\times \\mathop{\\sum }\\limits_{i=1}^{N}{({P}_{i}-{T}_{i})}^{2}},$$\\end{document}</tex-math><mml:math id=\"M112\"><mml:mi mathvariant=\"normal\">RMSE</mml:mi><mml:mo>=</mml:mo><mml:msqrt><mml:mrow><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mi>N</mml:mi></mml:mrow></mml:mfrac><mml:mo>×</mml:mo><mml:munderover accent=\"false\" accentunder=\"false\"><mml:mrow><mml:mo>∑</mml:mo></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mi>N</mml:mi></mml:mrow></mml:munderover><mml:msup><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mrow><mml:mi>P</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo>−</mml:mo><mml:msub><mml:mrow><mml:mi>T</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:msqrt><mml:mo>,</mml:mo></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ20\"><label>20</label><alternatives><tex-math id=\"M113\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} 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[ "<supplementary-material content-type=\"local-data\" id=\"MOESM1\"></supplementary-material>", "<supplementary-material content-type=\"local-data\" id=\"MOESM2\"></supplementary-material>", "<supplementary-material content-type=\"local-data\" id=\"MOESM3\"></supplementary-material>", "<supplementary-material content-type=\"local-data\" id=\"MOESM4\"></supplementary-material>" ]

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[ "<media xlink:href=\"41467_2023_44666_MOESM1_ESM.pdf\"><caption><p>Supplementary Information</p></caption></media>", "<media xlink:href=\"41467_2023_44666_MOESM2_ESM.pdf\"><caption><p>Peer review file</p></caption></media>", "<media xlink:href=\"41467_2023_44666_MOESM3_ESM.pdf\"><caption><p>Reporting Summary</p></caption></media>", "<media xlink:href=\"41467_2023_44666_MOESM4_ESM.zip\"><caption><p>Source data</p></caption></media>" ]

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[ "<title>Introduction</title>", "<p id=\"Par2\">Long fusion surgery for adult spinal deformity (ASD), performed only in a limited number of centers more than a decade ago, has rapidly spread and is now a standard and widely performed procedure^{##REF##31305254##1##}. ASD surgery was primarily performed for de novo scoliosis in the early days. Later, ASD became a broad disease concept that included sagittal imbalance as a surgical target. Thus, although ASD has complex conditions, patients with symptoms that warrant surgical treatment should have specific common problems.</p>", "<p id=\"Par3\">The Scoliosis Research Society-22 Patient Questionnaire (SRS-22) is a standard questionnaire used to evaluate the treatment of scoliosis^{##REF##12544958##2##}. The SRS-22 is sometimes used to assess ASD as well, because no ASD-specific scale currently exists. However, the questions in the SRS-22 were designed primarily for adolescent idiopathic scoliosis (AIS). AIS and ASD have different ages of onset, various pathologies, and main complaints. In addition, in AIS, the lowest end of fixation is usually more proximal than L3, whereas, in ASD, the level of fixation often includes the pelvis, which is often accompanied by postoperative mobility restrictions^{##REF##24785491##3##,##REF##29215502##4##} (Fig. ##FIG##0##1##). Recently, Hart et al. developed the lumbar stiffness disability index to evaluate the limitation of motion of the spine due to long fusion surgery^{##REF##25202930##5##}. They called the restriction for activities of daily living (ADL) due to long fusion the collateral outcome. There is a trade-off relationship, so to speak, between improving pain due to fusion and restriction of range of motion. This trade-off is considered to be well established if the patient’s needs are met^{##REF##27902557##6##}. Thus, ASD presents a unique condition among spinal disorders that has elements of scoliosis but also kyphosis, as well as pain and limited postoperative range of motion. Although surgery for ASD is becoming more widespread, some researchers are concerned about the cost of the procedure and the high complication rate^{##REF##25963608##7##}. Conversely, conservative treatment of ASD includes medication, orthotics, Nordic walking canes, and walkers. These conservative treatments have the advantage of being less risky and less expensive than surgery and do not cause a postoperative range of motion limitations. However, conservative treatment could be less effective with respect to improving posture and pain. Furthermore, the use of a cane may be inconvenient for household activities because both hands are occupied when walking^{##REF##34145338##8##}. Currently, there is no HR-PRO that evaluates these life inconveniences from the perspective of ASD patients.</p>", "<p id=\"Par4\">Therefore, we thought that a specific scale was needed to evaluate ASD. This study aimed to create a disease-specific patient-reported outcome measure (PROM) for ASD.</p>" ]

[ "<title>Methods</title>", "<title>Patients</title>", "<p id=\"Par5\">This study was a multicenter, self-report questionnaire survey conducted at two spine centers. In total, 106 patients were included: 97 patients who underwent long fusion surgery between 2007 and 2020 and nine patients who were undergoing conservative treatment and considering surgery for spinal deformity. The conservative patients had spinal deformities but preferred conservative treatment because their clinical symptoms were milder than those of the operative patients. A questionnaire consisting of 29 questions was mailed to these patients, and they were asked to complete and return it. Patients who had undergone surgery were asked to answer both preoperative and postoperative conditions. Conservatively treated patients were asked to answer questions about their current condition. A five-point satisfaction rating scale for surgery and Short-Form-8 (the physical component summary; PCS, and the mental component summary; MCS) were enclosed for criterion-related validation.</p>", "<p id=\"Par6\">Of the 106 patients, eight did not receive the mailing due to a change of address. The 98 patients (89 surgical patients) who responded were included in the study (Fig. ##FIG##1##2##). Long fusion was defined as the fusion of five or more vertebrae, including the lumbar spine. Fixation across the sacroiliac joint to the pelvis was counted as one vertebral segment. On imaging evaluation, all patients had a coronal plane Cobb angle &gt; 30°, SVA &gt; 40 mm, or pelvic tilt &gt; 20°^{##REF##22045006##9##}.</p>", "<title>Selection of 29 questions</title>", "<p id=\"Par7\">COnsensus-based Standards for the selection of health Measurement Instruments (COSMIN) aimed at improving the selection of PROM in research and clinical practice and some guidelines exist. We conducted this study in accordance with the COSMIN guidelines^{##REF##29435801##10##}. Content validity is the most important measurement property of PROM. It is the degree to which the content of an instrument is an adequate reflection of the construct to be measured. The criteria of content validity include the relevance, comprehensiveness, and comprehensibility of the PROM for the target population. We conducted a literature search to select questions relevant to ASD. We assumed that ADL, appearance, pain, mental health, and satisfaction would be the assessment items necessary to capture the disease concept of ASD^{##REF##31305254##1##,##REF##24785491##3##,##REF##27902557##6##,##REF##33822320##11##–##REF##30661199##13##}.</p>", "<p id=\"Par8\">To develop the comprehensive questions, we reviewed a wide variety of existing questionnaires (Table ##TAB##0##1##), including Short-Form-36^{##UREF##0##14##}, patient-reported outcomes measurement information system (PROMIS)^{##REF##30600494##15##}, Oswestry disability index (ODI)^{##REF##11064536##16##}, Roland–Morris questionnaire^{##REF##11124727##17##}, SRS-22^{##REF##12544958##2##}, Japanese Orthopedic association back pain evaluation questionnaire (JOABPEQ)^{##REF##29792993##18##}, Western Ontario and McMaster Universities Osteoarthritis Index (WOMAC)^{##REF##3068365##19##}, Knee Society Score^{##REF##22045067##20##}, Bath Ankylosing Spondylitis Functional Index (BASFI)^{##REF##7699629##21##}, Health Assessment Questionnaire (HAQ)^{##REF##12831398##22##}, pain disability assessment scale^{##REF##21178590##23##}, Zurich claudication questionnaire (ZCQ)^{##REF##8779009##24##}, EuroQol 5-dimensions 5-levels (EQ5D)^{##REF##23900659##25##}, lumbar stiffness disability index (LSDI)^{##REF##25202930##5##}, 25-question geriatric locomotive function scale (Locomo-25)^{##REF##22222445##26##}, gastroesophageal reflux disease questionnaire (GerdQ)^{##REF##19737151##27##}, and the Frequency Scale for the symptoms of gastroesophageal reflux disease (FSSG)^{##REF##15565409##28##}.</p>", "<p id=\"Par9\">In total, 390 items were placed into the following categories by content: (1) pain, (2) appearance, (3) sleeping, getting up from bed or floor and bedtime-related activities (4) sitting, standing up, and other sitting-related activities, (5) standing, walking, and stairs, (6) toilet and bathing-related activities, (7) dressing-related activities, (8) transportation, (10) housework, (11) sports, (12) social activities, (13) meals, and (14) mental health.</p>", "<p id=\"Par10\">From these categories, we extracted 114 items that were considered useful for assessing ASD (Fig. ##FIG##2##3##). Sexual life, although an important item, was not included because of the expected large number of non-responses^{##REF##15565409##28##}. To ensure the relevance of questions to ASD in content validity, eight surgeons with extensive experience in operating on patients with ASD gave these 114 items a score from 3 to 0 according to their level of importance. We used the total score as a reference and selected 29 question items after discussion among the senior surgeons (Table ##TAB##1##2##). We modified detailed wording partially modified as appropriate. To examine the results comprehensibility, the developed questionnaire was given to three patients and one nurse, who reviewed the items in terms of text, meaning, and ambiguity and who provided feedback. Responses were on a five-point scale^{##REF##26826003##29##}, with an additional free-text field.</p>", "<title>Ethics statement</title>", "<p id=\"Par11\">The study was conducted in accordance with the ethical standards of the Declaration of Helsinki. The study was approved by the local ethical review board (Osaka University Hospital Ethics Review Committee. No.11360). Written informed consent was obtained from each patient.</p>", "<title>Statistical analysis</title>", "<p id=\"Par12\">The COSMIN guidelines introduce classical test theory and Rasch analysis for construct validation. We used classical test theory and factor analysis. Factor analysis was used to reduce and group the questions in order to create a valid, simple, and easy-to-use questionnaire. An exploratory factor analysis was performed using the maximum likelihood method on data from a total of 98 patients, including 89 postoperative responses and nine conservative cases. The number of factors was determined using the scree method. Because correlations between factors can be assumed, oblique rotation was performed using the Promax method. Finally, reliability was evaluated for content consistency using Cronbach’s coefficient alpha.</p>", "<title>Score calculation formula</title>", "<p id=\"Par13\">Factor score coefficients obtained from factor analysis were used as a reference to correct the coefficients so that the scale’s total score ranged from 0 to 100. Specifically, individual items were weighted so that the difference between the minimum and maximum factor scores was approximately 100 depending on the choice of response^{##UREF##0##14##}. However, we provided greater weight to those questions that clinicians deemed important. For example, 0 represented a limited health status and 100 represented an excellent health status.</p>", "<title>Comparison of scores and responsiveness</title>", "<p id=\"Par14\">We compared the scores of the created scale, the PCS, and the MCS before and after surgery (paired t-test). Similarly, we compared the scale scores between the operated and conservative groups (unpaired t-test). We calculated Cohen’s d effect size by taking the difference between two means and dividing it by the standard deviation of the data. Cohen’s d effect size was used to evaluate the internal responsiveness of the scales. Next, we calculated Spearman’s correlation coefficients between the five satisfaction levels and the amount of score change on each scale. The external responsiveness of the scales was evaluated using Spearman’s correlation coefficients. An effect size of 0.2–0.49 was considered small, an effect size of 0.5–0.79 was considered moderate, and an effect size of 0.80 or greater was considered large^{##UREF##1##30##}. A correlation coefficient of 0.2–0.39 was considered weak, a correlation coefficient of 0.4–0.69 was considered moderate, and a correlation coefficient of 0.70 or greater was considered strong. A p-value &lt; 0.05 was considered statistically significant for two-tailed tests. SPSS Statistics (version 20; IBM, Armonk, NY, USA) was used for statistical analysis.</p>", "<title>External validation</title>", "<p id=\"Par15\">We collected new patients with ASD from another institution for external validation. We applied our ASD disease-specific scale for these patients and compared the results with the internal validation data.</p>" ]

[ "<title>Results</title>", "<title>Demographics of the patients</title>", "<p id=\"Par16\">Of a total of 98 patients, 88 were women. The mean age of the 89 operative patients was 68 ± 7 years, and the mean time since the last surgery was 56 ± 35 months (Table ##TAB##2##3##). The mean number of fixed vertebral segments was 10 ± 3, including the sacrum or pelvis, in 76 patients (85%). The preoperative PCS was 31 ± 7 and improved to 41 ± 8 postoperatively (p &lt; 0.0001). Postoperative satisfaction was 23 (26%) very satisfied, 42 (47%) satisfied, 18 (20%) neither satisfied nor dissatisfied, and 6 (7%) dissatisfied.</p>", "<title>Response of the patients</title>", "<p id=\"Par17\">The results of the responses to each question are shown in Table ##TAB##3##4##, and the correlation coefficients are shown in Table ##TAB##4##5##. Seven patients had a free-text response of not performing Q23 heavy housework. Therefore, Q23 heavy housework was deemed inappropriate and excluded from the factor analysis. Regarding Q16 walking distance, four patients answered that they did not know the distance. Because there was a strong correlation between Q16 walking distance and Q17 walking time, we considered that Q17 walking time could be substituted for Q16 walking distance and excluded Q16. Factor analysis was conducted on the remaining 27 questions.</p>", "<title>Factor analysis</title>", "<p id=\"Par18\">The two-factor solution was adopted based on the decay status of the eigenvalues (scree criteria). The proportion of the total variance of the 27 items explained by the two factors before rotation was 47%.</p>", "<p id=\"Par19\">Each item was ordered by factor loadings (Table ##TAB##5##6##). The first factor was named the main symptom because many of the symptoms were related to the patient’s primary complaints, such as the ability to do housework and walk, including Q25 dishwashing, Q21 laundry, Q20 shelving, and Q17 walking. The loadings for Q1 appearance, Q2 back pain, and Q29 anxiety were relatively low but were included because we considered these questions essential. We selected Q19 ride, Q24 garbage disposal, and Q15 standing as the remaining questions, according to factor loadings. Because Q22 light housework was strongly correlated with Q25 dishwashing (r = 0.82) and Q21 laundry (r = 0.80) and was considered to refer to the same thing, we excluded Q22. A total of 10 question items (Q1 appearance, Q2 back pain, Q15 standing, Q17 walking, Q19 ride, Q20 shelving, Q21 laundry, Q24 garbage disposal, Q25 dishwashing, Q29 anxiety) were used for the main symptom factor.</p>", "<p id=\"Par20\">The second factor was named the collateral symptom because many items were related to postoperative limitation of movement, such as Q12 socks wearing and Q9 picking up. Because wearing Q11 pants and Q12 socks were highly correlated (r = 0.76), we excluded Q11 because Q12 socks could be substituted for Q11 pants. According to factor loadings, we selected five question items (Q7 standing up floors, Q8 toilet, Q9 picking up, Q10 washing, Q12 socks) as collateral symptom factors.</p>", "<title>Reliability</title>", "<title>Internal consistency</title>", "<p id=\"Par21\">The Cronbach’s alpha coefficient was 0.90 for the main symptom and 0.84 for the collateral symptom.</p>", "<title>Calculation of scores</title>", "<p id=\"Par22\">The factor score coefficients were used as weighting coefficients for each question, rounding the factor score coefficients to whole numbers to distribute the total scale score was distributed from 0 to 100. Because Q1 appearance and Q2 back pain are particularly important items, we gave them the same coefficients as Q25 dishwashing, which had a higher factor score coefficient. The better symptoms were set to 100 and the worse symptoms were set to 0. The calculation formulas are shown below (Supplement File ##SUPPL##0##1##).</p>", "<title>Score and responsiveness</title>", "<title>Score change</title>", "<p id=\"Par23\">The scores calculated based on the above formula are shown in Table ##TAB##6##7##. Comparing the operative and conservative groups, the main symptom of the operative group was 47 ± 21 preoperatively, while the conservative group was 63 ± 15. The operative group had significantly worse preoperative main symptoms than the conservative group (p = 0.029).</p>", "<p id=\"Par24\">However, the main symptom of the surgical group significantly improved to 70 ± 22 after surgery (p &lt; 0.0001), exceeding those of the conservative group. As a result of the surgical improvement, there was no significant difference between the postoperative main symptom of the operative group and the main symptom of the conservative group (p = 0.3).</p>", "<p id=\"Par25\">The mean collateral symptom score in the operative group worsened from 76 ± 25 preoperatively to 60 ± 25 postoperatively (p &lt; 0.0001). The preoperative collateral symptom score in the operative group was significantly worse than that in the conservative group, 92 ± 12 (p = 0.005).</p>", "<title>Effect size</title>", "<p id=\"Par26\">The effect size measured by Cohen’s d was 1.09, indicating a large effect size, for the main symptom for comparison of the preoperative and the postoperative score (Table ##TAB##6##7##). In the same comparison, the effect size of the collateral symptom was 0.65 (moderate), and that of the PCS was 1.26 (large).</p>", "<p id=\"Par27\">In a comparison of operative and conservative groups, the effect size was 0.77 for the main symptom and 0.67 for the collateral symptom, indicating a moderate effect size.</p>", "<title>Correlation coefficient</title>", "<p id=\"Par28\">The Spearman’s correlation coefficient between satisfaction and the amount of score change was 0.48 (p &lt; 0.001) for the main symptom and 0.38 for the PCS, both showing a moderate correlation (Table ##TAB##7##8##). The correlation coefficient between the main symptom and the PCS was 0.43, indicating a moderate correlation (p = 0.002).</p>", "<title>Ceiling and floor effects</title>", "<p id=\"Par29\">The main symptoms had no floor or ceiling effect either preoperatively or postoperatively (Figs. ##FIG##3##4##, ##FIG##4##5##). Conversely, the collateral symptom had a ceiling effect preoperatively, but no floor effect postoperatively (Figs. ##FIG##5##6##, ##FIG##6##7##).</p>", "<title>External validation</title>", "<p id=\"Par30\">We added a new sample of 30 surgical patients with ASD in another facility for a disease-specific scale for ASD that we had created. This scale consisted of 10 main symptom and 5 collateral symptom questions, as described above. Total scores were calculated using the above formulas (Supplementary File ##SUPPL##0##1##). The SF-8 and satisfaction scale were enclosed, as well as the date when the scale was created.</p>", "<p id=\"Par31\">Twenty-five people responded (Table ##TAB##8##9##). There was a significant difference in the age and fixation range between 25 patients for external validation and 89 patients for internal validation. However, no other background information was significantly different. The main symptom improved from 56 ± 19 preoperatively to 76 ± 19 postoperatively with an effect size of 1.05. The collateral symptom worsened from 75 ± 23 preoperatively to 64 ± 24 postoperatively with an effect size of 0.48. In both domains, the effect size was not different from the effect size at the time of scale creation, indicating the robustness of the scale.</p>" ]

[ "<title>Discussion</title>", "<p id=\"Par32\">This study is the first to use factor analysis to create a disease-specific scale for ASD. The most important point in a scale is to be able to measure the construct it is trying to measure^{##REF##29435801##10##,##UREF##2##31##}. Factor analysis is a technique used to explore and validate constructs, and is often used to create scales. “Intelligence” and “health” are examples of constructs that cannot be observed or measured directly. However, it is considered that they can be measured indirectly through multiple behaviors and events related to the construct^{##UREF##2##31##}.</p>", "<p id=\"Par33\">In the present study, factor analysis allowed us to detect two factors that constitute the construct of ASD. The first factor was named the main symptom because it reflected the patient’s main problems, such as appearance, pain, and housework activities. The second factor was named the collateral symptom, and was related to postoperative movement limitations such as putting on socks, picking up, and using the toilet. We considered that these two factors could measure the construct of ASD. The Cronbach’s alpha coefficients for each were 0.90 and 0.84, respectively, and had reliabilities that were acceptable for a clinically used measure.</p>", "<p id=\"Par34\">In this study, the scale scores of the main symptom and the collateral symptom were calculated by weighting them according to the factor score coefficients. Both the main symptom and the collateral symptom showed significant differences in preoperative and postoperative comparisons of the surgery groups, and the effect size was large. Comparing preoperative scores of the surgery group and the conservative group also showed significant differences, and the effect size was moderate. In addition, the main symptom was significantly correlated with satisfaction and the PCS. These results indicated that the created scale had adequate responsiveness and criterion-related validity.</p>", "<p id=\"Par35\">The items included in the factor analysis in this study were selected from various representative scales by physicians with extensive experience in ASD surgery and had content validity.</p>", "<p id=\"Par36\">The SRS-22 is a commonly used outcome for assessing ASD, but several problems were noted^{##REF##35349122##32##}. Faraj et al. reviewed the strengths, weaknesses, and gaps of current outcomes in measuring ASD outcomes^{##REF##28534221##33##}. According to their study, the most frequently used outcome was the ODI, with the SRS-22s. However, they stated that both the ODI and the SRS-22 had weaknesses in their use to assess ASD. The ODI is a low back pain-specific questionnaire and does not necessarily include the concept of deformity. Conversely, the SRS-22 was developed for AIS, which is less functionally impaired and, therefore, is less relevant for ASD, which seeks to restore pain and quality of life. Faraj et al. stated that there was an overlap between the two outcomes and the need to develop a core outcome set that is more specific to the assessment of ASD.</p>", "<p id=\"Par37\">Mannion et al. performed a factor analysis of the SRS-22 on ASD patients^{##REF##28866740##34##}. They found a poor fit for four questions on the SRS-22: Q3 (nervous person), Q14 (personal relationship), Q15 (financial difficulties), and Q17 (sick days). They recommended the deletion of these four questions.</p>", "<p id=\"Par38\">Zaina et al. compared the newly developed Italian spine youth quality of life (ISYQOL) with the SRS-22 using Rasch analysis^{##REF##37568473##35##}. According to this group, Q15 (financial difficulties) in the SRS-22 was a poor fit, and they recommended 21 items except for that one. By excluding this item, the revised SRS-22 showed construct validity comparable with the ISYQOL.</p>", "<p id=\"Par39\">Scheer et al. devised a patient generated index, a questionnaire that patients were asked to fill out freely^{##REF##28414170##36##}. The top 10 concerns of patients with ASD were walking, activities, posture, pain, sports, housework, relationships, gardening, sleeping, and traveling. The 29 items we selected almost covered these items. Of these items, about sports, some patients in this study indicated in their free-text sections that they did not engage in these activities. The term “sports” covers an extensive range, from light gymnastics and walking to running and swimming. We did not select Q26 sports because the factor loading was small and also because different people perceived this item differently.</p>", "<p id=\"Par40\">Housework activities, conversely, are important for patients with ASD. In particular, as ASD is more common in older women, it is essential to include kitchen activities in the assessment. A kitchen elbow sign, for example, is a skin abnormality that develops on the elbow when working in the kitchen, as the patient must rest her elbow on a table to maintain a standing position^{##REF##28414170##36##}. In the current study, the factor loadings for washing dishes and laundry were large. Kitchen elbow sign is especially likely to occur when washing dishes because both hands are used, and the patient cannot hold a cane or walker during the task. Large factor loading of these two items suggests that patients with ASD have kyphosis, making it difficult for them to maintain an intermediate or dorsiflexed position.</p>", "<p id=\"Par41\">Restriction of lumbar spine mobility after long fusion is a concern for both surgeons and patients^{##REF##27902557##6##}. Ishikawa et al. conducted a study about ADL for 36 long fusion patients^{##REF##30661199##13##}. They found that patients after long fusion performed better than preoperatively in activities such as sleeping supine, standing upright, vacuuming, doing laundry, and reaching for objects placed at heights. Conversely, strenuous activities such as shoveling snow worsened postoperatively. Overall surgical satisfaction was 70%. Their report suggests that long fusion surgery for ASD requires evaluating both positive and negative aspects.</p>", "<p id=\"Par42\">Hart et al. investigated functional limitations due to lumbar stiffness in 62 patients^{##REF##25202930##5##}. They reported that 91% of the patients were satisfied with the trade-off between postoperative improvement in back pain and associated restriction of motion. In the present study, 73% of the patients were satisfied with their surgery. Their study included 24 cases (40%) of one vertebral fusion and only 19 cases (31%) of five or more vertebral fusion. Our patients had five or more intervertebral fusions, with an average of 10 fused vertebrae. This difference in fixation levels may have influenced the difference in satisfaction.</p>", "<p id=\"Par43\">One of the advantages of our scoring system was that factor analysis divided the questions into two domains. The effect of surgery on ASD resulted in improved ADLs associated with improved pain and posture, but also movement limitations. Simply adding up these improvements and any worsening could result in a total score of plus or minus 0. By dividing this score into two domains, we could assess each symptom with each domain having the appropriate responsiveness. This represents two aspects of surgery for ASD and is a necessary component for improving treatment efficacy and explaining surgery to patients.</p>", "<p id=\"Par44\">Another strength of this study was that the subject patients had an average of 10 long fixed vertebral intervertebral spaces, and 78% underwent fusion from the pelvic to the thoracic spine. Previous studies have focused on short lumbar intervertebral fusion procedures. Our patients are a more suitable population to assess ADLs for long fusion, especially as including L5/S in the fusion range would result in greater limitation.</p>", "<p id=\"Par45\">There were some limitations in this study. The number of the patients was limited. Factor analysis was performed on 98 patients, slightly less than 100 patients. However, considering the two factors that were found, this could be considered sufficient. Because this study was conducted in one country, the results may not be generalizable to other countries. The burden of housework activities may differ between developed and developing countries. Reliability was assessed by content consistency, and a test–retest was not conducted in this study. The preoperative score was based on memory and there may have been recall bias. Patients with a longer follow-up period become more accustomed to their current symptoms and may underestimate the difference between their preoperative and current conditions. These issues should be addressed in future studies.</p>" ]

[ "<title>Conclusion</title>", "<p id=\"Par46\">We developed a disease-specific outcome for ASD using factor analysis. This analysis is the first scientifically validated measure that could simultaneously assess the benefits and limitations of ASD surgery. This tool can complement existing outcomes and will be useful for explaining surgery to patients and for future clinical trials.</p>" ]

[ "<p id=\"Par1\">Adult spinal deformity (ASD) is a complex condition that combines scoliosis, kyphosis, pain, and postoperative range of motion limitation. The lack of a scale that can successfully capture this complex condition is a clinical challenge. We aimed to develop a disease-specific scale for ASD. The study included 106 patients (mean age; 68 years, 89 women) with ASD. We selected 29 questions that could be useful in assessing ASD and asked the patients to answer them. The factor analysis found two factors: the main symptom and the collateral symptom. The main symptom consisted of 10 questions and assessed activity of daily living (ADL), pain, and appearance. The collateral symptom consisted of five questions to assess ADL due to range of motion limitation. Cronbach’s alpha was 0.90 and 0.84, respectively. The Spearman’s correlation coefficient between the change of main symptom and satisfaction was 0.48 (p &lt; 0.001). The effect size of Cohen’s d for comparison between preoperative and postoperative scores was 1.09 in the main symptom and 0.65 in the collateral symptom. In conclusion, we have developed a validated disease-specific scale for ASD that can simultaneously evaluate the benefits and limitations of ASD surgery with enough responsiveness in clinical practice.</p>", "<title>Subject terms</title>" ]

[ "<title>Supplementary Information</title>", "<p>\n</p>" ]

[ "<title>Supplementary Information</title>", "<p>The online version contains supplementary material available at 10.1038/s41598-024-51783-4.</p>", "<title>Acknowledgements</title>", "<p>This work was supported by a Grant from JSPS KAKENHI No. JP21K20966.</p>", "<title>Author contributions</title>", "<p>T.F. prepared the manuscript. T.F. and Y.N. collected the data. S.T. provided a critical comment. T.K., Y.K., Y.U., M.F., T.M., Sh.O. provided research advices. M.I. and Se.O. supervised the entire project. All authors reviewed the paper.</p>", "<title>Data availability</title>", "<p>The datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.</p>", "<title>Competing interests</title>", "<p id=\"Par47\">The authors declare no competing interests.</p>" ]

[ "<fig id=\"Fig1\"><label>Figure 1</label><caption><p>Schematic of changes in a typical long fusion surgery. Preoperatively, the patient cannot maintain posture due to kyphotic deformity. Postoperatively, the patient can maintain posture, but has limited range of motion.</p></caption></fig>", "<fig id=\"Fig2\"><label>Figure 2</label><caption><p>Patient flowchart.</p></caption></fig>", "<fig id=\"Fig3\"><label>Figure 3</label><caption><p>Flowchart of question item selection.</p></caption></fig>", "<fig id=\"Fig4\"><label>Figure 4</label><caption><p>Histogram of the preoperative scores of the main symptom. The main symptom has no floor or ceiling effect.</p></caption></fig>", "<fig id=\"Fig5\"><label>Figure 5</label><caption><p>Histogram of the postoperative scores of the main symptom. The main symptom has no floor or ceiling effect.</p></caption></fig>", "<fig id=\"Fig6\"><label>Figure 6</label><caption><p>Histogram of the preoperative scores of the collateral symptom. The collateral symptom has no floor or ceiling effect.</p></caption></fig>", "<fig id=\"Fig7\"><label>Figure 7</label><caption><p>Histogram of the postoperative scores of the collateral symptom. The collateral symptom has no floor or ceiling effect.</p></caption></fig>" ]

[ "<table-wrap id=\"Tab1\"><label>Table 1</label><caption><p>Review list of the questionnaires.</p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th align=\"left\"/><th align=\"left\">Abbreviation</th><th align=\"left\">Number of question items</th></tr></thead><tbody><tr><td align=\"left\">The 36-item short form health survey</td><td align=\"left\">SF-36</td><td char=\".\" align=\"char\">36</td></tr><tr><td align=\"left\">Patient-reported outcomes measurement information system</td><td align=\"left\">PROMIS</td><td char=\".\" align=\"char\">121</td></tr><tr><td align=\"left\">Oswestry disability index</td><td align=\"left\">ODI</td><td char=\".\" align=\"char\">10</td></tr><tr><td align=\"left\">Roland–Morris questionnaire</td><td align=\"left\">RMQ</td><td char=\".\" align=\"char\">24</td></tr><tr><td align=\"left\">Zurich Claudication questionnaire</td><td align=\"left\">ZCQ</td><td char=\".\" align=\"char\">18</td></tr><tr><td align=\"left\">Scoliosis Research Society-22 Patient Questionnaire</td><td align=\"left\">SRS-22</td><td char=\".\" align=\"char\">22</td></tr><tr><td align=\"left\">Japanese orthopedic association back pain evaluation questionnaire</td><td align=\"left\">JOABPEQ</td><td char=\".\" align=\"char\">25</td></tr><tr><td align=\"left\">Western Ontario and McMaster Universities Arthritis Index</td><td align=\"left\">WOMAC</td><td char=\".\" align=\"char\">17</td></tr><tr><td align=\"left\">Knee Society scoring system</td><td align=\"left\">KSS</td><td char=\".\" align=\"char\">11</td></tr><tr><td align=\"left\">Bath ankylosing spondylitis functional index</td><td align=\"left\">BASFI</td><td char=\".\" align=\"char\">10</td></tr><tr><td align=\"left\">Health assessment questionnaire</td><td align=\"left\">HAQ</td><td char=\".\" align=\"char\">18</td></tr><tr><td align=\"left\">Pain disability assessment scale</td><td align=\"left\">PDAS</td><td char=\".\" align=\"char\">20</td></tr><tr><td align=\"left\">EuroQol 5-dimensions 5-levels</td><td align=\"left\">EQ5D-5L</td><td char=\".\" align=\"char\">5</td></tr><tr><td align=\"left\">Lumbar stiffness disability index</td><td align=\"left\">LSDI</td><td char=\".\" align=\"char\">10</td></tr><tr><td align=\"left\">The 25-question geriatric locomotive function scale</td><td align=\"left\">LOCOMO-25</td><td char=\".\" align=\"char\">25</td></tr><tr><td align=\"left\">Gastro-esophageal reflux disease questionnaire</td><td align=\"left\">GERDQ</td><td char=\".\" align=\"char\">6</td></tr><tr><td align=\"left\">Frequency scale for the symptoms of gastro-esophageal reflux disease</td><td align=\"left\">FSSG</td><td char=\".\" align=\"char\">12</td></tr><tr><td align=\"left\"/><td align=\"left\">Total</td><td char=\".\" align=\"char\">390</td></tr></tbody></table></table-wrap>", "<table-wrap id=\"Tab2\"><label>Table 2</label><caption><p>Twenty-nine items for factor analysis selected after discussion among the surgeons.</p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th align=\"left\" rowspan=\"3\">Question item number</th><th align=\"left\" rowspan=\"3\">Content</th><th align=\"left\" rowspan=\"3\">Questionnaire</th><th align=\"left\" colspan=\"5\">Answer options</th></tr><tr><th align=\"left\" colspan=\"5\">Score point</th></tr><tr><th align=\"left\">1</th><th align=\"left\">2</th><th align=\"left\">3</th><th align=\"left\">4</th><th align=\"left\">5</th></tr></thead><tbody><tr><td align=\"left\">1</td><td align=\"left\">Appearance</td><td align=\"left\">Are you concerned about the current appearance of your back?</td><td align=\"left\">Not at all</td><td align=\"left\">Not very much</td><td align=\"left\">Neither</td><td align=\"left\">Somewhat</td><td align=\"left\">Very much</td></tr><tr><td align=\"left\">2</td><td align=\"left\">Backpain</td><td align=\"left\">Which one of the following best describes the amount of back pain you have experienced?</td><td align=\"left\">No pain</td><td align=\"left\">A little pain</td><td align=\"left\">Moderate pain</td><td align=\"left\">Much pain</td><td align=\"left\">Severe pain</td></tr><tr><td align=\"left\">3</td><td align=\"left\">Leg pain</td><td align=\"left\">Which one of the following best describes the amount of buttock or leg pain you have experienced?</td><td align=\"left\">No pain</td><td align=\"left\">A little pain</td><td align=\"left\">Moderate pain</td><td align=\"left\">Much pain</td><td align=\"left\">Severe pain</td></tr><tr><td align=\"left\">4</td><td align=\"left\">Appetite</td><td align=\"left\">Do you have an appetite?</td><td align=\"left\">Yes, I have</td><td align=\"left\">A little</td><td align=\"left\">Neither</td><td align=\"left\">Not very much</td><td align=\"left\">None</td></tr><tr><td align=\"left\">5</td><td align=\"left\">Heartburn</td><td align=\"left\">In the last week, how many days have you had a burning sensation or burning pain in your chest?</td><td align=\"left\">Never</td><td align=\"left\">1 day</td><td align=\"left\">2 to 3 days</td><td align=\"left\">Between 4 and 6 days</td><td align=\"left\">7 days</td></tr><tr><td align=\"left\">6</td><td align=\"left\">Sleeping</td><td align=\"left\">Are you able to sleep on your back?</td><td align=\"left\">Can sleep without difficulty</td><td align=\"left\">Sometimes wake up</td><td align=\"left\">I have less than 6 h of sleep</td><td align=\"left\">I have less than 3 h of sleep</td><td align=\"left\">Cannot sleep</td></tr><tr><td align=\"left\">7</td><td align=\"left\">Standing up floors</td><td align=\"left\">Are you able to get up from the floor without help?</td><td align=\"left\">Can do easily</td><td align=\"left\">A little difficult</td><td align=\"left\">Somewhat difficult</td><td align=\"left\">Very difficult</td><td align=\"left\">Cannot do</td></tr><tr><td align=\"left\">8</td><td align=\"left\">Toilet</td><td align=\"left\">Are you able to wipe yourself after using the toilet?</td><td align=\"left\">Can do easily</td><td align=\"left\">A little difficult</td><td align=\"left\">Somewhat difficult</td><td align=\"left\">Very difficult</td><td align=\"left\">Cannot do</td></tr><tr><td align=\"left\">9</td><td align=\"left\">Picking up</td><td align=\"left\">Are you able to bend down and pick up something from the floor?</td><td align=\"left\">Can do easily</td><td align=\"left\">A little difficult</td><td align=\"left\">Somewhat difficult</td><td align=\"left\">Very difficult</td><td align=\"left\">Cannot do</td></tr><tr><td align=\"left\">10</td><td align=\"left\">Washing</td><td align=\"left\">Are you able to wash your body in the bath?</td><td align=\"left\">Can do easily</td><td align=\"left\">A little difficult</td><td align=\"left\">Somewhat difficult</td><td align=\"left\">Very difficult</td><td align=\"left\">Cannot do</td></tr><tr><td align=\"left\">11</td><td align=\"left\">Pants</td><td align=\"left\">Are you able to put on pants or trousers by yourself?</td><td align=\"left\">Can do easily</td><td align=\"left\">A little difficult</td><td align=\"left\">Somewhat difficult</td><td align=\"left\">Very difficult</td><td align=\"left\">Cannot do</td></tr><tr><td align=\"left\">12</td><td align=\"left\">Socks</td><td align=\"left\">Are you able to put on your socks?</td><td align=\"left\">Can do easily</td><td align=\"left\">A little difficult</td><td align=\"left\">Somewhat difficult</td><td align=\"left\">Very difficult</td><td align=\"left\">Cannot do</td></tr><tr><td align=\"left\">13</td><td align=\"left\">Sitting</td><td align=\"left\">Are you able to sit in a chair for a long time?</td><td align=\"left\">Can sit as long as I want</td><td align=\"left\">Can sit for 3 h</td><td align=\"left\">Can sit for 1 h</td><td align=\"left\">Can sit for 30 min</td><td align=\"left\">Can sit for 5 min</td></tr><tr><td align=\"left\">14</td><td align=\"left\">Standing up chairs</td><td align=\"left\">Are you able to stand up from a chair?</td><td align=\"left\">Can do easily</td><td align=\"left\">A little difficult</td><td align=\"left\">Somewhat difficult</td><td align=\"left\">Very difficult</td><td align=\"left\">Cannot do</td></tr><tr><td align=\"left\">15</td><td align=\"left\">Standing</td><td align=\"left\">How long can you keep standing without help?</td><td align=\"left\">Can stand more than 1 h</td><td align=\"left\">Can stand more than 30 min</td><td align=\"left\">Can stand more than 15 min</td><td align=\"left\">Can stand more than 5 min</td><td align=\"left\">Can stand more than one minute</td></tr><tr><td align=\"left\">16</td><td align=\"left\">Walking distance</td><td align=\"left\">How far can you keep walking without rest?</td><td align=\"left\">Can walk as far as I want</td><td align=\"left\">Can walk 1 km</td><td align=\"left\">Can walk 500 m</td><td align=\"left\">Can walk 300 m</td><td align=\"left\">Can walk 50 m</td></tr><tr><td align=\"left\">17</td><td align=\"left\">Walking time</td><td align=\"left\">How long can you keep walking without rest?</td><td align=\"left\">Can walk more than 1 h</td><td align=\"left\">Can walk more than 30 min</td><td align=\"left\">Can walk more than 15 min</td><td align=\"left\">Can walk more than 5 min</td><td align=\"left\">Can walk more than one minute</td></tr><tr><td align=\"left\">18</td><td align=\"left\">Stairs</td><td align=\"left\">Are you able to walk up and down the stairs?</td><td align=\"left\">Can do easily</td><td align=\"left\">A little difficult</td><td align=\"left\">Somewhat difficult</td><td align=\"left\">Very difficult</td><td align=\"left\">Cannot do</td></tr><tr><td align=\"left\">19</td><td align=\"left\">Ride</td><td align=\"left\">Are you able to get in and out of a car?</td><td align=\"left\">Can do easily</td><td align=\"left\">A little difficult</td><td align=\"left\">Somewhat difficult</td><td align=\"left\">Very difficult</td><td align=\"left\">Cannot do</td></tr><tr><td align=\"left\">20</td><td align=\"left\">Shelving</td><td align=\"left\">Are you able to reach to a high shelf?</td><td align=\"left\">Can do easily</td><td align=\"left\">A little difficult</td><td align=\"left\">Somewhat difficult</td><td align=\"left\">Very difficult</td><td align=\"left\">Cannot do</td></tr><tr><td align=\"left\">21</td><td align=\"left\">Hanging laundry</td><td align=\"left\">Are you able to hang your laundry on a clothesline?</td><td align=\"left\">Can do easily</td><td align=\"left\">A little difficult</td><td align=\"left\">Somewhat difficult</td><td align=\"left\">Very difficult</td><td align=\"left\">Cannot do</td></tr><tr><td align=\"left\">22</td><td align=\"left\">Light housework</td><td align=\"left\">Are you able to do simple tasks and housework (preparing meals, cleaning up, etc.)?</td><td align=\"left\">Can do easily</td><td align=\"left\">A little difficult</td><td align=\"left\">Somewhat difficult</td><td align=\"left\">Very difficult</td><td align=\"left\">Cannot do</td></tr><tr><td align=\"left\">23</td><td align=\"left\">Heavy housework</td><td align=\"left\">Are you able to do load-bearing tasks and housework (cleaning the yard, carrying heavy bedding, etc.)?</td><td align=\"left\">Can do easily</td><td align=\"left\">A little difficult</td><td align=\"left\">Somewhat difficult</td><td align=\"left\">Very difficult</td><td align=\"left\">Cannot do</td></tr><tr><td align=\"left\">24</td><td align=\"left\">Garbage</td><td align=\"left\">Are you able to take out the garbage?</td><td align=\"left\">Can do easily</td><td align=\"left\">A little difficult</td><td align=\"left\">Somewhat difficult</td><td align=\"left\">Very difficult</td><td align=\"left\">Cannot do</td></tr><tr><td align=\"left\">25</td><td align=\"left\">Dishwashing</td><td align=\"left\">Are you able to wash dishes, pots, and utensils by hand while standing at a sink?</td><td align=\"left\">Can do easily</td><td align=\"left\">A little difficult</td><td align=\"left\">Somewhat difficult</td><td align=\"left\">Very difficult</td><td align=\"left\">Cannot do</td></tr><tr><td align=\"left\">26</td><td align=\"left\">Sports</td><td align=\"left\">Are you able to play sports activity (jogging, swimming, gate ball, dancing, etc.)?</td><td align=\"left\">Can do easily</td><td align=\"left\">A little difficult</td><td align=\"left\">Somewhat difficult</td><td align=\"left\">Very difficult</td><td align=\"left\">Cannot do</td></tr><tr><td align=\"left\">27</td><td align=\"left\">Shopping</td><td align=\"left\">Are you able to carry objects weighting approximately 2 kg (2 standard milk bottles or 2 PET bottles each containing 1 L)?</td><td align=\"left\">Can do easily</td><td align=\"left\">A little difficult</td><td align=\"left\">Somewhat difficult</td><td align=\"left\">Very difficult</td><td align=\"left\">Cannot do</td></tr><tr><td align=\"left\">28</td><td align=\"left\">Community activity</td><td align=\"left\">Are you able to join social activities (meeting friends, playing sport, engaging in activities and hobbies, etc.)?</td><td align=\"left\">Can do easily</td><td align=\"left\">A little difficult</td><td align=\"left\">Somewhat difficult</td><td align=\"left\">Very difficult</td><td align=\"left\">Cannot do</td></tr><tr><td align=\"left\">29</td><td align=\"left\">Anxiety</td><td align=\"left\">Are you worried that you will not be able to walk in the future?</td><td align=\"left\">Not at all</td><td align=\"left\">A little anxious</td><td align=\"left\">Somewhat anxious</td><td align=\"left\">Fairly anxious</td><td align=\"left\">Very anxious</td></tr></tbody></table></table-wrap>", "<table-wrap id=\"Tab3\"><label>Table 3</label><caption><p>Demographics of the study patients.</p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th align=\"left\"/><th align=\"left\">Operative cases</th><th align=\"left\">Conservative cases</th></tr></thead><tbody><tr><td align=\"left\">Male/female (cases)</td><td align=\"left\">10/79</td><td align=\"left\">0/9</td></tr><tr><td align=\"left\">Mean age ± SD (years)</td><td align=\"left\">68 ± 7</td><td align=\"left\">68 ± 12</td></tr><tr><td align=\"left\">Mean post operative follow-up period ± SD (mos.)</td><td align=\"left\">56 ± 35</td><td align=\"left\">N.A.</td></tr><tr><td align=\"left\">Mean fusion intervertebral levels ± SD</td><td align=\"left\">10 ± 3</td><td align=\"left\">N.A.</td></tr><tr><td align=\"left\">Fixation to sacrum or pelvis (cases/%)</td><td align=\"left\">76/85</td><td align=\"left\">N.A.</td></tr><tr><td align=\"left\">Fixation from T8, T9, or T10 to pelvis (cases/%)</td><td align=\"left\">56/63</td><td align=\"left\">N.A.</td></tr><tr><td align=\"left\">Fixation from T3, T4, or T5 to pelvis (cases/%) 13 / 15 0 / 0 0.01*</td><td align=\"left\">13/15</td><td align=\"left\">N.A.</td></tr><tr><td align=\"left\" colspan=\"3\">Post operative satisfaction (cases/%)</td></tr><tr><td align=\"left\"> Very satisfied</td><td align=\"left\">23/26</td><td align=\"left\">N.A.</td></tr><tr><td align=\"left\"> Satisfied</td><td align=\"left\">42/47</td><td align=\"left\">N.A.</td></tr><tr><td align=\"left\"> Neither</td><td align=\"left\">18/20</td><td align=\"left\">N.A.</td></tr><tr><td align=\"left\"> Dissatisfied</td><td align=\"left\">6/7</td><td align=\"left\">N.A.</td></tr><tr><td align=\"left\"> Very dissatisfied</td><td align=\"left\">0/0</td><td align=\"left\">N.A.</td></tr></tbody></table></table-wrap>", "<table-wrap id=\"Tab4\"><label>Table 4</label><caption><p>Mean and standard deviation of raw data for each item.</p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th align=\"left\" rowspan=\"2\">Question item number</th><th align=\"left\" rowspan=\"2\">Content</th><th align=\"left\" colspan=\"2\">Preoperative</th><th align=\"left\" colspan=\"2\">Postoperative</th></tr><tr><th align=\"left\">Mean</th><th align=\"left\">SD</th><th align=\"left\">Mean</th><th align=\"left\">SD</th></tr></thead><tbody><tr><td align=\"left\">1</td><td align=\"left\">Appearance</td><td char=\".\" align=\"char\">4.3</td><td char=\".\" align=\"char\">1.0</td><td char=\".\" align=\"char\">2.8</td><td char=\".\" align=\"char\">1.4</td></tr><tr><td align=\"left\">2</td><td align=\"left\">Backpain</td><td char=\".\" align=\"char\">3.6</td><td char=\".\" align=\"char\">1.4</td><td char=\".\" align=\"char\">2.1</td><td char=\".\" align=\"char\">0.9</td></tr><tr><td align=\"left\">3</td><td align=\"left\">Leg pain</td><td char=\".\" align=\"char\">2.9</td><td char=\".\" align=\"char\">1.4</td><td char=\".\" align=\"char\">2.1</td><td char=\".\" align=\"char\">0.9</td></tr><tr><td align=\"left\">4</td><td align=\"left\">Appetite</td><td char=\".\" align=\"char\">1.9</td><td char=\".\" align=\"char\">1.2</td><td char=\".\" align=\"char\">1.5</td><td char=\".\" align=\"char\">0.9</td></tr><tr><td align=\"left\">5</td><td align=\"left\">Heartburn</td><td char=\".\" align=\"char\">1.6</td><td char=\".\" align=\"char\">1.1</td><td char=\".\" align=\"char\">1.4</td><td char=\".\" align=\"char\">0.9</td></tr><tr><td align=\"left\">6</td><td align=\"left\">Sleeping</td><td char=\".\" align=\"char\">2.6</td><td char=\".\" align=\"char\">1.6</td><td char=\".\" align=\"char\">2.0</td><td char=\".\" align=\"char\">1.4</td></tr><tr><td align=\"left\">7</td><td align=\"left\">Standing up floors</td><td char=\".\" align=\"char\">2.2</td><td char=\".\" align=\"char\">1.1</td><td char=\".\" align=\"char\">2.6</td><td char=\".\" align=\"char\">1.2</td></tr><tr><td align=\"left\">8</td><td align=\"left\">Toilet</td><td char=\".\" align=\"char\">1.6</td><td char=\".\" align=\"char\">0.8</td><td char=\".\" align=\"char\">2.0</td><td char=\".\" align=\"char\">1.1</td></tr><tr><td align=\"left\">9</td><td align=\"left\">Picking up</td><td char=\".\" align=\"char\">2.0</td><td char=\".\" align=\"char\">1.1</td><td char=\".\" align=\"char\">2.8</td><td char=\".\" align=\"char\">1.3</td></tr><tr><td align=\"left\">10</td><td align=\"left\">Washing</td><td char=\".\" align=\"char\">1.8</td><td char=\".\" align=\"char\">1.0</td><td char=\".\" align=\"char\">2.1</td><td char=\".\" align=\"char\">1.1</td></tr><tr><td align=\"left\">11</td><td align=\"left\">Pants</td><td char=\".\" align=\"char\">1.8</td><td char=\".\" align=\"char\">0.9</td><td char=\".\" align=\"char\">2.4</td><td char=\".\" align=\"char\">1.0</td></tr><tr><td align=\"left\">12</td><td align=\"left\">Socks</td><td char=\".\" align=\"char\">2.0</td><td char=\".\" align=\"char\">1.2</td><td char=\".\" align=\"char\">2.8</td><td char=\".\" align=\"char\">1.2</td></tr><tr><td align=\"left\">13</td><td align=\"left\">Sitting</td><td char=\".\" align=\"char\">2.6</td><td char=\".\" align=\"char\">1.2</td><td char=\".\" align=\"char\">2.3</td><td char=\".\" align=\"char\">1.1</td></tr><tr><td align=\"left\">14</td><td align=\"left\">Standing up chairs</td><td char=\".\" align=\"char\">2.0</td><td char=\".\" align=\"char\">1.1</td><td char=\".\" align=\"char\">2.0</td><td char=\".\" align=\"char\">1.1</td></tr><tr><td align=\"left\">15</td><td align=\"left\">Standing</td><td char=\".\" align=\"char\">2.9</td><td char=\".\" align=\"char\">1.4</td><td char=\".\" align=\"char\">2.5</td><td char=\".\" align=\"char\">1.2</td></tr><tr><td align=\"left\">16</td><td align=\"left\">Walking distance</td><td char=\".\" align=\"char\">3.4</td><td char=\".\" align=\"char\">1.4</td><td char=\".\" align=\"char\">2.5</td><td char=\".\" align=\"char\">1.2</td></tr><tr><td align=\"left\">17</td><td align=\"left\">Walking time</td><td char=\".\" align=\"char\">2.7</td><td char=\".\" align=\"char\">1.1</td><td char=\".\" align=\"char\">2.1</td><td char=\".\" align=\"char\">1.0</td></tr><tr><td align=\"left\">18</td><td align=\"left\">Stairs</td><td char=\".\" align=\"char\">2.1</td><td char=\".\" align=\"char\">1.1</td><td char=\".\" align=\"char\">1.8</td><td char=\".\" align=\"char\">0.9</td></tr><tr><td align=\"left\">19</td><td align=\"left\">Ride</td><td char=\".\" align=\"char\">2.3</td><td char=\".\" align=\"char\">1.3</td><td char=\".\" align=\"char\">2.0</td><td char=\".\" align=\"char\">1.3</td></tr><tr><td align=\"left\">20</td><td align=\"left\">Shelving</td><td char=\".\" align=\"char\">3.1</td><td char=\".\" align=\"char\">1.3</td><td char=\".\" align=\"char\">2.4</td><td char=\".\" align=\"char\">1.2</td></tr><tr><td align=\"left\">21</td><td align=\"left\">Hanging laundry</td><td char=\".\" align=\"char\">2.6</td><td char=\".\" align=\"char\">1.3</td><td char=\".\" align=\"char\">2.0</td><td char=\".\" align=\"char\">1.3</td></tr><tr><td align=\"left\">22</td><td align=\"left\">Light housework</td><td char=\".\" align=\"char\">2.2</td><td char=\".\" align=\"char\">1.1</td><td char=\".\" align=\"char\">1.7</td><td char=\".\" align=\"char\">1.1</td></tr><tr><td align=\"left\">23</td><td align=\"left\">Heavy housework</td><td char=\".\" align=\"char\">2.7</td><td char=\".\" align=\"char\">1.3</td><td char=\".\" align=\"char\">2.4</td><td char=\".\" align=\"char\">1.3</td></tr><tr><td align=\"left\">24</td><td align=\"left\">Garbage</td><td char=\".\" align=\"char\">2.5</td><td char=\".\" align=\"char\">1.4</td><td char=\".\" align=\"char\">2.3</td><td char=\".\" align=\"char\">1.4</td></tr><tr><td align=\"left\">25</td><td align=\"left\">Dishwashing</td><td char=\".\" align=\"char\">2.5</td><td char=\".\" align=\"char\">1.2</td><td char=\".\" align=\"char\">1.7</td><td char=\".\" align=\"char\">1.1</td></tr><tr><td align=\"left\">26</td><td align=\"left\">Sports</td><td char=\".\" align=\"char\">4.0</td><td char=\".\" align=\"char\">1.3</td><td char=\".\" align=\"char\">3.8</td><td char=\".\" align=\"char\">1.3</td></tr><tr><td align=\"left\">27</td><td align=\"left\">Shopping</td><td char=\".\" align=\"char\">3.4</td><td char=\".\" align=\"char\">1.4</td><td char=\".\" align=\"char\">2.9</td><td char=\".\" align=\"char\">1.5</td></tr><tr><td align=\"left\">28</td><td align=\"left\">Community activity</td><td char=\".\" align=\"char\">3.6</td><td char=\".\" align=\"char\">1.5</td><td char=\".\" align=\"char\">3.1</td><td char=\".\" align=\"char\">1.5</td></tr><tr><td align=\"left\">29</td><td align=\"left\">Anxiety</td><td char=\".\" align=\"char\">3.8</td><td char=\".\" align=\"char\">1.3</td><td char=\".\" align=\"char\">2.7</td><td char=\".\" align=\"char\">1.4</td></tr></tbody></table></table-wrap>", "<table-wrap id=\"Tab5\"><label>Table 5</label><caption><p>Spearman correlation coefficients between each item for postoperative answers.</p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th align=\"left\" rowspan=\"2\" colspan=\"2\">Question item number</th><th align=\"left\">1</th><th align=\"left\">2</th><th align=\"left\">3</th><th align=\"left\">4</th><th align=\"left\">5</th><th align=\"left\">6</th><th align=\"left\">7</th><th align=\"left\">8</th><th align=\"left\">9</th><th align=\"left\">10</th><th align=\"left\">11</th><th align=\"left\">12</th><th align=\"left\">13</th><th align=\"left\">14</th><th align=\"left\">15</th><th align=\"left\">16</th><th align=\"left\">17</th><th align=\"left\">18</th><th align=\"left\">19</th><th align=\"left\">20</th><th align=\"left\">21</th><th align=\"left\">22</th><th align=\"left\">23</th><th align=\"left\">24</th><th align=\"left\">25</th><th align=\"left\">26</th><th align=\"left\">27</th><th align=\"left\">28</th><th align=\"left\">29</th></tr><tr><th align=\"left\">Appearance</th><th align=\"left\">Back \npain</th><th align=\"left\">Leg pain</th><th align=\"left\">Appetite</th><th align=\"left\">Heartburn</th><th align=\"left\">Sleeping</th><th align=\"left\">Standing up floors</th><th align=\"left\">Toilet</th><th align=\"left\">Picking up</th><th align=\"left\">Washing</th><th align=\"left\">Pants</th><th align=\"left\">Socks</th><th align=\"left\">Sitting</th><th align=\"left\">Standing up chairs</th><th align=\"left\">Standing</th><th align=\"left\">Walking distance</th><th align=\"left\">Walking time</th><th align=\"left\">Stairs</th><th align=\"left\">Ride</th><th align=\"left\">Shelving</th><th align=\"left\">Hanging laundry</th><th align=\"left\">Light housework</th><th align=\"left\">Heavy housework</th><th align=\"left\">Garbage</th><th align=\"left\">Dishwashing</th><th align=\"left\">Sports</th><th align=\"left\">Shopping</th><th align=\"left\">Community activity</th><th align=\"left\">Anxiety</th></tr></thead><tbody><tr><td align=\"left\">1</td><td align=\"left\">Appearance</td><td char=\".\" align=\"char\">1.0</td><td char=\".\" align=\"char\">0.43*</td><td char=\".\" align=\"char\">0.22*</td><td char=\".\" align=\"char\">0.04</td><td char=\".\" align=\"char\">0.21*</td><td char=\".\" align=\"char\">0.26*</td><td char=\".\" align=\"char\">0.08</td><td char=\".\" align=\"char\"> − 0.11</td><td char=\".\" align=\"char\">0.05</td><td char=\".\" align=\"char\">0.08</td><td char=\".\" align=\"char\">0.07</td><td char=\".\" align=\"char\"> − 0.05</td><td char=\".\" align=\"char\">0.23*</td><td char=\".\" align=\"char\">0.25*</td><td char=\".\" align=\"char\">0.20*</td><td char=\".\" align=\"char\">0.23*</td><td char=\".\" align=\"char\">0.34*</td><td char=\".\" align=\"char\">0.13</td><td char=\".\" align=\"char\">0.20*</td><td char=\".\" align=\"char\">0.36*</td><td char=\".\" align=\"char\">0.36*</td><td char=\".\" align=\"char\">0.31*</td><td char=\".\" align=\"char\">0.23*</td><td char=\".\" align=\"char\">0.16</td><td char=\".\" align=\"char\">0.28*</td><td char=\".\" align=\"char\">0.13</td><td char=\".\" align=\"char\">0.10</td><td char=\".\" align=\"char\">0.21*</td><td char=\".\" align=\"char\">0.40*</td></tr><tr><td align=\"left\">2</td><td align=\"left\">Back pain</td><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\">1.0</td><td char=\".\" align=\"char\">0.41*</td><td char=\".\" align=\"char\">0.08</td><td char=\".\" align=\"char\">0.31*</td><td char=\".\" align=\"char\">0.32*</td><td char=\".\" align=\"char\">0.04</td><td char=\".\" align=\"char\">0.08</td><td char=\".\" align=\"char\">0.07</td><td char=\".\" align=\"char\">0.07</td><td char=\".\" align=\"char\">0.06</td><td char=\".\" align=\"char\">0.07</td><td char=\".\" align=\"char\">0.25*</td><td char=\".\" align=\"char\">0.25*</td><td char=\".\" align=\"char\">0.25*</td><td char=\".\" align=\"char\">0.31*</td><td char=\".\" align=\"char\">0.25*</td><td char=\".\" align=\"char\">0.05</td><td char=\".\" align=\"char\">0.24*</td><td char=\".\" align=\"char\">0.23*</td><td char=\".\" align=\"char\">0.23*</td><td char=\".\" align=\"char\">0.28*</td><td char=\".\" align=\"char\">0.23*</td><td char=\".\" align=\"char\">0.34*</td><td char=\".\" align=\"char\">0.24*</td><td char=\".\" align=\"char\">0.15</td><td char=\".\" align=\"char\">0.18</td><td char=\".\" align=\"char\">0.16</td><td char=\".\" align=\"char\">0.29*</td></tr><tr><td align=\"left\">3</td><td align=\"left\">leg Pain</td><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\">1.0</td><td char=\".\" align=\"char\">0.16</td><td char=\".\" align=\"char\">0.29*</td><td char=\".\" align=\"char\">0.33*</td><td char=\".\" align=\"char\">0.25*</td><td char=\".\" align=\"char\">0.13</td><td char=\".\" align=\"char\">0.18</td><td char=\".\" align=\"char\">0.19</td><td char=\".\" align=\"char\">0.25*</td><td char=\".\" align=\"char\">0.19</td><td char=\".\" align=\"char\">0.17</td><td char=\".\" align=\"char\">0.43*</td><td char=\".\" align=\"char\">0.22*</td><td char=\".\" align=\"char\">0.29*</td><td char=\".\" align=\"char\">0.33*</td><td char=\".\" align=\"char\">0.30*</td><td char=\".\" align=\"char\">0.35*</td><td char=\".\" align=\"char\">0.37*</td><td char=\".\" align=\"char\">0.32*</td><td char=\".\" align=\"char\">0.35*</td><td char=\".\" align=\"char\">0.33*</td><td char=\".\" align=\"char\">0.32*</td><td char=\".\" align=\"char\">0.30*</td><td char=\".\" align=\"char\">0.22*</td><td char=\".\" align=\"char\">0.30*</td><td char=\".\" align=\"char\">0.34*</td><td char=\".\" align=\"char\">0.35*</td></tr><tr><td align=\"left\">4</td><td align=\"left\">Appetite</td><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\">1.0</td><td char=\".\" align=\"char\">0.32*</td><td char=\".\" align=\"char\">0.17</td><td char=\".\" align=\"char\">0.14</td><td char=\".\" align=\"char\">0.16</td><td char=\".\" align=\"char\">0.15</td><td char=\".\" align=\"char\">0.19</td><td char=\".\" align=\"char\">0.13</td><td char=\".\" align=\"char\">0.08</td><td char=\".\" align=\"char\">0.17</td><td char=\".\" align=\"char\">0.04</td><td char=\".\" align=\"char\">0.06</td><td char=\".\" align=\"char\">0.23*</td><td char=\".\" align=\"char\">0.17</td><td char=\".\" align=\"char\">0.05</td><td char=\".\" align=\"char\">0.21*</td><td char=\".\" align=\"char\">0.15</td><td char=\".\" align=\"char\">0.13</td><td char=\".\" align=\"char\">0.31*</td><td char=\".\" align=\"char\">0.17</td><td char=\".\" align=\"char\">0.13</td><td char=\".\" align=\"char\">0.24*</td><td char=\".\" align=\"char\">0.25*</td><td char=\".\" align=\"char\">0.24*</td><td char=\".\" align=\"char\">0.12</td><td char=\".\" align=\"char\">0.08</td></tr><tr><td align=\"left\">5</td><td align=\"left\">Heartburn</td><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\">1.0</td><td char=\".\" align=\"char\">0.35*</td><td char=\".\" align=\"char\">0.25*</td><td char=\".\" align=\"char\">0.16</td><td char=\".\" align=\"char\">0.23*</td><td char=\".\" align=\"char\">0.19</td><td char=\".\" align=\"char\">0.19</td><td char=\".\" align=\"char\">0.14</td><td char=\".\" align=\"char\">0.18</td><td char=\".\" align=\"char\">0.28*</td><td char=\".\" align=\"char\">0.25*</td><td char=\".\" align=\"char\">0.42*</td><td char=\".\" align=\"char\">0.46*</td><td char=\".\" align=\"char\">0.21*</td><td char=\".\" align=\"char\">0.27*</td><td char=\".\" align=\"char\">0.21*</td><td char=\".\" align=\"char\">0.34*</td><td char=\".\" align=\"char\">0.30*</td><td char=\".\" align=\"char\">0.16</td><td char=\".\" align=\"char\">0.22*</td><td char=\".\" align=\"char\">0.35*</td><td char=\".\" align=\"char\">0.11</td><td char=\".\" align=\"char\">0.24*</td><td char=\".\" align=\"char\">0.18</td><td char=\".\" align=\"char\">0.14</td></tr><tr><td align=\"left\">6</td><td align=\"left\">Sleeping</td><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\">1.0</td><td char=\".\" align=\"char\">0.37*</td><td char=\".\" align=\"char\">0.32*</td><td char=\".\" align=\"char\">0.30*</td><td char=\".\" align=\"char\">0.15</td><td char=\".\" align=\"char\">0.24*</td><td char=\".\" align=\"char\">0.28*</td><td char=\".\" align=\"char\">0.30*</td><td char=\".\" align=\"char\">0.32*</td><td char=\".\" align=\"char\">0.28*</td><td char=\".\" align=\"char\">0.38*</td><td char=\".\" align=\"char\">0.37*</td><td char=\".\" align=\"char\">0.32*</td><td char=\".\" align=\"char\">0.40*</td><td char=\".\" align=\"char\">0.33*</td><td char=\".\" align=\"char\">0.23*</td><td char=\".\" align=\"char\">0.23*</td><td char=\".\" align=\"char\">0.30*</td><td char=\".\" align=\"char\">0.29*</td><td char=\".\" align=\"char\">0.25*</td><td char=\".\" align=\"char\">0.26*</td><td char=\".\" align=\"char\">0.35*</td><td char=\".\" align=\"char\">0.40*</td><td char=\".\" align=\"char\">0.29*</td></tr><tr><td align=\"left\">7</td><td align=\"left\">Standing up floors</td><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\">1.0</td><td char=\".\" align=\"char\">0.38*</td><td char=\".\" align=\"char\">0.61*</td><td char=\".\" align=\"char\">0.42*</td><td char=\".\" align=\"char\">0.50*</td><td char=\".\" align=\"char\">0.52*</td><td char=\".\" align=\"char\">0.24*</td><td char=\".\" align=\"char\">0.46*</td><td char=\".\" align=\"char\">0.42*</td><td char=\".\" align=\"char\">0.42*</td><td char=\".\" align=\"char\">0.40*</td><td char=\".\" align=\"char\">0.45*</td><td char=\".\" align=\"char\">0.55*</td><td char=\".\" align=\"char\">0.36*</td><td char=\".\" align=\"char\">0.37*</td><td char=\".\" align=\"char\">0.38*</td><td char=\".\" align=\"char\">0.53*</td><td char=\".\" align=\"char\">0.41*</td><td char=\".\" align=\"char\">0.35*</td><td char=\".\" align=\"char\">0.59*</td><td char=\".\" align=\"char\">0.49*</td><td char=\".\" align=\"char\">0.59*</td><td char=\".\" align=\"char\">0.32*</td></tr><tr><td align=\"left\">8</td><td align=\"left\">Toilet</td><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\">1.0</td><td char=\".\" align=\"char\">0.53*</td><td char=\".\" align=\"char\">0.58*</td><td char=\".\" align=\"char\">0.47*</td><td char=\".\" align=\"char\">0.61*</td><td char=\".\" align=\"char\">0.28*</td><td char=\".\" align=\"char\">0.22*</td><td char=\".\" align=\"char\">0.19</td><td char=\".\" align=\"char\">0.19</td><td char=\".\" align=\"char\">0.12</td><td char=\".\" align=\"char\">0.13</td><td char=\".\" align=\"char\">0.31*</td><td char=\".\" align=\"char\">0.19</td><td char=\".\" align=\"char\">0.19</td><td char=\".\" align=\"char\">0.24*</td><td char=\".\" align=\"char\">0.39*</td><td char=\".\" align=\"char\">0.27*</td><td char=\".\" align=\"char\">0.13</td><td char=\".\" align=\"char\">0.21*</td><td char=\".\" align=\"char\">0.23*</td><td char=\".\" align=\"char\">0.23*</td><td char=\".\" align=\"char\">0.17</td></tr><tr><td align=\"left\">9</td><td align=\"left\">Picking up</td><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\">1.0</td><td char=\".\" align=\"char\">0.61*</td><td char=\".\" align=\"char\">0.68*</td><td char=\".\" align=\"char\">0.75*</td><td char=\".\" align=\"char\">0.36*</td><td char=\".\" align=\"char\">0.44*</td><td char=\".\" align=\"char\">0.38*</td><td char=\".\" align=\"char\">0.30*</td><td char=\".\" align=\"char\">0.36*</td><td char=\".\" align=\"char\">0.28*</td><td char=\".\" align=\"char\">0.50*</td><td char=\".\" align=\"char\">0.23*</td><td char=\".\" align=\"char\">0.43*</td><td char=\".\" align=\"char\">0.38*</td><td char=\".\" align=\"char\">0.54*</td><td char=\".\" align=\"char\">0.39*</td><td char=\".\" align=\"char\">0.34*</td><td char=\".\" align=\"char\">0.47*</td><td char=\".\" align=\"char\">0.40*</td><td char=\".\" align=\"char\">0.57*</td><td char=\".\" align=\"char\">0.25*</td></tr><tr><td align=\"left\">10</td><td align=\"left\">Washing</td><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\">1.0</td><td char=\".\" align=\"char\">0.68*</td><td char=\".\" align=\"char\">0.61*</td><td char=\".\" align=\"char\">0.30*</td><td char=\".\" align=\"char\">0.33*</td><td char=\".\" align=\"char\">0.35*</td><td char=\".\" align=\"char\">0.24*</td><td char=\".\" align=\"char\">0.29*</td><td char=\".\" align=\"char\">0.24*</td><td char=\".\" align=\"char\">0.49*</td><td char=\".\" align=\"char\">0.29*</td><td char=\".\" align=\"char\">0.45*</td><td char=\".\" align=\"char\">0.52*</td><td char=\".\" align=\"char\">0.53*</td><td char=\".\" align=\"char\">0.39*</td><td char=\".\" align=\"char\">0.37*</td><td char=\".\" align=\"char\">0.31*</td><td char=\".\" align=\"char\">0.29*</td><td char=\".\" align=\"char\">0.38*</td><td char=\".\" align=\"char\">0.26*</td></tr><tr><td align=\"left\">11</td><td align=\"left\">Pants</td><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\">1.0</td><td char=\".\" align=\"char\">0.76*</td><td char=\".\" align=\"char\">0.51*</td><td char=\".\" align=\"char\">0.51*</td><td char=\".\" align=\"char\">0.46*</td><td char=\".\" align=\"char\">0.39*</td><td char=\".\" align=\"char\">0.41*</td><td char=\".\" align=\"char\">0.32*</td><td char=\".\" align=\"char\">0.63*</td><td char=\".\" align=\"char\">0.34*</td><td char=\".\" align=\"char\">0.48*</td><td char=\".\" align=\"char\">0.48*</td><td char=\".\" align=\"char\">0.57*</td><td char=\".\" align=\"char\">0.47*</td><td char=\".\" align=\"char\">0.40*</td><td char=\".\" align=\"char\">0.45*</td><td char=\".\" align=\"char\">0.43*</td><td char=\".\" align=\"char\">0.48*</td><td char=\".\" align=\"char\">0.29*</td></tr><tr><td align=\"left\">12</td><td align=\"left\">Socks</td><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\">1.0</td><td char=\".\" align=\"char\">0.37*</td><td char=\".\" align=\"char\">0.46*</td><td char=\".\" align=\"char\">0.40*</td><td char=\".\" align=\"char\">0.31*</td><td char=\".\" align=\"char\">0.35*</td><td char=\".\" align=\"char\">0.34*</td><td char=\".\" align=\"char\">0.49*</td><td char=\".\" align=\"char\">0.26*</td><td char=\".\" align=\"char\">0.36*</td><td char=\".\" align=\"char\">0.39*</td><td char=\".\" align=\"char\">0.50*</td><td char=\".\" align=\"char\">0.41*</td><td char=\".\" align=\"char\">0.31*</td><td char=\".\" align=\"char\">0.46*</td><td char=\".\" align=\"char\">0.44*</td><td char=\".\" align=\"char\">0.48*</td><td char=\".\" align=\"char\">0.25*</td></tr><tr><td align=\"left\">13</td><td align=\"left\">Sitting</td><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\">1.0</td><td char=\".\" align=\"char\">0.40*</td><td char=\".\" align=\"char\">0.44*</td><td char=\".\" align=\"char\">0.45*</td><td char=\".\" align=\"char\">0.39*</td><td char=\".\" align=\"char\">0.27*</td><td char=\".\" align=\"char\">0.44*</td><td char=\".\" align=\"char\">0.35*</td><td char=\".\" align=\"char\">0.37*</td><td char=\".\" align=\"char\">0.43*</td><td char=\".\" align=\"char\">0.42*</td><td char=\".\" align=\"char\">0.41*</td><td char=\".\" align=\"char\">0.30*</td><td char=\".\" align=\"char\">0.36*</td><td char=\".\" align=\"char\">0.38*</td><td char=\".\" align=\"char\">0.35*</td><td char=\".\" align=\"char\">0.22*</td></tr><tr><td align=\"left\">14</td><td align=\"left\">Standing up chairs</td><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\">1.0</td><td char=\".\" align=\"char\">0.60*</td><td char=\".\" align=\"char\">0.49*</td><td char=\".\" align=\"char\">0.59*</td><td char=\".\" align=\"char\">0.49*</td><td char=\".\" align=\"char\">0.68*</td><td char=\".\" align=\"char\">0.58*</td><td char=\".\" align=\"char\">0.64*</td><td char=\".\" align=\"char\">0.58*</td><td char=\".\" align=\"char\">0.61*</td><td char=\".\" align=\"char\">0.49*</td><td char=\".\" align=\"char\">0.61*</td><td char=\".\" align=\"char\">0.37*</td><td char=\".\" align=\"char\">0.49*</td><td char=\".\" align=\"char\">0.47*</td><td char=\".\" align=\"char\">0.52*</td></tr><tr><td align=\"left\">15</td><td align=\"left\">Standing</td><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\">1.0</td><td char=\".\" align=\"char\">0.71*</td><td char=\".\" align=\"char\">0.76*</td><td char=\".\" align=\"char\">0.56*</td><td char=\".\" align=\"char\">0.69*</td><td char=\".\" align=\"char\">0.59*</td><td char=\".\" align=\"char\">0.62*</td><td char=\".\" align=\"char\">0.57*</td><td char=\".\" align=\"char\">0.62*</td><td char=\".\" align=\"char\">0.63*</td><td char=\".\" align=\"char\">0.57*</td><td char=\".\" align=\"char\">0.55*</td><td char=\".\" align=\"char\">0.51*</td><td char=\".\" align=\"char\">0.55*</td><td char=\".\" align=\"char\">0.47*</td></tr><tr><td align=\"left\">16</td><td align=\"left\"><p>Walking</p><p>distance</p></td><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\">1.0</td><td char=\".\" align=\"char\">0.74*</td><td char=\".\" align=\"char\">0.51*</td><td char=\".\" align=\"char\">0.59*</td><td char=\".\" align=\"char\">0.54*</td><td char=\".\" align=\"char\">0.52*</td><td char=\".\" align=\"char\">0.50*</td><td char=\".\" align=\"char\">0.50*</td><td char=\".\" align=\"char\">0.54*</td><td char=\".\" align=\"char\">0.51*</td><td char=\".\" align=\"char\">0.58*</td><td char=\".\" align=\"char\">0.54*</td><td char=\".\" align=\"char\">0.59*</td><td char=\".\" align=\"char\">0.44*</td></tr><tr><td align=\"left\">17</td><td align=\"left\">Walking time</td><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\">1.0</td><td char=\".\" align=\"char\">0.49*</td><td char=\".\" align=\"char\">0.57*</td><td char=\".\" align=\"char\">0.52*</td><td char=\".\" align=\"char\">0.66*</td><td char=\".\" align=\"char\">0.56*</td><td char=\".\" align=\"char\">0.52*</td><td char=\".\" align=\"char\">0.49*</td><td char=\".\" align=\"char\">0.58*</td><td char=\".\" align=\"char\">0.50*</td><td char=\".\" align=\"char\">0.49*</td><td char=\".\" align=\"char\">0.56*</td><td char=\".\" align=\"char\">0.50*</td></tr><tr><td align=\"left\">18</td><td align=\"left\">Stairs</td><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\">1.0</td><td char=\".\" align=\"char\">0.52*</td><td char=\".\" align=\"char\">0.44*</td><td char=\".\" align=\"char\">0.52*</td><td char=\".\" align=\"char\">0.46*</td><td char=\".\" align=\"char\">0.39*</td><td char=\".\" align=\"char\">0.47*</td><td char=\".\" align=\"char\">0.52*</td><td char=\".\" align=\"char\">.450**</td><td char=\".\" align=\"char\">0.40*</td><td char=\".\" align=\"char\">0.57*</td><td char=\".\" align=\"char\">0.30*</td></tr><tr><td align=\"left\">19</td><td align=\"left\">Ride</td><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\">1.0</td><td char=\".\" align=\"char\">0.70*</td><td char=\".\" align=\"char\">0.70*</td><td char=\".\" align=\"char\">0.70*</td><td char=\".\" align=\"char\">0.72*</td><td char=\".\" align=\"char\">0.73*</td><td char=\".\" align=\"char\">0.68*</td><td char=\".\" align=\"char\">0.57*</td><td char=\".\" align=\"char\">0.61*</td><td char=\".\" align=\"char\">0.62*</td><td char=\".\" align=\"char\">0.56*</td></tr><tr><td align=\"left\">20</td><td align=\"left\">Shelving</td><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\">1.0</td><td char=\".\" align=\"char\">0.69*</td><td char=\".\" align=\"char\">0.65*</td><td char=\".\" align=\"char\">0.62*</td><td char=\".\" align=\"char\">0.66*</td><td char=\".\" align=\"char\">0.64*</td><td char=\".\" align=\"char\">0.41*</td><td char=\".\" align=\"char\">0.55*</td><td char=\".\" align=\"char\">0.55*</td><td char=\".\" align=\"char\">0.60*</td></tr><tr><td align=\"left\">21</td><td align=\"left\">Hanging laundry</td><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\">1.0</td><td char=\".\" align=\"char\">0.80*</td><td char=\".\" align=\"char\">0.64*</td><td char=\".\" align=\"char\">0.71*</td><td char=\".\" align=\"char\">0.81*</td><td char=\".\" align=\"char\">0.40*</td><td char=\".\" align=\"char\">0.50*</td><td char=\".\" align=\"char\">0.48*</td><td char=\".\" align=\"char\">0.55*</td></tr><tr><td align=\"left\">22</td><td align=\"left\">Light housework</td><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\">1.0</td><td char=\".\" align=\"char\">0.68*</td><td char=\".\" align=\"char\">0.64*</td><td char=\".\" align=\"char\">0.82*</td><td char=\".\" align=\"char\">0.46*</td><td char=\".\" align=\"char\">0.50*</td><td char=\".\" align=\"char\">0.42*</td><td char=\".\" align=\"char\">0.52*</td></tr><tr><td align=\"left\">23</td><td align=\"left\"><p>Heavy</p><p>housework</p></td><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\">1.0</td><td char=\".\" align=\"char\">0.69*</td><td char=\".\" align=\"char\">0.63*</td><td char=\".\" align=\"char\">0.53*</td><td char=\".\" align=\"char\">0.56*</td><td char=\".\" align=\"char\">0.51*</td><td char=\".\" align=\"char\">0.53*</td></tr><tr><td align=\"left\">24</td><td align=\"left\">Garbage</td><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\">1.0</td><td char=\".\" align=\"char\">0.68*</td><td char=\".\" align=\"char\">0.49*</td><td char=\".\" align=\"char\">0.61*</td><td char=\".\" align=\"char\">0.53*</td><td char=\".\" align=\"char\">0.46*</td></tr><tr><td align=\"left\">25</td><td align=\"left\">Dishwashing</td><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\">1.0</td><td char=\".\" align=\"char\">0.41*</td><td char=\".\" align=\"char\">0.53*</td><td char=\".\" align=\"char\">0.45*</td><td char=\".\" align=\"char\">0.49*</td></tr><tr><td align=\"left\">26</td><td align=\"left\">Sports</td><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\">1.0</td><td char=\".\" align=\"char\">0.67*</td><td char=\".\" align=\"char\">0.71*</td><td char=\".\" align=\"char\">0.45*</td></tr><tr><td align=\"left\">27</td><td align=\"left\">Shopping</td><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\">1.0</td><td char=\".\" align=\"char\">0.56*</td><td char=\".\" align=\"char\">0.49*</td></tr><tr><td align=\"left\">28</td><td align=\"left\">Community activity</td><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\">1.0</td><td char=\".\" align=\"char\">0.52*</td></tr><tr><td align=\"left\">29</td><td align=\"left\">Anxiety</td><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\">1.0</td></tr></tbody></table></table-wrap>", "<table-wrap id=\"Tab6\"><label>Table 6</label><caption><p>Factor loadings and factor score coefficients.</p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th align=\"left\" rowspan=\"2\">Question item number</th><th align=\"left\" rowspan=\"2\">Content</th><th align=\"left\" colspan=\"2\">Factor loading</th><th align=\"left\" colspan=\"2\">Factor score coefficients</th></tr><tr><th align=\"left\">Factor 1</th><th align=\"left\">Factor 2</th><th align=\"left\">Factor 1</th><th align=\"left\">Factor 2</th></tr></thead><tbody><tr><td align=\"left\">25</td><td align=\"left\">Dishwashing</td><td char=\".\" align=\"char\">0.941</td><td char=\".\" align=\"char\"> − 0.069</td><td char=\".\" align=\"char\">0.195</td><td char=\".\" align=\"char\"> − 0.011</td></tr><tr><td align=\"left\">21</td><td align=\"left\">Hanging laundry</td><td char=\".\" align=\"char\">0.888</td><td char=\".\" align=\"char\"> − 0.011</td><td char=\".\" align=\"char\">0.161</td><td char=\".\" align=\"char\">0.008</td></tr><tr><td align=\"left\">20</td><td align=\"left\">Shelving</td><td char=\".\" align=\"char\">0.873</td><td char=\".\" align=\"char\"> − 0.117</td><td char=\".\" align=\"char\">0.098</td><td char=\".\" align=\"char\"> − 0.016</td></tr><tr><td align=\"left\">22</td><td align=\"left\">Light housework</td><td char=\".\" align=\"char\">0.869</td><td char=\".\" align=\"char\">0.027</td><td char=\".\" align=\"char\">0.165</td><td char=\".\" align=\"char\">0.020</td></tr><tr><td align=\"left\">17</td><td align=\"left\">Walking time</td><td char=\".\" align=\"char\">0.753</td><td char=\".\" align=\"char\">0.058</td><td char=\".\" align=\"char\">0.082</td><td char=\".\" align=\"char\">0.016</td></tr><tr><td align=\"left\">24</td><td align=\"left\">Garbage disposal</td><td char=\".\" align=\"char\">0.679</td><td char=\".\" align=\"char\">0.154</td><td char=\".\" align=\"char\">0.074</td><td char=\".\" align=\"char\">0.033</td></tr><tr><td align=\"left\">29</td><td align=\"left\">Anxiety</td><td char=\".\" align=\"char\">0.670</td><td char=\".\" align=\"char\"> − 0.004</td><td char=\".\" align=\"char\">0.049</td><td char=\".\" align=\"char\">0.003</td></tr><tr><td align=\"left\">15</td><td align=\"left\">Standing</td><td char=\".\" align=\"char\">0.657</td><td char=\".\" align=\"char\">0.165</td><td char=\".\" align=\"char\">0.068</td><td char=\".\" align=\"char\">0.033</td></tr><tr><td align=\"left\">19</td><td align=\"left\">Ride</td><td char=\".\" align=\"char\">0.656</td><td char=\".\" align=\"char\">0.233</td><td char=\".\" align=\"char\">0.086</td><td char=\".\" align=\"char\">0.056</td></tr><tr><td align=\"left\">18</td><td align=\"left\">Stairs</td><td char=\".\" align=\"char\">0.610</td><td char=\".\" align=\"char\">0.105</td><td char=\".\" align=\"char\">0.047</td><td char=\".\" align=\"char\">0.017</td></tr><tr><td align=\"left\">14</td><td align=\"left\">Standing up chairs</td><td char=\".\" align=\"char\">0.512</td><td char=\".\" align=\"char\">0.240</td><td char=\".\" align=\"char\">0.041</td><td char=\".\" align=\"char\">0.034</td></tr><tr><td align=\"left\">27</td><td align=\"left\">Shopping</td><td char=\".\" align=\"char\">0.498</td><td char=\".\" align=\"char\">0.258</td><td char=\".\" align=\"char\">0.040</td><td char=\".\" align=\"char\">0.037</td></tr><tr><td align=\"left\">1</td><td align=\"left\">Appearance</td><td char=\".\" align=\"char\">0.479</td><td char=\".\" align=\"char\"> − 0.260</td><td char=\".\" align=\"char\">0.022</td><td char=\".\" align=\"char\"> − 0.019</td></tr><tr><td align=\"left\">5</td><td align=\"left\">Heartburn</td><td char=\".\" align=\"char\">0.453</td><td char=\".\" align=\"char\">0.003</td><td char=\".\" align=\"char\">0.023</td><td char=\".\" align=\"char\">0.002</td></tr><tr><td align=\"left\">2</td><td align=\"left\">Back pain</td><td char=\".\" align=\"char\">0.340</td><td char=\".\" align=\"char\"> − 0.122</td><td char=\".\" align=\"char\">0.015</td><td char=\".\" align=\"char\"> − 0.008</td></tr><tr><td align=\"left\">3</td><td align=\"left\">Leg pain</td><td char=\".\" align=\"char\">0.310</td><td char=\".\" align=\"char\">0.070</td><td char=\".\" align=\"char\">0.015</td><td char=\".\" align=\"char\">0.007</td></tr><tr><td align=\"left\">4</td><td align=\"left\">Appetite</td><td char=\".\" align=\"char\">0.207</td><td char=\".\" align=\"char\">0.063</td><td char=\".\" align=\"char\">0.009</td><td char=\".\" align=\"char\">0.005</td></tr><tr><td align=\"left\">12</td><td align=\"left\">Socks</td><td char=\".\" align=\"char\"> − 0.230</td><td char=\".\" align=\"char\">1.009</td><td char=\".\" align=\"char\"> − 0.028</td><td char=\".\" align=\"char\">0.294</td></tr><tr><td align=\"left\">9</td><td align=\"left\">Picking up</td><td char=\".\" align=\"char\"> − 0.156</td><td char=\".\" align=\"char\">0.933</td><td char=\".\" align=\"char\"> − 0.013</td><td char=\".\" align=\"char\">0.215</td></tr><tr><td align=\"left\">11</td><td align=\"left\">Pants</td><td char=\".\" align=\"char\"> − 0.012</td><td char=\".\" align=\"char\">0.803</td><td char=\".\" align=\"char\">0.005</td><td char=\".\" align=\"char\">0.150</td></tr><tr><td align=\"left\">8</td><td align=\"left\">Toilet</td><td char=\".\" align=\"char\"> − 0.157</td><td char=\".\" align=\"char\">0.667</td><td char=\".\" align=\"char\"> − 0.007</td><td char=\".\" align=\"char\">0.068</td></tr><tr><td align=\"left\">10</td><td align=\"left\">Washing</td><td char=\".\" align=\"char\">0.041</td><td char=\".\" align=\"char\">0.652</td><td char=\".\" align=\"char\">0.006</td><td char=\".\" align=\"char\">0.083</td></tr><tr><td align=\"left\">7</td><td align=\"left\">Standing up floors</td><td char=\".\" align=\"char\">0.064</td><td char=\".\" align=\"char\">0.607</td><td char=\".\" align=\"char\">0.007</td><td char=\".\" align=\"char\">0.072</td></tr><tr><td align=\"left\">28</td><td align=\"left\">Community activity</td><td char=\".\" align=\"char\">0.308</td><td char=\".\" align=\"char\">0.469</td><td char=\".\" align=\"char\">0.028</td><td char=\".\" align=\"char\">0.066</td></tr><tr><td align=\"left\">26</td><td align=\"left\">Sports</td><td char=\".\" align=\"char\">0.286</td><td char=\".\" align=\"char\">0.419</td><td char=\".\" align=\"char\">0.022</td><td char=\".\" align=\"char\">0.050</td></tr><tr><td align=\"left\">13</td><td align=\"left\">Sitting</td><td char=\".\" align=\"char\">0.247</td><td char=\".\" align=\"char\">0.325</td><td char=\".\" align=\"char\">0.015</td><td char=\".\" align=\"char\">0.032</td></tr><tr><td align=\"left\">6</td><td align=\"left\">Sleeping</td><td char=\".\" align=\"char\">0.111</td><td char=\".\" align=\"char\">0.282</td><td char=\".\" align=\"char\">0.006</td><td char=\".\" align=\"char\">0.023</td></tr></tbody></table></table-wrap>", "<table-wrap id=\"Tab7\"><label>Table 7</label><caption><p>Comparison of the final version scores between operative cases and conservative cases, and between preoperative condition and postoperative condition.</p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th align=\"left\"/><th align=\"left\" colspan=\"2\">Operative cases<break/><italic>N</italic> = 89</th><th align=\"left\" colspan=\"2\">Conservative cases<break/><italic>N</italic> = 9</th><th align=\"left\">p-value (operative vs. conservative)</th><th align=\"left\">Effect size (operative vs. conservative)</th></tr></thead><tbody><tr><td align=\"left\" rowspan=\"4\"><p>Factor 1</p><p>Main symptom (10 questions)</p></td><td align=\"left\">Pre-operative condition</td><td align=\"left\">47 ± 21</td><td align=\"left\" rowspan=\"2\">Current condition</td><td align=\"left\" rowspan=\"2\">63 ± 15</td><td align=\"left\">0.029</td><td align=\"left\">0.77</td></tr><tr><td align=\"left\">Post-operative condition</td><td align=\"left\">70 ± 22</td><td align=\"left\">0.3</td><td align=\"left\">0.35</td></tr><tr><td align=\"left\">p-value (pre vs. post)</td><td align=\"left\"> &lt; 0.0001</td><td align=\"left\" rowspan=\"2\"/><td align=\"left\" rowspan=\"2\">N.A.</td><td align=\"left\" rowspan=\"2\" colspan=\"2\">N.A.</td></tr><tr><td align=\"left\">Effect size (pre vs. post)</td><td align=\"left\">1.09</td></tr><tr><td align=\"left\" rowspan=\"4\"><p>Factor 2</p><p>Collateral symptom (5 questions)</p></td><td align=\"left\">Pre-operative condition</td><td align=\"left\">76 ± 25</td><td align=\"left\" rowspan=\"2\">Current condition</td><td align=\"left\" rowspan=\"2\">92 ± 12</td><td align=\"left\">0.005</td><td align=\"left\">0.67</td></tr><tr><td align=\"left\">Post-operative condition</td><td align=\"left\">60 ± 25</td><td align=\"left\">0.0001</td><td align=\"left\">1.34</td></tr><tr><td align=\"left\">p-value (pre vs. post)</td><td align=\"left\">0.0001</td><td align=\"left\" rowspan=\"2\"/><td align=\"left\" rowspan=\"2\">N.A.</td><td align=\"left\" rowspan=\"2\" colspan=\"2\">N.A.</td></tr><tr><td align=\"left\">Effect size (pre vs. post)</td><td align=\"left\">0.65</td></tr><tr><td align=\"left\" rowspan=\"4\">PCS (SF-8)</td><td align=\"left\">Pre-operative condition</td><td align=\"left\">31 ± 7</td><td align=\"left\" rowspan=\"2\">Current condition</td><td align=\"left\" rowspan=\"2\">35 ± 5</td><td align=\"left\">0.1</td><td align=\"left\">0.51</td></tr><tr><td align=\"left\">Post-operative condition</td><td align=\"left\">41 ± 8</td><td align=\"left\">0.03</td><td align=\"left\">0.76</td></tr><tr><td align=\"left\">p-value (pre vs. post)</td><td align=\"left\">0.0001</td><td align=\"left\" rowspan=\"2\"/><td align=\"left\" rowspan=\"2\">N.A.</td><td align=\"left\" rowspan=\"2\" colspan=\"2\">N.A.</td></tr><tr><td align=\"left\">Effect size (pre vs. post)</td><td align=\"left\">1.26</td></tr><tr><td align=\"left\" rowspan=\"4\">MCS (SF-8)</td><td align=\"left\">Pre-operative condition</td><td align=\"left\">43 ± 9</td><td align=\"left\" rowspan=\"2\">Current condition</td><td align=\"left\" rowspan=\"2\">47 ± 7</td><td align=\"left\">0.1</td><td align=\"left\">0.52</td></tr><tr><td align=\"left\">Post-operative condition</td><td align=\"left\">49 ± 7</td><td align=\"left\">0.6</td><td align=\"left\">0.16</td></tr><tr><td align=\"left\">p-value (pre vs. post)</td><td align=\"left\">0.0002</td><td align=\"left\" rowspan=\"2\"/><td align=\"left\" rowspan=\"2\">N.A.</td><td align=\"left\" rowspan=\"2\" colspan=\"2\">N.A.</td></tr><tr><td align=\"left\">Effect size (pre vs. post)</td><td align=\"left\">0.40</td></tr></tbody></table></table-wrap>", "<table-wrap id=\"Tab8\"><label>Table 8</label><caption><p>Spearman’s correlation coefficients between change scores and 5-point satisfaction rating scale.</p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th align=\"left\"/><th align=\"left\">Satisfaction</th><th align=\"left\">Main symptom</th><th align=\"left\">Collateral symptom</th><th align=\"left\">PCS (SF-8)</th><th align=\"left\">MCS (SF-8)</th></tr></thead><tbody><tr><td align=\"left\">Satisfaction</td><td align=\"left\">1</td><td align=\"left\">0.48*</td><td align=\"left\">0.16</td><td align=\"left\">0.38*</td><td align=\"left\">0.07</td></tr><tr><td align=\"left\">Main symptom</td><td align=\"left\"/><td align=\"left\">1</td><td align=\"left\">0.25*</td><td align=\"left\">0.43*</td><td align=\"left\">0.28*</td></tr><tr><td align=\"left\">Collateral symptom</td><td align=\"left\"/><td align=\"left\"/><td align=\"left\">1</td><td align=\"left\">0.11</td><td align=\"left\">0.22*</td></tr><tr><td align=\"left\">PCS (SF-8)</td><td align=\"left\"/><td align=\"left\"/><td align=\"left\"/><td align=\"left\">1</td><td align=\"left\">0.06</td></tr><tr><td align=\"left\">MCS (SF-8)</td><td align=\"left\"/><td align=\"left\"/><td align=\"left\"/><td align=\"left\"/><td align=\"left\">1</td></tr></tbody></table></table-wrap>", "<table-wrap id=\"Tab9\"><label>Table 9</label><caption><p>Validation data of the ASD disease-specific scale.</p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th align=\"left\"/><th align=\"left\">Study subjects from two sites</th><th align=\"left\">Cases collected for validation from another facility</th><th align=\"left\">p-value</th></tr></thead><tbody><tr><td align=\"left\">Male/female (cases)</td><td align=\"left\">10/79</td><td align=\"left\">2/23</td><td align=\"left\">1.0</td></tr><tr><td align=\"left\">Mean age ± SD (years)</td><td align=\"left\">68 ± 7</td><td align=\"left\">72 ± 8</td><td align=\"left\">0.02*</td></tr><tr><td align=\"left\">Mean post operative follow-up period ± SD (mos.)</td><td align=\"left\">56 ± 35</td><td align=\"left\">46 ± 22</td><td align=\"left\">0.1</td></tr><tr><td align=\"left\">Mean fusion intervertebral levels ± SD</td><td align=\"left\">10 ± 3</td><td align=\"left\">9 ± 1</td><td align=\"left\">0.1</td></tr><tr><td align=\"left\">Fixation to sacrum or pelvis (cases/%)</td><td align=\"left\">76/85</td><td align=\"left\">24/96</td><td align=\"left\">0.5</td></tr><tr><td align=\"left\">Fixation from T8, T9, or T10 to pelvis (cases/%)</td><td align=\"left\">56/63</td><td align=\"left\">25/100</td><td align=\"left\">0.01*</td></tr><tr><td align=\"left\">Fixation from T3, T4, or T5 to pelvis (cases/%) 13 / 15 0 / 0 0.01*</td><td align=\"left\"/><td align=\"left\">0/0</td><td align=\"left\">0.01*</td></tr><tr><td align=\"left\" colspan=\"4\">Post operative satisfaction (cases/%)</td></tr><tr><td align=\"left\"> Very satisfied</td><td align=\"left\">23/26</td><td align=\"left\">8/32</td><td align=\"left\">0.6</td></tr><tr><td align=\"left\"> Satisfied</td><td align=\"left\">42/47</td><td align=\"left\">11/44</td><td align=\"left\">0.8</td></tr><tr><td align=\"left\"> Neither</td><td align=\"left\">18/20</td><td align=\"left\">5/20</td><td align=\"left\">1.0</td></tr><tr><td align=\"left\"> Dissatisfied</td><td align=\"left\">6/7</td><td align=\"left\">1/4</td><td align=\"left\">1.0</td></tr><tr><td align=\"left\"> Very dissatisfied</td><td align=\"left\">0/0</td><td align=\"left\">0/0</td><td align=\"left\">1.0</td></tr><tr><td align=\"left\" colspan=\"4\">Main symptom (mean ± SD)</td></tr><tr><td align=\"left\"> Preoperative condition</td><td align=\"left\">47 ± 21</td><td align=\"left\">56 ± 19</td><td align=\"left\">0.05</td></tr><tr><td align=\"left\"> Postoperative condition</td><td align=\"left\">70 ± 22</td><td align=\"left\">76 ± 19</td><td align=\"left\">0.2</td></tr><tr><td align=\"left\"> Effect size</td><td align=\"left\">1.09</td><td align=\"left\">1.05</td><td align=\"left\">N.A.</td></tr><tr><td align=\"left\" colspan=\"4\">Collateral symptom (mean ± SD)</td></tr><tr><td align=\"left\"> Preoperative condition</td><td align=\"left\">76 ± 25</td><td align=\"left\">75 ± 23</td><td align=\"left\">0.9</td></tr><tr><td align=\"left\"> Postoperative condition</td><td align=\"left\">60 ± 25</td><td align=\"left\">64 ± 24</td><td align=\"left\">0.5</td></tr><tr><td align=\"left\"> Effect size</td><td align=\"left\">0.65</td><td align=\"left\">0.48</td><td align=\"left\">N.A.</td></tr><tr><td align=\"left\" colspan=\"4\">PCS (SF-8) (mean ± SD)</td></tr><tr><td align=\"left\"> Preoperative condition</td><td align=\"left\">31 ± 7</td><td align=\"left\">32 ± 7</td><td align=\"left\">0.6</td></tr><tr><td align=\"left\"> Postoperative condition</td><td align=\"left\">41 ± 8</td><td align=\"left\">42 ± 8</td><td align=\"left\">0.4</td></tr><tr><td align=\"left\"> Effect size</td><td align=\"left\">1.26</td><td align=\"left\">1.40</td><td align=\"left\">N.A.</td></tr><tr><td align=\"left\" colspan=\"4\">MCS (SF-8) (mean ± SD)</td></tr><tr><td align=\"left\"> Preoperative condition</td><td align=\"left\">43 ± 9</td><td align=\"left\">41 ± 9</td><td align=\"left\">0.4</td></tr><tr><td align=\"left\"> Postoperative condition</td><td align=\"left\">49 ± 7</td><td align=\"left\">46 ± 8</td><td align=\"left\">1.0</td></tr><tr><td align=\"left\"> Effect size</td><td align=\"left\">0.40</td><td align=\"left\">0.56</td><td align=\"left\">N.A.</td></tr></tbody></table></table-wrap>" ]

[ "<disp-formula id=\"Equ1\"><label>1</label><alternatives><tex-math id=\"M1\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$ {\\text{Main symptom score }}\\left( {\\text{first factor}} \\right) \\, = { 1}00 - ({\\text{Q1}} \\times {7 } + {\\text{ Q2}} \\times {7 } + {\\text{ Q15}} \\times {2 } + {\\text{ Q17}} \\times {3 } + {\\text{ Q19}} \\times {3 } + {\\text{ Q 2}}0 \\times {3} + {\\text{ Q 21}} \\times {6 } + {\\text{ Q 24}} \\times {3 } + {\\text{ Q 25}} \\times {7 } + {\\text{ Q 29}} \\times {2}) - {43})/{172} \\times {1}00, $$\\end{document}</tex-math><mml:math id=\"M2\" display=\"block\"><mml:mrow><mml:mrow><mml:mtext>Main symptom score</mml:mtext><mml:mspace width=\"0.333333em\"/></mml:mrow><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:mtext>first factor</mml:mtext></mml:mrow></mml:mfenced><mml:mrow><mml:mspace width=\"0.166667em\"/><mml:mo>=</mml:mo><mml:mn>100</mml:mn><mml:mo>-</mml:mo><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mtext>Q1</mml:mtext><mml:mo>×</mml:mo><mml:mn>7</mml:mn><mml:mo>+</mml:mo><mml:mrow><mml:mspace width=\"0.333333em\"/><mml:mtext>Q2</mml:mtext></mml:mrow><mml:mo>×</mml:mo><mml:mn>7</mml:mn><mml:mo>+</mml:mo><mml:mrow><mml:mspace width=\"0.333333em\"/><mml:mtext>Q15</mml:mtext></mml:mrow><mml:mo>×</mml:mo><mml:mn>2</mml:mn><mml:mo>+</mml:mo><mml:mrow><mml:mspace width=\"0.333333em\"/><mml:mtext>Q17</mml:mtext></mml:mrow><mml:mo>×</mml:mo><mml:mn>3</mml:mn><mml:mo>+</mml:mo><mml:mrow><mml:mspace width=\"0.333333em\"/><mml:mtext>Q19</mml:mtext></mml:mrow><mml:mo>×</mml:mo><mml:mn>3</mml:mn><mml:mo>+</mml:mo><mml:mrow><mml:mspace width=\"0.333333em\"/><mml:mtext>Q 2</mml:mtext></mml:mrow><mml:mn>0</mml:mn><mml:mo>×</mml:mo><mml:mn>3</mml:mn><mml:mo>+</mml:mo><mml:mrow><mml:mspace width=\"0.333333em\"/><mml:mtext>Q 21</mml:mtext></mml:mrow><mml:mo>×</mml:mo><mml:mn>6</mml:mn><mml:mo>+</mml:mo><mml:mrow><mml:mspace width=\"0.333333em\"/><mml:mtext>Q 24</mml:mtext></mml:mrow><mml:mo>×</mml:mo><mml:mn>3</mml:mn><mml:mo>+</mml:mo><mml:mrow><mml:mspace width=\"0.333333em\"/><mml:mtext>Q 25</mml:mtext></mml:mrow><mml:mo>×</mml:mo><mml:mn>7</mml:mn><mml:mo>+</mml:mo><mml:mrow><mml:mspace width=\"0.333333em\"/><mml:mtext>Q 29</mml:mtext></mml:mrow><mml:mo>×</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>-</mml:mo><mml:mn>43</mml:mn><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"false\">/</mml:mo><mml:mn>172</mml:mn><mml:mo>×</mml:mo><mml:mn>100</mml:mn><mml:mo>,</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ2\"><label>2</label><alternatives><tex-math id=\"M3\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$ {\\text{Collateral symptom score }}\\left( {\\text{second factor}} \\right) \\, = { 1}00 - ({\\text{Q 7}} \\times {3} + {\\text{ Q 8}} \\times {3} + {\\text{ Q 9}} \\times {9} + {\\text{ Q 1}}0 \\times {3} + {\\text{ Q 12}} \\times {12}) - {3}0)/{15}0 \\times {1}00. $$\\end{document}</tex-math><mml:math id=\"M4\" display=\"block\"><mml:mrow><mml:mrow><mml:mtext>Collateral symptom score</mml:mtext><mml:mspace width=\"0.333333em\"/></mml:mrow><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:mtext>second factor</mml:mtext></mml:mrow></mml:mfenced><mml:mrow><mml:mspace width=\"0.166667em\"/><mml:mo>=</mml:mo><mml:mn>100</mml:mn><mml:mo>-</mml:mo><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mrow><mml:mtext>Q 7</mml:mtext></mml:mrow><mml:mo>×</mml:mo><mml:mn>3</mml:mn><mml:mo>+</mml:mo><mml:mrow><mml:mspace width=\"0.333333em\"/><mml:mtext>Q 8</mml:mtext></mml:mrow><mml:mo>×</mml:mo><mml:mn>3</mml:mn><mml:mo>+</mml:mo><mml:mrow><mml:mspace width=\"0.333333em\"/><mml:mtext>Q 9</mml:mtext></mml:mrow><mml:mo>×</mml:mo><mml:mn>9</mml:mn><mml:mo>+</mml:mo><mml:mrow><mml:mspace width=\"0.333333em\"/><mml:mtext>Q 1</mml:mtext></mml:mrow><mml:mn>0</mml:mn><mml:mo>×</mml:mo><mml:mn>3</mml:mn><mml:mo>+</mml:mo><mml:mrow><mml:mspace width=\"0.333333em\"/><mml:mtext>Q 12</mml:mtext></mml:mrow><mml:mo>×</mml:mo><mml:mn>12</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>-</mml:mo><mml:mn>30</mml:mn><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"false\">/</mml:mo><mml:mn>150</mml:mn><mml:mo>×</mml:mo><mml:mn>100</mml:mn><mml:mo>.</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></disp-formula>" ]

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[ "<supplementary-material content-type=\"local-data\" id=\"MOESM1\"></supplementary-material>" ]

[ "<table-wrap-foot><p><italic>SD</italic> standard deviation.</p></table-wrap-foot>", "<table-wrap-foot><p>Raw data means the score points of the answer options. For raw data, higher numbers indicate more activity restrictions.</p><p><italic>SD</italic> standard deviation.</p></table-wrap-foot>", "<table-wrap-foot><p>*Means p &lt; 0.05.</p></table-wrap-foot>", "<table-wrap-foot><p>Scores are shown as mean ± SD.</p><p><italic>N.A.</italic> not available, <italic>SD</italic> standard deviation.</p></table-wrap-foot>", "<table-wrap-foot><p>*Means <italic>p</italic> &lt; 0.05.</p></table-wrap-foot>", "<table-wrap-foot><p><italic>ASD</italic> adult spinal deformity, <italic>N.A.</italic> not available, <italic>SD</italic> standard deviation.</p><p>*Means <italic>p</italic> &lt; 0.05.</p></table-wrap-foot>", "<fn-group><fn><p><bold>Publisher's note</bold></p><p>Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.</p></fn></fn-group>" ]

[ "<graphic xlink:href=\"41598_2024_51783_Fig1_HTML\" id=\"MO1\"/>", "<graphic xlink:href=\"41598_2024_51783_Fig2_HTML\" id=\"MO2\"/>", "<graphic xlink:href=\"41598_2024_51783_Fig3_HTML\" id=\"MO3\"/>", "<graphic xlink:href=\"41598_2024_51783_Article_Equ1.gif\" position=\"anchor\"/>", "<graphic xlink:href=\"41598_2024_51783_Article_Equ2.gif\" position=\"anchor\"/>", "<graphic xlink:href=\"41598_2024_51783_Fig4_HTML\" id=\"MO4\"/>", "<graphic xlink:href=\"41598_2024_51783_Fig5_HTML\" id=\"MO5\"/>", "<graphic xlink:href=\"41598_2024_51783_Fig6_HTML\" id=\"MO6\"/>", "<graphic xlink:href=\"41598_2024_51783_Fig7_HTML\" id=\"MO7\"/>" ]

[ "<media xlink:href=\"41598_2024_51783_MOESM1_ESM.xlsx\"><caption><p>Supplementary Information.</p></caption></media>" ]

{ "acronym": [], "definition": [] }

[ "<title>Introduction</title>", "<p id=\"Par2\">Sugarcane juice is a nutritious and energetic green brownish beverage with low acidity (pH 4.8-5.5) and high water activity (Aw~0.99). The juice’s composition depends on the variety, cultivar, maturation stage, soil, climatic and agricultural conditions. Cane juice has a limited shelf life due to its rapid microbiological and enzymic deterioration^{##UREF##0##1##–##UREF##2##3##}, and when processed in industrial plants is subjected to heat treatment in order to inactivate enzymes, spoilage and potentially pathogenic microorganisms. Thermal processing, depending on its intensity, may however damage the sensory, functional and nutritional juice’s quality^{##UREF##3##4##}. During extraction, the juice is exposed to oxygen, a reactant for enzymic browning, catalyzed by polyphenol oxidase (PPO) and peroxidase (POD), so the enzymic inactivation guarantees a higher quality product^{##UREF##0##1##}. Additionally, it is necessary to eliminate spoilage microorganisms, such as <italic>Leuconostoc mesenteroides</italic>, molds and yeasts. <italic>L. mesenteroids</italic> produces lactic acid and changes the juice’s viscosity during storage^{##UREF##0##1##}.</p>", "<p id=\"Par3\">The industrialization of cane juice is rapidly growing in Brazil, and over the last few years more than 10 brands of processed cane juice have been launched. The preservation technologies frequently combine acidification, chemical preservatives, heat treatment and refrigeration, and vary among brands. Nevertheless, the sensory quality of the products available on the market is questionable. As an alternative to heat treatment, the technology that employs supercritical carbon dioxide (SC-CO₂) is a promising intervention to inactivate pathogens and spoilage enzymes and microorganisms. This non-thermal technique consists of exposing food or beverages to high pressure (beyond 74 bar), and the main advantage is the preservation of the food sensory attributes^{##UREF##0##1##}. In contrast, conventional heat treatments may trigger the onset of nutritional and sensory losses. A supercritical fluid is defined as any substance maintained above its critical temperature and pressure. The critical temperature is the highest temperature at which a gas can be converted into a liquid by increasing pressure. Critical pressure is the highest pressure at which a liquid can be converted into a gas by increasing the temperature of the liquid^{##UREF##4##5##}. In SC-CO₂ treatment, the food is exposed to pressurized CO₂ for a certain period of time. The supercritical fluid diffuses through the food, showing a microbicidal effect, whose intensity depends on the pressure and holding time^{##UREF##5##6##–##UREF##7##8##}. The temperature and pressure that characterize the critical state/point of carbon dioxide are 31.1 °C and 73.8 bar^{##UREF##3##4##}.</p>", "<p id=\"Par4\">Gómez-López et al.^{##REF##33764212##9##} reported the specific energy required by the pressure change technology (PCT) application. The energy consumption per unit mass of treated product has been estimated and compared to that required by conventional indirect thermal technology. The estimated specific energy consumption was respectively 162.6 kJ/kg for indirect thermal and 26.3 kJ/kg for PCT. On the other hand, completely non-thermal processes such as PCT do not involve any energy costs for product and equipment cooling. Therefore, the energy consumption involved in this technology is only due to product and inert gas pumping and compression. Vignali et al.^{##UREF##8##10##} stated a comparison of typical specific working energy costs for thermal and non-thermal treatments in terms of KJ per kilograms of processed product. Non-thermal approaches seem to offer the most effective alternative in terms of nutrients and fresh-like characteristics preservation as well as working energy costs saving. Nevertheless, the SC-CO₂ treatment has not been considered in that study because the literature is scarce and the data available for microbial inactivation are very low in comparison to the other technologies. According to Bocker and Silva^{##UREF##9##11##}, CO₂ stands out as the best cost-benefit among supercritical fluids since it has a low cost and is non-toxic. The critical conditions of CO₂ (73,8 Bar and 31 °C) are moderate compared to those of other fluids used in the supercritical state. These moderate conditions reduce the process energy expenditure and promote less damage to the nutritional properties of the food matrices.</p>", "<p id=\"Par5\">Many studies address the direct injection application of SC-CO₂ in the inactivation of enzymes and microorganisms in fruit juices^{##UREF##0##1##,##REF##30064759##12##–##UREF##11##14##}. The SC-CO₂ proved to be efficient in the inactivation of microorganisms and enzymes. Additionally, the low toxicity and cost are important advantages of CO₂ since it is naturally found in the atmosphere. The use of SC-CO₂ under mild conditions is a technique that, when used in juices, allows greater preservation of thermally unstable constituents, such as phenolic compounds, flavonoids and anthocyanins^{##REF##25849158##15##}. Nevertheless, no work targeting the stabilization of SC-CO_2-treated cane juice has been found. This study was primarily conducted to evaluate the combined effect of mild temperatures and SC-CO₂ on microorganisms and enzymes in cane juice.</p>" ]

[ "<title>Methods</title>", "<p id=\"Par37\">The cane juice was procured from a local vendor, in the city of Pirassununga/SP-Brazil. The freshly extracted juice was collected in a plastic bottle by the vendor, kept on ice in an isothermal container, and rapidly transported to the laboratory in the Food Engineering Department at the University of Sao Paulo. The juice’s sample was divided into two parts and collected in previously sterilized glass bottles with screw caps. One fraction was used as a control (unprocessed juice) and the other one was treated with SC-CO₂. Figure ##FIG##3##4## illustrates the juice processing.</p>", "<p id=\"Par38\">The treatment of cane juice with direct injection of SC-CO₂ was carried out in a 100 mL-reactor of a supercritical fluid system (Thar Technologies SFE-500, Pittsburgh/USA) available at the Laboratory of High Pressure Technology and Natural Products. The juice sample (100 mL) was transferred to the reactor and kept under the pre-set conditions (subsequently described). At the end of the treatment the sample was moved out through rapid depressurization to a sterilized glass flask.</p>", "<p id=\"Par39\">Table ##TAB##5##6## points out the independent variables and their actual and coded levels tested according to Rodrigues and Iemma^{##UREF##24##29##}. Pressures (P) in the range of 74 to 351 bar, temperatures (T) between 33 and 67 °C, and holding times (t) varying from 20 to 70 min were tested in a central composite rotatable design. Seventeen trials were performed. This study aimed at exploring a wide range of CO₂ pressure (above the critical one – 73.8 bar). Also the operational limits of the equipment available to conduct this study were considered in the range of the investigated parameters set. Because no study carried out with cane juice was found, no reference is herein mentioned. As for the temperature, mild values were targeted to preserve the original quality of the juice.</p>", "<p id=\"Par40\">To get an approximate statistical inference, three trials were conducted at the central point of the experimental space; they can provide valuable information on the behavior of the responses between the levels attributed to the factors, and demonstrate the repeatability of the process (Rodrigues and Iemma)^{##UREF##24##29##}.</p>", "<p id=\"Par41\">The physicochemical, enzymic, microbiological analysis and instrumental determination of color parameters were carried out on raw and processed samples to evaluate the performance of multiple combinations of the processing’s parameters (pressure, temperature and holding time) and are as follows. All assays were performed in triplicate as described in Petrus and Simões^{##UREF##25##30##}.</p>", "<p id=\"Par42\">The physicochemical tests were performed according to the Association of Official Analytical Chemists (AOAC, 2010)^{##UREF##26##31##}. An Analyzer model 300 M was used to determine the pH. The soluble solids content (expressed in °Brix) was determined in a Reichert model AR 200 portable digital refractometer.</p>", "<p id=\"Par43\">Counts of mesophiles, molds and yeasts, lactic bacteria, and coliforms (at 45 °C) were conducted following the protocol described in the Compendium of Methods for the Microbiological Examination of Foods (Salfinger and Tortorello)^{##UREF##27##32##}.</p>", "<p id=\"Par44\">The protocols adapted from ref. ^{##UREF##28##33##} were used to determine the polyphenol oxidase (PPO) and peroxidase (POD) activities.</p>", "<p id=\"Par45\">Five and half milliliters of 0.2 M phosphate buffer solution (pH 6.0) and 1.5 mL of 0.2 M catechol were added into a test tube and maintained at 25 °C for 10 min. Then 1.0 mL of the diluted sample in deionized water (1:10) was added. The tube was stirred for 15 s and returned to the water bath at 25 °C for 30 min. The absorbance was read in a spectrophotometer at 425 nm. The blank was prepared by diluting the sample in deionized water.</p>", "<p id=\"Par46\">Seven milliliters of 0.2 M phosphate buffer solution (pH 5.5) and 1.0 mL of the diluted sample (juice) in deionized water (1:10) were added to a test tube and maintained in a heat bath at 35 °C for 10 min. Then 1.5 mL of 0.05% guaiacol and 0.5 mL of 0.1% hydrogen peroxide were added. The tube was magnetically stirred for 15 s and returned to the bath at 35 °C for 15 min. Finally, the absorbance was read in a spectrophotometer at 470 nm. One (1) unit of enzyme activity (U) was defined as the amount of enzymic extract capable of increasing absorbance at 425 and 470 nm for PPO and POD, respectively, at rates of 0.001 units per minute.</p>", "<p id=\"Par47\">The color parameters (L*, a* and b*) of unprocessed and treated juice’s samples were measured in a Hunterlab Ultra-Scan colorimeter (Hunter Associates Laboratory, Model SN7877 Reston, VA/USA). The iluminant D65 and observation angle at 10° were set up. The parameters a* and b* were used to determine chroma (C) and hue angle (°hue) (Eq. ##FORMU##3##3## and ##FORMU##4##4##). To compare the raw juice to the processed one, total color difference (TCD) was calculated by Eq. ##FORMU##5##5##. Also L*, a* and b* were inserted in the EasyRGB to convert them into color image (EasyRGB)^{##UREF##29##34##}.</p>", "<p id=\"Par48\">L*: lightness (0 a 100).</p>", "<p id=\"Par49\">a*: coordinate red (+60) / green (−60).</p>", "<p id=\"Par50\">B: coordinate yellow (+60) / blue (−60).</p>", "<p id=\"Par51\">ΔL*: lightness variation</p>", "<p id=\"Par52\">Δa*: red/green variation</p>", "<p id=\"Par53\">Δb*: yellow/blue variation</p>", "<p id=\"Par54\">Data from the central composite rotatable design were first subjected to the analysis of effects to identify the variable(s) (P, T and t) that had significant effect on the responses (PPO, POD, coliforms, mesophiles, molds and yeasts, and lactic bacteria reduction, total color difference, pH and soluble solids variation), at 10% of significance. Due to the high variability of processes involving microorganisms and enzymes, <italic>p</italic>-values below 10 percent (<italic>p</italic> ⩽ 0.1) are considered significant parameters, as stated by Rodrigues and Iemma (2015). The analysis of regression was performed for both 1st and 2nd (responses including axial points) orders. Then the mathematical model was re-parameterized considering only the statistically significant coefficients. The analysis of variance (ANOVA) was undertaken to evaluate if the model was statistically significant. If so, the response surface was generated. Statistical tests were performed using the software Protimiza Experimental Design (<ext-link ext-link-type=\"uri\" xlink:href=\"http://experimental-design.protimiza.com.br\">http://experimental-design.protimiza.com.br</ext-link>).</p>", "<title>Reporting summary</title>", "<p id=\"Par55\">Further information on research design is available in the ##SUPPL##0##Nature Research Reporting Summary## linked to this article.</p>" ]

[ "<title>Results and discussion</title>", "<title>Physicochemical tests</title>", "<p id=\"Par6\">Table ##TAB##0##1## exhibits the pH and soluble solids values determined in raw and processed cane juice.</p>", "<p id=\"Par7\">The pH values ranged from 4.6 to 6.0 in the raw juice and between 4.4 and 6.3 for the processed one. The treatments reduced up to 0.4 units in the pH; however, the pH from trials 3 (295 bar/40 °C/30 min), 6 (213 bar/50 °C/45 min) and 16 (213 bar/50 °C/20 min) remained unchanged. The variety of cane, type of soil, fertilization, climatic conditions, degree of maturity, harvesting and extraction methods are important factors to be considered in the variation of juice’s pH. Bomdespacho et al.^{##UREF##12##16##} evaluated different cultivars of raw cane juice and reported an average pH equivalent to 5.05. This data is close to the values found in most treatments performed in the present study, with the exception of trials 3, 9 and 12.</p>", "<p id=\"Par8\">Regarding the soluble solids content, variations between 18.5 and 25.3 °Brix were determined in the raw juice, and between 18.2 and 25.0 for the processed beverage. The variations (Δ) in this parameter caused by the treatment ranged between 0.0 and 0.4. With the exception of trials 15 and 16, there was a reduction in this parameter. These phenomena may be related to the variation of the treatments submitted, as well as to the batch used on the day of the respective trials. Bomdespacho et al.^{##UREF##12##16##} reported an average of 21.2 °Brix in fresh juice extracted from different cultivars. This result is in the range obtained in this study. In all 17 trials, no meaningful variations (Δ ≤ 0.4) were observed between processed and raw juice. These findings are positive as they lead to the hypothesis that there was no significant difference between the pH values and soluble solids after the treatments applied.</p>", "<title>Microbiological assays</title>", "<p id=\"Par9\">Table ##TAB##1##2## reports the microbial counts in raw and SC-CO₂-treated cane juice as well as the log reduction achieved in each trial.</p>", "<p id=\"Par10\">The results exhibited in Table ##TAB##1##2## show the potential of SC-CO₂ in the reduction of contaminants in raw cane juice. The reductions achieved by the different trinomials were 2.5 log for coliforms, 3.9 log for aerobic mesophiles, 2.1 log for lactic acid bacteria, and 4.1 log for molds and yeasts.</p>", "<p id=\"Par11\">The lactic bacteria counts in raw juice varied between 1.0 and 4.0 logCFU/mL. For the processed sample, counts ranged from &lt;1.0_est to 3.5 logCFU/mL; comparison with data from other studies was not possible, once counts were carried out after cane fermentation, as reported by Silva et al.^{##UREF##13##17##}, who performed counts after 3, 11 and 24 h of fermentation. The lactic bacteria contamination, such as <italic>Leuconostoc mesenteroides</italic> and some species of the <italic>Lactobacillus</italic>, can trigger the synthesis of dextrans (polysaccharides formed by glucose units) (Koblitz)^{##UREF##14##18##}, forming gums in the juice, leading to its rejection. The discrepancy among counts within the same group of microorganisms in raw juice may be attributed to failures in the hygiene procedures of the raw material, utensils and/or equipment used in the extraction. This event is usual when it comes to street vending.</p>", "<p id=\"Par12\">According to Prati, Moretti and Cardello^{##UREF##15##19##}, mesophilic counts above 6.0 logCFU/mL may be related to hygienic-sanitary deficiencies in the extraction and/or storage of cane juice. In this study both the raw and processed juice exhibited counts within the range 1.6–6.0 logCFU/mL. For molds and yeasts, counts were between 2.4 and 5.4 logCFU/mL; Jay^{##UREF##16##20##} holds that values above 3.0 logCFU/mL can cause undesirable changes.</p>", "<p id=\"Par13\">The efficiency of SC-CO₂ treatment on microbial inactivation is associated with the modification of intracellular and extracellular pH, and also the length of time CO₂ diffuses into the cells. Therefore, the holding time of treatment greatly impacts the microbial inactivation rate^{##UREF##3##4##}.</p>", "<p id=\"Par14\">Dhansu et al.^{##UREF##17##21##} pasteurized cane juice at 65 °C/25 min, and stored it under refrigeration, achieving a shelf life of 60 days. Oliveira et al.^{##UREF##0##1##} pasteurized the juice at 70 °C/25 min; the lactic acid bacteria counts in raw and processed cane juice were (5.9 and 1.3) logCFU/mL respectively, reaching 4.6 log reduction. The molds and yeasts’ counts in raw and processed juice were (6.1 and 1.7) logCFU/mL respectively. Gomes et al.^{##UREF##18##22##} optimized the time x temperature binomial used in the pasteurization of whole cane juice; temperatures and holding times ranging between 78 and 92 °C, and from 16 to 44 s, were tested. Regarding the reduction of microorganisms, the treatment at 90 °C/40 s was the most efficient, achieving 4.6 log reductions for mesophiles. For molds and yeasts, 3.2 log reductions were reached.</p>", "<p id=\"Par15\">Hart et al.^{##REF##35185167##23##} reported the application of SC-CO₂ in the inactivation of spores in foods, highlighting how this technique can be more efficient in preserving nutritional and sensory characteristics as compared to high hydrostatic pressure techniques and thermal methods at high temperatures. The action of SC-CO₂ occurs through disruption of the cell wall, coating, cortex and membranes, and degradation of proteins. More in-depth studies on larger scales are needed to disseminate this technology in the processing of fruits and juices.</p>", "<p id=\"Par16\">Eggleston^{##UREF##19##24##} reported the microbiological, enzymic and chemical deterioration (acid degradation) of sucrose in cane juice. The findings indicated that the growth of microorganisms is relevant for sucrose degradation. After 14 h, the largest contribution was microbiological, accounting for 93% of losses, while enzymatic degradation contributed with 5.7% of losses and chemical degradation with 1.3%.</p>", "<title>Enzymic tests</title>", "<p id=\"Par17\">The endogenous enzymes activity (polyphenol oxidase and peroxidase) as well as the percentages of reduction achieved by different treatments are shown in Table ##TAB##2##3##.</p>", "<p id=\"Par18\">The results exhibited in Table ##TAB##2##3## suggest the potential of SC-CO₂ combined with mild temperatures to inactivate the endogenous enzymes that are responsible for the degradation of the color, flavor and the nutritional values of cane juice. The percentages of PPO (3.3–64.5%) and POD (0.0–40.9%) reduction varied widely. The trinomial applied in trial 5 (295 bar/60 °C/30 min) reached the greatest PPO inactivation (64.5%), suggesting that temperature had a more significant effect on the percentage of reduction; however, this hypothesis will be confirmed in the light of the statistical analysis of the effects of the variables studied. POD exhibited greater resistance to the treatments in most trials. The trinomial applied in trial 15 (213 bar/67 °C/45 min) reached the highest percentage of inactivation (40.9%). Marszałek et al.^{##REF##30064759##12##} studied the effect of supercritical carbon dioxide on PPO and POD in mushroom and radish; surprisingly, PPO was more resistant to temperature and pressure than POD. In this study, similar result was observed in trials 7, 9, 10 and 15, i.e., PPO was more resistant to SC-CO₂. In most trials, however, the percentage of POD reduction in the juice was lower than PPO. This finding suggests that the type of food matrix also influences the impact of the technology that is applied. The food matrix can interfere with the intermolecular bonds of the two enzymes depending on the amount of water in the medium, since the impact of pressure on intra and intermolecular interactions can also be correlated with the ability of the functional groups of the enzymes to interact with water (Marszałek et al.)^{##UREF##20##25##}. The deactivation of enzymes exposed to high temperatures and prolonged times is explained by changes in the tertiary and secondary structures of the protein. The thermal stability of enzymes depends on a number of factors such as source, species, nature of the food matrix (Iqbal et al.)^{##UREF##21##26##}.</p>", "<title>Color parameters</title>", "<p id=\"Par19\">The color parameters instrumentally measured in cane juice are presented in Table ##TAB##3##4##.</p>", "<p id=\"Par20\">The parameter L*, which represents lightness, varied widely for raw (32.3–72.9) and processed (32.5–73.1) juice. Most treatments positively influenced the lightness of the juice samples, which is most likely related to enzymic inactivation (Table ##TAB##2##3##). This result could favor the juice’s sensory acceptance, assuming the consumers prefer a lighter drink. Similarly, there was a great variation in the a* parameter for fresh (1–16.1) and processed (4.0–14.7) juice. The b* parameter also varied considerably for raw (26.0–46.2) and processed (24.8–50.7) samples. Regarding the chroma parameter (C*), significant variations were also observed for raw (28.1–46.4) and processed (25.4–52.8) juice. Chroma correlates to saturation, characterizing the sample’s color as “vivid” or opaque (dull). This attribute is independent of lightness and °hue. Saturation ranges from purple-red to green, and increases from the center (0) to the edge of the color wheel. Oliveira et al.^{##UREF##0##1##} stated that low C* values represent gray, and values close to 60 represent vivid colors; they found C* values close to 9 (more neutral color) for raw cane juice, in contrast to the present study. Meerod^{##UREF##22##27##} studied different cultivars of raw material, which showed divergent colors at various levels. Following the same behavior as the previous parameters, the hue angle also showed great variation for raw (68.1–88.5) and processed (71.1–84.5°) juices. Hue, measured in degrees, classifies color (green, yellow, blue, etc.). The ranges determined in this study are positioned in the first quadrant of the color circle, and can be classified between yellow-red and yellow. The wide variation ranges in the color parameters can be explained by the variability inherent to the raw material; juice samples extracted on different days, from different stalks, were used during the course of this research.</p>", "<p id=\"Par21\">Figure ##FIG##0##1## illustrates the total color difference (TCD) between raw and processed cane juice.</p>", "<p id=\"Par22\">The highest (12.3) and lowest (2.0) values of total color difference (∆E*) were determined for juice treated at 295 bar/ 60 °C/ 60 min and (130 bar/ 40 °C/ 60 min and 213 bar/ 33 °C/45 min), respectively. Bernard et al.^{##UREF##23##28##} states that ∆E* values less than 3 cannot be easily detected by the human eye, and values greater than 12 represent different color “spaces”. Therefore, of the 17 tests carried out, only six preserved the original color of the juice, in terms of its sensory perception.</p>", "<title>Statistical analysis</title>", "<p id=\"Par23\">Because the log reduction in coliforms and lactic bacteria could not be calculated in some trials (Table ##TAB##2##3##), these responses were not subjected to the statistical analysis. Figure ##FIG##1##2## demonstrates the Pareto diagrams, built to investigate which parameters/variables (pressure/P/x₁, temperature/T/x₂, holding time/t/x₃) were significant (<italic>p</italic> ≤ 0.1) in the studied responses. The terms that were not statistically significant were incorporated into the lack-of-fit to calculate the coefficient of determination (R²).</p>", "<p id=\"Par24\">As for mesophiles, molds and yeasts reduction, and soluble solids variation, none of variables or their interactions were significant. In terms of polyphenol oxidase (PPO) reduction, only t (x₃) was significant; however, the parameters T (x₂), t (x₃), and the interaction between them (x₂.x₃) played a significant effect on the peroxidase (POD) reduction. In regards to pH variation, P (x₁) and the interaction between T and t (x₂.x₃) were significant. Finally, P, T, t, and the interaction between T and t were significant in the total color difference.</p>", "<p id=\"Par25\">Only the significant variables were encompassed into the mathematical model, whose statistical significance was evaluated through analysis of variance (ANOVA). Table ##TAB##4##5## exhibits the ANOVA carried out for the models (1st and 2nd orders) generated for the responses <italic>POD reduction</italic> and <italic>total color difference</italic> (TDC); the models for other responses were not statistically significant (<italic>p</italic> &gt; 0.1). The coded predicted models obtained for the aforementioned responses are represented by Eqs. ##FORMU##1##1## and ##FORMU##2##2##.</p>", "<p id=\"Par26\">Y₁ – POD reduction (%)</p>", "<p id=\"Par27\">x₂ – Temperature (T)</p>", "<p id=\"Par28\">x₃ – holding time (t)</p>", "<p id=\"Par29\">Y₂ – Total color difference</p>", "<p id=\"Par30\">x₁ – Pressure (P)</p>", "<p id=\"Par31\">x₂ – Temperature (T)</p>", "<p id=\"Par32\">x₃ – holding time (t)</p>", "<p id=\"Par33\">For practical purposes, it is desirable that the fitted model be as simple as possible and contain the smallest possible number of parameters without giving up the quality assured in the careful selection of the experimental design. The models herein presented were re-parameterized/reduced because the parameters with little or no influence on the outcome of the final fit were excluded.</p>", "<p id=\"Par34\">Regarding the response <italic>POD reduction</italic>, Table ##TAB##4##5## shows that the 1^st order model (R² = 0.86) better fitted to experimental data than the 2^nd order model (R² = 0.47). For both orders, F_calc was greater than F_tab. Similarly, as for the <italic>total color difference</italic> (TCD), the 1st order mathematical model (R² = 0.90) best fitted to experimental data. The coded Eqs. ##FORMU##1##1## and ##FORMU##2##2## can be used to predict the percentage of POD reduction and the TCD that can be achieved in cane juice processed under the same conditions of this study. The coded model is that whose regression coefficients are obtained from the matrix of coded variables (−α, −1, 0, +1, +α). Given this, to obtain a predicted value from the model one must replace the values in the coded equation. In contrast, if using real values for the variables in the model, the predicted value may be incorrect and even absurd. Of particular relevance is the claim that the first order mathematical models hereby presented (Eqs. ##FORMU##1##1## and ##FORMU##2##2##) are only valid in a range of pressure from 130 to 295 bar, temperature from 40 to 60 min, and holding time between 30 and 60 min (Table ##TAB##5##6##).</p>", "<p id=\"Par35\">Figure ##FIG##2##3## depicts the response surfaces and contour curves that represent Eqs. ##FORMU##1##1## and ##FORMU##2##2##. By analyzing the surface for POD reduction, one can identify the existence of an optimal range for the temperature (57–60 °C) and holding time (56–60 min), regardless the pressure (in the range 130–295 bar). As for TDC, the ranges 130–150 bar, 40–43 °C and 30–35 min, within which the color difference between raw and processed juice is minimal, represent the optimal conditions in this experiment. This is of much greater interest than a simple point value, because it provides information about the “robustness” of the process, and most notably, it is the variation in pressure, temperature and holding time that may be permitted around optimal values which still maintains the process under optimized conditions. This finding is fundamental for the control engineer to define and maintain the pressure, temperature and time sensors and controller levels. This directly affects viability and process implementation (Rodrigues and Iemma)^{##UREF##24##29##}.</p>", "<p id=\"Par36\">A fact worth highlighting is that studies addressing the use of SC-CO₂ in cane juice processing have not been found. In this way, data comparison could not be made. The combination of supercritical carbon dioxide and mild temperatures exhibited a meaningful effect on microorganism’s reduction in sugarcane juice, under the conditions of this study. Endogenous enzymes that deteriorate the juice’s quality were partially inactivated. None of variables (pressure/P, temperature/T, holding time/t) or their interactions were significant in mesophiles, molds and yeasts reduction, or soluble solids variation. In terms of polyphenol oxidase (PPO) reduction, only t was significant; however, T, t and the interaction between them played a significant effect on the peroxidase (POD) reduction. In regards to pH variation, P and the interaction between T and t were significant. Finally, P, T, t, and the interaction between T and t were significant in the total color difference. The optimal parameters (P, T and t) determined in this study varied for different responses. The combination of mild temperatures and SC-CO₂ can be potentially used for cane juice preservation.</p>" ]

[ "<title>Results and discussion</title>", "<title>Physicochemical tests</title>", "<p id=\"Par6\">Table ##TAB##0##1## exhibits the pH and soluble solids values determined in raw and processed cane juice.</p>", "<p id=\"Par7\">The pH values ranged from 4.6 to 6.0 in the raw juice and between 4.4 and 6.3 for the processed one. The treatments reduced up to 0.4 units in the pH; however, the pH from trials 3 (295 bar/40 °C/30 min), 6 (213 bar/50 °C/45 min) and 16 (213 bar/50 °C/20 min) remained unchanged. The variety of cane, type of soil, fertilization, climatic conditions, degree of maturity, harvesting and extraction methods are important factors to be considered in the variation of juice’s pH. Bomdespacho et al.^{##UREF##12##16##} evaluated different cultivars of raw cane juice and reported an average pH equivalent to 5.05. This data is close to the values found in most treatments performed in the present study, with the exception of trials 3, 9 and 12.</p>", "<p id=\"Par8\">Regarding the soluble solids content, variations between 18.5 and 25.3 °Brix were determined in the raw juice, and between 18.2 and 25.0 for the processed beverage. The variations (Δ) in this parameter caused by the treatment ranged between 0.0 and 0.4. With the exception of trials 15 and 16, there was a reduction in this parameter. These phenomena may be related to the variation of the treatments submitted, as well as to the batch used on the day of the respective trials. Bomdespacho et al.^{##UREF##12##16##} reported an average of 21.2 °Brix in fresh juice extracted from different cultivars. This result is in the range obtained in this study. In all 17 trials, no meaningful variations (Δ ≤ 0.4) were observed between processed and raw juice. These findings are positive as they lead to the hypothesis that there was no significant difference between the pH values and soluble solids after the treatments applied.</p>", "<title>Microbiological assays</title>", "<p id=\"Par9\">Table ##TAB##1##2## reports the microbial counts in raw and SC-CO₂-treated cane juice as well as the log reduction achieved in each trial.</p>", "<p id=\"Par10\">The results exhibited in Table ##TAB##1##2## show the potential of SC-CO₂ in the reduction of contaminants in raw cane juice. The reductions achieved by the different trinomials were 2.5 log for coliforms, 3.9 log for aerobic mesophiles, 2.1 log for lactic acid bacteria, and 4.1 log for molds and yeasts.</p>", "<p id=\"Par11\">The lactic bacteria counts in raw juice varied between 1.0 and 4.0 logCFU/mL. For the processed sample, counts ranged from &lt;1.0_est to 3.5 logCFU/mL; comparison with data from other studies was not possible, once counts were carried out after cane fermentation, as reported by Silva et al.^{##UREF##13##17##}, who performed counts after 3, 11 and 24 h of fermentation. The lactic bacteria contamination, such as <italic>Leuconostoc mesenteroides</italic> and some species of the <italic>Lactobacillus</italic>, can trigger the synthesis of dextrans (polysaccharides formed by glucose units) (Koblitz)^{##UREF##14##18##}, forming gums in the juice, leading to its rejection. The discrepancy among counts within the same group of microorganisms in raw juice may be attributed to failures in the hygiene procedures of the raw material, utensils and/or equipment used in the extraction. This event is usual when it comes to street vending.</p>", "<p id=\"Par12\">According to Prati, Moretti and Cardello^{##UREF##15##19##}, mesophilic counts above 6.0 logCFU/mL may be related to hygienic-sanitary deficiencies in the extraction and/or storage of cane juice. In this study both the raw and processed juice exhibited counts within the range 1.6–6.0 logCFU/mL. For molds and yeasts, counts were between 2.4 and 5.4 logCFU/mL; Jay^{##UREF##16##20##} holds that values above 3.0 logCFU/mL can cause undesirable changes.</p>", "<p id=\"Par13\">The efficiency of SC-CO₂ treatment on microbial inactivation is associated with the modification of intracellular and extracellular pH, and also the length of time CO₂ diffuses into the cells. Therefore, the holding time of treatment greatly impacts the microbial inactivation rate^{##UREF##3##4##}.</p>", "<p id=\"Par14\">Dhansu et al.^{##UREF##17##21##} pasteurized cane juice at 65 °C/25 min, and stored it under refrigeration, achieving a shelf life of 60 days. Oliveira et al.^{##UREF##0##1##} pasteurized the juice at 70 °C/25 min; the lactic acid bacteria counts in raw and processed cane juice were (5.9 and 1.3) logCFU/mL respectively, reaching 4.6 log reduction. The molds and yeasts’ counts in raw and processed juice were (6.1 and 1.7) logCFU/mL respectively. Gomes et al.^{##UREF##18##22##} optimized the time x temperature binomial used in the pasteurization of whole cane juice; temperatures and holding times ranging between 78 and 92 °C, and from 16 to 44 s, were tested. Regarding the reduction of microorganisms, the treatment at 90 °C/40 s was the most efficient, achieving 4.6 log reductions for mesophiles. For molds and yeasts, 3.2 log reductions were reached.</p>", "<p id=\"Par15\">Hart et al.^{##REF##35185167##23##} reported the application of SC-CO₂ in the inactivation of spores in foods, highlighting how this technique can be more efficient in preserving nutritional and sensory characteristics as compared to high hydrostatic pressure techniques and thermal methods at high temperatures. The action of SC-CO₂ occurs through disruption of the cell wall, coating, cortex and membranes, and degradation of proteins. More in-depth studies on larger scales are needed to disseminate this technology in the processing of fruits and juices.</p>", "<p id=\"Par16\">Eggleston^{##UREF##19##24##} reported the microbiological, enzymic and chemical deterioration (acid degradation) of sucrose in cane juice. The findings indicated that the growth of microorganisms is relevant for sucrose degradation. After 14 h, the largest contribution was microbiological, accounting for 93% of losses, while enzymatic degradation contributed with 5.7% of losses and chemical degradation with 1.3%.</p>", "<title>Enzymic tests</title>", "<p id=\"Par17\">The endogenous enzymes activity (polyphenol oxidase and peroxidase) as well as the percentages of reduction achieved by different treatments are shown in Table ##TAB##2##3##.</p>", "<p id=\"Par18\">The results exhibited in Table ##TAB##2##3## suggest the potential of SC-CO₂ combined with mild temperatures to inactivate the endogenous enzymes that are responsible for the degradation of the color, flavor and the nutritional values of cane juice. The percentages of PPO (3.3–64.5%) and POD (0.0–40.9%) reduction varied widely. The trinomial applied in trial 5 (295 bar/60 °C/30 min) reached the greatest PPO inactivation (64.5%), suggesting that temperature had a more significant effect on the percentage of reduction; however, this hypothesis will be confirmed in the light of the statistical analysis of the effects of the variables studied. POD exhibited greater resistance to the treatments in most trials. The trinomial applied in trial 15 (213 bar/67 °C/45 min) reached the highest percentage of inactivation (40.9%). Marszałek et al.^{##REF##30064759##12##} studied the effect of supercritical carbon dioxide on PPO and POD in mushroom and radish; surprisingly, PPO was more resistant to temperature and pressure than POD. In this study, similar result was observed in trials 7, 9, 10 and 15, i.e., PPO was more resistant to SC-CO₂. In most trials, however, the percentage of POD reduction in the juice was lower than PPO. This finding suggests that the type of food matrix also influences the impact of the technology that is applied. The food matrix can interfere with the intermolecular bonds of the two enzymes depending on the amount of water in the medium, since the impact of pressure on intra and intermolecular interactions can also be correlated with the ability of the functional groups of the enzymes to interact with water (Marszałek et al.)^{##UREF##20##25##}. The deactivation of enzymes exposed to high temperatures and prolonged times is explained by changes in the tertiary and secondary structures of the protein. The thermal stability of enzymes depends on a number of factors such as source, species, nature of the food matrix (Iqbal et al.)^{##UREF##21##26##}.</p>", "<title>Color parameters</title>", "<p id=\"Par19\">The color parameters instrumentally measured in cane juice are presented in Table ##TAB##3##4##.</p>", "<p id=\"Par20\">The parameter L*, which represents lightness, varied widely for raw (32.3–72.9) and processed (32.5–73.1) juice. Most treatments positively influenced the lightness of the juice samples, which is most likely related to enzymic inactivation (Table ##TAB##2##3##). This result could favor the juice’s sensory acceptance, assuming the consumers prefer a lighter drink. Similarly, there was a great variation in the a* parameter for fresh (1–16.1) and processed (4.0–14.7) juice. The b* parameter also varied considerably for raw (26.0–46.2) and processed (24.8–50.7) samples. Regarding the chroma parameter (C*), significant variations were also observed for raw (28.1–46.4) and processed (25.4–52.8) juice. Chroma correlates to saturation, characterizing the sample’s color as “vivid” or opaque (dull). This attribute is independent of lightness and °hue. Saturation ranges from purple-red to green, and increases from the center (0) to the edge of the color wheel. Oliveira et al.^{##UREF##0##1##} stated that low C* values represent gray, and values close to 60 represent vivid colors; they found C* values close to 9 (more neutral color) for raw cane juice, in contrast to the present study. Meerod^{##UREF##22##27##} studied different cultivars of raw material, which showed divergent colors at various levels. Following the same behavior as the previous parameters, the hue angle also showed great variation for raw (68.1–88.5) and processed (71.1–84.5°) juices. Hue, measured in degrees, classifies color (green, yellow, blue, etc.). The ranges determined in this study are positioned in the first quadrant of the color circle, and can be classified between yellow-red and yellow. The wide variation ranges in the color parameters can be explained by the variability inherent to the raw material; juice samples extracted on different days, from different stalks, were used during the course of this research.</p>", "<p id=\"Par21\">Figure ##FIG##0##1## illustrates the total color difference (TCD) between raw and processed cane juice.</p>", "<p id=\"Par22\">The highest (12.3) and lowest (2.0) values of total color difference (∆E*) were determined for juice treated at 295 bar/ 60 °C/ 60 min and (130 bar/ 40 °C/ 60 min and 213 bar/ 33 °C/45 min), respectively. Bernard et al.^{##UREF##23##28##} states that ∆E* values less than 3 cannot be easily detected by the human eye, and values greater than 12 represent different color “spaces”. Therefore, of the 17 tests carried out, only six preserved the original color of the juice, in terms of its sensory perception.</p>", "<title>Statistical analysis</title>", "<p id=\"Par23\">Because the log reduction in coliforms and lactic bacteria could not be calculated in some trials (Table ##TAB##2##3##), these responses were not subjected to the statistical analysis. Figure ##FIG##1##2## demonstrates the Pareto diagrams, built to investigate which parameters/variables (pressure/P/x₁, temperature/T/x₂, holding time/t/x₃) were significant (<italic>p</italic> ≤ 0.1) in the studied responses. The terms that were not statistically significant were incorporated into the lack-of-fit to calculate the coefficient of determination (R²).</p>", "<p id=\"Par24\">As for mesophiles, molds and yeasts reduction, and soluble solids variation, none of variables or their interactions were significant. In terms of polyphenol oxidase (PPO) reduction, only t (x₃) was significant; however, the parameters T (x₂), t (x₃), and the interaction between them (x₂.x₃) played a significant effect on the peroxidase (POD) reduction. In regards to pH variation, P (x₁) and the interaction between T and t (x₂.x₃) were significant. Finally, P, T, t, and the interaction between T and t were significant in the total color difference.</p>", "<p id=\"Par25\">Only the significant variables were encompassed into the mathematical model, whose statistical significance was evaluated through analysis of variance (ANOVA). Table ##TAB##4##5## exhibits the ANOVA carried out for the models (1st and 2nd orders) generated for the responses <italic>POD reduction</italic> and <italic>total color difference</italic> (TDC); the models for other responses were not statistically significant (<italic>p</italic> &gt; 0.1). The coded predicted models obtained for the aforementioned responses are represented by Eqs. ##FORMU##1##1## and ##FORMU##2##2##.</p>", "<p id=\"Par26\">Y₁ – POD reduction (%)</p>", "<p id=\"Par27\">x₂ – Temperature (T)</p>", "<p id=\"Par28\">x₃ – holding time (t)</p>", "<p id=\"Par29\">Y₂ – Total color difference</p>", "<p id=\"Par30\">x₁ – Pressure (P)</p>", "<p id=\"Par31\">x₂ – Temperature (T)</p>", "<p id=\"Par32\">x₃ – holding time (t)</p>", "<p id=\"Par33\">For practical purposes, it is desirable that the fitted model be as simple as possible and contain the smallest possible number of parameters without giving up the quality assured in the careful selection of the experimental design. The models herein presented were re-parameterized/reduced because the parameters with little or no influence on the outcome of the final fit were excluded.</p>", "<p id=\"Par34\">Regarding the response <italic>POD reduction</italic>, Table ##TAB##4##5## shows that the 1^st order model (R² = 0.86) better fitted to experimental data than the 2^nd order model (R² = 0.47). For both orders, F_calc was greater than F_tab. Similarly, as for the <italic>total color difference</italic> (TCD), the 1st order mathematical model (R² = 0.90) best fitted to experimental data. The coded Eqs. ##FORMU##1##1## and ##FORMU##2##2## can be used to predict the percentage of POD reduction and the TCD that can be achieved in cane juice processed under the same conditions of this study. The coded model is that whose regression coefficients are obtained from the matrix of coded variables (−α, −1, 0, +1, +α). Given this, to obtain a predicted value from the model one must replace the values in the coded equation. In contrast, if using real values for the variables in the model, the predicted value may be incorrect and even absurd. Of particular relevance is the claim that the first order mathematical models hereby presented (Eqs. ##FORMU##1##1## and ##FORMU##2##2##) are only valid in a range of pressure from 130 to 295 bar, temperature from 40 to 60 min, and holding time between 30 and 60 min (Table ##TAB##5##6##).</p>", "<p id=\"Par35\">Figure ##FIG##2##3## depicts the response surfaces and contour curves that represent Eqs. ##FORMU##1##1## and ##FORMU##2##2##. By analyzing the surface for POD reduction, one can identify the existence of an optimal range for the temperature (57–60 °C) and holding time (56–60 min), regardless the pressure (in the range 130–295 bar). As for TDC, the ranges 130–150 bar, 40–43 °C and 30–35 min, within which the color difference between raw and processed juice is minimal, represent the optimal conditions in this experiment. This is of much greater interest than a simple point value, because it provides information about the “robustness” of the process, and most notably, it is the variation in pressure, temperature and holding time that may be permitted around optimal values which still maintains the process under optimized conditions. This finding is fundamental for the control engineer to define and maintain the pressure, temperature and time sensors and controller levels. This directly affects viability and process implementation (Rodrigues and Iemma)^{##UREF##24##29##}.</p>", "<p id=\"Par36\">A fact worth highlighting is that studies addressing the use of SC-CO₂ in cane juice processing have not been found. In this way, data comparison could not be made. The combination of supercritical carbon dioxide and mild temperatures exhibited a meaningful effect on microorganism’s reduction in sugarcane juice, under the conditions of this study. Endogenous enzymes that deteriorate the juice’s quality were partially inactivated. None of variables (pressure/P, temperature/T, holding time/t) or their interactions were significant in mesophiles, molds and yeasts reduction, or soluble solids variation. In terms of polyphenol oxidase (PPO) reduction, only t was significant; however, T, t and the interaction between them played a significant effect on the peroxidase (POD) reduction. In regards to pH variation, P and the interaction between T and t were significant. Finally, P, T, t, and the interaction between T and t were significant in the total color difference. The optimal parameters (P, T and t) determined in this study varied for different responses. The combination of mild temperatures and SC-CO₂ can be potentially used for cane juice preservation.</p>" ]

[]

[ "<p id=\"Par1\">Sugarcane juice is a nutritious and energetic drink. For its processing, the use of supercritical carbon dioxide (SC-CO₂) technology as an intervention potentially capable of rendering a high quality product can be considered. This study evaluated the combined effect of SC-CO₂ and mild temperatures, primarily aiming for the reduction of endogenous microorganisms and enzymes in sugarcane juice (pH~5.5). Pressures (P) ranging from 74 to 351 bar, temperatures (T) between 33 and 67 °C, and holding times (t) between 20 and 70 min were tested in a central composite rotational design. Seventeen trials were performed, comprising three replicates at the central points. Counts of aerobic mesophiles, molds and yeasts, lactic acid bacteria and coliforms at 45 °C, determination of polyphenol oxidase (PPO) and peroxidase (POD) activities, and measurement of color parameters in freshly extracted and processed juice’s samples were carried out. The pH of fresh and processed juice varied between 4.6 and 6.0, and between 4.6 and 6.3, respectively. The number of decimal reductions achieved in mesophiles, molds and yeasts, lactic acid bacteria and coliforms varied between 0.1 and 3.9, 2.1 and 4.1, 0.0 and 2.1, and 0.3 to 2.5, respectively. The percentages of PPO reduction ranged from 3.51% to 64.18%. Regarding the POD, reductions between 0.27% and 41.42% were obtained. Color variations between fresh and processed samples varied between 2.0 and 12.3. As for mesophiles, molds and yeasts reduction, and soluble solids variation, none of the variables or their interactions were significant. In terms of polyphenol oxidase (PPO) reduction, only t was significant; however, T, t, and the interaction between them significantly affected the peroxidase (POD) reduction. In regards to pH variation, P, and the interaction between T and t were significant. P, T, t, and the interaction between T and t played a significant effect on color. The combination of mild temperatures and SC-CO₂ can be potentially used for cane juice preservation.</p>", "<title>Subject terms</title>" ]

[ "<title>Supplementary information</title>", "<p>\n\n</p>" ]

[ "<title>Supplementary information</title>", "<p>The online version contains supplementary material available at 10.1038/s41538-023-00242-x.</p>", "<title>Acknowledgements</title>", "<p>Grateful acknowledgment is made for the grant (2020/08011-5) provided by the Sao Paulo Research Foundation/Brazil (Fundação de Amparo à Pesquisa do Estado de São Paulo), and support offered by the Laboratory of High Pressure Technology and Natural Products.</p>", "<title>Author contributions</title>", "<p>F.C.P. performed the assays and reviewed the manuscript. T.C.K.M. performed the assays and reviewed the manuscript. G.C.D. reviewed the manuscript. A.L.dO. helped to design the study and reviewed the manuscript. R.P. designed the study and wrote the manuscript.</p>", "<title>Data availability</title>", "<p>The datasets generated during and/or analyzed during the current study are available from the corresponding author upon reasonable request.</p>", "<title>Competing interests</title>", "<p id=\"Par56\">The authors declare no competing interests.</p>" ]

[ "<fig id=\"Fig1\"><label>Fig. 1</label><caption><p>Total color difference (TCD) between raw and SC-CO₂-processed cane juice.</p></caption></fig>", "<fig id=\"Fig2\"><label>Fig. 2</label><caption><title>Pareto diagrams for cane juice treated with SC-CO₂.</title><p>Pareto diagrams for mesophiles reduction (2a), molds and yeasts reduction (2b), polyphenol oxidase reduction (2c), peroxidase reduction (2d), pH variation (2e), soluble solids variation (2 f) and total color difference (2 g).</p></caption></fig>", "<fig id=\"Fig3\"><label>Fig. 3</label><caption><title>Response surfaces and contour curves for peroxidase reduction (3a), and total difference color (3b, 3c, 3d) in cane juice treated with SC-CO₂.</title><p>Response surfaces.</p></caption></fig>", "<fig id=\"Fig4\"><label>Fig. 4</label><caption><title>Cane juice processing (4a. 100 mL-feeding vessel, 4b. supercritical processing equipment, 4c. outlet valve).</title><p>Juice processing.</p></caption></fig>" ]

[ "<table-wrap id=\"Tab1\"><label>Table 1</label><caption><p>Physicochemical parameters of raw and SC-CO₂-treated cane juice.</p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th>Trial</th><th>Treatment</th><th>pH</th><th>Soluble solids (°Brix)</th></tr></thead><tbody><tr><td rowspan=\"3\">1</td><td>raw</td><td>5.4 ± 0.1</td><td>25.3 ± 0.0</td></tr><tr><td>213 bar/50 °C/45 min</td><td>5.3 ± 0.0</td><td>24.9 ± 0.0</td></tr><tr><td>∆</td><td>−0.1</td><td>−0.4</td></tr><tr><td rowspan=\"3\">2</td><td>raw</td><td>5.2 ± 0.1</td><td>24.2 ± 0.1</td></tr><tr><td>130 bar/40 °C/30 min</td><td>5.0 ± 0.1</td><td>24.0 ± 0.1</td></tr><tr><td>∆</td><td>−0.2</td><td>−0.2</td></tr><tr><td rowspan=\"3\">3</td><td>raw</td><td>4.6 ± 0.0</td><td>22.1 ± 0.1</td></tr><tr><td>295 bar/40 °C/30 min</td><td>4.6 ± 0.1</td><td>21.8 ± 0.1</td></tr><tr><td>∆</td><td>0.0</td><td>−0.3</td></tr><tr><td rowspan=\"3\">4</td><td>raw</td><td>5.4 ± 0.1</td><td>25.2 ± 0.1</td></tr><tr><td>130 bar/60 °C/30 min</td><td>5.1 ± 0.1</td><td>25.0 ± 0.1</td></tr><tr><td>∆</td><td>−0.3</td><td>−0.2</td></tr><tr><td rowspan=\"3\">5</td><td>raw</td><td>5.4 ± 0.1</td><td>24.3 ± 0.0</td></tr><tr><td>295 bar/60 °C/30 min</td><td>5.2 ± 0.1</td><td>24.0 ± 0.1</td></tr><tr><td>∆</td><td>−0.2</td><td>−0.3</td></tr><tr><td rowspan=\"3\">6</td><td>raw</td><td>5.4 ± 0.0</td><td>23.5 ± 0.1</td></tr><tr><td>213 bar/50 °C/45 min</td><td>5.4 ± 0.0</td><td>23.2 ± 0.1</td></tr><tr><td>∆</td><td>0.0</td><td>−0.3</td></tr><tr><td rowspan=\"3\">7</td><td>raw</td><td>5.4 ± 0.1</td><td>23.6 ± 0.1</td></tr><tr><td>130 bar/40 °C/60 min</td><td>5.0 ± 0.1</td><td>23.2 ± 0.1</td></tr><tr><td>∆</td><td>−0.4</td><td>−0.4</td></tr><tr><td rowspan=\"3\">8</td><td>raw</td><td>5.3 ± 0.1</td><td>23.7 ±</td></tr><tr><td>295 bar/40 °C/60 min</td><td>5.1 ±</td><td>23.6 ±</td></tr><tr><td>∆</td><td>−0.2</td><td>-0.1</td></tr><tr><td rowspan=\"3\">9</td><td>raw</td><td>4.6 ± 0.1</td><td>21.4 ± 0.1</td></tr><tr><td>130 bar/60 °C/60 min</td><td>4.4 ± 0.0</td><td>21.2 ± 0.1</td></tr><tr><td>∆</td><td>−0.2</td><td>−0.2</td></tr><tr><td rowspan=\"3\">10</td><td>raw</td><td>5.3 ± 0.1</td><td>18.7 ± 0.1</td></tr><tr><td>295 bar/60 °C/60 min</td><td>5.5 ± 0.0</td><td>18.7 ± 0.1</td></tr><tr><td>∆</td><td>0.2</td><td>0.0</td></tr><tr><td rowspan=\"3\">11</td><td>raw</td><td>5.5 ± 0.1</td><td>20.6 ± 0.1</td></tr><tr><td>213 bar/50 °C/45 min</td><td>5.6 ± 0.1</td><td>20.4 ± 0.0</td></tr><tr><td>∆</td><td>0.1</td><td>−0.2</td></tr><tr><td rowspan=\"3\">12</td><td>raw</td><td>6.0 ± 0.1</td><td>19.0 ± 0.1</td></tr><tr><td>74 bar/ 50 °C/ 45 min</td><td>6.3 ± 0.0</td><td>18.7 ± 0.1</td></tr><tr><td>∆</td><td>0.3</td><td>−0.3</td></tr><tr><td rowspan=\"3\">13</td><td>raw</td><td>5.6 ± 0.1</td><td>20.9 ± 0.1</td></tr><tr><td>351 bar/50 °C/45 min</td><td>5.4 ± 0.0</td><td>20.8 ± 0.1</td></tr><tr><td>∆</td><td>−0.2</td><td>−0.1</td></tr><tr><td rowspan=\"3\">14</td><td>raw</td><td>5.4 ± 0.1</td><td>20.4 ± 0.1</td></tr><tr><td>213 bar/33 °C/45 min</td><td>5.2 ± 0.0</td><td>20.1 ± 0.1</td></tr><tr><td>∆</td><td>−0.2</td><td>−0.3</td></tr><tr><td rowspan=\"3\">15</td><td>raw</td><td>5.6 ± 0.1</td><td>19.9 ± 0.1</td></tr><tr><td>213 bar/67 °C/45 min</td><td>5.5 ± 0.1</td><td>19.9 ± 0.0</td></tr><tr><td>∆</td><td>−0.1</td><td>0.0</td></tr><tr><td rowspan=\"3\">16</td><td>raw</td><td>5.6 ± 0.1</td><td>18.6 ± 0.1</td></tr><tr><td>213 bar/50 °C/20 min</td><td>5.6 ± 0.0</td><td>18.6 ± 0.1</td></tr><tr><td>∆</td><td>0.0</td><td>0.0</td></tr><tr><td rowspan=\"3\">17</td><td>raw</td><td>5.3 ± 0.1</td><td>18.5 ± 0.1</td></tr><tr><td>213 bar/50 °C/70 min</td><td>5.4 ± 0.0</td><td>18.2 ± 0.1</td></tr><tr><td>∆</td><td>0.1</td><td>−0.3</td></tr></tbody></table></table-wrap>", "<table-wrap id=\"Tab2\"><label>Table 2</label><caption><p>Microbial counts (logCFU/mL) in raw and SC-CO₂-treated cane juice.</p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th>Trial</th><th>Treatment</th><th>Coliforms (45 °C)</th><th>Mesophiles</th><th>Lactic bacteria</th><th>Molds and yeasts</th></tr></thead><tbody><tr><td/><td>raw</td><td>1.5 ± 0.2</td><td>3.6 ± 0.1</td><td>1.0 ± 0.0</td><td>3.5 ± 0.3</td></tr><tr><td>1</td><td>213 bar/50 °C/45 min</td><td>&lt;1_est</td><td>1.8 ± 0.1</td><td>&lt;1_est</td><td>1.1 ± 0.1</td></tr><tr><td/><td><bold>log red</bold></td><td>&gt;0.5 _est</td><td>1.8</td><td>-</td><td>2.4</td></tr><tr><td/><td>raw</td><td>&lt;1_est</td><td>3.1 ± 0.4</td><td>2.6 ± 0.3</td><td>3.5 ± 0.3</td></tr><tr><td>2</td><td>130 bar/40 °C/30 min</td><td>&lt;1_est</td><td>2.7 ± 0.1</td><td>1.2 ± 0.2</td><td>1.4 ± 0.2</td></tr><tr><td/><td><bold>log red</bold></td><td>-</td><td>0.4</td><td>1.4</td><td>2.1</td></tr><tr><td/><td>raw</td><td>&lt;1_est</td><td>4.6 ± 0.3</td><td>1.3 ± 0.1</td><td>2.4 ± 0.2</td></tr><tr><td>3</td><td>295 bar/40 °C/30 min</td><td>&lt;1_est</td><td>2.6 ± 0.3</td><td>&lt; 1_est</td><td>0.3 ± 01</td></tr><tr><td/><td><bold>log red</bold></td><td>-</td><td>2.0</td><td>&gt; 0.3 _est</td><td>2.1</td></tr><tr><td/><td>raw</td><td>2.5 ± 0.1</td><td>4.6 ± 0.2</td><td>3.0 ± 0.2</td><td>4.2 ± 0.4</td></tr><tr><td>4</td><td>130 bar/60 °C/30 min</td><td>0.7 ± 0.2</td><td>1.6 ± 0.1</td><td>1.0 ± 0.2</td><td>1.0 ± 0.1</td></tr><tr><td/><td><bold>log red</bold></td><td>1.8</td><td>3.0</td><td>2.0</td><td>3.2</td></tr><tr><td/><td>raw</td><td>1.3 ± 0.3</td><td>3.7 ± 0.1</td><td>3.1 ± 0.2</td><td>3.8 ± 0.5</td></tr><tr><td>5</td><td>295 bar/60 °C/30 min</td><td>&lt;1_est</td><td>1.5 ± 0.1</td><td>1.5 ± 0.2</td><td>1.6 ± 0.3</td></tr><tr><td/><td><bold>log red</bold></td><td>&gt;0.3</td><td>2.2</td><td>1.6</td><td>2.2</td></tr><tr><td/><td>raw</td><td>&lt;1_est</td><td>4.9 ± 0.3</td><td>3.0 ± 0.3</td><td>4.1 ± 0.3</td></tr><tr><td>6</td><td>213 bar/50 °C/45 min</td><td>&lt;1_est</td><td>1.7 ± 0.2</td><td>1.6 ± 0.3</td><td>0.9 ± 0.2</td></tr><tr><td/><td><bold>log red</bold></td><td>-</td><td>3.2</td><td>1.4</td><td>3.2</td></tr><tr><td/><td>raw</td><td>2.3 ± 0.1</td><td>2.7 ± 0.2</td><td>2.2 ± 0.2</td><td>4.9 ± 0.3</td></tr><tr><td>7</td><td>130 bar/40 °C/60 min</td><td>&lt;1_est</td><td>&lt;1_est</td><td>2.1 ± 0.2</td><td>1.7 ± 0.2</td></tr><tr><td/><td><bold>log red</bold></td><td>&gt;1.3_est</td><td>&gt;1.7_est</td><td>0.1</td><td>3.2</td></tr><tr><td/><td>raw</td><td>&lt;1_est</td><td>6.0 ± 0.3</td><td>3.5 ± 0.2</td><td>4.1 ± 0.3</td></tr><tr><td>8</td><td>295 bar/40 °C/60 min</td><td>&lt;1_est</td><td>2.1 ± 0.2</td><td>1.4 ± 0.2</td><td>1.8 ± 0.2</td></tr><tr><td/><td><bold>log red</bold></td><td>-</td><td>3.9</td><td>2.1</td><td>2.3</td></tr><tr><td/><td>raw</td><td>3.5 ± 0.3</td><td>4.7 ± 0.3</td><td>&lt; 1_est</td><td>5.4 ± 0.2</td></tr><tr><td>9</td><td>130 bar/60 °C/60 min</td><td>2.2 ± 0.1</td><td>1.7 ± 0.1</td><td>&lt; 1_est</td><td>2.6 ± 0.1</td></tr><tr><td/><td><bold>log red</bold></td><td>1.3</td><td>3.0</td><td>-</td><td>2.8</td></tr><tr><td/><td>raw</td><td>5.2 ± 0.1</td><td>3.0 ± 0.1</td><td>4.0 ± 0.1</td><td>4.9 ± 0.6</td></tr><tr><td>10</td><td>295 bar/60 °C/60 min</td><td>2.9 ± 0.3</td><td>2.0 ± 0.3</td><td>2.2 ± 0.1</td><td>1.5 ± 0.1</td></tr><tr><td/><td><bold>log red</bold></td><td>2.3</td><td>1.0</td><td>1.8</td><td>3.4</td></tr><tr><td/><td>raw</td><td>3.9 ± 0.1</td><td>2.0 ± 0.2</td><td>2.1 ± 0.2</td><td>4.9 ± 0.4</td></tr><tr><td>11</td><td>213 bar/50 °C/45 min</td><td>1.4 ± 0.3</td><td>1.9 ± 0.1</td><td>0.7 ± 0.1</td><td>2.7 ± 0.2</td></tr><tr><td/><td><bold>log red</bold></td><td>2.5</td><td>0.1</td><td>1.4</td><td>2.2</td></tr><tr><td/><td>raw</td><td>3.3 ± 0.1</td><td>5.5 ± 0.6</td><td>0.7 ± 0.2</td><td>4.8 ± 0.4</td></tr><tr><td>12</td><td>74 bar/ 50 °C/ 45 min</td><td>2.3 ± 0.4</td><td>3.2 ± 0.2</td><td>0.4 ± 0.2</td><td>1.9 ± 0.1</td></tr><tr><td/><td><bold>log red</bold></td><td>1.0</td><td>2.3</td><td>0.3</td><td>2.9</td></tr><tr><td/><td>raw</td><td>3.4 ± 0.2</td><td>5.6 ± 0.1</td><td>3.9 ± 0.2</td><td>4.2 ± 0.4</td></tr><tr><td>13</td><td>351 bar/50 °C/45 min</td><td>2.5 ± 0.2</td><td>3.6 ± 0.3</td><td>2.8 ± 0.2</td><td>1.8 ± 0.3</td></tr><tr><td/><td><bold>log red</bold></td><td>0.9</td><td>2.0</td><td>1.1</td><td>2.4</td></tr><tr><td/><td>raw</td><td>3.3 ± 0.1</td><td>5.6 ± 0.3</td><td>1.2 ± 0.1</td><td>5.1 ± 0.5</td></tr><tr><td>14</td><td>213 bar/33 °C/45 min</td><td>2.8 ± 0.2</td><td>2.8 ± 0.1</td><td>&lt; 1_est</td><td>2.6 ± 0.3</td></tr><tr><td/><td><bold>log red</bold></td><td>0.5</td><td>2.8</td><td>&gt; 0.2_est</td><td>2.5</td></tr><tr><td/><td>raw</td><td>1.8 ± 0.2</td><td>5.9 ± 0.5</td><td>2.5 ± 0.4</td><td>4.7 ± 0.1</td></tr><tr><td>15</td><td>213 bar/67 °C/45 min</td><td>&lt;1_est</td><td>4.1</td><td>&lt; 1_est</td><td>0.6 ±</td></tr><tr><td/><td><bold>log red</bold></td><td>&gt;0.8_est</td><td>1.8</td><td>0.0</td><td>4.1</td></tr><tr><td/><td>raw</td><td>2.8 ± 0.3</td><td>5.3 ± 0.5</td><td>3.7 ± 0.1</td><td>5.3 ± 0.2</td></tr><tr><td>16</td><td>213 bar/50 °C/20 min</td><td>0.8 ± 0.1</td><td>3.2 ± 0.2</td><td>3.5 ± 0.3</td><td>1.8 ± 0.2</td></tr><tr><td/><td><bold>log red</bold></td><td>2.0</td><td>2.1</td><td>0.2</td><td>3.5</td></tr><tr><td/><td>raw</td><td>2.9 ± 0.2</td><td>4.8 ± 0.1</td><td>3.6 ± 0.3</td><td>4.3 ± 0.2</td></tr><tr><td>17</td><td>213 bar/50 °C/70 min</td><td>0.8 ± 0.1</td><td>2.8 ± 0.2</td><td>2.8 ± 0.1</td><td>1.6 ± 0.3</td></tr><tr><td/><td><bold>log red</bold></td><td>2.1</td><td>2.0</td><td>0.8</td><td>2.7</td></tr></tbody></table></table-wrap>", "<table-wrap id=\"Tab3\"><label>Table 3</label><caption><p>Polyphenol oxidase (PPO) e peroxidase (POD) activities (U) in raw and SC-CO₂-treated cane juice.</p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th>Trial</th><th>Treatment</th><th>PPO</th><th>POD</th></tr></thead><tbody><tr><td/><td>raw</td><td>11.4 ± 0.0</td><td>226.7 ± 0.1</td></tr><tr><td>1</td><td>213 bar/50 °C/45 min</td><td>7.7 ± 0.2</td><td>198.0 ± 0.5</td></tr><tr><td/><td><bold>red (%)</bold></td><td>32.5</td><td>12.7</td></tr><tr><td/><td>raw</td><td>11.0 ± 0.5</td><td>245.1 ± 0.5</td></tr><tr><td>2</td><td>130 bar/40 °C/30 min</td><td>5.4 ± 0.5</td><td>202.2 ± 2</td></tr><tr><td/><td><bold>red (%)</bold></td><td>50.9</td><td>17.5</td></tr><tr><td/><td>raw</td><td>11.0 ± 0.1</td><td>238.4 ± 1</td></tr><tr><td>3</td><td>295 bar/40 °C/30 min</td><td>6.6 ± 0.1</td><td>202.2 ± 2</td></tr><tr><td/><td><bold>red (%)</bold></td><td>40.0</td><td>15.2</td></tr><tr><td/><td>raw</td><td>11.0 ± 0.1</td><td>231.6 ± 7</td></tr><tr><td>4</td><td>130 bar/60 °C/30 min</td><td>6.1 ± 0.3</td><td>202.2 ± 2</td></tr><tr><td/><td><bold>red (%)</bold></td><td>44.5</td><td>12.7</td></tr><tr><td/><td>raw</td><td>11.0 ± 0.1</td><td>202.2 ± 2</td></tr><tr><td>5</td><td>295 bar/60 °C/30 min</td><td>3.9 ± 0.0</td><td>192.9 ± 0.3</td></tr><tr><td/><td><bold>red (%)</bold></td><td>64.5</td><td>4.6</td></tr><tr><td/><td>raw</td><td>11.4 ± 0.0</td><td>235.6 ± 2</td></tr><tr><td>6</td><td>213 bar/50 °C/45 min</td><td>7.7 ± 0.3</td><td>190.7 ± 0.0</td></tr><tr><td/><td><bold>red (%)</bold></td><td>32.5</td><td>19.1</td></tr><tr><td/><td>raw</td><td>11.4 ± 0.0</td><td>232.2 ± 6</td></tr><tr><td>7</td><td>130 bar/40 °C/60 min</td><td>9.4 ± 0.3</td><td>183.8 ± 4.1</td></tr><tr><td/><td><bold>red (%)</bold></td><td>17.5</td><td>20.8</td></tr><tr><td/><td>raw</td><td>11.4 ± 0.0</td><td>231.1 ± 8</td></tr><tr><td>8</td><td>295 bar/40 °C/60 min</td><td>8.0 ± 0.2</td><td>215.1 ± 3</td></tr><tr><td/><td><bold>red (%)</bold></td><td>29.8</td><td>6.9</td></tr><tr><td/><td>raw</td><td>10.0 ± 0.2</td><td>56.1 ± 4</td></tr><tr><td>9</td><td>130 bar/60 °C/60 min</td><td>9.3 ± 2.0</td><td>34.0 ± 7</td></tr><tr><td/><td><bold>red (%)</bold></td><td>7.0</td><td>39.4</td></tr><tr><td/><td>raw</td><td>3.4 ± 0.6</td><td>56.1 ± 4</td></tr><tr><td>10</td><td>295 bar/60 °C/60 min</td><td>2.5 ± 0.4</td><td>34.0 ± 7</td></tr><tr><td/><td><bold>red (%)</bold></td><td>26.5</td><td>39.4</td></tr><tr><td/><td>raw</td><td>2.3 ± 0.6</td><td>8.4 ± 2</td></tr><tr><td>11</td><td>213 bar/50 °C/45 min</td><td>0.8 ± 0.4</td><td>7.2 ± 2</td></tr><tr><td/><td><bold>red (%)</bold></td><td>63.2</td><td>14.3</td></tr><tr><td/><td>raw</td><td>1.8 ± 0.6</td><td>8.1 ± 2</td></tr><tr><td>12</td><td>74 bar/ 50 °C/ 45 min</td><td>0.7 ± 0.4</td><td>6.7 ± 2</td></tr><tr><td/><td><bold>red (%)</bold></td><td>61.1</td><td>17.3</td></tr><tr><td/><td>raw</td><td>2.4 ± 0.5</td><td>8.1 ± 0.2</td></tr><tr><td>13</td><td>351 bar/50 °C/45 min</td><td>0.9 ± 0.5</td><td>5.6 ± 2</td></tr><tr><td/><td><bold>red (%)</bold></td><td>62.5</td><td>30.9</td></tr><tr><td/><td>raw</td><td>18.0 ± 0.7</td><td>7.3 ± 2</td></tr><tr><td>14</td><td>213 bar/33 °C/45 min</td><td>17.4 ± 0.5</td><td>7.2 ± 2</td></tr><tr><td/><td><bold>red (%)</bold></td><td>3.3</td><td>1.4</td></tr><tr><td/><td>raw</td><td>21.9 ± 0.7</td><td>8.8 ± 2</td></tr><tr><td>15</td><td>213 bar/67 °C/45 min</td><td>20.5 ± 0.3</td><td>5.2 ± 0.1</td></tr><tr><td/><td><bold>red (%)</bold></td><td>6.4</td><td>40.9</td></tr><tr><td/><td>raw</td><td>2.7 ± 0.2</td><td>7.1 ± 2</td></tr><tr><td>16</td><td>213 bar/50 °C/20 min</td><td>1.2 ± 0.0</td><td>7.1 ± 2</td></tr><tr><td/><td><bold>red (%)</bold></td><td>55.6</td><td>0.0</td></tr><tr><td/><td>raw</td><td>3.4 ± 0.6</td><td>5.1 ± 0.1</td></tr><tr><td>17</td><td>213 bar/50 °C/70 min</td><td>2.1 ± 0.4</td><td>5.0 ± 0.4</td></tr><tr><td/><td><bold>red (%)</bold></td><td>38.2</td><td>2.0</td></tr></tbody></table></table-wrap>", "<table-wrap id=\"Tab4\"><label>Table 4</label><caption><p>Color parameters determined in raw and SC-CO₂-treated cane juice.</p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th>Trial</th><th>Treatment</th><th>L*</th><th>a*</th><th>b*</th><th>chroma</th><th>°hue</th></tr></thead><tbody><tr><td>1</td><td>raw</td><td>33.67 ± 0.02</td><td>8.71 ± 0.01</td><td>29.95 ± 0.02</td><td>31.2</td><td>73.8</td></tr><tr><td/><td>213 bar/50 °C/45 min</td><td>37.91 ± 0.01</td><td>6.25 ± 0.02</td><td>27.92 ± 0.02</td><td>28.6</td><td>77.4</td></tr><tr><td>2</td><td>raw</td><td>34.17 ± 0.02</td><td>10.5 ± 0.01</td><td>32.55 ± 0.01</td><td>34.2</td><td>72.1</td></tr><tr><td/><td>130 bar/40 °C/30 min</td><td>35.75 ± 0.03</td><td>9.37 ± 0.02</td><td>30.02 ± 0.01</td><td>31.4</td><td>72.7</td></tr><tr><td>3</td><td>raw</td><td>32.26 ± 0.02</td><td>10.5 ± 0.02</td><td>26.03 ± 0.01</td><td>28.1</td><td>68.1</td></tr><tr><td/><td>295 bar/40 °C/30 min</td><td>34.92 ± 0.01</td><td>9.02 ± 0.01</td><td>26.40 ± 0.01</td><td>27.9</td><td>71.1</td></tr><tr><td>4</td><td>raw</td><td>36.75 ± 0.01</td><td>6.19 ± 0.02</td><td>36.71 ± 0.03</td><td>37.2</td><td>80.4</td></tr><tr><td/><td>130 bar/60 °C/30 min</td><td>37.77 ± 0.02</td><td>6.27 ± 0.02</td><td>34.55 ± 0.02</td><td>35.1</td><td>79.7</td></tr><tr><td>5</td><td>raw</td><td>38.93 ± 0.03</td><td>4.61 ± 0.01</td><td>35.44 ± 0.05</td><td>35.7</td><td>82.6</td></tr><tr><td/><td>295 bar/60 °C/30 min</td><td>44.14 ± 0.02</td><td>3.96 ± 0.05</td><td>37.62 ± 0.03</td><td>37.6</td><td>84.0</td></tr><tr><td>6</td><td>raw</td><td>69.03 ± 0.10</td><td>0.96 ± 0.0</td><td>36.60 ± 0.02</td><td>36.6</td><td>88.5</td></tr><tr><td/><td>213 bar/50 °C/45 min</td><td>63.14 ± 0.02</td><td>4.62 ± 0.01</td><td>43.95 ± 0.01</td><td>44.2</td><td>84.0</td></tr><tr><td>7</td><td>raw</td><td>40.67 ± 0.01</td><td>4.89 ± 0.01</td><td>38.58 ± 0.02</td><td>38.9</td><td>82.8</td></tr><tr><td/><td>130 bar/ 40 °C/ 60 min</td><td>42.55 ± 0.02</td><td>4.89 ± 0.01</td><td>37.84 ± 0.01</td><td>38.2</td><td>82.6</td></tr><tr><td>8</td><td>raw</td><td>72.92 ± 0.01</td><td>1.22 ± 0.02</td><td>36.71 ± 0.02</td><td>36.7</td><td>88.1</td></tr><tr><td/><td>295 bar/40 °C/60 min</td><td>73.06 ± 0.01</td><td>4.04 ± 0.01</td><td>42.28 ± 0.02</td><td>42.5</td><td>84.5</td></tr><tr><td>9</td><td>raw</td><td>71.09 ± 0.02</td><td>2.64 ± 0.04</td><td>35.90 ± 0.03</td><td>36.0</td><td>85.8</td></tr><tr><td/><td>130 bar/60 °C/60 min</td><td>60.91 ± 0.02</td><td>3.97 ± 0.02</td><td>39.33 ± 0.02</td><td>39.5</td><td>84.2</td></tr><tr><td>10</td><td>raw</td><td>37.02 ± 0.01</td><td>16.09 ± 0.0</td><td>45.84 ± 0.01</td><td>40.6</td><td>70.7</td></tr><tr><td/><td>295 bar/60 °C/60 min</td><td>50.14 ± 0.02</td><td>14.74 ± 0.0</td><td>50.67 ± 0.02</td><td>52.8</td><td>73.8</td></tr><tr><td>11</td><td>raw</td><td>31.48 ± 0.02</td><td>4.60 ± 0.03</td><td>30.29 ± 0.03</td><td>30.6</td><td>81.4</td></tr><tr><td/><td>213 bar/50 °C/45 min</td><td>32.53 ± 0.01</td><td>5.80 ± 0.03</td><td>24.75 ± 0.02</td><td>25.4</td><td>76.8</td></tr><tr><td>12</td><td>raw</td><td>55.16 ± 0.02</td><td>5.14 ± 0.02</td><td>41.59 ± 0.02</td><td>41.9</td><td>83.0</td></tr><tr><td/><td>74 bar/ 50 °C/ 45 min</td><td>49.73 ± 0.02</td><td>6.78 ± 0.02</td><td>34.34 ± 0.01</td><td>35.0</td><td>78.8</td></tr><tr><td>13</td><td>raw</td><td>49.15 ± 0.01</td><td>5.65 ± 0.01</td><td>44.44 ± 0.01</td><td>44.8</td><td>82.8</td></tr><tr><td/><td>351 bar/50 °C/45 min</td><td>51.37 ± 0.01</td><td>4.33 ± 0.02</td><td>37.39 ± 0.05</td><td>37.6</td><td>83.4</td></tr><tr><td>14</td><td>raw</td><td>60.81 ± 0.02</td><td>4.94 ± 0.01</td><td>44.73 ± 0.01</td><td>44.9</td><td>83.7</td></tr><tr><td/><td>213 bar/33 °C/45 min</td><td>66.36 ± 0.02</td><td>4.05 ± 0.04</td><td>38.75 ± 0.02</td><td>39.0</td><td>84.0</td></tr><tr><td>15</td><td>raw</td><td>58.49 ± 0.01</td><td>3.05 ± 0.05</td><td>40.05 ± 0.02</td><td>40.2</td><td>85.6</td></tr><tr><td/><td>213 bar/67 °C/45 min</td><td>52.32 ± 0.02</td><td>4.25 ± 0.03</td><td>36.51 ± 0.01</td><td>36.8</td><td>83.4</td></tr><tr><td>16</td><td>raw</td><td>52.49 ± 0.03</td><td>6.53 ± 0.05</td><td>45.88 ± 0.02</td><td>46.3</td><td>81.9</td></tr><tr><td/><td>213 bar/50 °C/20 min</td><td>52.82 ± 0.02</td><td>6.63 ± 0.04</td><td>41.09 ± 0.02</td><td>41.6</td><td>80.8</td></tr><tr><td>17</td><td>raw</td><td>63.63 ± 0.01</td><td>4.84 ± 0.03</td><td>46.17 ± 0.05</td><td>46.4</td><td>84.0</td></tr><tr><td/><td>213 bar/50 °C/70 min</td><td>65.70 ± 0.04</td><td>4.58 ± 0.02</td><td>42.14 ± 0.02</td><td>42.4</td><td>83.8</td></tr></tbody></table></table-wrap>", "<table-wrap id=\"Tab5\"><label>Table 5</label><caption><p>Analysis of variance (<italic>p</italic> ≤ 0.1) for peroxidase reduction and total color difference in cane juice.</p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th>Response</th><th>Variation source</th><th>Sum of squares</th><th>Degrees of freedom</th><th>Mean square</th><th><italic>F</italic> value</th><th/></tr></thead><tbody><tr><td/><td/><td/><td/><td/><td>F_calc</td><td>F_tab</td></tr><tr><td rowspan=\"4\">Peroxidase reduction 1^st order</td><td>Regression</td><td>1112.0</td><td>3</td><td>370.7</td><td>13.8</td><td>3.1</td></tr><tr><td>Residual</td><td>188.4</td><td>7</td><td>26.9</td><td/><td/></tr><tr><td>Total</td><td>1300.4</td><td/><td/><td/><td/></tr><tr><td>R²</td><td>0.86</td><td/><td/><td/><td/></tr><tr><td rowspan=\"4\">Peroxidase reduction 2^nd order</td><td>Regression</td><td>1356.7</td><td>2</td><td>678.3</td><td>6.2</td><td>2.73</td></tr><tr><td>Residual</td><td>1536.4</td><td>14</td><td>109.7</td><td/><td/></tr><tr><td>Total</td><td>2893.1</td><td/><td/><td/><td/></tr><tr><td>R²</td><td>0.47</td><td/><td/><td/><td/></tr><tr><td rowspan=\"4\">Total color difference 1^st order</td><td>Regression</td><td>99.1</td><td>4</td><td>24.8</td><td>13.1</td><td>3.2</td></tr><tr><td>Residual</td><td>11.5</td><td>6</td><td>1.9</td><td/><td/></tr><tr><td>Total</td><td>110.6</td><td/><td/><td/><td/></tr><tr><td>R²</td><td>0.90</td><td/><td/><td/><td/></tr><tr><td rowspan=\"4\">Total color difference 2^nd order</td><td>Regression</td><td>99.8</td><td>4</td><td>25.0</td><td>10.0</td><td>2.5</td></tr><tr><td>Residual</td><td>29.5</td><td>12</td><td>2.5</td><td/><td/></tr><tr><td>Total</td><td>129.3</td><td/><td/><td/><td/></tr><tr><td>R²</td><td>0.77</td><td/><td/><td/><td/></tr></tbody></table></table-wrap>", "<table-wrap id=\"Tab6\"><label>Table 6</label><caption><p>Actual and coded levels tested in the central composite rotational design (CCRD) for cane juice processing with SC-CO₂.</p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th>Variable</th><th>code</th><th>−1.68 (-α)</th><th>−1</th><th>0</th><th>+1</th><th>+1.68 ( + α)</th></tr></thead><tbody><tr><td>Pressure (bar)</td><td>x₁</td><td>74</td><td>130</td><td>213</td><td>295</td><td>351</td></tr><tr><td>Temperature (°C)</td><td>x₂</td><td>33</td><td>40</td><td>50</td><td>60</td><td>67</td></tr><tr><td>Time (min)</td><td>x₃</td><td>20</td><td>30</td><td>45</td><td>60</td><td>70</td></tr></tbody></table></table-wrap>" ]

[ "<inline-formula id=\"IEq1\"><alternatives><tex-math id=\"M1\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\rm{Chroma}}={({{\\rm{a}}}^{* 2}+{{\\rm{b}}}^{* 2})}^{\\frac{1}{2}}\\, ^\\circ {\\rm{hue}}=\\arctan \\left(\\frac{{{\\rm{b}}}^{* }}{{{\\rm{a}}}^{* }}\\right)$$\\end{document}</tex-math><mml:math id=\"M2\"><mml:mrow><mml:mi mathvariant=\"normal\">Chroma</mml:mi><mml:mo>=</mml:mo><mml:msup><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msup><mml:mrow><mml:mi mathvariant=\"normal\">a</mml:mi></mml:mrow><mml:mrow><mml:mo>*</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo>+</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant=\"normal\">b</mml:mi></mml:mrow><mml:mrow><mml:mo>*</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:msup><mml:mrow/><mml:mrow><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:mfrac></mml:mrow></mml:msup><mml:mspace width=\"0.25em\"/></mml:mrow><mml:mrow><mml:mo>∘</mml:mo></mml:mrow></mml:msup><mml:mi mathvariant=\"normal\">hue</mml:mi><mml:mo>=</mml:mo><mml:mi>arctan</mml:mi><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:mfrac><mml:mrow><mml:msup><mml:mrow><mml:mi mathvariant=\"normal\">b</mml:mi></mml:mrow><mml:mrow><mml:mo>*</mml:mo></mml:mrow></mml:msup></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mi mathvariant=\"normal\">a</mml:mi></mml:mrow><mml:mrow><mml:mo>*</mml:mo></mml:mrow></mml:msup></mml:mrow></mml:mfrac></mml:mrow></mml:mfenced></mml:mrow></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ1\"><label>1</label><alternatives><tex-math id=\"M3\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${{\\rm{Y}}}_{1}=18.56+4.46\\,{x}_{2}+7.09\\,{x}_{3}+8.30\\,{x}_{2}\\,{x}_{3}$$\\end{document}</tex-math><mml:math id=\"M4\"><mml:mrow><mml:msub><mml:mrow><mml:mi mathvariant=\"normal\">Y</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mn>18.56</mml:mn><mml:mo>+</mml:mo><mml:mn>4.46</mml:mn><mml:mspace width=\"0.25em\"/><mml:msub><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:mn>7.09</mml:mn><mml:mspace width=\"0.25em\"/><mml:msub><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:mn>8.30</mml:mn><mml:mspace width=\"0.25em\"/><mml:msub><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mspace width=\"0.25em\"/><mml:msub><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ2\"><label>2</label><alternatives><tex-math id=\"M5\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${{\\rm{Y}}}_{2}=5.15+1.36\\,{x}_{1}+2.06\\,{x}_{2}+2.09\\,{x}_{3}+1.39\\,{x}_{2}\\,{x}_{3}$$\\end{document}</tex-math><mml:math id=\"M6\"><mml:mrow><mml:msub><mml:mrow><mml:mi mathvariant=\"normal\">Y</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mn>5.15</mml:mn><mml:mo>+</mml:mo><mml:mn>1.36</mml:mn><mml:mspace width=\"0.25em\"/><mml:msub><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:mn>2.06</mml:mn><mml:mspace width=\"0.25em\"/><mml:msub><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:mn>2.09</mml:mn><mml:mspace width=\"0.25em\"/><mml:msub><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:mn>1.39</mml:mn><mml:mspace width=\"0.25em\"/><mml:msub><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mspace width=\"0.25em\"/><mml:msub><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ3\"><label>3</label><alternatives><tex-math id=\"M7\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$C={\\left({a}^{\\ast 2}+{b}^{\\ast 2}\\right)}^{\\frac{1}{2}}$$\\end{document}</tex-math><mml:math id=\"M8\"><mml:mrow><mml:mi>C</mml:mi><mml:mo>=</mml:mo><mml:msup><mml:mrow><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:msup><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mo>*</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo>+</mml:mo><mml:msup><mml:mrow><mml:mi>b</mml:mi></mml:mrow><mml:mrow><mml:mo>*</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:mfenced></mml:mrow><mml:mrow><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:mfrac></mml:mrow></mml:msup></mml:mrow></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ4\"><label>4</label><alternatives><tex-math id=\"M9\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$^\\circ hue=arctan\\left(\\frac{{b}^{\\ast }}{{a}^{\\ast }}\\right)$$\\end{document}</tex-math><mml:math id=\"M10\"><mml:mrow><mml:msup><mml:mrow/><mml:mrow><mml:mo>∘</mml:mo></mml:mrow></mml:msup><mml:mi>h</mml:mi><mml:mi>u</mml:mi><mml:mi>e</mml:mi><mml:mo>=</mml:mo><mml:mi>a</mml:mi><mml:mi>r</mml:mi><mml:mi>c</mml:mi><mml:mi>t</mml:mi><mml:mi>a</mml:mi><mml:mi>n</mml:mi><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:mfrac><mml:mrow><mml:msup><mml:mrow><mml:mi>b</mml:mi></mml:mrow><mml:mrow><mml:mo>*</mml:mo></mml:mrow></mml:msup></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mo>*</mml:mo></mml:mrow></mml:msup></mml:mrow></mml:mfrac></mml:mrow></mml:mfenced></mml:mrow></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ5\"><label>5</label><alternatives><tex-math id=\"M11\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$TCD={\\left(\\varDelta {L}^{\\ast 2}+\\varDelta {a}^{\\ast 2}+\\varDelta {b}^{\\ast 2}\\right)}^{\\frac{1}{2}}$$\\end{document}</tex-math><mml:math id=\"M12\"><mml:mrow><mml:mi>T</mml:mi><mml:mi>C</mml:mi><mml:mi>D</mml:mi><mml:mo>=</mml:mo><mml:msup><mml:mrow><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:mi>Δ</mml:mi><mml:msup><mml:mrow><mml:mi>L</mml:mi></mml:mrow><mml:mrow><mml:mo>*</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo>+</mml:mo><mml:mi>Δ</mml:mi><mml:msup><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mo>*</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo>+</mml:mo><mml:mi>Δ</mml:mi><mml:msup><mml:mrow><mml:mi>b</mml:mi></mml:mrow><mml:mrow><mml:mo>*</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:mfenced></mml:mrow><mml:mrow><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:mfrac></mml:mrow></mml:msup></mml:mrow></mml:math></alternatives></disp-formula>" ]

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[ "<supplementary-material content-type=\"local-data\" id=\"MOESM1\"></supplementary-material>" ]

[ "<table-wrap-foot><p>Mean values of 3 replicates ± standard deviation.</p><p>Δ - variation.</p></table-wrap-foot>", "<table-wrap-foot><p>Mean values of 3 replicates ± standard deviation.</p><p>est – estimate count (under detection limit).</p></table-wrap-foot>", "<table-wrap-foot><p>Mean values of 3 replicates ± standard deviation.</p><p>Red reduction.</p></table-wrap-foot>", "<table-wrap-foot><p>L* (lightness) = 0 (black). 100 (white). +a* = red. −a* = green. +b* = yellow. −b* = blue.</p><p>Mean values of 3 replicates ± standard deviation.</p><p>.</p></table-wrap-foot>", "<table-wrap-foot><p>(−1.68) lower axial point, (−1) lower level, (0) central point, (+1) upper level, (+1.68) upper axial point. α = (2ⁿ)^1/4 = 1.68. <italic>n</italic> = number of variables (3).</p></table-wrap-foot>", "<fn-group><fn><p><bold>Publisher’s note</bold> Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.</p></fn></fn-group>" ]

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[ "<media xlink:href=\"41538_2023_242_MOESM1_ESM.pdf\"><caption><p>Reporting summary</p></caption></media>" ]

{ "acronym": [], "definition": [] }

[ "<title>Introduction</title>", "<p id=\"Par3\">Proton exchange membrane fuel cells (PEMFCs) have shown great potential as power sources for electric vehicle applications owing to their high energy conversion efficiency and environmental benign characteristics^{##REF##34677946##1##–##REF##33479539##4##}. However, the widespread uptake of such PEMFCs in automotive vehicles is largely impeded at present because a high loading amount of precious Pt on the cathode (~0.4 mg cm⁻²) is needed to promote the sluggish kinetics of oxygen reduction reaction (ORR) and compensate the insufficient catalyst durability^{##REF##23808919##5##–##REF##26206931##8##}. Alloying Pt with a transition metal, such as Fe, Co, Ni, and Cu, has been identified as an effective way to boost the catalytic activity toward ORR through the advantageous ligand and strain effects^{##REF##17218494##9##–##REF##20489713##14##}, which has thus been regarded as the most promising solution to reduce the excessive Pt usage. As a convincing example, the spherical PtCo alloy nanocatalysts are adopted in the Toyota Mirai fuel cell vehicle at present^{##REF##33479539##4##}. Unfortunately, the long-term durability of these active PtM (M = transition metals) alloy catalysts still remains a serious problem, which can be generally ascribed to the rapid leaching of the reactive transition metal under detrimentally corrosive ORR conditions and “start-up/shutdown” process (the applied potential is generally over 1.0 V)^{##REF##23770725##15##–##REF##36196871##17##}. Under this context, how to stabilize the active PtM alloy catalyst for long-term fuel cell operation has become the most concerned research topic for the industrial application of PEMFCs.</p>", "<p id=\"Par4\">Doping a specific element, with notable examples of Mo, Au, and Rh, into active PtM alloy catalysts and metal oxide support has been discovered as a straightforward and potential strategy to enhance the catalytic durability of PtM alloy catalysts for ORR^{##REF##26068847##18##–##UREF##8##25##}. Mechanistically, the prior studies suggested that these dopants may stabilize the alloy catalysts via suppressing the surface Pt migration and/or mitigating the outward diffusion of reactive transition metal^{##REF##26068847##18##}. However, it needs to be pointed out that these two main reasons (i.e., surface Pt migration and outward diffusion of reactive transition metal) are essentially entangled^{##REF##26068847##18##,##REF##31639295##26##–##REF##26854940##28##}, making great difficulty to gain a clear mechanistic understanding. Moreover, those arguments on the stabilization mechanism of dopants were mostly established via theoretical simulations, probably because of the challenges in constructing the model catalysts, more or less leading to the discrepancy with a realistic situation. As a result, the atomistic understanding of the role of dopants that is concluded from a comprehensive combining of convincing experiments and theoretical computations is still lacking presently, which indeed greatly limits the rational design of active and stable PtM alloy catalysts for practical fuel cells.</p>", "<p id=\"Par5\">Because Mo and Au elements are resistant to corrosion under acidic ORR, these elements offer ideal cases for the stability study. As a result, we systematically investigate the roles of Mo and Au dopants in stabilizing PtNi alloy catalysts based on the well-defined PtNi-based nanowires (NWs) model catalysts. Combining ex-situ experimental characterizations and density of functional theory (DFT) calculations, the distinct roles of Mo and Au in stabilizing PtNi NWs catalysts are identified. We find that the Mo dopant plays a major role in suppressing the outward diffusion of Ni atoms and the Au dopant mainly stabilizes the surface Pt overlayer. Based on this atomistic understanding, PtNiMoAu NWs are rationally designed and constructed by integrating the different functions of Mo and Au into PtNi NWs. As expected, the as-synthesized PtNiMoAu NWs catalysts present an unprecedented activity and stability toward ORR, with a 16.2% loss in its mass activity (MA) after 80,000 (80 K) cycles of durability test. When assembling the PtNiMoAu NWs catalysts into the fuel cell cathode, a high MA retention of 77.4% (H₂-O₂, 0.9 V_iR-free) and a low voltage loss of 25 mV (H₂-Air, 0.8 A cm⁻²) after 30 K cycles of durability test are output, proving the highly durable fuel cell performance.</p>" ]

[ "<title>Methods</title>", "<title>Chemicals</title>", "<p id=\"Par20\">Platinum (II) acetylacetonate (Pt(acac)₂, 97%), nickel (II) acetylacetonate (Ni(acac)₂, 97%), molybdenum (III) acetylacetonate (Mo(acac)₃, 97%), gold (III) chloride trihydrate (HAuCl₄·3H₂O, 99%), cetyltrimethylammonium bromide (CTAB, 99%), tungsten hexacarbonyl (W(CO)₆, 97%), oleylamine (OAm, 70%) and Nafion (5 wt%) were purchased from Sigma-Aldrich. Ethanol (CH₃CH₂OH, 99%) and perchloric acid (HClO₄, 70–72%) were purchased from Sinopharm Chemical Reagent Co. Ltd. (Shanghai, China). The water (18.2 MΩ/cm) was freshly prepared through an ultra-pure purification system (Master-515Q, HHitech). All the chemicals were used without further purification.</p>", "<title>Synthesis of ultrathin Pt-based nanowires</title>", "<p id=\"Par21\">In a typical synthesis of ultrathin PtNiMoAu NWs, Pt(acac)₂ (20 mg), Ni(acac)₂ (10 mg), Mo(acac)₃ (8 mg), HAuCl₄·4H₂O (2.7 mg), CTAB (75 mg), and 4 mL OAm were added into a 30 mL glass vial. After sonicated for 30 min, W(CO)₆ (20 mg) was added into the pre-dispersed solution and capped. The resulting homogeneous mixture was then heated and kept at 175 °C for 2 h in an oil bath. Finally, the products were collected by centrifugation at 13000 × g and cleaned four times with a hexane/ethanol mixture (v/v = 2/1), then dried under vacuum. The preparation of other ultrathin Pt-based nanostructures was similar to that of PtNiMoAu NWs except that Mo(acac)₃ or HAuCl₄·4H₂O were taken out and the reaction temperature was changed. The detailed synthetic parameters for other Pt-based NWs have been listed in Supplementary Table ##SUPPL##0##1##.</p>", "<title>Characterization techniques</title>", "<p id=\"Par22\">XRD patterns were collected to analyze the crystal structures of NWs by X-ray diffractometer (Rigaku Miniflex-600) with Cu K<italic>α</italic> radiation (<italic>λ</italic> = 0.15406 nm, 40 kV). TEM images were carried out on a JEOL 2100-Plus operating at 120 kV with the samples deposited on carbon-coated copper TEM grids. HAADF-STEM images and EDS line-scan file were taken on a JEOL ARM-200F microscope with a spherical aberration corrector operating at 200 kV. Elemental analysis of ultrathin NWs was quantitatively determined by ICP-MS with a SPECTRO BLUE SOP. The XPS spectra were collected using an Escalab 250Xi equipped with an Al Ka (1486.6 eV) excitation source. All the spectra collected were corrected using a Shirley background. All XAFS data were collected at BL14B station of Shanghai Synchrotron Radiation Facility (SSRF), China. The storage ring of SSRF operates at 3.5 GeV with a maximum current of 210 mA. The electrochemical in situ XAFS measurements were carried out in a custom-fabricated three-electrode system with a 1.4 × 0.7 cm² carbon cloth loaded with PtNi and PtNiMoAu NWs catalysts as the working electrode and 0.1 M continuously O₂-saturated HClO₄ solution as the electrolyte. For the in situ XAFS spectral data acquisition of Pt L₃-edge (11564 eV), we calibrated the positions of absorption edges (E₀) by using Pt foil and Fe foil standard samples, respectively, and all spectra were collected in the same beam time by fluorescence mode to ensure comparability.</p>", "<title>Preparation of working electrode</title>", "<p id=\"Par23\">For different NWs/C catalysts, 4 mg of the prepared NWs was added to a chloroform solution (8 mL) and sonicated for 1 h. The above-dispersed solution was dropwise added to an ethanol solution containing 16 mg of carbon support (Vulcan XC-72) under vigorous stirring for 30 min. After centrifugation and washing twice with hexanes by centrifugation, the NWs/C catalysts were re-dispersed in acetic acid and then heated at 70 °C for 12 h to remove the surfactants on the surface of NWs. A certain amount of as-obtained catalyst was mixed with 0.5 mL isopropanol, 0.495 mL ethanol, and 0.005 mL Nafion (5 wt%) and sonicated for 1 h to form the homogenously mixed catalyst ink solution. For the commercial Pt/C catalysts (20 wt% loading on Vulcan XC-72 carbon support, Johnson Matthey), the ink solution (2 mg/mL) was prepared and sonicated for 1 h. Finally, prepared catalyst ink was dropped onto the glassy carbon rotating disk electrode (GC, RDE with geometric area of 0.196 cm²) with the loading amount of Pt at 2.0 µg, 2.1 µg, 2.2 µg, 2.0 µg, 2.3 µg, and 4.0 µg for PtNiMoAu NWs/C, PtNiAu NWs/C, PtNiMo NWs/C, PtNi NWs/C, Pt NWs/C, and commercial Pt/C catalysts, respectively (based on ICP-MS).</p>", "<title>Electrochemical testing</title>", "<p id=\"Par24\">Electrochemical tests were conducted using a three-electrode cell on a CHI760e electrochemical workstation (Chenhua Instrument, China). A glassy carbon Rotating Disk Electrode (RDE, diameter: 5 mm) was used as the working electrode, the Ag/AgCl (3 M KCl) electrode was used as a reference electrode, and a platinum wire was used as a counter electrode. All potentials were converted to the reversible hydrogen electrode reference. The CVs were tested in a N₂-saturated 0.1 M HClO₄ electrolyte with a sweep rate of 50 mV s⁻¹. The ECSAs were calculated by integrating the hydrogen adsorption/desorption charge area between 0.05 and 0.38 V_RHE from the CVs. The equation for the calculation of ECSAs is shown as follows:</p>", "<p id=\"Par25\">The charge density for the adsorption of one monolayer hydrogen on Pt (Q_H) was assumed to be 210 μC cm⁻². L_Pt (mg_Pt cm⁻²) was the working electrode Pt loading. A_g (cm²) was the geometric surface area of the glassy carbon electrode (0.196 cm²). The ORR polarization curves were measured in an O₂-saturated 0.1 M HClO₄ solution between 0.05 and 1.05 V_RHE using a sweep rate of 10 mV s⁻¹ at a rotation rate of 1600 rpm. The ADTs were performed via cyclic sweeps between 0.6 and 1.0 V_RHE in an O2-saturated 0.1 M HClO₄ electrolyte at a sweep rate of 100 mV s⁻¹ for different cycles.</p>", "<title>MEA fabrication and test</title>", "<p id=\"Par26\">The catalytic activity was also evaluated under PEMFC operating conditions. Specifically, each catalyst was mixed with isopropanol, deionized water, and Nafion solution by ultrasonicating for 1 h to form homogeneous ink with the ratio of ionomer to carbon (I/C) of 0.8. The ink was then sprayed the ink onto the Nafion® 211 membrane (DuPont). The catalyst-coated-membrane (CCM) with an active geometric area of 12.25 cm² was applied to the gas diffusion layer (GDL, Toray TGP-H-060). The compression ratio of GDL was calculated to be 31.5%. PtNiMoAu NWs/C, PtNiAu NWs/C, PtNiMo NWs/C, PtNi NWs/C, and commercial Pt/C were employed as the cathode catalysts and Pt/C (20 wt% loading, Johnson Matthey) was used for the anode. The Pt loading at the cathode was 0.10 mg_Pt cm⁻² for PtNi-based NWs/C and 0.12 mg_Pt cm⁻² for commercial Pt/C, respectively. As for the anode, Pt/C was used with a loading of 0.05 mg_Pt cm⁻². Fuel cell testing was performed in a single cell using a commercial fuel cell test system (Scribner 850e, Hephas Energy Corporation). The MEA was sandwiched between two graphite plates with single serpentine flow channels. The corresponding pressure drop across the cathode and anode was measured using two pressure sensors connected to the inlet and outlet of the cell, respectively, while the cell was operated at 80 ^oC, 150 kPa (absolute, abs) back pressure, relative humidity (RH) 100%, H₂ flow rate of 200 mL min⁻¹ and O₂/Air flow rate of 500 mL min⁻¹. Fuel cell polarization curves were recorded using potential step mode with 50 mV/point (holding 2 min for each point). The MA was calculated by normalizing the measuring currents with Pt loading amount in 150 kPa_abs H₂/O₂ (80 ^oC, 100% RH, 200_anode/500_cathode sccm) at 0.9 V_iR-free. The ADTs were conducted by the standard 30 K-cycle square-wave protocol to evaluate the durability of catalysts. Specifically, the cathode was held at 0.60 V for 3 s and 0.95 V for 3 s in each cycle.</p>", "<title>DFT calculations</title>", "<p id=\"Par27\">All of the DFT calculations were performed using the Vienna Ab-initio simulation package program^{##UREF##15##40##–##UREF##17##42##}, which uses a plane-wave basis set and a projector augmented wave method (PAW) for the treatment of core electrons^{##UREF##16##41##}. The Perdew, Burke, and Ernzerhof exchange-correlation functional within a generalized gradient approximation (GGA-PBE)^{##REF##10062328##43##} was used in our calculations, and the van der Waals (vdW) correction proposed by Grimme (DFT-D3)^{##REF##20423165##44##} was employed due to its good description of long-range vdW interactions. For the expansion of wavefunctions over the plane-wave basis set, a converged cutoff was set to 450 eV.</p>", "<p id=\"Par28\">In order to simulate the Pt NWs and PtNi bimetallic NWs, seven-layer 2×2 Pt (111) (a = b = 5.456 Å) and 2 × 2 Pt₃Ni (111) (a = b = 5.383 Å) slabs with periodical boundary conditions were used, respectively. Besides, to model the PtNiMo trimetallic NWs and PtNiMoAu tetrametallic NWs, a Pt atom in the subsurface layers of the Pt₃Ni (111) slab was substituted by a Mo atom and two Pt atoms in the subsurface layers of the Pt₃Ni (111) slab were substituted by a Mo and an Au atom. Moreover, considering the surface Ni and Mo atoms will be leached into an acidic solution, the outmost layers of the Pt₃Ni (111), Mo-doped Pt₃Ni (111), and Mo- and Au-doped Pt₃Ni (111) slabs were all replaced as shown in Supplementary Fig. ##SUPPL##0##10##. The vacuum space was set to 17 Å in the z direction to avoid interactions between periodic images. In geometry optimizations, all the atomic coordinates were fully relaxed up to the residual atomic forces smaller than 1 × 10⁻⁴ eV/Å, and the total energy was converged to 10⁻⁵ eV. The Brillouin zone integration was performed on the (9 × 9× 1) Monkhorst–Pack k-point mesh^{##UREF##18##45##}.</p>", "<p id=\"Par29\">The ORR pathways on various NWs were calculated in detail according to the electrochemical framework developed by Nørskov and his co-workers^{##UREF##19##46##,##UREF##20##47##}. For ORR, the four-electron reaction mechanism follows several elementary steps:where the represents the active site on the electrocatalyst surface, and refer to liquid and gas phases, respectively, and , and are adsorbed intermediates.</p>", "<p id=\"Par30\">The binding energies of , and were obtained by DFT calculations as follows:^{##UREF##19##46##,##UREF##20##47##}in which, , , , and are the ground state energies of a clean surface and surfaces adsorbed with , and , respectively. and are the calculated DFT energies of H₂O and H₂ molecules in the gas phase. If we consider the zero point energy (ZPE) and entropy correction, the free energies of adsorption, ΔG_ads, can be transformed from DFT binding energies, ΔE_ads, as follows:where ΔE_ads is the binding energy of adsorption species OOH^*, O^* and OH^*. ΔZPE, ΔS, U and e are the ZPE changes, entropy changes, applied potential at the electrode, and charge transferred.</p>", "<p id=\"Par31\">Using the adsorption free energies obtained from (10) and (7)-(9), the reaction free energies of ORR reactions (2)-(6) can be calculated as:</p>", "<p id=\"Par32\">Thus, for the ORR reactions, the onset potential, , and the overpotential, η^ORR, can be expressed as:^{##UREF##19##46##,##UREF##20##47##}</p>", "<p id=\"Par33\">The vacancy formation energy is calculated as:where E_slab-VPt represents the energy of a slab with one Pt vacancy, µ_Pt represents the chemical energy of a Pt atom, E_slab represents the energy of a perfect slab.</p>", "<p id=\"Par34\">We also calculated the vacancy formation energy of Ni (E_slab-_VNi) in PtNi slab, PtNiMo slab, PtNiAu slab and PtNiMoAu slab by formula (17), where E_slab-VNi represents the energy of a slab with one Ni vacancy on the subsurface layer, <italic>µ</italic>_Ni represents the chemical energy of a Ni atom, E_slab represents the energy of a perfect slab. In addition, we constructed 2 × 2 Pt₃Ni conventional cells with a Ni vacancy model to calculate the Ni diffusion energy. The bulk Ni diffusion was proposed to diffuse to the nearest Ni vacancy site and the corresponding transition state searches were conducted with the climbing image nudged elastic band method (CI-NEB) method. For the Ni diffusion energy in the PtNiMo slab and PtNiAu slab, a Pt atom near the Ni diffusion pathway of the Pt₃Ni (111) slab was substituted by the Mo atom and Au atom, respectively. For the Ni diffusion energy in the PtNiMoAu slab, two Pt atoms near the Ni diffusion pathway of the Pt₃Ni (111) slab were substituted by the Mo atom and Au atom.</p>" ]

[ "<title>Results</title>", "<title>Synthesis and characterizations of PtNi-based model catalysts</title>", "<p id=\"Par6\">To explore the effect of Mo and Au dopants on the ORR performance of the PtNi catalyst, we synthesized a series of PtNi-based NWs, containing PtNi NWs, PtNiMo NWs, and PtNiAu NWs with the same diameter, as model catalysts by slightly adjusting the synthetic method (Supplementary Table ##SUPPL##0##1##). The detailed structure and composition of the three NWs were carefully analyzed. The low-magnification transmission electron microscopy (TEM) images in Supplementary Fig. ##SUPPL##0##1a–c## shows the successful preparation of PtNi NWs, PtNiMo NWs, and PtNiAu NWs in high purity and uniformity. The average diameters of these three NWs are all about 1.2 nm, which were determined by counting 100 NWs (Supplementary Fig. ##SUPPL##0##1d–f##). To acquire the atomic-level structural information of three NWs, we captured the atomically resolved high-angle annular dark-field scanning transmission electron microscopy (HAADF-STEM) image of a single NW (Fig. ##FIG##0##1a–c##). Specifically, for all these three NWs, a typical NW is made up of about six atomic layers, agreeing well with the average diameter. And the interplanar spacings of {111} planes for three NWs are almost the same, suggesting that the negligible strain is introduced with the doping of Mo or Au atoms into PtNi NWs. As shown in Fig. ##FIG##0##1d##, the X-ray diffraction (XRD) pattern of three NWs presented the same features in terms of peak number and peak position, being consistent with the data from its interplanar spacing. Figure. ##FIG##0##1##a–c and Supplementary Fig. ##SUPPL##0##2## show the energy-dispersive X-ray spectroscopy (EDS) elemental mapping and EDS line-scanning profile, clearly demonstrating the homogeneous distribution of Pt/Ni in PtNi NWs, Pt/Ni/Mo in PtNiMo NWs, and Pt/Ni/Au in PtNiAu NWs. According to the inductively coupled plasma mass spectrometry (ICP-MS), the atomic ratio of Pt/Ni in PtNi NWs, Pt/Ni/Mo in PtNiMo NWs, and Pt/Ni/Au in PtNiAu NWs were determined to be 3.03:1.00, 2.99:1.00:0.10, and 3.01:1.00:0.11, indicting the similar atomic ratio of Pt:Ni in three NWs. Taken together, the as-prepared PtNi-based NWs possess almost identical morphological structure, diameter, ratio of Pt/Ni, and interplanar spacing, thus offering the ideal platform to study the effects of Mo and Au dopants on the catalytic stability of PtNi NWs toward ORR.</p>", "<title>Atomistic Insights into the Roles of Mo and Au Dopants for Catalytic Stability</title>", "<p id=\"Par7\">Prior to ORR measurements, three PtNi-based NWs, as well as the Pt NWs using as the reference catalyst (Supplementary Fig. ##SUPPL##0##3##), were first loaded on Vulcan XC-7 carbon to prepare the carbon-supported NWs (NWs/C) catalysts (Supplementary Fig. ##SUPPL##0##4##). The cyclic voltammograms (CVs) and positive-going polarization curves of all NWs/C catalysts were further recorded to obtain the initial electrochemically active surface areas (ECSAs) and MA, respectively (Supplementary Fig. ##SUPPL##0##5## and Fig. ##FIG##0##1e, f##). Clearly, the MA of PtNi NWs/C (1.96 A mg⁻¹_Pt) catalyst displayed an obvious increase with respect to the Pt NWs/C (0.84 A mg⁻¹_Pt), in line with the prior studies^{##REF##30384601##29##}. When the Mo and Au dopants are introduced, the MA could be further increased to 2.54 A mg⁻¹_Pt (PtNiMo NWs/C) and 2.24 A mg⁻¹_Pt (PtNiAu NWs/C), respectively. The long-term catalytic stability for these three PtNi-based NWs catalysts was then assessed via an accelerated durability test (ADT) between 0.6 and 1.0 V versus reversible hydrogen electrode (V_RHE) in O₂-saturated 0.1 M HClO₄ at room temperature. The CVs curves and polarization curves of NWs catalysts after certain cycles of ADT were then recorded, where the corresponding specific activity (SA), ECSAs, and MA were derived (Supplementary Figs. ##SUPPL##0##6## and ##SUPPL##0##7##). The results demonstrate that both the Mo and Au dopants could significantly enhance the catalytic stability of PtNi NWs/C catalyst for ORR. Specifically, the PtNiMo NWs/C and PtNiAu NWs/C only show a drop of 22.8% and 18.3% in MA after 20 K cycles of ADT, respectively, contrasting with the big decrease of 53.5% for PtNi NWs/C catalyst (Fig. ##FIG##0##1e##). The changes in structure for three PtNi-based catalysts after 20 K cycles of ADT were further examined to support the observations on durability trend. Self-consistently, the PtNiMo NWs/C and PtNiAu NWs/C catalysts both present smaller changes in ECSAs and NWs diameters after ADT measurements relative to PtNi NWs/C catalyst (Fig. ##FIG##0##1f##, Supplementary Figs. ##SUPPL##0##8## and ##SUPPL##0##9##). These results together manifest that both the Mo and Au dopants could improve the ORR performance of the PtNi NWs/C catalyst (Supplementary Table ##SUPPL##0##2##).</p>", "<p id=\"Par8\">The in-depth insights into how Mo and Au dopants stabilize the PtNi catalyst for ORR operation were further explored. In principle, the Pt-based ORR catalysts degradation can be traced back to the following four possible reasons: (i) degradation of carbon support; (ii) dissolution of reactive metal components, e.g. the Pt and Ni for PtNi catalyst; (iii) detachment of catalyst nanoparticles; (iV) loss of active surface area induced by electrochemical Ostwald ripening and nanoparticle coalesce^{##REF##29745393##30##,##UREF##9##31##}. Because our PtNi-based model catalysts have almost identical structures and the same carbon support, we can safely exclude the carbon degradation and catalyst detachment as the reasons for the different stability performance of PtNi-based model catalysts. Besides, it should be noted that the dissolution of reactive metal components can trigger the electrochemical Ostwald ripening and nanoparticle coalesce^{##UREF##10##32##}. In this case, we could reasonably assume that the dissolution of Pt and Ni elements should be the main cause for the PtNi-based catalysts degradation under long-term operation^{##UREF##11##33##,##REF##29578723##34##}. Consequently, we systematically detected the dissolution process of Pt and Ni elements for these catalysts during ADT measurements by a stationary rotating disk electrode coupled with an ex-situ ICP-MS. Fig. ##FIG##0##1g, h## shows the leaching amounts of Pt and Ni in the electrolyte after cyclic testing at intervals, showing the continuous leaching of Pt and Ni for each catalyst. Intriguingly, it is found that the Pt dissolution is maximally suppressed when the Au dopant is introduced, as indicated by the lowest dissolution rate of Pt in PtNiAu after 20 K cycles (Fig. ##FIG##0##1i##). The difference is that the Ni dissolution is suppressed most when the Mo dopant is adopted. These experimental evidences definitely indicate that the Mo and Au dopants play distinct roles in improving the stability of PtNi NWs/C catalyst. Specifically, the Mo dopant mainly contributes to the Ni stabilization while the Au dopant dominantly restrains the Pt dissolution.</p>", "<p id=\"Par9\">To further verify the distinct roles of Mo and Au dopants, the DFT calculations were performed. In light of the corresponding structural and compositional parameters, we first constructed the PtNiAu (111) slab, PtNiMo (111) slab, PtNi (111) slab (Supplementary Fig. ##SUPPL##0##10##) as the models to represent the PtNiAu NWs/C, PtNiMo NWs/C, and PtNi NWs/C catalysts, respectively. Since the vacancy formation energies (E_vac) of Pt and Ni atoms could reflect the tendency of Pt atoms dissolution and Ni atoms leaching^{##UREF##12##35##,##REF##26670103##36##}, the E_vac of Pt and Ni atoms for PtNi-based (111) slabs were calculated (Supplementary Fig. ##SUPPL##0##11##). In comparison with PtNi (111) slab, both PtNiMo (111) slab and PtNiAu (111) slab present increased E_vac of Pt and Ni atoms, self-supporting the mitigated dissolution of Pt and Ni for PtNiMo NWs/C and PtNiAu NWs/C catalysts. We further found that the E_vac of Pt atoms for the PtNiAu (111) slab and E_vac of Ni atoms for PtNiMo (111) slab present the largest increment with respect to the PtNi-based slab, respectively (Fig. ##FIG##1##2a##). The results thus confirm that the Mo and Au dopants could separately mitigate the leaching of Ni and Pt atoms, in accord with experimental observations from the ICP-MS. Besides the E_vac, the diffusion energy barrier of Ni is also crucial for Ni leaching. We thus calculated the Ni diffusion energy diagrams for PtNi-based slabs (Fig. ##FIG##1##2b##, Supplementary Fig. ##SUPPL##0##12##). It could be seen that the diffusion energy barrier of Ni for PtNiMo (111) slab (0.68 eV) was the largest, followed by PtNiAu (111) slab (0.57 eV) and PtNi (111) slab (0.40 eV), indicating that Mo dopant could obviously inhibit the outward diffusion of bulk Ni atoms (Fig. ##FIG##1##2b##). All these preceding results together confirm that, in improving the durability of PtNi NWs, the Mo dopant mainly increases E_vac and diffusion energy of Ni atoms, while the Au dopant mainly increases E_vac of Pt atoms.</p>", "<p id=\"Par10\">We further attempted to understand how the Mo and Au dopants play distinct roles in stabilizing Ni and Pt, which should be, in principle, originated from the different interaction strengths between dopants and Pt/Ni^{##REF##26068847##18##,##REF##31639295##26##}. To explore the interaction strength for the atomic pair of Mo-Pt, Au-Pt, Mo-Ni, and Au-Ni, the partial density of states (PDOS) of PtNiMo (111) and PtNiAu (111) slabs were analyzed (Fig. ##FIG##1##2c, d##). In general, the degree of overlaps between <italic>d</italic>-orbitals could reflect bonding strength between different metals. The results show that the degree of overlap between Pt <italic>d</italic>-orbital and Au <italic>d</italic>-orbital is larger than that of Pt <italic>d</italic>-orbital and Mo <italic>d</italic>-orbital, indicating the reinforced interaction between Pt and Au. Meanwhile, the stronger interaction between Mo <italic>d</italic>-orbital and Ni <italic>d</italic>-orbital is found relative to that between Au <italic>d</italic>-orbital and Ni <italic>d</italic>-orbital. These insights into the interaction between dopants and Pt/Ni atoms reasonably underpin the distinct roles of Mo and Au dopants in stabilizing the PtNi catalyst.</p>", "<p id=\"Par11\">Besides, the improved activities of PtNiMo NWs/C and PtNiAu NWs/C catalysts were also rationalized by DFT calculations (Supplementary Fig. ##SUPPL##0##13##). Supplementary Fig. ##SUPPL##0##14a, b## shows the Gibbs free energies of reaction intermediates for the ORR on different slabs. The results testify that the overpotential on Pt (111), PtNi (111), PtNiMo (111), and PtNiAu (111) slabs follows the trend of PtNiMo (111) &lt; PtNiAu (111) &lt; PtNi (111) &lt; Pt (111), well-consistent with our experimental observations on the activity trend. Relative to PtNi NWs/C catalyst, we could conclude that the weakened adsorption of oxygenated species on PtNiMo NWs/C and PtNiAu NWs/C catalysts arising from the ligand effect is the intrinsic cause for the improved activity, which is further supported by the downshifted <italic>d</italic>-band center on PtNiMo (111) and PtNiAu (111) slabs (Supplementary Fig. ##SUPPL##0##14c##)^{##UREF##12##35##,##UREF##13##37##,##REF##21378936##38##}.</p>", "<title>Rational design and synthesis of PtNiMoAu NWs</title>", "<p id=\"Par12\">Since Au and Mo dopants could respectively stabilize the surface Pt overlayer and suppress the leaching of Ni atoms, it is reasonably hypothesized that integrating these two dopants into PtNi would further improve the ORR stability. The hypothesis is further verified by the DFT calculations, as indicated by the largest E_vac of Pt and Ni atoms, as well as the highest diffusion energy barrier (0.76 eV) of Ni atoms, on PtNiMoAu (111) slab (Supplementary Figs. ##SUPPL##0##10##–##SUPPL##0##12## and ##SUPPL##0##15a##). The strong coupling degrees between Mo, Pt, and Au <italic>d</italic> orbitals, as well as Mo, Ni, and Au <italic>d</italic> orbitals may account for the increased E_vac and diffusion energy barrier (Supplementary Fig. ##SUPPL##0##15b, c##). Moreover, it is predicted that the PtNiMoAu would present the enhanced ORR activity, as indicated by the lowest overpotential and downshifted <italic>d</italic>-band center of the PtNiMoAu (111) slab (Supplementary Fig. ##SUPPL##0##14##). Inspired by this conceptual design, we subsequently synthesized PtNiMoAu NWs catalysts by using a modified synthetic procedure (see Methods for detail).</p>", "<p id=\"Par13\">Figure ##FIG##2##3a,b## shows low-magnification TEM and STEM images of the as-synthesized PtNiMoAu NWs, respectively. The NWs with a diameter of 1.1 ± 0.3 nm and length of 62 ± 21 nm are identified by statistic counting (Fig. ##FIG##2##3c##). The atomic-level structural information was analyzed based on the HAADF-STEM image of a single NW (Fig. ##FIG##2##3d##). Specifically, the well-defined lattice spacing can be assigned to the {200} and {111} planes, matching with the fast Fourier transform (FFT) pattern in the inset. From the distinguished lattice planes, the grown direction of &lt;110&gt; orientation could be deduced. The EDS elemental mapping (Fig. ##FIG##2##3e##) and line-scanning profile (Fig. ##FIG##2##3f##) both indicate the homogeneous distribution of Pt, Ni, Mo, and Au elements throughout a single NW. The Pt:Ni:Mo:Au atomic ratio was estimated to be 3.02:1.00:0.11:0.13 by ICP-MS, very close to the value (3.10:1.00:0.15:0.14) determined by the XPS spectrum (Supplementary Fig. ##SUPPL##0##16##). These structural characterizations prove the successful formation of ultrathin PtNiMoAu tetrametallic NWs.</p>", "<p id=\"Par14\">The chemical states for the constituent elements of PtNi NWs and PtNiMoAu NWs were then analyzed by XPS. Compared with the Pt 4 <italic>f</italic> of PtNi NWs, the decreased binding energy for Pt 4 <italic>f</italic> of PtNiMoAu NWs implies more electrons accumulate around Pt in PtNiMoAu NWs (Fig. ##FIG##2##3g##). To examine the local coordination and electronic structures of the PtNi NWs and PtNiMoAu NWs, we further employed X-ray absorption spectroscopy (XAS), in comparison with a bulk Pt foil and PtO₂. Figure ##FIG##2##3h## shows the X-ray absorption near-edge structure (XANES) spectra of Pt L₃-edge. Agreeing well with XPS results and Bader analysis (Supplementary Table ##SUPPL##0##3##), the Pt white line of PtNiMoAu NWs is found to possess lower intensity and be close to the metallic state (Pt foil) when compared to the PtNi NWs. These results manifest that, after introducing the Mo and Au atoms, Pt atoms in PtNiMoAu NWs could gain more electrons. The local structure was further derived from the Fourier transform of the phase-corrected extended X-ray absorption fine structure (EXAFS). Fig. ##FIG##2##3i## and Supplementary Fig. ##SUPPL##0##17## show the fitted R-space and K-space data of the Pt edge for PtNiMoAu NWs and PtNi NWs. Clearly, the first-shell Pt-Pt length of PtNi NWs and PtNiMoAu NWs was obviously shorter than that of Pt/C, attributing to the smaller radius of Ni/Mo atoms. Besides, the PtNiMoAu NWs present a similar Pt-Pt coordination number (5.5 ± 0.4) with that of PtNi NWs (6.3 ± 0.4), due to the similar NWs diameter (Supplementary Table ##SUPPL##0##4##). These results together demonstrate that the Mo and Au dopants could redistribute the electrons and induce the compressive strain, which are believed as the structural foundations for the improved ORR performance as predicted.</p>", "<title>ORR Performance of PtNiMoAu NWs</title>", "<p id=\"Par15\">To verify the validity of our rational design for a practical catalyst, we further assessed the ORR performance of PtNiMoAu NWs/C catalyst by recording the CVs and polarization curves in 0.1 M HClO₄ solution (Supplementary Fig. ##SUPPL##0##18## and ##SUPPL##0##19##). Specifically, the PtNiMoAu NWs/C catalyst exhibits specific activity (SA) and MA of 3.13 mA cm⁻² and 2.89 A mg⁻¹_Pt at 0.9 V_RHE, much higher than those of Pt/C (0.27 mA cm⁻² and 0.18 A mg⁻¹_Pt), PtNi NWs/C (2.34 mA cm⁻² and 1.96 A mg⁻¹_Pt), PtNiMo NWs/C (2.93 mA cm⁻² and 2.54 A mg⁻¹_Pt) and PtNiAu NWs/C catalysts (2.61 mA cm⁻² and 2.24 A mg⁻¹_Pt) (Fig. ##FIG##3##4a## and Supplementary Fig. ##SUPPL##0##19##). These results prove that the co-doping of Mo and Au elements into PtNi NWs evidently boost the ORR activity, which is in accordance with DFT prediction.</p>", "<p id=\"Par16\">The long-term stability of PtNiMoAu NWs/C catalyst was further evaluated by ADT. Figure ##FIG##3##4b## shows the polarization curves of PtNiMoAu NWs/C catalyst after different cycles of ADT. Remarkably, the ECSAs (86.1 m² g⁻¹_Pt) and MA (2.42 A mg⁻¹_Pt) of PtNiMoAu/C catalyst only displayed a drop of 6.7% and 16.2% after 80 K cycles (Fig. ##FIG##3##4c## and Supplementary Fig. ##SUPPL##0##20##), respectively. Of note, such excellent long-term stability of PtNiMoAu NWs surpasses the recently-reported Pt-based ORR electrocatalysts at different cycles of ADT (Fig. ##FIG##3##4d## and Supplementary Table ##SUPPL##0##5##). By contrast, after only 20 K cycles, the ECSAs and MA for commercial Pt/C, Pt NWs/C, PtNi NWs/C, PtNiMo NWs/C, and PtNiAu NWs/C catalysts severally decrease 68.4% and 70.2%, 30.8% and 34.5%, 49.4% and 53.5%, 19.2% and 22.8%, 14.4% and 18.3% (Fig. ##FIG##0##1e## and Supplementary Fig. ##SUPPL##0##21##). The structure and composition of PtNiMoAu NWs/C catalyst after 80 K of ADT were further probed. The STEM image demonstrates the well-maintained morphology of PtNiMoAu NWs/C after 80 K cycles (Fig. ##FIG##4##5a##), whereas the commercial Pt/C and other Pt-based NWs/C catalysts exhibit varying degrees of sintering after only 20 K cycles (Supplementary Fig. ##SUPPL##0##8##, ##SUPPL##0##22a## and ##SUPPL##0##23##). Its diameter only increases from 1.1 nm (initial) to 1.4 nm (after 80 K cycles) (Fig. ##FIG##4##5a## and Supplementary Fig. ##SUPPL##0##22b##), consistent with the ECSAs losses. And the EDS mapping and line-scan profiles further confirm its well-reserved elemental distribution and composition (Fig. ##FIG##4##5b, c##). The atomic ratio of Pt:Ni:Mo:Au (3.65:1.00:0.16:0.20) in PtNiMoAu NWs/C catalyst after 80 K cycles is relatively close to its initial composition (3.02:1.00:0.11:0.13), contrasting with the big change for PtNi NWs/C catalyst after only 20 K cycles from 3.03:1.00 to 11.01:1.00. The dissolution of Pt and Ni for PtNiMoAu NWs/C catalyst at different ADT cycles was further examined based on ex-situ ICP-MS measurements. Self-consistent with our expectation from the distinct roles of Mo and Au dopants, the PtNiMoAu NWs/C presents the lowest leaching mass of Pt and Ni metal among all PtNi-based NWs (Fig. ##FIG##4##5d##).</p>", "<p id=\"Par17\">Besides, in-situ XANES was carried out to experimentally support the remarkable stability of the PtNiMoAu NWs/C catalyst for ORR. From XANES spectra of the Pt L₃ edge, we can find that the normalized white line intensity (μ_norm) of PtNiMoAu NWs slightly increases with the potential (Fig. ##FIG##4##5e##), while the PtNi NWs display a much larger increase at the same operational condition (Supplementary Fig. ##SUPPL##0##24##). The difference is more clear in the relative change of the white line intensity [(Δμ_E-Δμ_0.42VRHE)/Δμ_0.42VRHE)] of the Pt L₃ edge spectra for PtNiMoAu NWs/C and PtNi NWs/C catalysts as a function of applied potential^{##REF##17218522##20##,##UREF##14##39##}. As shown in Fig. ##FIG##4##5f##, the increase in white line intensity for PtNiMoAu NWs/C catalyst occurs at a higher applied potential compared with that for PtNi NWs/C catalyst, corroborating the elevated Pt oxidation potential for PtNiMoAu NWs/C catalyst. In addition, anodic shift of Pt/Pt²⁺ redox potential for PtNiMoAu NWs/C catalyst also supports this result (Supplementary Fig. ##SUPPL##0##25##). This finding is indeed consistent with the electron-rich Pt atoms for PtNiMoAu NWs as demonstrated by the XANES spectrum (Fig. ##FIG##2##3h##) and Bader analysis (Supplementary Table ##SUPPL##0##3##), which could curb the Pt dissolution and in turn stabilize the catalyst. All these preceding evidences strongly support our conceptual design that integrating the different functions of Mo and Au into PtNi NWs would maximize the catalyst stability.</p>", "<title>Fuel cell performance</title>", "<p id=\"Par18\">We further fabricated the membrane electrode assembly (MEA) to evaluate the fuel cell performance of PtNi NWs/C, PtNiMo NWs/C, PtNiAu NWs/C, and PtNiMoAu NWs/C. The polarization curves and power density curves were recorded in H₂-O₂ (Fig. ##FIG##5##6a##) and H₂-Air environment (Fig. ##FIG##5##6b##) (80 °C and 150 kPa absolute pressure), in which the Pt loadings for Pt/C as anode catalyst and PtNi-based NWs/C as cathode catalyst were 0.05 mg_Pt cm⁻² and 0.10 mg_Pt cm⁻², respectively (Pt/C catalyst with Pt loading of 0.12 mg_Pt cm⁻² on the cathode was also employed as the reference). Before evaluating MEA performance of catalysts, we first tested the pressure drop between the inlet and outlet at different flow rates (Supplementary Fig. ##SUPPL##0##26##), in which the gap pressure drop of anode (H₂, 200 mL min⁻¹) and cathode (O₂, 500 mL min⁻¹) is severally 3 kPa and 6 kPa. Given the relatively low pressure drop, we reasonably believe that it has a negligible impact on MEA performance. The H₂–O₂/H₂-Air fuel cell assembled with PtNiMoAu NWs/C cathode shows a peak power density of 1.71/0.93 W cm⁻², exceeding the corresponding values of 1.24/0.82 W cm⁻² for commercial Pt/C. Besides, the PtNiMoAu NWs/C presents a 6.6-, 2.0-, 1.3-, and 1.7-fold enhancement in the beginning-of-life (BOL) MA (0.93 A mg⁻¹_Pt) relative to the Pt/C (0.14 A mg⁻¹_Pt), PtNi NWs/C (0.46 A mg⁻¹_Pt), PtNiMo NWs/C (0.70 A mg⁻¹_Pt), and PtNiAu NWs/C (0.55 A mg⁻¹_Pt) at 0.9 V_iR-free (Fig. ##FIG##5##6c## and Supplementary Fig. ##SUPPL##0##27##). Even comparing with the value that was set in the 2025 targets by U.S. Department of Energy (DOE) (MA of 0.44 A mg⁻¹_Pt), the remarkable mass activity of PtNiMoAu NWs/C catalyst presents 2.1-fold increasement, showing the great potential for PEMFCs. Moreover, the PtNiMoAu NWs/C under the H₂-Air condition delivers the improved current density of 0.35 A cm⁻² at the kinetic dominant region of 0.8 V when compared to the Pt/C (0.16 A cm⁻² at 0.8 V) and the DOE 2025 target (0.30 A cm⁻²) (Supplementary Fig. ##SUPPL##0##28##). For ADT, the standard square-wave protocol by holding the cathode at 0.60 V for 3 s and 0.95 V for 3 s was employed to evaluate the durability of catalysts (see Methods). The end-of-life (EOL) MA at 0.9 V_iR-free (H₂-O₂) and a voltage loss at 0.8 A cm⁻² (H₂-Air) for PtNiMoAu NWs/C are 0.72 A mg⁻¹_Pt (22.6% MA loss) and 25 mV after 30 K cycles of ADT, respectively, which comprehensively exceed the DOE’s 2025 durability goal (MA loss &lt;40% and voltage loss &lt;30 mV). As a contrast, the Pt/C, PtNi NWs/C, PtNiMo NWs/C, and PtNiAu NWs/C show substantial MA loss of 45.7%, 58.7%, 41.4%, and 32.7%, respectively. Besides, the variation trend of ECSAs, voltages loss (H₂-O₂), Ni content, and morphology of PtNi NWs/C, PtNiAu NWs/C, PtNiMo NWs/C, and PtNiMoAu NWs/C catalysts during the MEA durability tests also was in accordance with the results obtained under the rotating disk electrode measurements (Fig. ##FIG##5##6d, e##, Supplementary Figs. ##SUPPL##0##29##–##SUPPL##0##31##, and Supplementary Table ##SUPPL##0##6##). Even comparing with the Pt-based catalysts reported recently, the excellent BOL MA and durability of PtNiMoAu NWs/C indeed render it as the top catalyst for durable fuel cells (Fig. ##FIG##5##6f## and Supplementary Table ##SUPPL##0##7##). Beyond this, we can envision our delicate design would also enhance the catalyst durability a more challenging start-up and shutdown process of the fuel cells (generally under the applied potential over 1.0 V) through mitigation of the sintering process. All these preceding results together strongly evidence the successful design of active and durable PtNiMoAu NWs/C catalyst for practical fuel cells.</p>" ]

[ "<title>Discussion</title>", "<p id=\"Par19\">To sum up, we have identified the distinct roles of Mo and Au dopants in inhibiting the dissolution of Ni and Pt metal, respectively, in terms of PtNi-based NWs/C as model catalysts, by combining the experimental evidences and DFT calculations. Insightful studies indicated that the distinct roles of Mo and Au dopants are essentially derived from the stronger interaction of atomic orbital for Pt-Au (<italic>d</italic>-<italic>d</italic>) and Ni-Mo (<italic>d</italic>-<italic>d</italic>). Based on this atomistic understanding, we delicately designed a remarkable PtNiMoAu NWs/C catalyst that possesses a concurrently high MA of 2.89 A mg⁻¹_Pt and preeminent stability, with only 16.2% loss of MA after 80 K cycles of ADT. By contrast, PtNi NWs/C catalyst showed a 53.5% loss of MA after only 20 K cycles of ADT, persuasively confirming the advantage of Mo/Au co-doping in improving its ORR stability. Moreover, when assembling the PtNiMoAu NWs/C catalyst into the fuel cell cathode, a high MA retention of 77.4% (H₂-O₂, 0.9 V_iR-free) and a low voltage loss of 25 mV (H₂-Air, 0.8 A cm⁻²) after ADT were output, proving the highly durable fuel cell performance. This work advances the design of robust PtM catalysts to a more precise stage by providing insights into the functions of dopants, which is thus of general importance for the fields of catalysis, sensing, and even beyond.</p>" ]

[]

[ "<p id=\"Par1\">Stabilizing active PtNi alloy catalyst toward oxygen reduction reaction is essential for fuel cell. Doping of specific metals is an empirical strategy, however, the atomistic insight into how dopant boosts the stability of PtNi catalyst still remains elusive. Here, with typical examples of Mo and Au dopants, we identify the distinct roles of Mo and Au in stabilizing PtNi nanowires catalysts. Specifically, due to the stronger interaction between atomic orbital for Ni-Mo and Pt-Au, the Mo dopant mainly suppresses the outward diffusion of Ni atoms while the Au dopant contributes to the stabilization of surface Pt overlayer. Inspired by this atomistic understanding, we rationally construct the PtNiMoAu nanowires by integrating the different functions of Mo and Au into one entity. Such catalyst assembled in fuel cell cathode thus presents both remarkable activity and durability, even surpassing the United States Department of Energy technical targets for 2025.</p>", "<p id=\"Par2\">Doping strategies have been shown to stabilize the active platinum-nickel (PtNi) catalyst in fuel cells, however, the atomistic mechanism is less known. Here, the authors identify the roles of Mo and Au dopants in improving the durability of a PtNi nanowire catalyst for fuel cells.</p>", "<title>Subject terms</title>" ]

[ "<title>Supplementary information</title>", "<p>\n\n\n</p>", "<title>Source data</title>", "<p>\n\n</p>" ]

[ "<title>Supplementary information</title>", "<p>The online version contains supplementary material available at 10.1038/s41467-024-44788-0.</p>", "<title>Acknowledgements</title>", "<p>This work was supported by the National Key Research and Development Program of China (No. 2021YFA1502000 to H.H.), NSFC (Nos. 22322902, U22A20396, 22211540385, and 22309050 to H.H. and L.G.), the Science and Technology Innovation Program of Hunan Province (No. 2021RC3065, 2021RC2053, and 2023JJ40117 to H.H. and L.G.), the Jiebang Guashuai Project of Changsha City (Grant No. kq2301009 to H.H.), the China Postdoctoral Science Foundation (2023T160205 and 2023M741118 to L.G.), Shenzhen Science and Technology Program (Nos. JCYJ20210324120800002 and JCYJ20220818100012025 to H.H.). The X-ray ab-sorption structure (XAS) spectra were performed at the BL11B beam line of the Shanghai Synchrotron Radiation Facility (SSRF).</p>", "<title>Author contributions</title>", "<p>L.G. and H.H. conceived the idea and designed experiments. L.G. performed the preparation of samples, carried out the electrochemical measurements, and analyzed the experimental data. Z.Y. and W.Z. conducted the DFT simulation and theoretical analyses. T.S. and M.L. conducted the TEM and EDX characterizations. X.C., W.L. and Q.Y. helped with the analysis and discussion of experimental data. H.H. wrote the manuscript. All the authors were involved in the discussion and analysis of the manuscript.</p>", "<title>Peer review</title>", "<title>Peer review information</title>", "<p id=\"Par35\"><italic>Nature Communications</italic> thanks the anonymous reviewers for their contribution to the peer review of this work. A peer review file is available.</p>", "<title>Data availability</title>", "<p>The data that support the plots within this paper and other finding of this study are available from the corresponding authors upon request. <xref ref-type=\"sec\" rid=\"Sec18\">Source data</xref> are provided with this paper.</p>", "<title>Competing interests</title>", "<p id=\"Par36\">The authors declare no competing interests.</p>" ]

[ "<fig id=\"Fig1\"><label>Fig. 1</label><caption><title>Structural characterizations and ORR performance.</title><p>HAADF-STEM images and EDS EDS line-scanning profiles of <bold>a</bold> PtNi NWs, <bold>b</bold> PtNiMo NWs, and <bold>c</bold> PtNiAu NWs. <bold>d</bold> XRD patterns of PtNi-bassed NWs. <bold>e</bold> MA and <bold>f</bold> ECSAs for different NWs/C catalysts after different cycles. Dissolving mass of <bold>g</bold> Pt and <bold>h</bold> Ni metals for PtNi-based NWs/C catalysts after different cycles. <bold>i</bold> Dissolving rate of Pt and Ni metals for PtNi-based NWs/C catalysts after 20 K cycles. All dissolving mass and dissolving rate values are means with error bars (standard deviations) from three replicates.</p></caption></fig>", "<fig id=\"Fig2\"><label>Fig. 2</label><caption><title>Stability of catalysts based on DFT calculations.</title><p><bold>a</bold> The change of Pt and Ni vacancy formation energies (E_V(Pt)) for PtNiMo (111) and PtNiAu (111) slabs relation to PtNi slab. <bold>b</bold> Ni diffusion pathway energy diagram for different slabs. The inset showed the proposed Ni diffusion pathway in the bulk alloy. <bold>c</bold>, <bold>d</bold> The PDOSs. Site-dependent PDOSs of Pt-5<italic>d</italic>, Ni-3<italic>d</italic>, Mo-3<italic>d</italic> and Au-3<italic>d</italic> on PtNiMo (111) and PtNiAu (111) slabs.</p></caption></fig>", "<fig id=\"Fig3\"><label>Fig. 3</label><caption><title>Structural and compositional characterizations of ultrathin PtNiMoAu tetrametallic NWs.</title><p><bold>a</bold> Low-magnification TEM image. <bold>b</bold> Low-magnification STEM image. <bold>c</bold> Histogram of diameter and length distributions. <bold>d</bold> Atomically resolved HAADF-STEM image. The inset shows the corresponding FFT pattern. <bold>e</bold> STEM-EDS elemental mapping profiles. <bold>f</bold> EDS line-scanning profiles. <bold>g</bold> High-resolution XPS spectra of Pt 4 <italic>f</italic> for PtNi NWs and PtNiMoAu NWs. <bold>h</bold> Pt L₃-edge XANES spectra and <bold>i</bold> the Fourier transforms of EXAFS spectra for different samples.</p></caption></fig>", "<fig id=\"Fig4\"><label>Fig. 4</label><caption><title>Electrochemical properties of different catalysts.</title><p><bold>a</bold> The SA and MA of different catalysts at 0.9 V_RHE. <bold>b</bold> ORR polarization curves of PtNiMoAu NWs/C catalyst before and after ADT for different cycles. The inset showed an enlarged view of the area marked by the orange square. <bold>c</bold> The comparison of ECSAs and MA for PtNiMoAu NWs/C catalyst at 0.9 V_RHE before and after ADT for different cycles. <bold>d</bold> Comparison of MA retention for recently reported catalysts after different cycles of ADT.</p></caption></fig>", "<fig id=\"Fig5\"><label>Fig. 5</label><caption><title>Structural characterizations and in-situ XANES analysis.</title><p><bold>a</bold> Low-magnification STEM image after 80 K of ADT. The inset showed atomically resolved HAADF-STEM image. <bold>b</bold> STEM-EDS elemental mapping profiles and <bold>c</bold> EDS line-scanning profiles after 80 K of ADT. <bold>d</bold> Dissolving mass of Pt/Ni for PtNi-based NWs after different cycles. The dissolving mass values are means with error bars (standard deviations) from three replicates. <bold>e</bold> In situ Pt L₃-edge XANES spectra of PtNiMoAu NWs at different potentials. The inset showed an enlarged view of the area marked by the red square. <bold>f</bold> Comparison of the change of the Pt adsorption edge peaks (Δμ) of the XANES spectra (relative to Δμ at 0.42 V_RHE) for PtNi NWs and PtNiMoAu NWs as a function of potential obtained in 0.1 M HClO₄.</p></caption></fig>", "<fig id=\"Fig6\"><label>Fig. 6</label><caption><title>Fuel cell performance of the PtNiMoAu NWs/C catalyst.</title><p><bold>a</bold> H₂-O₂ fuel cell polarization curves and power density curves before and after 30 K cycles of ADT. Anode: H₂ flow rate = 200 mL min⁻¹, 0.05 mg_Pt cm⁻² for Pt/C; Cathode: O₂ flow rate = 500 mL min⁻¹, 0.10 mg_Pt cm⁻² for PtNiMoAu NWs/C and 0.12 mg_Pt cm⁻² for Pt/C. <bold>b</bold> H₂-Air fuel cell polarization curves and power density curves of PtNiMoAu NWs/C and Pt/C before and after 30 K cycles of ADT. Anode: H₂ flow rate = 200 mL min⁻¹; Cathode: Air flow rate = 500 mL min⁻¹. <bold>c</bold> Changes of MA (left, H₂-O₂, black and orange dash lines indicate BOL and EOL of DOE targets, respectively) and cell voltage at 0.8 A cm⁻¹ (right, H₂-Air) of PtNiMoAu NWs/C and Pt/C before and after 30 K cycles of ADT. <bold>d</bold> The changes of MEA ECSAs and <bold>e</bold> atomic percent of Ni for PtNiMoAu NWs/C, PtNiAu NWs/C, PtNiMo NWs/C, and PtNi NWs/C before and after 30 K cycles of ADT. <bold>f</bold> Comparison of BOL MA and MA retention (after 30K-cycle ADT) with those typical catalysts reported recently.</p></caption></fig>" ]

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[ "<disp-formula id=\"Equ1\"><label>1</label><alternatives><tex-math id=\"M1\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${{{{\\rm{ECSAs}}}}}\\left({{{{{\\rm{m}}}}}}^{2}/{{{{{\\rm{g}}}}}}_{{{{{\\rm{Pt}}}}}}\\right)=\\left[\\frac{{{{{{\\rm{Q}}}}}}_{{{{{\\rm{H}}}}}-{{{{\\rm{adsorption}}}}}}({{{{\\rm{C}}}}})}{210\\mu {{{{{\\rm{C}}}}}}/{{{{{{\\rm{c}}}}}}{{{{{\\rm{m}}}}}}}^{2}{{{{{\\rm{L}}}}}}_{{{{{\\rm{Pt}}}}}}\\left({{{{{{\\rm{mg}}}}}}}_{{{{{{\\rm{Pt}}}}}}}/{{{{{{\\rm{cm}}}}}}}^{2}\\right){{{{{\\rm{A}}}}}}_{{{{{\\rm{g}}}}}}\\left({{{{{{\\rm{cm}}}}}}}^{2}\\right)}\\right]\\times {10}^{5}$$\\end{document}</tex-math><mml:math id=\"M2\"><mml:mi mathvariant=\"normal\">ECSAs</mml:mi><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:msup><mml:mrow><mml:mi mathvariant=\"normal\">m</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo>/</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant=\"normal\">g</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">Pt</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:mfenced><mml:mo>=</mml:mo><mml:mfenced close=\"]\" open=\"[\"><mml:mrow><mml:mfrac><mml:mrow><mml:msub><mml:mrow><mml:mi mathvariant=\"normal\">Q</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">H</mml:mi><mml:mo>−</mml:mo><mml:mi mathvariant=\"normal\">adsorption</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi mathvariant=\"normal\">C</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mrow><mml:mn>210</mml:mn><mml:mi>μ</mml:mi><mml:mi mathvariant=\"normal\">C</mml:mi><mml:mo>/</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant=\"normal\">c</mml:mi><mml:mi mathvariant=\"normal\">m</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:msub><mml:mrow><mml:mi mathvariant=\"normal\">L</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">Pt</mml:mi></mml:mrow></mml:msub><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:msub><mml:mrow><mml:mi mathvariant=\"normal\">mg</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">Pt</mml:mi></mml:mrow></mml:msub><mml:mo>/</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant=\"normal\">cm</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:mfenced><mml:msub><mml:mrow><mml:mi mathvariant=\"normal\">A</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">g</mml:mi></mml:mrow></mml:msub><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:msup><mml:mrow><mml:mi mathvariant=\"normal\">cm</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:mfenced></mml:mrow></mml:mfrac></mml:mrow></mml:mfenced><mml:mo>×</mml:mo><mml:msup><mml:mrow><mml:mn>10</mml:mn></mml:mrow><mml:mrow><mml:mn>5</mml:mn></mml:mrow></mml:msup></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ2\"><label>2</label><alternatives><tex-math id=\"M3\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${{{{{\\rm{O}}}}}}_{2}({{{{\\rm{g}}}}})+\\ast \\to {{{{{\\rm{O}}}}}}_{2}^{*}$$\\end{document}</tex-math><mml:math id=\"M4\"><mml:msub><mml:mrow><mml:mi mathvariant=\"normal\">O</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi mathvariant=\"normal\">g</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>+</mml:mo><mml:mo>*</mml:mo><mml:mo>→</mml:mo><mml:msubsup><mml:mrow><mml:mi mathvariant=\"normal\">O</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow><mml:mrow><mml:mo>*</mml:mo></mml:mrow></mml:msubsup></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ3\"><label>3</label><alternatives><tex-math id=\"M5\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${{{{{\\rm{O}}}}}}_{2}^{*}+{{{{{\\rm{H}}}}}}^{+}+{{{{{\\rm{e}}}}}}^{-}\\to {{{{\\rm{OO}}}}}{{{{{\\rm{H}}}}}}^{*}$$\\end{document}</tex-math><mml:math id=\"M6\"><mml:msubsup><mml:mrow><mml:mi mathvariant=\"normal\">O</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow><mml:mrow><mml:mo>*</mml:mo></mml:mrow></mml:msubsup><mml:mo>+</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant=\"normal\">H</mml:mi></mml:mrow><mml:mrow><mml:mo>+</mml:mo></mml:mrow></mml:msup><mml:mo>+</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant=\"normal\">e</mml:mi></mml:mrow><mml:mrow><mml:mo>−</mml:mo></mml:mrow></mml:msup><mml:mo>→</mml:mo><mml:mi mathvariant=\"normal\">OO</mml:mi><mml:msup><mml:mrow><mml:mi mathvariant=\"normal\">H</mml:mi></mml:mrow><mml:mrow><mml:mo>*</mml:mo></mml:mrow></mml:msup></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ4\"><label>4</label><alternatives><tex-math id=\"M7\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${{{{\\rm{OO}}}}}{{{{{\\rm{H}}}}}}^{*}+{{{{{\\rm{H}}}}}}^{+}+{{{{{\\rm{e}}}}}}^{-}\\to {{{{{\\rm{O}}}}}}^{*}+{{{{{\\rm{H}}}}}}_{2}{{{{\\rm{O}}}}}({{{{\\rm{l}}}}})$$\\end{document}</tex-math><mml:math id=\"M8\"><mml:mi mathvariant=\"normal\">OO</mml:mi><mml:msup><mml:mrow><mml:mi mathvariant=\"normal\">H</mml:mi></mml:mrow><mml:mrow><mml:mo>*</mml:mo></mml:mrow></mml:msup><mml:mo>+</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant=\"normal\">H</mml:mi></mml:mrow><mml:mrow><mml:mo>+</mml:mo></mml:mrow></mml:msup><mml:mo>+</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant=\"normal\">e</mml:mi></mml:mrow><mml:mrow><mml:mo>−</mml:mo></mml:mrow></mml:msup><mml:mo>→</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant=\"normal\">O</mml:mi></mml:mrow><mml:mrow><mml:mo>*</mml:mo></mml:mrow></mml:msup><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant=\"normal\">H</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mi mathvariant=\"normal\">O</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi mathvariant=\"normal\">l</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ5\"><label>5</label><alternatives><tex-math id=\"M9\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${{{{{\\rm{O}}}}}}^{*}+{{{{{\\rm{H}}}}}}^{+}+{{{{{\\rm{e}}}}}}^{-}\\to {{{{\\rm{O}}}}}{{{{{\\rm{H}}}}}}^{*}$$\\end{document}</tex-math><mml:math id=\"M10\"><mml:msup><mml:mrow><mml:mi mathvariant=\"normal\">O</mml:mi></mml:mrow><mml:mrow><mml:mo>*</mml:mo></mml:mrow></mml:msup><mml:mo>+</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant=\"normal\">H</mml:mi></mml:mrow><mml:mrow><mml:mo>+</mml:mo></mml:mrow></mml:msup><mml:mo>+</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant=\"normal\">e</mml:mi></mml:mrow><mml:mrow><mml:mo>−</mml:mo></mml:mrow></mml:msup><mml:mo>→</mml:mo><mml:mi mathvariant=\"normal\">O</mml:mi><mml:msup><mml:mrow><mml:mi mathvariant=\"normal\">H</mml:mi></mml:mrow><mml:mrow><mml:mo>*</mml:mo></mml:mrow></mml:msup></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ6\"><label>6</label><alternatives><tex-math id=\"M11\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${{{{\\rm{O}}}}}{{{{{\\rm{H}}}}}}^{*}+{{{{{\\rm{H}}}}}}^{+}+{{{{{\\rm{e}}}}}}^{-}\\to {{{{{\\rm{H}}}}}}_{2}{{{{\\rm{O}}}}}({{{{\\rm{l}}}}})+\\ast$$\\end{document}</tex-math><mml:math id=\"M12\"><mml:mi mathvariant=\"normal\">O</mml:mi><mml:msup><mml:mrow><mml:mi mathvariant=\"normal\">H</mml:mi></mml:mrow><mml:mrow><mml:mo>*</mml:mo></mml:mrow></mml:msup><mml:mo>+</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant=\"normal\">H</mml:mi></mml:mrow><mml:mrow><mml:mo>+</mml:mo></mml:mrow></mml:msup><mml:mo>+</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant=\"normal\">e</mml:mi></mml:mrow><mml:mrow><mml:mo>−</mml:mo></mml:mrow></mml:msup><mml:mo>→</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant=\"normal\">H</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mi mathvariant=\"normal\">O</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi mathvariant=\"normal\">l</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>+</mml:mo><mml:mo>*</mml:mo></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq1\"><alternatives><tex-math id=\"M13\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$*$$\\end{document}</tex-math><mml:math id=\"M14\"><mml:mo>*</mml:mo></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq2\"><alternatives><tex-math id=\"M15\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\left({{{{\\rm{l}}}}}\\right)$$\\end{document}</tex-math><mml:math id=\"M16\"><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:mi mathvariant=\"normal\">l</mml:mi></mml:mrow></mml:mfenced></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq3\"><alternatives><tex-math id=\"M17\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\left({{{{\\rm{g}}}}}\\right)$$\\end{document}</tex-math><mml:math id=\"M18\"><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:mi mathvariant=\"normal\">g</mml:mi></mml:mrow></mml:mfenced></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq4\"><alternatives><tex-math id=\"M19\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${{{{\\rm{OO}}}}}{{{{{\\rm{H}}}}}}^{*}$$\\end{document}</tex-math><mml:math id=\"M20\"><mml:mi mathvariant=\"normal\">OO</mml:mi><mml:msup><mml:mrow><mml:mi mathvariant=\"normal\">H</mml:mi></mml:mrow><mml:mrow><mml:mo>*</mml:mo></mml:mrow></mml:msup></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq5\"><alternatives><tex-math id=\"M21\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${{{{{\\rm{O}}}}}}^{*}$$\\end{document}</tex-math><mml:math id=\"M22\"><mml:msup><mml:mrow><mml:mi mathvariant=\"normal\">O</mml:mi></mml:mrow><mml:mrow><mml:mo>*</mml:mo></mml:mrow></mml:msup></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq6\"><alternatives><tex-math id=\"M23\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${{{{\\rm{O}}}}}{{{{{\\rm{H}}}}}}^{*}$$\\end{document}</tex-math><mml:math id=\"M24\"><mml:mi mathvariant=\"normal\">O</mml:mi><mml:msup><mml:mrow><mml:mi mathvariant=\"normal\">H</mml:mi></mml:mrow><mml:mrow><mml:mo>*</mml:mo></mml:mrow></mml:msup></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq7\"><alternatives><tex-math id=\"M25\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${{{{\\rm{OO}}}}}{{{{{\\rm{H}}}}}}^{*}$$\\end{document}</tex-math><mml:math id=\"M26\"><mml:mi mathvariant=\"normal\">OO</mml:mi><mml:msup><mml:mrow><mml:mi mathvariant=\"normal\">H</mml:mi></mml:mrow><mml:mrow><mml:mo>*</mml:mo></mml:mrow></mml:msup></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq8\"><alternatives><tex-math id=\"M27\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${{{{{\\rm{O}}}}}}^{*}$$\\end{document}</tex-math><mml:math id=\"M28\"><mml:msup><mml:mrow><mml:mi mathvariant=\"normal\">O</mml:mi></mml:mrow><mml:mrow><mml:mo>*</mml:mo></mml:mrow></mml:msup></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq9\"><alternatives><tex-math id=\"M29\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${{{{\\rm{O}}}}}{{{{{\\rm{H}}}}}}^{*}$$\\end{document}</tex-math><mml:math id=\"M30\"><mml:mi mathvariant=\"normal\">O</mml:mi><mml:msup><mml:mrow><mml:mi mathvariant=\"normal\">H</mml:mi></mml:mrow><mml:mrow><mml:mo>*</mml:mo></mml:mrow></mml:msup></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ7\"><label>7</label><alternatives><tex-math id=\"M31\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\Delta {{{{{\\rm{E}}}}}}_{{{{{\\rm{OO}}}}}{{{{{\\rm{H}}}}}}^{*}}={{{{\\rm{E}}}}}({{{{\\rm{OO}}}}}{{{{{\\rm{H}}}}}}^{*})-{{{{\\rm{E}}}}}(*)-\\left(2{{{{{\\rm{E}}}}}}_{{{{{{\\rm{H}}}}}}_{2}{{{{\\rm{O}}}}}}-3/2{{{{{\\rm{E}}}}}}_{{{{{{\\rm{H}}}}}}_{2}}\\right)$$\\end{document}</tex-math><mml:math id=\"M32\"><mml:mi mathvariant=\"normal\">Δ</mml:mi><mml:msub><mml:mrow><mml:mi mathvariant=\"normal\">E</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">OO</mml:mi><mml:msup><mml:mrow><mml:mi mathvariant=\"normal\">H</mml:mi></mml:mrow><mml:mrow><mml:mo>*</mml:mo></mml:mrow></mml:msup></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mi mathvariant=\"normal\">E</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi mathvariant=\"normal\">OO</mml:mi><mml:msup><mml:mrow><mml:mi mathvariant=\"normal\">H</mml:mi></mml:mrow><mml:mrow><mml:mo>*</mml:mo></mml:mrow></mml:msup></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>−</mml:mo><mml:mi mathvariant=\"normal\">E</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mo>*</mml:mo></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>−</mml:mo><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:mn>2</mml:mn><mml:msub><mml:mrow><mml:mi mathvariant=\"normal\">E</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi mathvariant=\"normal\">H</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mi mathvariant=\"normal\">O</mml:mi></mml:mrow></mml:msub><mml:mo>−</mml:mo><mml:mn>3</mml:mn><mml:mo>/</mml:mo><mml:mn>2</mml:mn><mml:msub><mml:mrow><mml:mi mathvariant=\"normal\">E</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi mathvariant=\"normal\">H</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:msub></mml:mrow></mml:mfenced></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ8\"><label>8</label><alternatives><tex-math id=\"M33\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\Delta {{{{{\\rm{E}}}}}}_{{{{{{\\rm{O}}}}}}^{*}}={{{{\\rm{E}}}}}({{{{{\\rm{O}}}}}}^{*})-{{{{\\rm{E}}}}}(*)-\\left({{{{{\\rm{E}}}}}}_{{{{{{\\rm{H}}}}}}_{2}{{{{\\rm{O}}}}}}-{{{{{\\rm{E}}}}}}_{{{{{{\\rm{H}}}}}}_{2}}\\right)$$\\end{document}</tex-math><mml:math id=\"M34\"><mml:mi mathvariant=\"normal\">Δ</mml:mi><mml:msub><mml:mrow><mml:mi mathvariant=\"normal\">E</mml:mi></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mi mathvariant=\"normal\">O</mml:mi></mml:mrow><mml:mrow><mml:mo>*</mml:mo></mml:mrow></mml:msup></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mi mathvariant=\"normal\">E</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msup><mml:mrow><mml:mi mathvariant=\"normal\">O</mml:mi></mml:mrow><mml:mrow><mml:mo>*</mml:mo></mml:mrow></mml:msup></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>−</mml:mo><mml:mi mathvariant=\"normal\">E</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mo>*</mml:mo></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>−</mml:mo><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:msub><mml:mrow><mml:mi mathvariant=\"normal\">E</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi mathvariant=\"normal\">H</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mi mathvariant=\"normal\">O</mml:mi></mml:mrow></mml:msub><mml:mo>−</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant=\"normal\">E</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi mathvariant=\"normal\">H</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:msub></mml:mrow></mml:mfenced></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ9\"><label>9</label><alternatives><tex-math id=\"M35\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\Delta {{{{{\\rm{E}}}}}}_{{{{{\\rm{O}}}}}{{{{{\\rm{H}}}}}}^{*}}={{{{\\rm{E}}}}}({{{{\\rm{O}}}}}{{{{{\\rm{H}}}}}}^{*})-{{{{\\rm{E}}}}}(*)-\\left({{{{{\\rm{E}}}}}}_{{{{{{\\rm{H}}}}}}_{2}{{{{\\rm{O}}}}}}-1/2{{{{{\\rm{E}}}}}}_{{{{{{\\rm{H}}}}}}_{2}}\\right)$$\\end{document}</tex-math><mml:math id=\"M36\"><mml:mi mathvariant=\"normal\">Δ</mml:mi><mml:msub><mml:mrow><mml:mi mathvariant=\"normal\">E</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">O</mml:mi><mml:msup><mml:mrow><mml:mi mathvariant=\"normal\">H</mml:mi></mml:mrow><mml:mrow><mml:mo>*</mml:mo></mml:mrow></mml:msup></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mi mathvariant=\"normal\">E</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi mathvariant=\"normal\">O</mml:mi><mml:msup><mml:mrow><mml:mi mathvariant=\"normal\">H</mml:mi></mml:mrow><mml:mrow><mml:mo>*</mml:mo></mml:mrow></mml:msup></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>−</mml:mo><mml:mi mathvariant=\"normal\">E</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mo>*</mml:mo></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>−</mml:mo><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:msub><mml:mrow><mml:mi mathvariant=\"normal\">E</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi mathvariant=\"normal\">H</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mi mathvariant=\"normal\">O</mml:mi></mml:mrow></mml:msub><mml:mo>−</mml:mo><mml:mn>1</mml:mn><mml:mo>/</mml:mo><mml:mn>2</mml:mn><mml:msub><mml:mrow><mml:mi mathvariant=\"normal\">E</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi mathvariant=\"normal\">H</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:msub></mml:mrow></mml:mfenced></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq10\"><alternatives><tex-math id=\"M37\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${{{{\\rm{E}}}}}(*)$$\\end{document}</tex-math><mml:math id=\"M38\"><mml:mi mathvariant=\"normal\">E</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mo>*</mml:mo></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq11\"><alternatives><tex-math id=\"M39\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${{{{\\rm{E}}}}}({{{{\\rm{OO}}}}}{{{{{\\rm{H}}}}}}^{*})$$\\end{document}</tex-math><mml:math id=\"M40\"><mml:mi mathvariant=\"normal\">E</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi mathvariant=\"normal\">OO</mml:mi><mml:msup><mml:mrow><mml:mi mathvariant=\"normal\">H</mml:mi></mml:mrow><mml:mrow><mml:mo>*</mml:mo></mml:mrow></mml:msup></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq12\"><alternatives><tex-math id=\"M41\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${{{{\\rm{E}}}}}({{{{{\\rm{O}}}}}}^{*})$$\\end{document}</tex-math><mml:math id=\"M42\"><mml:mi mathvariant=\"normal\">E</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msup><mml:mrow><mml:mi mathvariant=\"normal\">O</mml:mi></mml:mrow><mml:mrow><mml:mo>*</mml:mo></mml:mrow></mml:msup></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq13\"><alternatives><tex-math id=\"M43\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${{{{\\rm{E}}}}}({{{{\\rm{O}}}}}{{{{{\\rm{H}}}}}}^{*})$$\\end{document}</tex-math><mml:math id=\"M44\"><mml:mi mathvariant=\"normal\">E</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi mathvariant=\"normal\">O</mml:mi><mml:msup><mml:mrow><mml:mi mathvariant=\"normal\">H</mml:mi></mml:mrow><mml:mrow><mml:mo>*</mml:mo></mml:mrow></mml:msup></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq14\"><alternatives><tex-math id=\"M45\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${{{{\\rm{OO}}}}}{{{{{\\rm{H}}}}}}^{*}$$\\end{document}</tex-math><mml:math id=\"M46\"><mml:mi mathvariant=\"normal\">OO</mml:mi><mml:msup><mml:mrow><mml:mi mathvariant=\"normal\">H</mml:mi></mml:mrow><mml:mrow><mml:mo>*</mml:mo></mml:mrow></mml:msup></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq15\"><alternatives><tex-math id=\"M47\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${{{{{\\rm{O}}}}}}^{*}$$\\end{document}</tex-math><mml:math id=\"M48\"><mml:msup><mml:mrow><mml:mi mathvariant=\"normal\">O</mml:mi></mml:mrow><mml:mrow><mml:mo>*</mml:mo></mml:mrow></mml:msup></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq16\"><alternatives><tex-math id=\"M49\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${{{{\\rm{O}}}}}{{{{{\\rm{H}}}}}}^{*}$$\\end{document}</tex-math><mml:math id=\"M50\"><mml:mi mathvariant=\"normal\">O</mml:mi><mml:msup><mml:mrow><mml:mi mathvariant=\"normal\">H</mml:mi></mml:mrow><mml:mrow><mml:mo>*</mml:mo></mml:mrow></mml:msup></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq17\"><alternatives><tex-math id=\"M51\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${{{{{\\rm{E}}}}}}_{{{{{{\\rm{H}}}}}}_{2}{{{{\\rm{O}}}}}}$$\\end{document}</tex-math><mml:math id=\"M52\"><mml:msub><mml:mrow><mml:mi mathvariant=\"normal\">E</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi mathvariant=\"normal\">H</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mi mathvariant=\"normal\">O</mml:mi></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq18\"><alternatives><tex-math id=\"M53\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${{{{{\\rm{E}}}}}}_{{{{{{\\rm{H}}}}}}_{2}}$$\\end{document}</tex-math><mml:math id=\"M54\"><mml:msub><mml:mrow><mml:mi mathvariant=\"normal\">E</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi mathvariant=\"normal\">H</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ10\"><label>10</label><alternatives><tex-math id=\"M55\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\Delta {{{{{{\\rm{G}}}}}}}_{{{{{{\\rm{ads}}}}}}}=\\Delta {{{{{{\\rm{E}}}}}}}_{{{{{{\\rm{ads}}}}}}}+\\Delta {{{{{\\rm{ZPE}}}}}}-{{{{{\\rm{T}}}}}}\\Delta {{{{{\\rm{S}}}}}}+{{{{{\\rm{eU}}}}}}$$\\end{document}</tex-math><mml:math id=\"M56\"><mml:mi mathvariant=\"normal\">Δ</mml:mi><mml:msub><mml:mrow><mml:mi mathvariant=\"normal\">G</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">ads</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mi mathvariant=\"normal\">Δ</mml:mi><mml:msub><mml:mrow><mml:mi mathvariant=\"normal\">E</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">ads</mml:mi></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:mi mathvariant=\"normal\">Δ</mml:mi><mml:mi mathvariant=\"normal\">ZPE</mml:mi><mml:mo>−</mml:mo><mml:mi mathvariant=\"normal\">T</mml:mi><mml:mi mathvariant=\"normal\">Δ</mml:mi><mml:mi mathvariant=\"normal\">S</mml:mi><mml:mo>+</mml:mo><mml:mi mathvariant=\"normal\">eU</mml:mi></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ11\"><label>11</label><alternatives><tex-math id=\"M57\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\Delta {{{{{\\rm{G}}}}}}_{1}=\\Delta {{{{{\\rm{G}}}}}}_{{{{{\\rm{OO}}}}}{{{{{\\rm{H}}}}}}^{*}}-4.92$$\\end{document}</tex-math><mml:math id=\"M58\"><mml:mi mathvariant=\"normal\">Δ</mml:mi><mml:msub><mml:mrow><mml:mi mathvariant=\"normal\">G</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mi mathvariant=\"normal\">Δ</mml:mi><mml:msub><mml:mrow><mml:mi mathvariant=\"normal\">G</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">OO</mml:mi><mml:msup><mml:mrow><mml:mi mathvariant=\"normal\">H</mml:mi></mml:mrow><mml:mrow><mml:mo>*</mml:mo></mml:mrow></mml:msup></mml:mrow></mml:msub><mml:mo>−</mml:mo><mml:mn>4.92</mml:mn></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ12\"><label>12</label><alternatives><tex-math id=\"M59\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\Delta {{{{{\\rm{G}}}}}}_{2}=\\Delta {{{{{\\rm{G}}}}}}_{{{{{{\\rm{O}}}}}}^{*}}-\\Delta {{{{{\\rm{G}}}}}}_{{{{{\\rm{OO}}}}}{{{{{\\rm{H}}}}}}^{*}}$$\\end{document}</tex-math><mml:math id=\"M60\"><mml:mi mathvariant=\"normal\">Δ</mml:mi><mml:msub><mml:mrow><mml:mi mathvariant=\"normal\">G</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mi mathvariant=\"normal\">Δ</mml:mi><mml:msub><mml:mrow><mml:mi mathvariant=\"normal\">G</mml:mi></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mi mathvariant=\"normal\">O</mml:mi></mml:mrow><mml:mrow><mml:mo>*</mml:mo></mml:mrow></mml:msup></mml:mrow></mml:msub><mml:mo>−</mml:mo><mml:mi mathvariant=\"normal\">Δ</mml:mi><mml:msub><mml:mrow><mml:mi mathvariant=\"normal\">G</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">OO</mml:mi><mml:msup><mml:mrow><mml:mi mathvariant=\"normal\">H</mml:mi></mml:mrow><mml:mrow><mml:mo>*</mml:mo></mml:mrow></mml:msup></mml:mrow></mml:msub></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ13\"><label>13</label><alternatives><tex-math id=\"M61\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\Delta {{{{{\\rm{G}}}}}}_{3}=\\Delta {{{{{\\rm{G}}}}}}_{{{{{\\rm{O}}}}}{{{{{\\rm{H}}}}}}^{*}}-\\Delta {{{{{\\rm{G}}}}}}_{{{{{{\\rm{O}}}}}}^{*}}$$\\end{document}</tex-math><mml:math id=\"M62\"><mml:mi mathvariant=\"normal\">Δ</mml:mi><mml:msub><mml:mrow><mml:mi mathvariant=\"normal\">G</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mi mathvariant=\"normal\">Δ</mml:mi><mml:msub><mml:mrow><mml:mi mathvariant=\"normal\">G</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">O</mml:mi><mml:msup><mml:mrow><mml:mi mathvariant=\"normal\">H</mml:mi></mml:mrow><mml:mrow><mml:mo>*</mml:mo></mml:mrow></mml:msup></mml:mrow></mml:msub><mml:mo>−</mml:mo><mml:mi mathvariant=\"normal\">Δ</mml:mi><mml:msub><mml:mrow><mml:mi mathvariant=\"normal\">G</mml:mi></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mi mathvariant=\"normal\">O</mml:mi></mml:mrow><mml:mrow><mml:mo>*</mml:mo></mml:mrow></mml:msup></mml:mrow></mml:msub></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ14\"><label>14</label><alternatives><tex-math id=\"M63\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\Delta {{{{{\\rm{G}}}}}}_{4}=-\\Delta {{{{{\\rm{G}}}}}}_{{{{{\\rm{O}}}}}{{{{{\\rm{H}}}}}}^{*}}$$\\end{document}</tex-math><mml:math id=\"M64\"><mml:mi mathvariant=\"normal\">Δ</mml:mi><mml:msub><mml:mrow><mml:mi mathvariant=\"normal\">G</mml:mi></mml:mrow><mml:mrow><mml:mn>4</mml:mn></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mo>−</mml:mo><mml:mi mathvariant=\"normal\">Δ</mml:mi><mml:msub><mml:mrow><mml:mi mathvariant=\"normal\">G</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">O</mml:mi><mml:msup><mml:mrow><mml:mi mathvariant=\"normal\">H</mml:mi></mml:mrow><mml:mrow><mml:mo>*</mml:mo></mml:mrow></mml:msup></mml:mrow></mml:msub></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq19\"><alternatives><tex-math id=\"M65\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${{{{{\\rm{U}}}}}}_{{{{{\\rm{RHE}}}}}}^{{{{{\\rm{onset}}}}}}$$\\end{document}</tex-math><mml:math id=\"M66\"><mml:msubsup><mml:mrow><mml:mi mathvariant=\"normal\">U</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">RHE</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">onset</mml:mi></mml:mrow></mml:msubsup></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ15\"><label>15</label><alternatives><tex-math id=\"M67\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${{{{{\\rm{U}}}}}}_{{{{{\\rm{RHE}}}}}}^{{{{{\\rm{onset}}}}}}=-\\max \\left\\{\\Delta {{{{{\\rm{G}}}}}}_{1},\\Delta {{{{{\\rm{G}}}}}}_{2},\\Delta {{{{{\\rm{G}}}}}}_{3},\\Delta {{{{{\\rm{G}}}}}}_{4}\\right\\}$$\\end{document}</tex-math><mml:math id=\"M68\"><mml:msubsup><mml:mrow><mml:mi mathvariant=\"normal\">U</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">RHE</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">onset</mml:mi></mml:mrow></mml:msubsup><mml:mo>=</mml:mo><mml:mo>−</mml:mo><mml:mi>max</mml:mi><mml:mfenced close=\"}\" open=\"{\"><mml:mrow><mml:mi mathvariant=\"normal\">Δ</mml:mi><mml:msub><mml:mrow><mml:mi mathvariant=\"normal\">G</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:mi mathvariant=\"normal\">Δ</mml:mi><mml:msub><mml:mrow><mml:mi mathvariant=\"normal\">G</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:mi mathvariant=\"normal\">Δ</mml:mi><mml:msub><mml:mrow><mml:mi mathvariant=\"normal\">G</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:mi mathvariant=\"normal\">Δ</mml:mi><mml:msub><mml:mrow><mml:mi mathvariant=\"normal\">G</mml:mi></mml:mrow><mml:mrow><mml:mn>4</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:mfenced></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ16\"><label>16</label><alternatives><tex-math id=\"M69\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\eta }^{{{{{\\rm{ORR}}}}}}=1.23{{{{\\rm{V}}}}}-{{{{{\\rm{U}}}}}}_{{{{{\\rm{RHE}}}}}}^{{{{{\\rm{onset}}}}}}/{{{{\\rm{e}}}}}$$\\end{document}</tex-math><mml:math id=\"M70\"><mml:msup><mml:mrow><mml:mi>η</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">ORR</mml:mi></mml:mrow></mml:msup><mml:mo>=</mml:mo><mml:mn>1.23</mml:mn><mml:mi mathvariant=\"normal\">V</mml:mi><mml:mo>−</mml:mo><mml:msubsup><mml:mrow><mml:mi mathvariant=\"normal\">U</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">RHE</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">onset</mml:mi></mml:mrow></mml:msubsup><mml:mo>/</mml:mo><mml:mi mathvariant=\"normal\">e</mml:mi></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ17\"><label>17</label><alternatives><tex-math id=\"M71\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${{{{{\\rm{E}}}}}}_{{{{{{{\\rm{V}}}}}}_{{{{{\\rm{P}}}}}}}_{{{{{\\rm{t}}}}}}}={{{{{{{\\rm{E}}}}}}_{{{{{\\rm{slab}}}}}}}_{-}}_{{{{{{{\\rm{V}}}}}}_{{{{{\\rm{P}}}}}}}_{{{{{\\rm{t}}}}}}}+{\\mu }_{{{{{\\rm{Pt}}}}}}-{{{{{\\rm{E}}}}}}_{{{{{\\rm{slab}}}}}}$$\\end{document}</tex-math><mml:math id=\"M72\"><mml:msub><mml:mrow><mml:mi mathvariant=\"normal\">E</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:msub><mml:mrow><mml:mi mathvariant=\"normal\">V</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">P</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">t</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:msub><mml:mrow><mml:msub><mml:mrow><mml:msub><mml:mrow><mml:mi mathvariant=\"normal\">E</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">slab</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:mo>−</mml:mo></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:msub><mml:mrow><mml:mi mathvariant=\"normal\">V</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">P</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">t</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mi>μ</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">Pt</mml:mi></mml:mrow></mml:msub><mml:mo>−</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant=\"normal\">E</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">slab</mml:mi></mml:mrow></mml:msub></mml:math></alternatives></disp-formula>" ]

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[ "<supplementary-material content-type=\"local-data\" id=\"MOESM1\"></supplementary-material>", "<supplementary-material content-type=\"local-data\" id=\"MOESM2\"></supplementary-material>", "<supplementary-material content-type=\"local-data\" id=\"MOESM3\"></supplementary-material>" ]

[ "<fn-group><fn><p><bold>Publisher’s note</bold> Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.</p></fn><fn><p>These authors contributed equally: Lei Gao, Tulai Sun, Xuli Chen.</p></fn></fn-group>" ]

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[ "<media xlink:href=\"41467_2024_44788_MOESM1_ESM.pdf\"><caption><p>Supplementary Information</p></caption></media>", "<media xlink:href=\"41467_2024_44788_MOESM2_ESM.pdf\"><caption><p>Peer Review File</p></caption></media>", "<media xlink:href=\"41467_2024_44788_MOESM3_ESM.xlsx\"><caption><p>Source Data</p></caption></media>" ]

{ "acronym": [], "definition": [] }

[ "<title>Introduction</title>", "<p id=\"Par2\">Abiotic factors play a pivotal role in the decline of agricultural and allied sector production. Climate change, pollution, and degraded water quality stand out as major abiotic influencers shaping the life patterns of aquatic organisms, including fish^{##UREF##0##1##,##UREF##1##2##}. Ammonia holds a crucial position in the nitrogen cycle, undergoing conversion into nitrite (NO₂) by Nitrosospira and Nitrosomonas bacteria in aquatic systems through the nitrification process. Additionally, it originates from fish waste, high-protein diets, and metabolic processes, contributing to the presence of ammonia in aquatic systems^{##REF##21423375##3##,##UREF##2##4##}. The breakdown of amino acids, pyrimidines, and purines also generates ammonia^{##UREF##3##5##}, existing in two forms: unionized ammonia (NH₃) and ionized ammonium (NH₄⁺)^{##REF##12398363##6##}. Ammonia toxicity can lead to noticeable reductions in growth performance^{##UREF##1##2##}, immunity, tissue erosion, neurotoxicity, oxidative stress, and ultimately result in high mortality^{##REF##24657724##7##}.</p>", "<p id=\"Par3\">Naturally occurring arsenic, typically harmless in its natural state, can undergo transformation into inorganic arsenic, thereby contaminating groundwater sources used for drinking or irrigating crops. The accumulation of arsenic from one trophic level to another level, depends not only on the total arsenic content but also significantly on its bioavailability^{##UREF##4##8##}. The chemical forms of arsenic, such as inorganic and organic forms present in crops, vegetables, and fish, play a crucial role in determining bioavailability, which is essential for estimating its toxicity. Humans can also uptake arsenic from contaminated sources such as rice, vegetables, milk, and meat. Consequently, 'plant–human' and 'plant–animal–human' represent potential food chain pathways for arsenic accumulation^{##UREF##5##9##,##REF##14987870##10##}.</p>", "<p id=\"Par4\">It is also considered to be consumption of even low dose of arsenic can cause deadly diseases, including cancer^{##UREF##6##11##,##REF##15613666##12##}. In all around the world, such as in India, Bangladesh, Argentina, China, Ghana, USA, and Vietnam, more than 200 million peoples are at high risk^{##REF##32022256##13##,##UREF##7##14##}. Further, in Bangladesh, 43,000 peoples die annually due to arsenic pollution^{##UREF##7##14##}.</p>", "<p id=\"Par5\">Fish are classified as poikilothermic animals; however, even with slight temperature variations, their physiology undergoes abrupt changes. These changes include alterations in growth, metabolism, food consumption, thermal tolerance, and an inability to maintain internal homeostasis in response to the fluctuating external environment^{##UREF##8##15##,##UREF##9##16##}. Moreover, elevated temperatures diminish the availability of oxygen to aquatic animals, creating challenges in meeting metabolic demands, especially as the water flow rate increases across the gills^{##UREF##0##1##}.</p>", "<p id=\"Par6\">Interestingly, manganese (Mn) plays a vital role as an essential micronutrient in the growth and development of the vertebral column, serving as an antioxidant and acting as a cofactor for numerous enzymes^{##UREF##10##17##}. Typically, the requirement for Mn is met through waterborne sources, but additional supplementation is necessary to fulfil the physiological needs of the fish^{##UREF##11##18##,##UREF##12##19##}. Therefore, Mn supplements are provided to meet the physiological requirements and metabolic scope of the fish. A deficiency in Mn for fish can lead to retarded growth performance, skeletal deformities (dwarfism), eye lens cataracts, decreased activities of copper-zinc superoxide dismutase (Cu–Zn-SOD), manganese superoxide dismutase complex (Mn-SOD), and reduced reproductive performance^{##UREF##12##19##–##UREF##14##21##}. Mn is primarily located in the mitochondria and plays a crucial role in activating several enzymes, including decarboxylases, kinases, hydrolases, and transferases. Key manganese metallo-enzymes, such as pyruvate carboxylase, catalyze the conversion of pyruvate to oxaloacetate^{##REF##5919680##22##}.</p>", "<p id=\"Par7\">Apoptosis is a programmed cell death crucial for regular cell repair, cellular function, immune and hormone-related gene development, and chemical cell death in all organisms, including fish^{##REF##17562483##23##}. Cytokines, serving as essential signaling molecules, are released during both physiological and pathological conditions. They play a role in stress responses and modulate the host's inflammatory response and immunobiological mechanisms^{##UREF##15##24##,##UREF##16##25##}. Furthermore, NF-kB regulates and controls the transcription of genes related to immune cells, inflammation, proliferation, the cell cycle, and cell death^{##REF##26999213##26##}.</p>", "<p id=\"Par8\"><italic>Pangasianodon hypophthalmus</italic> exhibits great potential as a fish species suitable for cultivation in challenging conditions, displaying tolerance to high abiotic and biotic stress^{##UREF##9##16##,##REF##33414495##27##,##REF##29332272##28##}. Moreover, it is an ideal species for studying gene regulation involved in both abiotic and biotic stress^{##UREF##1##2##}. Consequently, the present investigation aims to study the role of manganese in mitigating arsenic and ammonia toxicity, as well as high-temperature stress. This study also explores gene regulation associated with abiotic and biotic stress in response to dietary manganese in <italic>P. hypophthalmus</italic>.</p>" ]

[ "<title>Materials and methods</title>", "<title>Ethics statement</title>", "<p id=\"Par9\">The Research Advisory Committee (RAC) of the Institute (ICAR-National Institute of Abiotic Stress Management, Baramati, Pune) has approved the experimental procedures, and this study adheres to the Animal Research: Reporting of In Vivo Experiments (ARRIVE) guidelines. All methods were conducted in strict accordance with the relevant guidelines and regulations.</p>", "<title>Experimental animal and design</title>", "<p id=\"Par10\"><italic>Pangasianodon hypophthalmus</italic> specimens with an average weight of 6.71 ± 0.52 g and a length of 5.12 ± 0.17 cm was utilized in the current investigation. These fish were sourced from the NIASM farm pond and were in a healthy condition. At Prior to the commencement of the experiment, a two-week acclimatization period was provided to the fish in a Fiberglass Reinforced Plastic (FRP) tank. During this acclimatization period, the fish received regular feeding and other necessary maintenance. Subsequently, the eighteen fish were evenly distributed in a plastic rectangular tank with a capacity of 150 L. The experiment was designed with 12 treatments, each replicated in triplicate, employing a Completely Randomized Design (CRD). The treatments followed as 1. Control 2. As exposed group 3. Ammonia exposed group 4. Concurrent exposure to arsenic and ammonia group 5. Concurrent exposure to ammonia and high temperature group 6. Concurrent exposure to arsenic, ammonia and high temperature group 7. Group fed with Mn at 4 mg kg⁻¹ diet 8. Group fed with Mn at 8 mg kg⁻¹ diet 9. Group fed with Mn at 12 mg kg⁻¹ diet 10. Group fed with Mn at 4 mg kg⁻¹ diet and concurrently exposed to arsenic, ammonia and high temperature 11. Group fed with Mn at 8 mg kg⁻¹ diet and concurrently exposed to arsenic, ammonia and high temperature 12. Group fed with Mn at 12 mg kg⁻¹ diet and concurrently exposed to arsenic, ammonia and high temperature. The details of the treatment is shown in Table ##TAB##0##1##. The experimental diets were administered twice daily to the fish at 9:00 AM and 5:00 PM. Continuous aeration was maintained throughout the experiment using an aerator. Daily removal of uneaten feed and faecal matter was carried out through siphoning. Periodic analysis of water quality parameters was conducted using the APHA method^{##UREF##17##29##}, and the results consistently fell within acceptable ranges throughout the experiment. Every alternate day, 2/3rd of the water in the tank was manually replaced. Additionally, (NH₄)₂SO₄ was added as a source of ammonia toxicity (NH₃), and sodium arsenite, NaAsO₂, was introduced as a source of arsenic. The concentrations used were Ammonium sulfate (1/10th of LC₅₀, 2.0 mg L⁻¹ of (NH₄)₂SO₄)² and As (1/10th of LC₅₀, 2.68 mg L⁻¹ of arsenic)¹. The water temperature was maintained at a high level (34 °C) throughout the experiment to induce stress. Four iso-caloric (365 kcal/100 g) and iso-nitrogenous (35% crude protein) pelleted diets containing manganese were prepared. The feed ingredients included wheat flour, groundnut meal, soybean meal, and fish meal. Cod liver oil, lecithin, vitamin C, and other labile nutrients were added after heating the feed ingredients. A manganese-free mineral mixture was manually prepared. Proximate analysis was conducted using the AOAC method^{##UREF##18##30##}, while ether extract (EE) was determined through solvent extraction, crude protein by nitrogen content, and ash content by using a muffle furnace at 550 °C. Total carbohydrate content was calculated using the formula 100—(CP% + EE% + Ash %+moisture). Gross energy was determined using the Halver method^{##UREF##19##31##} (Table ##TAB##1##2##).</p>", "<title>Tissue homogenate preparation and blood collection</title>", "<p id=\"Par11\">The gill, muscle, brain, liver, and kidney were dissected from anesthetized fish (clove oil, 50 µl L⁻¹) under aseptic conditions. The chilled sucrose (5% w/v, 0.25 M) and EDTA solution (1 mM) were used as homogenates for tissue homogenization using a homogenizer (Omni Tissue Master Homogenize, Kennesaw, GA) for enzyme analysis. The gene expression and quantification, the liver and muscle tissues samples were processed with liquid nitrogen. For the enzymes analysis, the tissues were homogenized and centrifuged at 5,000 × g for 15 min at 4 °C to homogenated samples. The supernatants were collected and stored at -20 °C until further analysis. During dissection, the blood (3 fish) was also collected from the same fish of each tank and serum (5 fish) was processed from the same collected blood. Lowry protein assay^{##REF##14907713##32##} was used for tissue protein analysis.</p>", "<title>RNA isolation and quantification and cDNA synthesis and quantitative PCR</title>", "<p id=\"Par12\">The TRIzol method was employed for total RNA isolation from the liver tissue of <italic>P. hypophthalmus</italic>. Liquid nitrogen was utilized for the homogenization of the liver tissue. Subsequently, chloroform was added to the homogenized samples, and the mixture was incubated for 5 min to allow for phase separation. The resulting solution was then centrifuged to separate the RNA, followed by the addition of 75% ethanol and air drying. The RNA pellet was dissolved in free water and stored at -80 °C for future use. To assess RNA integrity, 1% agarose gel electrophoresis was conducted, and the RNA bands were visualized using a Gel documentation system (ChemiDocTM MP imaging system, Bio-Rad). RNA quantification was performed using a NanoDrop spectrophotometer (Thermo Scientific). Revert Aid First Strand cDNA synthesis kit (Thermo Scientific) was utilized. DNase I was employed to remove trace amounts of DNA. The reaction mixture, consisting of oligo dT primers (15 pmol) and RNA template (100 ng) in 12 µl, was heated for 5 min at 65 °C and then chilled on ice. Subsequently, 1.0 µl of reverse transcriptase enzyme, 2 µl dNTP Mix (10 mM), and 1 µl Ribo Lock RNase Inhibitor (20 U/µL) were added to the chilled mixture, followed by a brief centrifugation. The reaction mixture was incubated for 42 min at 60 °C, then at 70 °C for 5 min, and the synthesized cDNA was stored at -20 °C. β-actin was used as a reference for confirming the synthesized cDNA. Real-time PCR was conducted using SYBR green and gene-specific primers (Bio-Rad). The quantification protocol included an initial denaturation for 10 min at 95 °C, followed by 39 cycles of cDNA amplification, denaturation at 95 °C for 15 s, and annealing at 60 °C for 1 minute^{##REF##11328886##33##}. Details of the primers are recorded in Table ##TAB##2##3##.</p>", "<title>Genes</title>", "<p id=\"Par13\">The genes were investigated in liver tissues in this study viz. catalase (CAT), glutathione-s-transferase (GST), superoxide dismutase (SOD), nitric oxide synthase (iNOS), heat shock protein (HSP 70), Caspase 3a (CAS 3a and 3b), cytochrome P450 (CYP 450), tumor necrosis factor (TNFα), toll like receptor (TLR), metallothionine (MT), growth hormone receptor (Ghr1 and Ghrb), interleukin (IL), immunoglobulin (Ig), insulin like growth factor 1 and 2 (IGF1X1 and IGF1X2), somatostatin (SMT), myostatin (MYST), and growth hormone (GH), studied for real-time quantification.</p>", "<title>Antioxidant enzyme activities</title>", "<p id=\"Par14\">Superoxide dismutase (SOD) (EC 1.15.1.1) activities in different fish tissues were determined by Misra and Fridovich^{##REF##4623845##34##}. Catalase (EC 1.11.1.6) was determined as followed as a procedure of Takahara et al^{##UREF##20##35##}. The glutathione S-transferase (GST) (EC 2.5.1.18) was determined as per the procedure of Habing et al^{##REF##4436300##36##}. Glutathione peroxidase (GPx) (EC 1.11.1.9) activity was accomplished following the method of Paglia and Valentine^{##REF##6066618##37##}.</p>", "<title>Neurotransmitter enzyme activities</title>", "<p id=\"Par15\">Hestrin modified by Augustinsson^{##REF##18133390##38##} method was applied to determine the acetylcholine esterase activities (AChE) (EC. 3.1.1.7) in brain tissue.</p>", "<title>Lipid peroxidation (LPO) and Vitamin C</title>", "<p id=\"Par16\">Uchiyama and Mihara^{##REF##655387##39##} method was followed to determine the LPO in liver and kidney tissues. Similarly, Roe and Keuther^{##UREF##21##40##} used to determine the Vitamin C in brain and muscle tissues.</p>", "<title>Hematological parameters</title>", "<p id=\"Par17\">Blood was drawn from the caudal peduncle region of the fish using heparinised syringe. Indices measured included erythrocyte count (RBC), hemoglobin concentration (Hb), WBC (total leucocyte count) and the procedures were based on unified methods for hematological examination of fish.</p>", "<title>Immunological attributes</title>", "<p id=\"Par18\">Total serum protein, albumin, globulin, and A:G ratio was determined using the protein estimation kit. Secombes^{##UREF##22##41##}, with some modification by Stasiack and Baumann^{##UREF##23##42##} used for the estimation of respiratory burst activity. The blood glucose was determined using Nelson^{##UREF##24##43##} and Somoyogi^{##UREF##25##44##}. Moreover, Quade and Roth^{##REF##9436268##45##}, with some modifications^{##UREF##26##46##} and Anderson and Siwicki^{##UREF##27##47##} were applied for the determination of myeloperoxidase and total immunoglobulin.</p>", "<title>Cortisol</title>", "<p id=\"Par19\">Serum cortisol was determined using ELISA kit (Commercially available Cortisol EIA kit, catalogue no. 500360, Cayman Chemicals, USA). The assay was performed as per instruction provided with the kit using ELISA plate reader (Biotek India Pvt. Ltd.).</p>", "<title>Arsenic and manganese analysis from fish tissues and experimental water</title>", "<p id=\"Par20\">Liver, muscle, gill, brain, and kidney were collected to determine in arsenic concentration. Whereas, Mn concentration was determined in the feed and fish muscle. The tissues and diets were processed in a microwave digestion system (Microwave Reaction System, Multiwave PRO, Anton Paar GmbH, Austria, Europe) using Inductively Coupled Plasma Mass Spectrometry (ICP-MS) (Agilent 7700 series, Agilent Technologies, USA) as followed the method of Kumar et al.^{##REF##28027471##48##,##REF##27992807##49##}.</p>", "<title>Alkaline single-cell gel electrophoresis (SCGE)/Comet assay</title>", "<p id=\"Par21\">Alkaline single cell gel electrophoresis/comet assay was applied for determination of DNA damage in kidney tissue using Ali et al^{##REF##18359502##50##}. with slight modification^{##REF##36977939##51##}. The slides coating and other procedure were followed the above method. Then prepared slides for genotoxicity were analysed in fluorescent microscope (Leica Microsystems Ltd, DM 2000, Heerbrugg, Switzerland). The position of DNA damage was captured using the microscope and analyzed using Open comet. The parameter selected for quantification of DNA damage was percent tail DNA (i.e., % tail DNA = 100% head DNA) as determined by the software.</p>", "<title>Aspartate aminotransferase (AST) and alanine aminotransferase (ALT), Lactate dehydrogenase (LDH), and malate dehydrogenase (MDH)</title>", "<p id=\"Par22\">AST (E.C.2.6.1.1) and ALT (E.C.2.6.1.2) were determined using Wooten^{##UREF##28##52##} method. Similarly, LDH activities were determined using Wroblewski and Ladue^{##REF##13273400##53##}. Similarly, MDH was determined using Ochoa^{##UREF##29##54##}. A similar reaction mixture was used except for substrate oxaloacetate instead of sodium pyruvate.</p>", "<title>Growth performance</title>", "<p id=\"Par23\">The growth performance was determined by evaluating the following method. The sampling/weighing of the fish was observed by every 15 days up to 105 days.</p>", "<title>Challenge study with <italic>Aeromonas hydrophila</italic></title>", "<p id=\"Par24\">After 105 days of the feeding trial, 8 fishes per replicates in each treatment were challenged with virulent <italic>A. hydrophilla</italic> (Lot no. 637–51-5 and Ref 0637P, HiMedia, Mumbai). <italic>A. hydrophilla</italic> was grown on a nutrient broth for 24 h at 37 °C in a BOD incubator and harvested by centrifuging the culture broth at 10,000 × g for 10 min at 4 °C. The cells were then washed thrice in sterile PBS (pH 7.2), and the final concentration was maintained at 10⁸ CFU ml⁻¹. The fish were intraperitoneally injected with 0.15 ml of bacterial suspension in each treatment group. The fish mortality in each treatment group was recorded up to 7 days of challenge study. The tissues were dissected out from morbid fish for confirmation of <italic>A. hydrophilla</italic> as a causative agent for death.</p>", "<title>Statistics</title>", "<p id=\"Par25\">The data were analysed using Statistical Package for the Social Sciences (SPSS) version 16 software. The data were tested for normality and homogeneity of variance using Shapiro–Wilk’s and Levene's test and Shapiro–Wilk's test, respectively. One way ANOVA (analysis of variance) using Duncan’s multiple range tests were applied in the present study. The data were analysed and significant at <italic>p</italic> &lt; 0.05.</p>", "<title>Ethics approval</title>", "<p id=\"Par26\">The Institute (ICAR-National Institute of Abiotic Stress Management, Baramati, Pune) Research Advisory Committee (RAC) has approved the experimental procedures and this study compliance with Animal Research: Reporting of In Vivo Experiments (ARRIVE) guidelines.</p>", "<title>Consent to participate</title>", "<p id=\"Par27\">All authors are aware and agree with this submission for publication.</p>" ]

[ "<title>Results</title>", "<p id=\"Par28\">3.1. Effect of Mn on cortisol levels.</p>", "<p id=\"Par29\">In the present investigation, cortisol levels were assessed in response to dietary manganese (Mn) at 4, 8, and 12 mg kg⁻¹ diet fed to <italic>P. hypophthalmus</italic>. The fish were reared under normal conditions as well as under the arsenic and ammonia pollution, coupled with high-temperature stress, over a period of 105 days. The corresponding data are illustrated in Fig. ##FIG##0##1##A. Cortisol levels exhibited a noticeable increase (<italic>p</italic> = 0.0025) in the group subjected to concurrent exposure to arsenic, ammonia toxicity, and high-temperature stress, followed by the group exposed to arsenic and ammonia, when compared to the control and other groups. Furthermore, dietary manganese at 8 and 4 mg kg⁻¹ diet, with or without stressors (As + NH₃ + T), significantly reduced cortisol levels (<italic>p</italic> = 0.0025) compared to the control and other groups. However, manganese at 12 mg kg⁻¹ diet did not exhibit an inhibitory effect on cortisol levels in fish reared under both control and stressor conditions.</p>", "<title>Effect of Mn on Heat shock protein (HSP 70) and cytochrome P450 (CYP P450)</title>", "<p id=\"Par30\">The expression of the HSP70 gene in liver tissue exhibited a significant increase (<italic>p</italic> = 0.0017) under concurrent exposure to ammonia and arsenic toxicity, along with high-temperature stress. This was followed by the group exposed to ammonia and high temperature, then arsenic and ammonia, ammonia alone, and arsenic alone groups, as compared to the control and Mn-supplemented groups. This observation held true for fish reared both in control conditions and under multiple stressors (As + NH₃ + T). Interestingly, the group fed with a Mn-containing diet at 8 mg kg⁻¹ with stressors, followed by the same group without stressors, and then the Mn at 4 mg kg⁻¹ diet, exhibited a significant difference compared to the control and other groups (see Fig. ##FIG##0##1##B). Intriguingly, the expression of the CYP 450 gene in liver tissue was significantly upregulated (<italic>p</italic> = 0.0013) in response to a combination of different stressors (As + NH₃ + T, NH₃, As, As + NH₃, and NH₃ + T) compared to the control and Mn-supplemented groups (Mn at 4, 8, and 12 mg kg⁻¹ in the diet). Conversely, the supplementation of dietary Mn at 8 and 4 mg kg⁻¹, with or without stressors, resulted in a substantial downregulation of CYP 450 gene expression compared to the control and other groups (Fig. ##FIG##0##1##C).</p>", "<title>Effect of Mn on DNA damage-inducible protein (DDIP), DNA damage and metallothionine (MT)</title>", "<p id=\"Par31\">DNA damage inducible protein (DDIP) exhibited a noticeable upregulation (<italic>p</italic> = 0.0011) with concurrent exposure to ammonia, arsenic, and high-temperature stress. This was followed by the group exposed to ammonia and high temperature, and then the arsenic and ammonia, ammonia alone, and arsenic alone groups, in comparison to the control and other groups. Surprisingly, dietary Mn at 8 mg kg⁻¹ in the diet, with or without stressors, demonstrated the ability to substantially downregulated DDIP gene expression. This effect was followed by Mn at 4 mg kg⁻¹ in the diet, as compared to the control and other groups. However, Mn at 12 mg kg⁻¹ in the diet was not as effective in modulating DDIP gene expression against multiple stresses (Fig. ##FIG##0##1##C). Similarly, this study determined DNA damage in terms of tail DNA %, head DNA %, comet length, comet DNA, and head area, with the data recorded in Table ##TAB##3##4##. The tail DNA % was significantly highest in the group exposed to As + NH₃ + T, NH₃ + T, As + NH₃, NH₃, followed by the As alone group, compared to the control and Mn-supplemented groups. Further, the noticeably least tail DNA % was determined in the group fed with Mn at 4, 8, and 12 mg kg⁻¹ in the diet without stressors, followed by the same feeding group but with stressors. Similarly, the results of head DNA % were inverse to tail DNA %. Interestingly, the expression of the metallothionein (MT) gene was substantially downregulated (<italic>p</italic> = 0.0002) with dietary Mn at 8 mg kg⁻¹ in the diet, followed by Mn at 4 mg kg⁻¹ in the diet, with or without stressors, in comparison to the control and other groups. However, concurrent exposure to As, NH₃, and high temperature noticeably upregulated MT gene expression compared to arsenic and ammonia alone groups, in comparison to the control and Mn-supplemented groups (Fig. ##FIG##0##1##C).</p>", "<title>Effect of Mn on caspase 3a and 3b (Cas 3a and 3b)</title>", "<p id=\"Par32\">The gene regulation of caspase 3a and 3b (Cas 3a and 3b) in liver tissue was significantly upregulated by concurrent exposure to ammonia, arsenic, and high-temperature stress. This upregulation was followed by the arsenic and ammonia alone group compared to the control and other treatments. Furthermore, the supplementation of Mn at 8 mg kg⁻¹ diet, followed by Mn at 4 mg kg⁻¹ diet, noticeably downregulated the Cas 3a (<italic>p</italic> = 0.0052) and 3b (<italic>p</italic> = 0.0003) gene regulations compared to the control and stressors group. However, Mn at 12 mg kg⁻¹ in the diet was not effective for the gene regulation of Cas 3a and 3b (Fig. ##FIG##1##2##A).</p>", "<title>Effect of Mn on tumor necrosis factor (TNFα) and immunoglobulin (Ig), (TLR), and interleukin (IL) gene regulation</title>", "<p id=\"Par33\">In the present investigation, the gene regulation of tumor necrosis factor (TNFα) and immunoglobulin (Ig) is presented in Fig. ##FIG##1##2##B. The gene regulation of TNFα was significantly downregulated (<italic>p</italic> = 0.0006) with the supplementation of dietary Mn at 8 mg kg⁻¹ diet, with or without stressors. In contrast, TNFα was significantly upregulated, compared to the control and other groups, by concurrent exposure to As + NH₃ + T, As, NH₃, As + NH₃, and As + T groups. Moreover, the Ig gene expression was noticeably upregulated (<italic>p</italic> = 0.0012) by the supplementation of Mn at 8 mg kg⁻¹ diet, compared to other Mn-supplemented, control, and stressors groups. Further, the Ig gene expression was significantly downregulated with concurrent exposure to As + NH₃ + T and NH₃ + T, followed by As + NH₃, NH₃, and As alone groups, compared to the control and other groups. The toll-like receptors (TLR) (<italic>p</italic> = 0.0046) and interleukin (IL) (0.0029) were substantially upregulated by concurrent exposure to ammonia, arsenic, and high temperature, followed by NH₃ + T, As + NH₃, NH₃, and As alone, in comparison to the control and Mn-supplemented groups. Furthermore, the gene regulation of TLR and IL was noticeably downregulated by Mn at 8 mg kg⁻¹ diet, compared to the control and other groups (Fig. ##FIG##1##2##C).</p>", "<p id=\"Par34\">3.6. Effect of Mn on catalase (CAT), superoxide dismutase (SOD), glutathione-s-transferase (GST) and glutathione peroxidase of biochemical activities and gene expressions. This is subtitle caption, Please make the subtitled like others.</p>", "<p id=\"Par35\">In the present investigation, the activities of anti-oxidative enzymes such as catalase (CAT), superoxide dismutase (SOD), glutathione-s-transferase (GST), and glutathione peroxidase were determined in the liver and gill tissues of <italic>P. hypophthalmus</italic> reared under arsenic and ammonia toxicity, along with high-temperature stress. The corresponding data are recorded in Table ##TAB##4##5##. CAT, GST, and GPx activities in the liver and gill were notably higher (<italic>p</italic> &lt; 0.01) in the group concurrently exposed to As + NH₃ + T, followed by NH₃ + T, compared to the control and supplemented groups. Similarly, CAT, GPx, and GST activities in the liver and gill were also higher in the group exposed to arsenic and ammonia, compared to the control and Mn-supplemented groups (4 and 8 mg kg⁻¹ diet). Interestingly, the supplementation of dietary Mn at 8 mg kg⁻¹ diet, with or without stressors (As + NH₃ + T), noticeably reduced CAT, GPx, and GST activities compared to the control and other treatment groups. The supplemented group with Mn at 4 mg kg⁻¹ in the diet also effectively controlled CAT, GPx, and GST activities. Regarding SOD activities in the liver (<italic>p</italic> = 0.018) and gill (<italic>p</italic> = 0.037), they were significantly higher in the group treated with all stressors (As + NH₃ + T, NH₃ + T, As + NH₃, NH₃, and As) compared to the control and Mn-supplemented groups. The supplemented groups of Mn at 4, 8, and 12 mg kg⁻¹ diet exhibited SOD activities similar to the control group in both liver and gill tissues. Interestingly, the gene expression of CAT, SOD, and GPx was also quantified in the present investigation, and the related data are noted in Fig. ##FIG##2##3##A,B. The CAT (<italic>p</italic> = 0.002), SOD (<italic>p</italic> = 0.0061), and GPx (<italic>p</italic> = 0.014) gene expressions were substantially upregulated with concurrent exposure to arsenic, ammonia, and high-temperature stress, followed by other stressor groups, in comparison to the control and Mn-supplemented groups. Furthermore, the gene expressions of CAT, SOD, and GPx were noticeably downregulated by Mn at 8 mg kg⁻¹ diet with stressors (As + NH₃ + T), followed by the same diet group without stressors, Mn at 4 mg kg⁻¹ diet, compared to the control and other treatment groups.</p>", "<title>Effect of Mn on inducible nitric oxide synthase (iNOS) and Na⁺K⁺ATPase gene expression</title>", "<p id=\"Par36\">In the present investigation, the inducible nitric oxide synthase (iNOS) in liver tissue was quantified, and the results are presented in Fig. ##FIG##2##3##B. The iNOS gene expression was remarkably upregulated (<italic>p</italic> = 0.0036) by the As + NH₃ + T group, followed by arsenic alone, compared to the control and other groups. Exposure to NH₃, As + NH₃, and NH₃ + T groups showed similar iNOS gene expression levels. Dietary supplementation of Mn at 8 mg kg⁻¹ group noticeably downregulated iNOS gene expression, with or without stressors, followed by Mn at 4 mg kg⁻¹ diet, compared to the control and other treatments. Additionally, the Na⁺K⁺ATPase gene expression was quantified in the present investigation, and the data are presented in Fig. ##FIG##2##3##C. The gene expression of Na⁺K⁺ATPase was noticeably upregulated (<italic>p</italic> = 0.0022) by As + NH₃ and As + NH₃ + T, followed by As alone and NH₃ + T, compared to the control and other groups. Moreover, Na⁺K⁺ATPase was significantly downregulated with the supplementation of Mn at 8 mg kg⁻¹ diet, with or without stressors, compared to the control and other treatment groups.</p>", "<title>Effect of Mn on growth performance attributes and gene regulation</title>", "<p id=\"Par37\">In the present investigation, genes related to growth performance, such as growth hormone (GH), myostatin (MYST), somatostatin (SMT), growth hormone regulatory (GHR1 and GHRβ), and insulin-like growth factors (IGF1X1 and IGF1X2), were quantified, and the data are presented in Figs. ##FIG##2##3##C and ##FIG##3##4##A-C. GH was significantly upregulated (<italic>p</italic> = 0.0016) by the supplementation of Mn at 8 mg kg⁻¹ diet, with or without stressors, compared to the control and other treatment groups, including other Mn-supplemented diets. Moreover, GH gene regulation was noticeably downregulated by As + NH₃ + T, As + NH₃, and NH₃ + T, in comparison to the control and supplemented groups. Furthermore, Mn at 4 and 12 mg kg⁻¹ in the diet did not effectively regulate GH gene expression (Fig. ##FIG##2##3##C). On the other hand, MYST (<italic>p</italic> = 0.0023) and SMT (<italic>p</italic> = 0.0042) gene regulations were remarkably upregulated by concurrent exposure to the As + NH₃ + T stress group, in comparison to the control and other treatment groups. Moreover, dietary supplementation of Mn at 8 mg kg⁻¹ diet significantly downregulated MYST and SMT in the liver tissue of <italic>P. hypophthalmus</italic>, compared to the control and other treatment groups (Fig. ##FIG##3##4##A). Surprisingly, GHR1 (<italic>p</italic> = 0.0027), GHRβ (<italic>p</italic> = 0.0033), IGF1X1 (<italic>p</italic> = 0.015), and IGF1X2 (<italic>p</italic> = 0.0072) genes were substantially upregulated with the supplementation of Mn at 8 mg kg⁻¹ in the diet, with or without stressors (As + NH₃ + T), followed by Mn at 4 mg kg⁻¹ in the diet, in comparison to the control and other treatment groups. In contrast, GHR1, GHRβ, IGF1X1, and IGF1X2 genes were significantly downregulated by stressors (As + NH₃ + T, As + NH₃, NH₃ + T, As, and NH₃), compared to the control and other treatment groups (Fig. ##FIG##3##4## B,C).</p>", "<p id=\"Par38\">In the present investigation, the growth performance indicators of <italic>P. hypophthalmus</italic> viz. final weight gain %, FCR, SGR, PER, DGI, TGC and RFI were presented in Table ##TAB##5##6##. Results of final weight gain and SGR, PER, DGI and RFI were significantly reduced with concurrent exposure to arsenic, ammonia, and high-temperature stress, followed by NH₃ + T, respectively As + NH₃, NH₃, and As groups compared to control and Mn-supplemented groups. Further, the supplementation of Mn at 8 mg kg⁻¹ diet with or without stressors was noticeably enhanced, followed by Mn at 4 mg kg⁻¹ diet compared to control and other treatments. Whereas, the results of FCR were significantly inverse to SGR and PER. The Mn diet at 8 mg kg⁻¹ diet was observed significantly lowest FCR followed by Mn at 4 mg kg⁻¹ diet with or without stressors. However, the highest FCR was observed in the concurrent exposure to As + NH₃ + T and control fed group.</p>", "<title>Effect of Mn on LPO, Vit C and haematological parameters</title>", "<p id=\"Par39\">The results of lipid peroxidation (LPO) in the liver and kidney, and Vitamin C levels in muscle and brain, as well as the counts of RBC, WBC, and Hb in <italic>P. hypophthalmus</italic> reared under control conditions and multiple stressors (As, NH₃, As + NH₃, NH₃ + T, As + NH₃ + T), and fed with control and Mn-supplemented diets, are presented in Table ##TAB##6##7##. LPO levels in the liver (<italic>p</italic> = 0.0022) and kidney (<italic>p</italic> = 0.0053) were significantly higher with concurrent exposure to ammonia, arsenic, and high temperature, followed by other stressors, compared to control and Mn-supplemented diets. Interestingly, Mn at 8 mg kg⁻¹ diet, with or without stressors, significantly reduced LPO levels in the liver and kidney, followed by Mn at 4 mg kg⁻¹ diet, compared to the control and other treatment groups. Vitamin C levels in muscle (<italic>p</italic> = 0.015) and brain (<italic>p</italic> = 0.021) were noticeably elevated with dietary Mn at 8 mg kg⁻¹ diet, with or without stressors, compared to control and other treatment groups. Further, Vitamin C levels in muscle and brain were significantly lowered with concurrent exposure to arsenic, ammonia, and high temperature, followed by other stressor groups, compared to control and Mn-supplemented groups. Surprisingly, total blood counts such as RBC and Hb were significantly elevated with stressors (As + NH₃ + T, As + T, As + NH₃, NH₃, and As) compared to control and Mn-supplemented groups. Moreover, RBC (<italic>p</italic> = 0.0066) and Hb (<italic>p</italic> = 0.017) counts were noticeably reduced with dietary Mn at 8 mg kg⁻¹ diet, followed by Mn at 4 mg kg⁻¹ diet, with or without stressors, in comparison to control and other treatment groups. In contrast, WBC counts (<italic>p</italic> = 0.019) were inversely related to RBC and Hb, as they were significantly reduced with different stressors such as arsenic, ammonia, and high temperature. Moreover, Mn supplementation considerably enhanced WBC counts (Mn at 8 and 4 mg kg⁻¹ diet).</p>", "<title>Effect of Mn on immunological attributes</title>", "<p id=\"Par40\">The data on immunological attributes, such as nitroblue tetrazolium (NBT), blood glucose (BG), total protein, albumin, globulin, A:G ratio, and myeloperoxidase (MPO) in <italic>P. hypophthalmus</italic> reared under arsenic, ammonia, and high temperature, and fed with different levels of Mn, are presented in Table ##TAB##7##8##. The levels of NBT (<italic>p</italic> = 0.018), total protein (<italic>p</italic> = 0.023), and globulin (<italic>p</italic> = 0.011) were noticeably inhibited with concurrent exposure to ammonia, arsenic, and high-temperature groups, in comparison to all other groups. However, NBT was significantly reduced in the ammonia and high-temperature groups. Further, Mn at 8 mg kg⁻¹ diet was noticeably elevated, with supplementation of Mn at 8 mg kg⁻¹ diet, followed by other Mn-supplemented groups, in comparison to the control and other treatment groups. Whereas BG (<italic>p</italic> = 0.0046) and A:G ratio (<italic>p</italic> = 0.029) were significantly reduced with supplementation of Mn at 8 mg kg⁻¹ diet, compared to the control, Mn at 4 and 12 mg kg⁻¹ diet, and stressor groups. Similarly, levels of MPO were significantly elevated with Mn at 8 mg kg⁻¹ diet, with or without stressors, followed by Mn at 4 mg kg⁻¹ diet, compared to the control and other treatment groups.</p>", "<title>Effect of Mn on protein and carbohydrate metabolic enzymes</title>", "<p id=\"Par41\">In the present investigation, the data on ALT, AST, LDH, and MDH activities in the liver and gill, as well as acetylcholine esterase (AChE) in the brain of <italic>P. hypophthalmus</italic>, are recorded in Table ##TAB##8##9##. ALT and AST activities in the liver and gill were noticeably elevated (<italic>p</italic> &lt; 0.01) with concurrent exposure to As, NH₃, and high temperature, followed by As + NH₃, NH₃ + T, and other stressor groups, compared to the control and Mn-supplemented groups. Moreover, ALT, AST, LDH, and MDH activities in the liver and gill were remarkably reduced with Mn at 8 mg kg⁻¹ diet, with or without stressors, followed by Mn at 4 mg kg⁻¹ diet, compared to the control and other treatment groups. Furthermore, Mn at 12 mg kg⁻¹ diet was not effective in modulating the activities of ALT, AST, LDH, and MDH against multiple stressors.</p>", "<title>Effect of Mn on neurotransmitter</title>", "<p id=\"Par42\">Interestingly, the AChE activities in the brain were remarkably inhibited (<italic>p</italic> = 0.0039) with concurrent exposure to arsenic, ammonia, and high temperature, followed by NH₃ + T, As + NH₃, NH₃, and As groups, in comparison to the control and Mn-supplemented groups. Conversely, AChE activities were noticeably elevated with Mn at 8 mg kg⁻¹ in the diet, compared to the control, Mn at 4 and 12 mg kg⁻¹ in the diet, and stressor groups (Table ##TAB##8##9##).</p>", "<title>Effect of Mn on bioaccumulation of arsenic</title>", "<p id=\"Par43\">The results of bioaccumulation and concentration of arsenic in different fish tissues and experimental water are presented in Table ##TAB##9##10##. The arsenic concentration in water was determined to be the highest in the group treated under arsenic, ammonia, and high temperature and fed with a control diet (1776 µg L⁻¹), followed by Mn at 12 mg kg⁻¹ diet with stressors (As + NH₃ + T) (1337 µg L⁻¹), the arsenic alone group (1186 µg L⁻¹), and Mn at 4 mg kg⁻¹ diet with stressors (1040 µg L⁻¹) groups. Meanwhile, the bioaccumulation of arsenic was found to be highest in the liver and kidney tissues treated under As + NH₃ + T. Arsenic was below the detection limit in the groups treated with Mn at 4 and 8 mg kg⁻¹ diet, as well as in the control groups, in muscle and brain tissues. Moreover, in the same diets but exposed to As + NH₃ + T, the arsenic concentration was the least in muscle and brain tissues. Furthermore, Mn bioaccumulation was highest in the Mn-12 mg kg⁻¹ diet, followed by Mn at 4 and 8 mg kg⁻¹ in the diet.</p>", "<title>Effect of Mn on bacterial infection</title>", "<p id=\"Par44\">After the experiment, the fish were infected with the <italic>Aeromonas hydrophila</italic>. Cumulative and relative % survival was determined up to seven days after the fish were infected. Cumulative mortality was observed to be higher in the group treated with concurrent exposure to As, NH₃, and T, followed by As + NH₃ (63), NH₃ + T (61), and Mn at 12 mg kg⁻¹ in the diet with stressors (61). In contrast, the least mortality was observed in Mn at 8 mg kg⁻¹ in the diet (27) with stressors (38). Similarly, the relative % survival was observed as -37.5, -37.5, -48.3, -37.5, -56, 0, 27, -25, -12.5, 12.5, -37% for As, NH₃, As + NH₃, NH₃ + T, As + NH₃ + T, Mn at 4, 8, and 12 mg kg⁻¹ in the diet, and Mn at 4, 8, and 12 mg kg⁻¹ diet with stressors, respectively (Fig. ##FIG##4##5##).</p>" ]

[ "<title>Discussion</title>", "<p id=\"Par45\">Cortisol is secreted from the inter-renal cell of head kidney released directly into the blood^{##UREF##30##55##}. It is an important stress hormone plays a role in growth, reproduction, and osmoregulation^{##REF##30667059##56##}. The multiple stressors (As + NH₃ + T) induces stress in the fish which needed energy to combat the stress which compensated through glucogenic pathway. The stress indues by As + NH₃ + T, that elevated cortisol levels could be due to the stressors targeting the multiple sites in the hypothalamus-pituitary-interrenal axis and altered the adrenocorticotropic hormone (ACTH) secretion^{##REF##29702234##57##–##REF##28528485##59##}. Moreover, the dietary Mn at 8 mg kg⁻¹ diet followed by 4 mg kg⁻¹ diet noticeably reduced the cortisol levels which could be due to Mn support the energy provided to glucogenic pathway^{##UREF##31##60##}. During this stage the physiological change also occurred and adapted by fish and homeostasis return which was supported by Mn diet.</p>", "<p id=\"Par46\">HSP 70 are chaperones protein, maintain the normal structure and function of cell protein and help in folding protein into unfolded protein^{##REF##30388847##61##} and indicating physiological conditions of fish during stress. HSPs protein expression is generally upregulated during temperature and metal stress^{##UREF##32##62##,##REF##31657758##63##}. Indeed, the Mn-containing diet downregulated the HSP 70 expression might be due to its role in ApoA-1 gene regulation. It is also proven that if brain could not control oxidative challenges, heat shock protein upregulates the HSPs during stress. The present study showed that Mn at 8 mg kg⁻¹ diet helps in maintenance of the cellular homeostasis through correct folding of nascent and stress-accumulated misfolded proteins in the cell^{##UREF##33##64##}. This might be due to activating the transcription factor of HSP by Mn as it loses the binding activity of heat shock elements, and thus Mn downregulates the HSP70 expression^{##REF##9710578##65##,##REF##9727490##66##}.</p>", "<p id=\"Par47\">The cytochrome P450 is the heme-thiolate protein, is a major component of the membrane-bound microsomal monooxygenase system (MMO), which helps in catalyzes the oxygenation of exogenous and endogenous compounds (Xenobiotics, drugs, and carcinogens)^{##REF##15364547##67##}. Moreover, the toxicity of arsenic, ammonia, and high-temperature stress upregulated the CYP 450 gene expression as it generates ROS, which potentially causes lipid peroxidation, cell toxicity, and death. Interestingly, it is inferred that CYP 450 involved in the arsenic, ammonia toxicity and high temperature stress which take part for apoptosis and upregulated the transcription of bcl2-associated X (Bax)^{##UREF##34##68##}. Bax is the important cell death promoting gene in fish which induce release of cytochrome c, leading to caspase activation^{##REF##11326099##69##}. Surprisingly, Mn at 8 mg kg⁻¹ diet was remarkably downregulated the CYP 450 gene expression might be due to it regulating and controlling the generated reactive oxygen systems (ROS), cytokines regulation, and lipid peroxidation. In the present study, results of ROS, LPO, and cytokines regulation supported the role of Mn in the control CYP 450 regulation.</p>", "<p id=\"Par48\">DNA damage and DNA damage inducible protein (DDIP) gene expression was upregulated by ammonia, arsenic and high temperature stress could be due to extensive generation of reactive oxygen species, dysregulation of cell proliferation, apoptosis, diminished DNA repair, aberrant in histone post-translational modification and DNA methylation^{##REF##12076506##70##}. However, the dietary Mn at 8 mg kg⁻¹ diet protect against DNA damage and downregulated DDIP might be due to at this lower dose of Mn enhances the viability of SH-SY5Y cells, reduced the ROS production and LPO levels as well as enhances GSH levels^{##REF##17351332##71##}. Interestingly, the MT gene expression was highly upregulated by arsenic, ammonia, and high temperature stress and downregulated by Mn diet. Moreover, the higher dietary Mn induces overexpression of MT gene^{##REF##23282023##72##}.</p>", "<p id=\"Par49\">Apoptosis indicates cell programming death in which Cas 3a and 3b belong to the apoptosis gene. The Cas 3a and 3b were upregulated due to arsenic, ammonia, and high-temperature stress could incur apoptosis using p53 and regulated the apoptosis in the liver tissue^{##REF##10102818##73##} and upregulated gene related to oxidative stress and inflammatory response as shown in the present study. Indeed, the supplementation of Mn at 8 and 4 mg kg⁻¹ diet help in controlling the regulation of Cas 3a and 3b, which might be due to it has a role in the activation of the caspase cascade and DNA fragmentation in the liver cell^{##UREF##35##74##}.</p>", "<p id=\"Par50\">The present study revealed that stressors (As + NH₃ + T) reduced the immunity of the fish through cytokines gene upregulation such as TNFα, TLR and IL and downregulated the Ig gene. Zhang et al^{##REF##29802884##75##}. reported that the ammonia toxicity altered the immunity of the fish. The stressors induced the stress and showed higher inflammation rate in liver tissue in fish and hence the higher upregulation of TNFα, IL and TLR was determined in the present study^{##UREF##1##2##}. The TNFα, IL and TLR acts as an essential pro-inflammatory cytokine that enhances the immunity in aquatic animals including fish^{##REF##30253258##76##}. Notably, the Mn at 8 mg kg⁻¹ diet was improved the immunity of the fish using the strengthening/downregulating the TNFα, IL and TLR. This might be due to Mn have an important role in immunostimulants and activating the NF-κB signaling pathways to enhance immunity of the fish against multiple stresses. In contrast to the results of TNFα, IL, TLR and Ig was downregulated with stressors (As + NH₃ + T, NH₃ + T, As + NH₃, NH₃ and As) and upregulated by Mn diet. This could be due to the role of Mn in enhancing humoral and cell-mediated immunity and improving antibody affinity, early β cell development, complement system, cell mediated immunity, phagocytose activity, and antibody reaction.</p>", "<p id=\"Par51\">The present study pointed out the remarkable reduction of oxidative stress enzymes (SOD, CAT, GST, and GPx activities) and gene expression of SOD, CAT, and GPx through Mn diet at 8 mg kg⁻¹. This could be due to the role of manganese in substituting as a cofactor for iron in certain enzymes which is responsible for oxidative stress elevation^{##UREF##36##77##}. Mn is also a cofactor for many enzymes, including pyruvate carboxylase and manganese superoxide dismutase (Mn-SOD). It also protects the cell against reactive oxygen species (ROS) as Mn is part of metalloenzyme by catalyzing the one-electron reduction of peroxide anion to hydrogen peroxide^{##UREF##37##78##}. Mn is found in the Mn-SOD complex, which is useful in maintaining the structure of antioxidant enzymes^{##REF##2835081##79##,##UREF##38##80##}, affecting the Fenton reaction. Mn also enhances the organism's anti-oxidative status through synthesizing and activating certain enzymes such as oxidoreductase transferase, hydrolases, and ligase, as well as vitamins C and B. It also involved metalloenzymes such as arginase, glutamine synthetase, phosphoenol pyruvate, and decarboxylase^{##UREF##39##81##}. Moreover, Mn is the essential mineral nutrient for managing aquatic animals' oxidative stress.</p>", "<p id=\"Par52\">Interestingly, iNOS gene expression was notably highly upregulated by stressors (As + NH₃ + T, NH₃ + T, As + NH₃, NH₃ and As) could be due to higher accumulation of NH₃ in fish tissues. Moreover, the NH₃ is converted into urea via ornithine-urea cycle (OUC) and then converted into glutamine via the glutamine synthetase including non-essential amino acids^{##REF##29864497##82##}. Similarly, the blood carrying the high ammonia concentration and affecting the liver tissue^{##UREF##40##83##}. Moreover, nitric oxide provided protection to cellular system against oxidative stress^{##REF##8570706##84##}. Moreover, the dietary Mn at 8 mg kg⁻¹ diet remarkably downregulated the iNOS gene expression in liver tissue. Further, during stress condition, the organism needs more energy in the form of ATPase, therefore the gene expression of Na⁺K⁺ATPase was highly upregulated. Notably, the Mn diet help in formation of more ATPase and supplied to the fish reread under multiple stress condition.</p>", "<p id=\"Par53\">The growth performance related gene expression viz. GH, GHR1, GHRβ, IGF1X1 and IGF1X2 were remarkably downregulated by stressors (As + NH₃ + T, NH₃ + T, As + NH₃, NH₃ and As) could be due to disruption of endocrine receptor which control the growth related gene expression. The GH gene bind with GHR and controlled by hypothalamic regulation viz. GH-releasing hormone, ghrelin, dopamine and somatostatin^{##REF##17919810##85##,##UREF##41##86##}. The growth-related genes mainly regulated by genetically, endocrinologically and environmentally. It is also related with better nutrition, optimum temperature, good husbandry condition and better functioning of endocrine regulation^{##UREF##42##87##}. It is also observed that the Mn diet notably downregulated MYST and SMT at 8 mg kg⁻¹ diet. It might be due to the role of MYST in decreasing the myoblast, which results in terminal differentiation and division of fiber enlargement^{##REF##10527122##88##}. Further, the IGF1X1 and IGF1X2 gene expressions have important role in biomolecular regulation such as carbohydrates, lipid, protein, and mineral metabolism, differentiation and proliferation of the cell and ultimately growth^{##REF##10993139##89##}. As the GH bind to the receptor in the liver cell to stimulate, release and synthesize IGF gene expression and dietary Mn help in this process, the Mn diet is responsible for growth enhancement and biomolecular function in the cell of aquatic organism. The stressors (As + NH₃ + T, NH₃ + T, As + NH₃, NH₃, and As) drastically inhibited the growth performance (final weight gain %, FCR, SGR, PER, DGI, TGC, and RFI) of the fish might be due to arsenic and ammonia toxicity, and high-temperature stress reduces the feed intake and metabolic rate, which was reported by our previous study². Interestingly, the Mn diet improved growth performance could be due to the role of Mn in improving feed efficiency, feed utilization, growth rate, and immunity of the fish. It also improved the specific growth rate, daily growth index %, relative feed intake and protein efficiency in the fish^{##UREF##43##90##}. Moreover, the deficiency and inadequate supply of Mn result in reduced growth rate, reduced feed intake, skeletal abnormalities such as dwarfism and cataract in fish^{##UREF##44##91##}. Mn has also provided the uptake of glucose, insulin receptors and triglyceride synthesis^{##REF##2204694##92##}. However, the dietary Mn at optimum levels is beneficial for growth enhancement of fish reread under control and stressed environment.</p>", "<p id=\"Par54\">The present study revealed that ammonia and arsenic toxicity and high-temperature stress elevated the LPO level in the liver and kidney tissues might be due to the formation of ROS by stressors. A free radical producing system mainly generates it. Excessive ROS generation may cause oxidative stress and damage critical biomolecules, resulting in deleterious biological effects^{##REF##16557614##93##}. Moreover, the dietary Mn at 4 and 8 mg kg⁻¹ diet remarkably reduced the ROS and LPO levels. Similarly, the muscle and brain Vit C were noticeably elevated by dietary Mn at 4 and 8 mg kg⁻¹ diet. It is crucial for collagen synthesis and in metabolism of biomolecules, including steroids and detoxification of xenobiotics^{##REF##12569111##94##}. Therefore, Mn is important in maintaining the Vit C in fish tissues. The blood profiling viz. Hb, WBC and RBC were important component which were altered by arsenic, ammonia and high temperature, whereas, the dietary Mn at 8 mg kg⁻¹ diet was corrected the count of Hb, WBC and RBC. Hb helps in aerobic metabolism, distribution of the gases and maintenance of the physiological attributes in the fish via fish growth and health^{##REF##5042150##95##}. The RBC helped absorb oxygen through gill and circulated in the different tissues in the body. Stress changed not only the RBC count but also the shape of the RBC. Further, the WBC is an important component for acquired and innate immune response. It constituted eosinophils, neutrophils, lymphocytes, monocytes, and basophils. Hence, the dietary Mn enhances the WBC count in fish reared in control or stress conditions.</p>", "<p id=\"Par55\">NBT, blood glucose, total protein, albumin, globulin, A:G ratio, and MPO are important attributes of immunity. In the present study, stressors altered the immunity, whereas the dietary Mn approved the immunity in the fish. NBT indicates the health of the fish as elevated levels mention higher immunity. It involved the phagocytes for intercellular superoxide radicals produced by leucocytes^{##UREF##45##96##}. Moreover, the globulin are also major component and four types such as α₁, α₂, β and γ^{##UREF##46##97##}, which the gamma globulin is essential for blood immunological protein^{##REF##9040980##98##}. Further, albumin helps in transportation of hormones, metal, bilirubin, drug and vitamin. It also regulates the free available hormones^{##REF##15820128##99##} and fat metabolism. Interestingly, the supplementation of Mn diet enhances the production of B-lymphocytes and it maintained the higher immunity of the fish. However, MPO is the haemoprotein and important during respiratory burst using H₂O₂ to produce hypochlorous acid^{##REF##14698223##100##}. Hypochlorous acid is a potent oxidant that elicit the cytotoxic effect on bacterial cells^{##UREF##47##101##}. Moreover, the Mn containing diets helps in released of neutrophils and O₂ derived species (H₂O₂) and H₂O₂ to oxidize Cl^- ions to form HOCl. Moreover, the blood glucose are indicators for good health and Mn diet improved the BG level after exposure to stressors. The role of Mn in regulating blood glucose might be due to, it enhances the gluconeogenesis viz. synthesis of glucose from non-carbohydrate source mainly protein and amino acid, and the enhancement of secretion of catecholamine^{##REF##11504339##102##}.</p>", "<p id=\"Par56\">The carbohydrate and protein metabolic enzymes viz. LDH, MDH, ALT and AST activities were notably elevated whereas the dietary Mn at 4 and 8 mg kg⁻¹ diet reduced the activities. This might be due to Mn fulfilled the energy demand during stress conditions, and hence, it reduces the activities of LDH and MDH. Notably, the LDH is the glycolytic enzyme that catalyzes the interconversion of pyruvate and lactate using the nicotinamide adenine dinucleotide (NAD) as a coenzyme. Moreover, the MDH is the limiting enzyme for the oxidative catabolism of carbohydrates^{##REF##31732893##103##}. The ALT and AST activities were also reduced by the Mn diet, possibly because Mn is required for many cofactors for biomolecular enzymes.</p>", "<p id=\"Par57\">The stressors (As + NH₃ + T, As + T, As + NH₃, NH₃, and As) significantly inhibited AChE activities might be due to arsenic, ammonia, and high temperature preventing the hydrolysis of acetylcholine^{##REF##33087779##58##}. Moreover, acetylcholine helps dominate cholinergic synapses and neuromuscular junctions in the fish's central nervous system (CNS). This results in the hydrolysis of acetylcholine and choline after the activation of acetylcholine receptors at the postsynaptic membrane^{##UREF##48##104##}. Surprisingly, the AChE activities were improved by dietary Mn. It also showed that a higher Mn diet at 12 mg kg⁻¹ diet significantly inhibited AChE activities, which might be due to its nature to induced the toxicity to the postsynaptic membrane.</p>", "<p id=\"Par58\">The stressors group, such as arsenic, ammonia toxicity, and high-temperature stress, enhances arsenic bioaccumulation in the fish, whereas the Mn diet at 8 mg kg⁻¹ diet reduced the arsenic bioaccumulation. This might be due to the ability of Mn to enhance the detoxification of arsenic in all tissues. Moreover, the kidney and liver tissues had higher arsenic bioaccumulation reported in the present investigation. These results revealed that Mn could detoxify arsenic efficiently in all the tissues.</p>", "<p id=\"Par59\">The present study also showed that dietary Mn at 8 mg kg⁻¹ diet enhanced fish survival after an infection of a bacterial pathogen. The results of the present study showed the Mn diet improved the antioxidant and immunity of the fish. However, this might be the reason for the higher survival of the fish against pathogenic infection after the dietary application of Mn. It is also reported that Mn helps generate neutrophils, which enhances the effector cells function for defencing the fish against pathogenic bacteria^{##REF##22080185##105##}.</p>" ]

[ "<title>Conclusion</title>", "<p id=\"Par60\">The present study is the first report on the role of manganese (Mn) on gene regulations and biochemical regulators in response to arsenic and ammonia toxicity and high-temperature stress in <italic>P. hypophthalmus</italic>. The immunity, anti-oxidative status, growth performance, genotoxicity, and other stress-responsive genes were controlled and regulated by dietary Mn at 8 mg kg⁻¹ diet. Mn at 8 mg kg⁻¹ diet efficiently regulates cortisol, HSP 70, and apoptosis and protects against genotoxicity. Mn at 8 mg kg⁻¹ diet is also efficient in enhancing the detoxification of arsenic in different fish tissues. Moreover, the results revealed that Mn at 8 mg kg⁻¹ efficiently controls the gene regulation involved in the multiple stressors (As + NH₃ + T). Indeed, dietary Mn at 8 mg kg⁻¹ diet improved gene regulation, maintained fish hemostasis, and noticeably reduced the bioaccumulation of arsenic in fish tissues. Overall results of the present investigation concluded that Mn at 8 mg kg⁻¹ diet should be included in the fish diet to maintain gene regulation of the NFkB signaling pathway and mitigate the multiple stresses in fish.</p>" ]

[ "<p id=\"Par1\">The ongoing challenges of climate change and pollution are major factors disturbing ecosystems, including aquatic systems. They also have an impact on gene regulation and biochemical changes in aquatic animals, including fish. Understanding the mechanisms of gene regulation and biochemical changes due to climate change and pollution in aquatic animals is a challenging task. However, with this backdrop, the present investigation was conducted to explore the effects of arsenic (As) and ammonia (NH₃) toxicity and high-temperature (T) stress on gene regulation and biochemical profiles, mitigated by dietary manganese (Mn) in <italic>Pangasianodon hypophthalmus</italic>. The fish were exposed to different combinations of As, NH₃, and T, and fed with dietary Mn at 4, 8, and 12 mg kg⁻¹ to evaluate the gene expression of immunity, antioxidative status, cytokine, and NfKB signaling pathway genes. HSP 70, cytochrome P450 (CYP 450), metallothionein (MT), DNA damage-inducible protein (DDIP), caspase (CAS), tumor necrosis factor (TNFα), toll-like receptor (TLR), interleukin (IL), inducible nitric oxide synthase (iNOS), catalase (CAT), superoxide dismutase (SOD), and glutathione peroxidase (GPx) were noticeably highly upregulated by As + NH₃ + T stress, whereas Mn diet at 8 mg kg⁻¹ downregulated these genes. Further, total immunoglobulin (Ig), myostatin (MYST), somatostatin (SMT), growth hormone (GH), growth hormone regulator 1 and β, insulin-like growth factors (IGF1X1 and IGF1X2) were significantly upregulated by Mn diets. The biochemical profiles were highly affected by stressors (As + NH₃ + T). The bioaccumulation of arsenic in different tissues was also notably reduced by Mn diets. Furthermore, the infectivity of the fish was reduced, and survival against pathogenic bacteria was enhanced by Mn diet at 8 mg kg⁻¹. The results of the present investigation revealed that dietary Mn at 8 mg kg⁻¹ controls gene regulation against multiple stressors (As, NH₃, As + NH₃, NH₃ + T, As + NH₃ + T) in fish.</p>", "<title>Subject terms</title>" ]

[]

[ "<title>Acknowledgements</title>", "<p>The present work was supported by Indian Council of Agricultural Research (ICAR), New Delhi, India under Project “Lal bahadur Shastri Young Scientist Award (Project code: OXX5181). Authors also sincerely acknowledged to Director ICAR-NIASM for providing research facilities for this work.</p>", "<title>Author contributions</title>", "<p>N.K., Conceived and designed the experiments; performed the experiments; analysed the data; contributed reagents/materials/analysis tools; wrote the paper S.T.T. Perform gene analysis S.A.K. Data Validation K.S.R. Supervision and editing.</p>", "<title>Funding</title>", "<p>The present work was supported by Indian Council of Agricultural Research (ICAR), New Delhi, India under Project “Lal bahadur Shastri Young Scientist Award (Project code: OXX5181).</p>", "<title>Data availability</title>", "<p>The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.</p>", "<title>Competing interests</title>", "<p id=\"Par61\">The authors declare no competing interests.</p>" ]

[ "<fig id=\"Fig1\"><label>Figure 1</label><caption><p>Manganese diets control the cortisol and gene expression of HSP 70, CYP 450, DNA damage inducible protein (DDIP) and metallothionine (MT) against multiple stressors in fish. Within endpoints and groups, bars with different superscripts differ significantly (a–h) Cortisol (<italic>p</italic> = 0.0025), HSP-L (<italic>p</italic> = 0.0017), CYP 450 <italic>p</italic> = 0.0013), DDIP <italic>p</italic> = 0.0011), MT <italic>p</italic> = 0.0002). Data expressed as Mean ± SE (<italic>n</italic> = 3).</p></caption></fig>", "<fig id=\"Fig2\"><label>Figure 2</label><caption><p>Manganese diets regulate the gene expression of Caspase 3a and 3b, TNFα, Ig, TLR and IL against multiple stressors in fish. Within endpoints and groups, bars with different superscripts differ significantly (a–g) Cas 3a <italic>p</italic> = 0.0052), Cas 3b <italic>p</italic> = 0.0003), TNFα <italic>p</italic> = 0.0023), Ig <italic>p</italic> = 0.0012), TLR <italic>p</italic> = 0.0046), IL <italic>p</italic> = 0.0029). Data expressed as Mean ± SE (<italic>n</italic> = 3).</p></caption></fig>", "<fig id=\"Fig3\"><label>Figure 3</label><caption><p>Manganese diets regulate the gene expression of Na⁺ K⁺ ATPase, GH, CAT, GPx, iNOS and SOD against multiple stress in fish. Within endpoints and groups, bars with different superscripts differ significantly (a–i) Na⁺ K⁺ ATPase <italic>p</italic> = 0.0022), GH <italic>p</italic> = 0.0016), CAT <italic>p</italic> = 0.001), GPx <italic>p</italic> = 0.014), iNOS <italic>p</italic> = 0.0036), SOD <italic>p</italic> = 0.0061). Data expressed as Mean ± SE (<italic>n</italic> = 3).</p></caption></fig>", "<fig id=\"Fig4\"><label>Figure 4</label><caption><p>Manganese diets regulate the gene expression of MYST, SMT, GHR1, GHRβ, IGF1X1, and IGF1X2 against multiple stressors in fish. Within endpoints and groups, bars with different superscripts differ significantly (a–g) MYST <italic>p</italic> = 0.0023), SMT <italic>p</italic> = 0.0042), GHR1 <italic>p</italic> = 0.0027), GHRβ <italic>p</italic> = 0.0033), IGF1X1 <italic>p</italic> = 0.015), IGD1X2 <italic>p</italic> = 0.0072). Data expressed as Mean ± SE (<italic>n</italic> = 3).</p></caption></fig>", "<fig id=\"Fig5\"><label>Figure 5</label><caption><p>Manganese diets reduces cumulative mortality and enhances relative percentage survival against pathogenic bacteria in fish reared under multiple stressors.</p></caption></fig>" ]

[ "<table-wrap id=\"Tab1\"><label>Table 1</label><caption><p>experimental design of present investigation.</p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th align=\"left\">S. No</th><th align=\"left\">Details of the treatments</th><th align=\"left\">Notation</th></tr></thead><tbody><tr><td align=\"left\">1</td><td align=\"left\">Control</td><td align=\"left\">Ctr</td></tr><tr><td align=\"left\">2</td><td align=\"left\">Fed with control diet and exposure to arsenic</td><td align=\"left\">As</td></tr><tr><td align=\"left\">3</td><td align=\"left\">Fed with control diet and exposure to ammonia</td><td align=\"left\">NH₃</td></tr><tr><td align=\"left\">4</td><td align=\"left\">Fed with control diet and concurrently exposure to arsenic and ammonia</td><td align=\"left\">As + NH₃</td></tr><tr><td align=\"left\">5</td><td align=\"left\">Fed with control diet and concurrent exposure to ammonia and high temperature</td><td align=\"left\">NH₃ + T</td></tr><tr><td align=\"left\">6</td><td align=\"left\">Fed with control diet and concurrent exposure to arsenic, ammonia and high temperature</td><td align=\"left\">As + NH₃ + T</td></tr><tr><td align=\"left\">7</td><td align=\"left\">Fed with manganese at 4 mg kg⁻¹ diet</td><td align=\"left\">Mn at 4 mg kg⁻¹ diet</td></tr><tr><td align=\"left\">8</td><td align=\"left\">Fed with manganese at 8 mg kg⁻¹ diet</td><td align=\"left\">Mn at 8 mg kg⁻¹ diet</td></tr><tr><td align=\"left\">9</td><td align=\"left\">Fed with manganese at 12 mg kg⁻¹ diet</td><td align=\"left\">Mn at 12 mg kg⁻¹ diet</td></tr><tr><td align=\"left\">10</td><td align=\"left\">Fed with manganese at 4 mg kg⁻¹ diet and concurrent exposure to arsenic, ammonia and high temperature</td><td align=\"left\">Mn at 4 mg kg⁻¹ diet + As + NH₃ + T</td></tr><tr><td align=\"left\">11</td><td align=\"left\">Fed with manganese at 8 mg kg⁻¹ diet and concurrent exposure to arsenic, ammonia and high temperature</td><td align=\"left\">Mn at 8 mg kg⁻¹ diet + As + NH₃ + T</td></tr><tr><td align=\"left\">12</td><td align=\"left\">Fed with manganese at 12 mg kg⁻¹ diet and concurrent exposure to arsenic, ammonia and high temperature and</td><td align=\"left\">Mn at 12 mg kg⁻¹ diet + As + NH₃ + T</td></tr></tbody></table></table-wrap>", "<table-wrap id=\"Tab2\"><label>Table 2</label><caption><p>Ingredients composition and proximate analysis of experimental diets (% dry matter) of manganese (Mn), fed to <italic>Pangasianodon hypophthalmus</italic> for 105 days.</p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th align=\"left\">Feed ingredients</th><th align=\"left\">Mn-0 mg kg⁻¹ diet</th><th align=\"left\">Mn-4 mg kg⁻¹ diet</th><th align=\"left\">Mn-8 mg kg⁻¹ diet</th><th align=\"left\">Mn-12 mg kg⁻¹ diet</th></tr></thead><tbody><tr><td align=\"left\">Soybean meal^a</td><td align=\"left\">35.5</td><td align=\"left\">35.5</td><td align=\"left\">35.5</td><td align=\"left\">35.5</td></tr><tr><td align=\"left\">Fish meal^a</td><td align=\"left\">25</td><td align=\"left\">25</td><td align=\"left\">25</td><td align=\"left\">25</td></tr><tr><td align=\"left\">Groundnut meal^a</td><td align=\"left\">10</td><td align=\"left\">10</td><td align=\"left\">10</td><td align=\"left\">10</td></tr><tr><td align=\"left\">Wheat flour^a</td><td align=\"left\">17.2</td><td align=\"left\">17.196</td><td align=\"left\">17.192</td><td align=\"left\">17.188</td></tr><tr><td align=\"left\">Sunflower oil^a</td><td align=\"left\">4.5</td><td align=\"left\">4.5</td><td align=\"left\">4.5</td><td align=\"left\">4.5</td></tr><tr><td align=\"left\">Cod liver oil^a</td><td align=\"left\">1.5</td><td align=\"left\">1.5</td><td align=\"left\">1.5</td><td align=\"left\">1.5</td></tr><tr><td align=\"left\">CMC^b</td><td align=\"left\">2</td><td align=\"left\">2</td><td align=\"left\">2</td><td align=\"left\">2</td></tr><tr><td align=\"left\">Vitamin and mineral mix*</td><td align=\"left\">2</td><td align=\"left\">2</td><td align=\"left\">2</td><td align=\"left\">2</td></tr><tr><td align=\"left\">Vitamin C^c</td><td align=\"left\">0.3</td><td align=\"left\">0.3</td><td align=\"left\">0.3</td><td align=\"left\">0.3</td></tr><tr><td align=\"left\">Lecithin^b</td><td align=\"left\">2</td><td align=\"left\">2</td><td align=\"left\">2</td><td align=\"left\">2</td></tr><tr><td align=\"left\">Mn</td><td align=\"left\">0</td><td align=\"left\">0.004</td><td align=\"left\">0.008</td><td align=\"left\">0.012</td></tr><tr><td align=\"left\" colspan=\"5\">Proximate composition of the diets</td></tr><tr><td align=\"left\">Crude protein (CP)</td><td align=\"left\">35.34 ± 0.39</td><td align=\"left\">35.16 ± 0.08</td><td align=\"left\">35.43 ± 0.18</td><td align=\"left\">35.17 ± 0.02</td></tr><tr><td align=\"left\">Ether extract (EE)</td><td align=\"left\">8.23 ± 0.09</td><td align=\"left\">8.57 ± 0.22</td><td align=\"left\">8.72 ± 0.31</td><td align=\"left\">8.34 ± 0.10</td></tr><tr><td align=\"left\">Total carbohydrate (TC)</td><td align=\"left\">40.37 ± 0.36</td><td align=\"left\">40.89 ± 0.68</td><td align=\"left\">40.58 ± 0.41</td><td align=\"left\">40.13 ± 0.08</td></tr><tr><td align=\"left\">Organic matter (OM)</td><td align=\"left\">92.05 ± 0.08</td><td align=\"left\">92.26 ± 0.28</td><td align=\"left\">91.98 ± 0.07</td><td align=\"left\">91.81 ± 0.02</td></tr><tr><td align=\"left\">Dry matter (DM)</td><td align=\"left\">91.90 ± 0.13</td><td align=\"left\">92.36 ± 0.30</td><td align=\"left\">92.74 ± 0.04</td><td align=\"left\">91.83 ± 0.13</td></tr><tr><td align=\"left\">Digestible energy (DE)</td><td align=\"left\">363.47 ± 0.93</td><td align=\"left\">364.20 ± 1.95</td><td align=\"left\">365.03 ± 0.99</td><td align=\"left\">364.59 ± 0.85</td></tr></tbody></table></table-wrap>", "<table-wrap id=\"Tab3\"><label>Table 3</label><caption><p>Details of primer for relative quantitative real-time PCR.</p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th align=\"left\">Gene</th><th align=\"left\">Primer sequence (5′–3′)</th><th align=\"left\">Accession number</th></tr></thead><tbody><tr><td align=\"left\">SOD</td><td align=\"left\"><p>F-GTCCATCTTACCCGGTGCCC</p><p>R-CGAGAGAAGACCCGGAACGC</p></td><td align=\"left\">XM_034299545.1</td></tr><tr><td align=\"left\">CAT</td><td align=\"left\"><p>F-AGCAGGCGGAGAAGTACCCA</p><p>R-GCTGCTCCACCTCAGCGAAA</p></td><td align=\"left\">XM_026919141.2</td></tr><tr><td align=\"left\">GPx</td><td align=\"left\"><p>F- GTCACTGCAGGATGCAACAC</p><p>R- TTGGAATTCCGCTCATTGAT</p></td><td align=\"left\">XM_026947312.2</td></tr><tr><td align=\"left\">HSP 70</td><td align=\"left\"><p>F- CTCCTCCTAAACCCCGAGTC</p><p>R- CCACCAGCACGTTAAACACA</p></td><td align=\"left\">XM_026934573.2</td></tr><tr><td align=\"left\">iNOS</td><td align=\"left\"><p>F-ACACCACGGAGTGTGTTCGT</p><p>R-GGATGCATGGGACGTTGCTG</p></td><td align=\"left\">XM_026931613.2</td></tr><tr><td align=\"left\">DNA damage inducible protein</td><td align=\"left\"><p>F-CACCTTCGCCCTCGAAGTCT</p><p>R-GCTCGGGTGAGGTCTCTCAG</p></td><td align=\"left\">XM_026938137.2</td></tr><tr><td align=\"left\">TNFα</td><td align=\"left\"><p>F-TGGAGTTCTGCTTGCCGTGG</p><p>R-GCAGCCTTTGCAGTCTCGGA</p></td><td align=\"left\">XM_026942329.2</td></tr><tr><td align=\"left\">TLR</td><td align=\"left\"><p>F: TCACCACGAACGAGACTTCATCC</p><p>R : GACAGCACGAAGACACAGCATC</p></td><td align=\"left\">XM_026916808.2</td></tr><tr><td align=\"left\">Ghr1</td><td align=\"left\"><p>FTATTGGCTACAGCTCGCCGC</p><p>R-AATCACCCCGACTGTGCTGC</p></td><td align=\"left\">XM_034306157.1</td></tr><tr><td align=\"left\">Ghrb</td><td align=\"left\"><p>F-TTGAGCTTTGGGACTCGGAC</p><p>R-CGTCGATCTTCTCGGTGAGG</p></td><td align=\"left\">XM_026942987.2</td></tr><tr><td align=\"left\">IGF-1X1</td><td align=\"left\"><p>F-GCAACGGCACACAGACACGC</p><p>R-CAGACGTTCCCTCACCATCCTCT</p></td><td align=\"left\">XM_034313382.2</td></tr><tr><td align=\"left\">IGF-1X2</td><td align=\"left\"><p>F-CGAGAGCAACGGCACACAGA</p><p>R-TTCTGATGGACCTCCTTACAAGATG</p></td><td align=\"left\">XM_034313383.2</td></tr><tr><td align=\"left\">IL</td><td align=\"left\"><p>F-AGCAGGATCCATCAAAGTGG</p><p>R-GTGCTCCAGCTCTCTGGGTA</p></td><td align=\"left\">XM_026918084.2</td></tr><tr><td align=\"left\">Ig</td><td align=\"left\"><p>F-GGCCAGTAATCGTACCTCCA</p><p>R-CTTCGTAAGGTCCCCACTGA</p></td><td align=\"left\">XM_026923540.2</td></tr><tr><td align=\"left\">MYST</td><td align=\"left\"><p>F-GGGAAAGACCTGGCCGTGAC</p><p>R-TCGAGGCCGGATTCTCGTCT</p></td><td align=\"left\">XM_026910492.2</td></tr><tr><td align=\"left\">SMT</td><td align=\"left\"><p>F-CTCTGGGTGGCAGAATGAAT</p><p>R-AACATGAAGAGAACGTTTTCCAG</p></td><td align=\"left\">XM_026921272.2</td></tr><tr><td align=\"left\">GH</td><td align=\"left\"><p>F-CCCAGCAAGAACCTCGGCAA</p><p>R-GCGGAGCCAGAGAGTCGTTC</p></td><td align=\"left\">GQ859589.1</td></tr><tr><td align=\"left\">Cas3b</td><td align=\"left\"><p>F-AGCTTTCCGTGAGCTGGGCT</p><p>R-TGGCTGACTTGCTGTGGTCCT</p></td><td align=\"left\">NC_047601.1</td></tr><tr><td align=\"left\">Na⁺K⁺ATPase</td><td align=\"left\"><p>F-AACTACAAGCCCACGTACCA</p><p>R-CTTGCCAGCCTTAAAGCCAA</p></td><td align=\"left\">XM_026923907.3</td></tr><tr><td align=\"left\">β-Actin</td><td align=\"left\"><p>F-CAGCAAGCAGGAGTACGATG</p><p>R-TGTGTGGTGTGTGGTTGTTTTG</p></td><td align=\"left\">XM_031749543.1</td></tr></tbody></table></table-wrap>", "<table-wrap id=\"Tab4\"><label>Table 4</label><caption><p>Effect of manganese (Mn) on DNA damage in <italic>P. hypophthalmus</italic> reared under control and multi stress condition.</p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th align=\"left\">Exposure/diets (mg kg⁻¹)</th><th align=\"left\">Comet length</th><th align=\"left\">Comet DNA</th><th align=\"left\">Head area</th><th align=\"left\">Head DNA</th><th align=\"left\">Head DNA (%)</th><th align=\"left\">Tail DNA</th><th align=\"left\">Tail DNA (%)</th></tr></thead><tbody><tr><td align=\"left\">Ctr/Ctr</td><td char=\".\" align=\"char\">35.0d ± 1.55</td><td char=\".\" align=\"char\">140,301 ± 23.98</td><td char=\".\" align=\"char\">942 ± 7.67</td><td char=\".\" align=\"char\">133,230 ± 26.67</td><td char=\".\" align=\"char\">94.96 g ± 1.98</td><td char=\".\" align=\"char\">7071 ± 21.87</td><td char=\".\" align=\"char\">5.04a ± 0.44</td></tr><tr><td align=\"left\">As/Ctr</td><td char=\".\" align=\"char\">30.0c ± 1.05</td><td char=\".\" align=\"char\">62,845 ± 15.34</td><td char=\".\" align=\"char\">156 ± 2.76</td><td char=\".\" align=\"char\">15,004 ± 27.09</td><td char=\".\" align=\"char\">23.87c ± 2.65</td><td char=\".\" align=\"char\">47,841 ± 20.71</td><td char=\".\" align=\"char\">76.13e ± 1.65</td></tr><tr><td align=\"left\">NH₃/Ctr</td><td char=\".\" align=\"char\">28.0bc ± 0.67</td><td char=\".\" align=\"char\">46,116 ± 6.47</td><td char=\".\" align=\"char\">52 ± 1.45</td><td char=\".\" align=\"char\">4851 ± 11.45</td><td char=\".\" align=\"char\">10.52ab ± 1.22</td><td char=\".\" align=\"char\">41,265 ± 31.87</td><td char=\".\" align=\"char\">89.48f. ± 1.99</td></tr><tr><td align=\"left\">As + NH₃/Ctr</td><td char=\".\" align=\"char\">24.0bc ± 0.93</td><td char=\".\" align=\"char\">48,760 ± 8.94</td><td char=\".\" align=\"char\">52 ± 1.21</td><td char=\".\" align=\"char\">5973 ± 16.32</td><td char=\".\" align=\"char\">12.25b ± 1.17</td><td char=\".\" align=\"char\">42,787 ± 16.69</td><td char=\".\" align=\"char\">87.75f. ± 2.67</td></tr><tr><td align=\"left\">NH₃ + T/Ctr</td><td char=\".\" align=\"char\">48.0e ± 1.32</td><td char=\".\" align=\"char\">58,087 ± 9.27</td><td char=\".\" align=\"char\">51 ± 1.43</td><td char=\".\" align=\"char\">6723 ± 14.36</td><td char=\".\" align=\"char\">11.57b ± 1.05</td><td char=\".\" align=\"char\">51,364 ± 13.46</td><td char=\".\" align=\"char\">88.43f. ± 3.61</td></tr><tr><td align=\"left\">As + T + NH₃/Ctr</td><td char=\".\" align=\"char\">23.0b ± 0.56</td><td char=\".\" align=\"char\">57,594 ± 11.54</td><td char=\".\" align=\"char\">52 ± 1.87</td><td char=\".\" align=\"char\">5573 ± 11.34</td><td char=\".\" align=\"char\">9.68a ± 0.56</td><td char=\".\" align=\"char\">52,021 ± 11.94</td><td char=\".\" align=\"char\">90.32f. ± 4.87</td></tr><tr><td align=\"left\">Ctr/Mn-4</td><td char=\".\" align=\"char\">9.93a ± 0.15</td><td char=\".\" align=\"char\">23,179 ± 6.34</td><td char=\".\" align=\"char\">40 ± 1.81</td><td char=\".\" align=\"char\">3886 ± 11.34</td><td char=\".\" align=\"char\">93.73 g ± 0.37</td><td char=\".\" align=\"char\">22,630 ± 22.76</td><td char=\".\" align=\"char\">6.27a ± 0.11</td></tr><tr><td align=\"left\">Ctr/Mn-8</td><td char=\".\" align=\"char\">22.0b ± 1.89</td><td char=\".\" align=\"char\">52,922 ± 7.93</td><td char=\".\" align=\"char\">377 ± 11.98</td><td char=\".\" align=\"char\">49,464 ± 25.87</td><td char=\".\" align=\"char\">93.47 g ± 1.85</td><td char=\".\" align=\"char\">3458 ± 11.33</td><td char=\".\" align=\"char\">6.53a ± 0.21</td></tr><tr><td align=\"left\">Ctr/Mn-12</td><td char=\".\" align=\"char\">30.0c ± 1.48</td><td char=\".\" align=\"char\">58,541 ± 17.45</td><td char=\".\" align=\"char\">798 ± 13.76</td><td char=\".\" align=\"char\">54,224 ± 16.74</td><td char=\".\" align=\"char\">92.63 g ± 1.25</td><td char=\".\" align=\"char\">4317 ± 17.56</td><td char=\".\" align=\"char\">7.37a ± 0.15</td></tr><tr><td align=\"left\">As + T + NH_3/Mn-4</td><td char=\".\" align=\"char\">22.0b ± 1.01</td><td char=\".\" align=\"char\">48,292 ± 28.65</td><td char=\".\" align=\"char\">329 ± 11.71</td><td char=\".\" align=\"char\">40,212 ± 11.34</td><td char=\".\" align=\"char\">83.27e ± 1.67</td><td char=\".\" align=\"char\">8080 ± 9.32</td><td char=\".\" align=\"char\">16.73c ± 1.77</td></tr><tr><td align=\"left\">As + T + NH_3/Mn-8</td><td char=\".\" align=\"char\">25.0bc ± 1.36</td><td char=\".\" align=\"char\">49,287 ± 27.65</td><td char=\".\" align=\"char\">380 ± 7.56</td><td char=\".\" align=\"char\">42,443 ± 11.87</td><td char=\".\" align=\"char\">86.11f. ± 1.15</td><td char=\".\" align=\"char\">6844 ± 13.23</td><td char=\".\" align=\"char\">13.89b ± 0.76</td></tr><tr><td align=\"left\">As + T + NH₃/Mn-12</td><td char=\".\" align=\"char\">47.0 ± 1.76</td><td char=\".\" align=\"char\">71,698 ± 24.45</td><td char=\".\" align=\"char\">438 ± 8.36</td><td char=\".\" align=\"char\">41,374 ± 11.87</td><td char=\".\" align=\"char\">57.71d ± 1.09</td><td char=\".\" align=\"char\">30,324 ± 14.87</td><td char=\".\" align=\"char\">42.29d ± 0.65</td></tr></tbody></table></table-wrap>", "<table-wrap id=\"Tab5\"><label>Table 5</label><caption><p>Effect of dietary manganese (Mn) on CAT, SOD, GST and GPx enzymatic activities against multiple stressors in fish.</p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th align=\"left\" rowspan=\"2\">Exposure/diets (mg kg⁻¹)</th><th align=\"left\" colspan=\"2\">Catalase (CAT)</th><th align=\"left\" colspan=\"2\">Super-oxide dismutase (SOD)</th><th align=\"left\" colspan=\"2\">Glutathione-s-transferase (GST)</th><th align=\"left\" colspan=\"2\">Glutathione peroxidase (GPx)</th></tr><tr><th align=\"left\">CAT-L</th><th align=\"left\">CAT-G</th><th align=\"left\">SOD-L</th><th align=\"left\">SOD-G</th><th align=\"left\">GST-L</th><th align=\"left\">GST-G</th><th align=\"left\">GPx-L</th><th align=\"left\">GPx-G</th></tr></thead><tbody><tr><td align=\"left\">Ctr/Ctr</td><td align=\"left\">9.67c ± 0.38</td><td align=\"left\">7.54b ± 0.60</td><td align=\"left\">45.22a ± 0.97</td><td align=\"left\">35.25a ± 1.44</td><td align=\"left\">0.32b ± 0.02</td><td align=\"left\">0.34b ± 0.04</td><td align=\"left\">0.55c ± 0.06</td><td align=\"left\">0.44b ± 0.02</td></tr><tr><td align=\"left\">As/Ctr</td><td align=\"left\">16.95d ± 0.36</td><td align=\"left\">12.76d ± 1.21</td><td align=\"left\">48.63b ± 0.56</td><td align=\"left\">40.37b ± 0.33</td><td align=\"left\">0.48c ± 0.06</td><td align=\"left\">0.49c ± 0.03</td><td align=\"left\">0.81d ± 0.14</td><td align=\"left\">0.68d ± 0.06</td></tr><tr><td align=\"left\">NH₃/Ctr</td><td align=\"left\">17.48d ± 1.28</td><td align=\"left\">12.82d ± 0.83</td><td align=\"left\">47.02b ± 1.18</td><td align=\"left\">40.94b ± 0.61</td><td align=\"left\">0.57d ± 0.03</td><td align=\"left\">0.48c ± 0.02</td><td align=\"left\">0.78d ± 0.08</td><td align=\"left\">0.61d ± 0.08</td></tr><tr><td align=\"left\">As + NH₃/Ctr</td><td align=\"left\">22.52e ± 0.87</td><td align=\"left\">17.73d ± 1.16</td><td align=\"left\">51.02c ± 0.34</td><td align=\"left\">41.79b ± 0.32</td><td align=\"left\">0.70e ± 0.09</td><td align=\"left\">0.63d ± 0.04</td><td align=\"left\">0.94e ± 0.14</td><td align=\"left\">0.73de ± 0.07</td></tr><tr><td align=\"left\">NH₃ + T/Ctr</td><td align=\"left\">23.65e ± 0.89</td><td align=\"left\">18.90d ± 1.54</td><td align=\"left\">48.52b ± 0.27</td><td align=\"left\">42.33b ± 1.21</td><td align=\"left\">0.59d ± 0.05</td><td align=\"left\">0.63d ± 0.03</td><td align=\"left\">0.86de ± 0.04</td><td align=\"left\">0.79e ± 0.05</td></tr><tr><td align=\"left\">As + T + NH₃/Ctr</td><td align=\"left\">31.51f. ± 1.20</td><td align=\"left\">22.94e ± 1.47</td><td align=\"left\">49.07b ± 0.29</td><td align=\"left\">45.08c ± 0.77</td><td align=\"left\">0.94f. ± 0.06</td><td align=\"left\">0.82e ± 0.05</td><td align=\"left\">1.24f. ± 0.15</td><td align=\"left\">1.29f. ± 0.07</td></tr><tr><td align=\"left\">Ctr/Mn-4</td><td align=\"left\">7.38b ± 0.88</td><td align=\"left\">5.81b ± 0.56</td><td align=\"left\">46.77a ± 1.0</td><td align=\"left\">36.18a ± 1.14</td><td align=\"left\">0.22a ± 0.02</td><td align=\"left\">0.35b ± 0.02</td><td align=\"left\">0.42b ± 0.01</td><td align=\"left\">0.44b ± 0.06</td></tr><tr><td align=\"left\">Ctr/Mn-8</td><td align=\"left\">4.60a ± 0.70</td><td align=\"left\">3.98a ± 0.23</td><td align=\"left\">44.47a ± 0.56</td><td align=\"left\">36.07a ± 1.88</td><td align=\"left\">0.18a ± 0.01</td><td align=\"left\">0.17a ± 0.01</td><td align=\"left\">0.34a ± 0.02</td><td align=\"left\">0.29a ± 0.02</td></tr><tr><td align=\"left\">Ctr/Mn-12</td><td align=\"left\">11.44 cd ± 0.57</td><td align=\"left\">10.32c ± 1.04</td><td align=\"left\">45.11a ± 0.81</td><td align=\"left\">36.46a ± 2.22</td><td align=\"left\">0.45c ± 0.02</td><td align=\"left\">0.36b ± 0.03</td><td align=\"left\">0.81d ± 0.11</td><td align=\"left\">0.55c ± 0.08</td></tr><tr><td align=\"left\">As + T + NH_3/Mn-4</td><td align=\"left\">6.99b ± 0.76</td><td align=\"left\">6.59b ± 0.35</td><td align=\"left\">44.36a ± 0.87</td><td align=\"left\">37.29a ± 1.79</td><td align=\"left\">0.35b ± 0.03</td><td align=\"left\">0.40bc ± 0.02</td><td align=\"left\">0.58c ± 0.07</td><td align=\"left\">0.51c ± 0.02</td></tr><tr><td align=\"left\">As + T + NH_3/Mn-8</td><td align=\"left\">4.71a ± 0.90</td><td align=\"left\">4.13a ± 0.48</td><td align=\"left\">45.19a ± 1.01</td><td align=\"left\">35.37a ± 0.80</td><td align=\"left\">0.18a ± 0.01</td><td align=\"left\">0.20a ± 0.01</td><td align=\"left\">0.40ab ± 0.03</td><td align=\"left\">0.24a ± 0.01</td></tr><tr><td align=\"left\">As + T + NH₃/Mn-12</td><td align=\"left\">9.07c ± 0.79</td><td align=\"left\">10.95c ± 1.68</td><td align=\"left\">44.81a ± 0.92</td><td align=\"left\">37.29a ± 0.69</td><td align=\"left\">0.49c ± 0.07</td><td align=\"left\">0.45c ± 0.02</td><td align=\"left\">0.75d ± 0.05</td><td align=\"left\">0.56c ± 0.11</td></tr><tr><td align=\"left\"><italic>P</italic>_Value</td><td align=\"left\">0.0005</td><td align=\"left\">0.0001</td><td align=\"left\">0.018</td><td align=\"left\">0.037</td><td align=\"left\">0.0024</td><td align=\"left\">0.0047</td><td align=\"left\">0.0018</td><td align=\"left\">0.0013</td></tr></tbody></table></table-wrap>", "<table-wrap id=\"Tab6\"><label>Table 6</label><caption><p>Effect of dietary manganese (Mn) on growth performance viz. final body weight gain (%), FCR, SGR, PER, DGI (%), TGC and RFI against multiple stressors in fish.</p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th align=\"left\">Exposure/diets (mg kg⁻¹)</th><th align=\"left\">Final body weight gain %</th><th align=\"left\">FCR</th><th align=\"left\">SGR</th><th align=\"left\">PER</th><th align=\"left\">DGI (%)</th><th align=\"left\">TGC</th><th align=\"left\">RFI</th></tr></thead><tbody><tr><td align=\"left\">Ctr/Ctr</td><td char=\".\" align=\"char\">95.20d ± 3.26</td><td char=\".\" align=\"char\">3.22c ± 0.09</td><td char=\".\" align=\"char\">0.57c ± 0.01</td><td char=\".\" align=\"char\">0.89d ± 0.02</td><td char=\".\" align=\"char\">1.00de ± 0.03</td><td char=\".\" align=\"char\">0.0396</td><td char=\".\" align=\"char\">305.58d ± 2.38</td></tr><tr><td align=\"left\">As/Ctr</td><td char=\".\" align=\"char\">51.83b ± 3.27</td><td char=\".\" align=\"char\">5.25d ± 0.28</td><td char=\".\" align=\"char\">0.33b ± 0.02</td><td char=\".\" align=\"char\">0.55b ± 0.05</td><td char=\".\" align=\"char\">0.58b ± 0.02</td><td char=\".\" align=\"char\">0.0404</td><td char=\".\" align=\"char\">270.46b ± 2.52</td></tr><tr><td align=\"left\">NH₃/Ctr</td><td char=\".\" align=\"char\">49.89b ± 0.63</td><td char=\".\" align=\"char\">5.40d ± 0.06</td><td char=\".\" align=\"char\">0.35b ± 0.01</td><td char=\".\" align=\"char\">0.53b ± 0.02</td><td char=\".\" align=\"char\">0.56b ± 0.01</td><td char=\".\" align=\"char\">0.0389</td><td char=\".\" align=\"char\">269.22b ± 0.48</td></tr><tr><td align=\"left\">As + NH₃/Ctr</td><td char=\".\" align=\"char\">50.25b ± 3.67</td><td char=\".\" align=\"char\">5.39d ± 0.33</td><td char=\".\" align=\"char\">0.34b ± 0.01</td><td char=\".\" align=\"char\">0.55b ± 0.05</td><td char=\".\" align=\"char\">0.56b ± 0.03</td><td char=\".\" align=\"char\">0.0390</td><td char=\".\" align=\"char\">268.47b ± 2.56</td></tr><tr><td align=\"left\">NH₃ + T/Ctr</td><td char=\".\" align=\"char\">49.91b ± 0.54</td><td char=\".\" align=\"char\">5.36e ± 0.05</td><td char=\".\" align=\"char\">0.35b ± 0.02</td><td char=\".\" align=\"char\">0.58b ± 0.01</td><td char=\".\" align=\"char\">0.56b ± 0.01</td><td char=\".\" align=\"char\">0.0308</td><td char=\".\" align=\"char\">267.28b ± 2.11</td></tr><tr><td align=\"left\">As + T + NH₃/Ctr</td><td char=\".\" align=\"char\">42.03a ± 0.77</td><td char=\".\" align=\"char\">6.16f. ± 0.09</td><td char=\".\" align=\"char\">0.27a ± 0.03</td><td char=\".\" align=\"char\">0.50a ± 0.01</td><td char=\".\" align=\"char\">0.48a ± 0.02</td><td char=\".\" align=\"char\">0.0306</td><td char=\".\" align=\"char\">258.66a ± 0.91</td></tr><tr><td align=\"left\">Ctr/Mn-4</td><td char=\".\" align=\"char\">152.16f. ± 3.48</td><td char=\".\" align=\"char\">2.27b ± 0.04</td><td char=\".\" align=\"char\">0.86d ± 0.02</td><td char=\".\" align=\"char\">1.37f. ± 0.02</td><td char=\".\" align=\"char\">1.40f. ± 0.02</td><td char=\".\" align=\"char\">0.0391</td><td char=\".\" align=\"char\">344.68f. ± 2.36</td></tr><tr><td align=\"left\">Ctr/Mn-8</td><td char=\".\" align=\"char\">191.60 g ± 6.10</td><td char=\".\" align=\"char\">2.06a ± 0.06</td><td char=\".\" align=\"char\">1.08e ± 0.01</td><td char=\".\" align=\"char\">1.57 h ± 0.04</td><td char=\".\" align=\"char\">1.66 g ± 0.05</td><td char=\".\" align=\"char\">0.0382</td><td char=\".\" align=\"char\">394.17 h ± 1.78</td></tr><tr><td align=\"left\">Ctr/Mn-12</td><td char=\".\" align=\"char\">75.54c ± 2.69</td><td char=\".\" align=\"char\">3.87c ± 0.12</td><td char=\".\" align=\"char\">0.57c ± 0.02</td><td char=\".\" align=\"char\">0.74c ± 0.03</td><td char=\".\" align=\"char\">0.82c ± 0.03</td><td char=\".\" align=\"char\">0.0400</td><td char=\".\" align=\"char\">291.94c ± 1.84</td></tr><tr><td align=\"left\">As + T + NH_3/Mn-4</td><td char=\".\" align=\"char\">136.94e ± 4.85</td><td char=\".\" align=\"char\">2.42b ± 0.06</td><td char=\".\" align=\"char\">0.82d ± 0.03</td><td char=\".\" align=\"char\">1.21e ± 0.01</td><td char=\".\" align=\"char\">1.33e ± 0.03</td><td char=\".\" align=\"char\">0.0305</td><td char=\".\" align=\"char\">331.24e ± 3.92</td></tr><tr><td align=\"left\">As + T + NH_3/Mn-8</td><td char=\".\" align=\"char\">197.42 g ± 2.38</td><td char=\".\" align=\"char\">1.98a ± 0.03</td><td char=\".\" align=\"char\">1.11e ± 0.04</td><td char=\".\" align=\"char\">1.49 g ± 0.05</td><td char=\".\" align=\"char\">1.72 h ± 0.01</td><td char=\".\" align=\"char\">0.0308</td><td char=\".\" align=\"char\">389.88 g ± 2.59</td></tr><tr><td align=\"left\">As + T + NH₃/Mn-12</td><td char=\".\" align=\"char\">86.73c ± 5.91</td><td char=\".\" align=\"char\">3.44 ± c0.23</td><td char=\".\" align=\"char\">0.60c ± 0.03</td><td char=\".\" align=\"char\">0.93d ± 0.07</td><td char=\".\" align=\"char\">0.92d ± 0.05</td><td char=\".\" align=\"char\">0.0308</td><td char=\".\" align=\"char\">295.46c ± 1.87</td></tr><tr><td align=\"left\">P-Value</td><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/></tr></tbody></table></table-wrap>", "<table-wrap id=\"Tab7\"><label>Table 7</label><caption><p>Effect of dietary manganese (Mn) on LPO, Vit C, RBC, WBC and Hb against multiple stress in fish.</p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th align=\"left\">Exposure/diets (mg kg⁻¹)</th><th align=\"left\">LPO-L</th><th align=\"left\">LPO-K</th><th align=\"left\">Vit C-M</th><th align=\"left\">Vit C-B</th><th align=\"left\">RBC</th><th align=\"left\">WBC</th><th align=\"left\">Hb</th></tr></thead><tbody><tr><td align=\"left\">Ctr/Ctr</td><td align=\"left\">1.94c ± 0.44</td><td align=\"left\">20.77b ± 0.54</td><td align=\"left\">18.94e ± 0.54</td><td align=\"left\">16.50d ± 0.77</td><td align=\"left\">1.29c ± 0.0058</td><td align=\"left\">66.21e ± 0.90</td><td align=\"left\">5.09b ± 0.15</td></tr><tr><td align=\"left\">As/Ctr</td><td align=\"left\">2.93e ± 0.46</td><td align=\"left\">25.91c ± 0.37</td><td align=\"left\">12.82c ± 0.75</td><td align=\"left\">10.99c ± 0.72</td><td align=\"left\">1.36d ± 0.011</td><td align=\"left\">54.96c ± 0.99</td><td align=\"left\">5.78c ± 0.10</td></tr><tr><td align=\"left\">NH₃/Ctr</td><td align=\"left\">3.31ef ± 0.82</td><td align=\"left\">25.39c ± 1.96</td><td align=\"left\">12.73c ± 1.0</td><td align=\"left\">11.25c ± 1.03</td><td align=\"left\">1.34c ± 0.0053</td><td align=\"left\">54.43c ± 1.87</td><td align=\"left\">5.69c ± 0.03</td></tr><tr><td align=\"left\">As + NH₃/Ctr</td><td align=\"left\">3.25ef ± 0.41</td><td align=\"left\">32.34d ± 1.38</td><td align=\"left\">10.68b ± 0.76</td><td align=\"left\">8.24b ± 0.52</td><td align=\"left\">1.39e ± 0.051</td><td align=\"left\">50.69b ± 0.73</td><td align=\"left\">6.11d ± 0.15</td></tr><tr><td align=\"left\">NH₃ + T/Ctr</td><td align=\"left\">4.77f. ± 0.38</td><td align=\"left\">30.24d ± 0.37</td><td align=\"left\">9.65b ± 0.45</td><td align=\"left\">12.41c ± 3.28</td><td align=\"left\">1.40e ± 0.0087</td><td align=\"left\">50.19b ± 1.16</td><td align=\"left\">5.99c ± 0.25</td></tr><tr><td align=\"left\">As + T + NH₃/Ctr</td><td align=\"left\">5.46 g ± 0.55</td><td align=\"left\">40.89e ± 0.86</td><td align=\"left\">7.48a ± 0.52</td><td align=\"left\">5.79a ± 0.33</td><td align=\"left\">1.46f. ± 0.0084</td><td align=\"left\">45.98a ± 0.85</td><td align=\"left\">6.39e ± 0.02</td></tr><tr><td align=\"left\">Ctr/Mn-4</td><td align=\"left\">1.86c ± 0.47</td><td align=\"left\">20.94b ± 1.48</td><td align=\"left\">19.62e ± 0.57</td><td align=\"left\">16.22d ± 0.56</td><td align=\"left\">1.28b ± 0.047</td><td align=\"left\">65.02e ± 0.41</td><td align=\"left\">4.97a ± 0.14</td></tr><tr><td align=\"left\">Ctr/Mn-8</td><td align=\"left\">0.94a ± 0.02</td><td align=\"left\">12.77a ± 0.52</td><td align=\"left\">26.31 g ± 0.87</td><td align=\"left\">23.68f. ± 0.89</td><td align=\"left\">1.22a ± 0.0076</td><td align=\"left\">72.43f. ± 1.58</td><td align=\"left\">4.65a ± 0.03</td></tr><tr><td align=\"left\">Ctr/Mn-12</td><td align=\"left\">2.35 cd ± 0.28</td><td align=\"left\">24.82c ± 0.59</td><td align=\"left\">14.34d ± 0.90</td><td align=\"left\">9.88c ± 0.78</td><td align=\"left\">1.32c ± 0.011</td><td align=\"left\">57.42d ± 0.80</td><td align=\"left\">5.19b ± 0.15</td></tr><tr><td align=\"left\">As + T + NH_3/Mn-4</td><td align=\"left\">1.79bc ± 0.056</td><td align=\"left\">20.26b ± 0.97</td><td align=\"left\">18.69e ± 0.78</td><td align=\"left\">15.80d ± 0.66</td><td align=\"left\">1.31c ± 0.008</td><td align=\"left\">60.82e ± 1.09</td><td align=\"left\">4.82a ± 0.08</td></tr><tr><td align=\"left\">As + T + NH_3/Mn-8</td><td align=\"left\">1.27b ± 0.02</td><td align=\"left\">13.23a ± 1.02</td><td align=\"left\">23.89f. ± 1.03</td><td align=\"left\">21.15e ± 0.69</td><td align=\"left\">1.20a ± 0.02</td><td align=\"left\">71.03f. ± 0.87</td><td align=\"left\">4.65a ± 0.05</td></tr><tr><td align=\"left\">As + T + NH₃/Mn-12</td><td align=\"left\">2.78d ± 0.025</td><td align=\"left\">20.76b ± 0.62</td><td align=\"left\">14.59d ± 0.56</td><td align=\"left\">16.50d ± 0.15</td><td align=\"left\">1.33c ± 0.01</td><td align=\"left\">58.93d ± 1.08</td><td align=\"left\">5.20b ± 0.17</td></tr><tr><td align=\"left\"><italic>P</italic>_Value</td><td align=\"left\">0.0022</td><td align=\"left\">0.0053</td><td align=\"left\">0.015</td><td align=\"left\">0.021</td><td align=\"left\">0.0066</td><td align=\"left\">0.019</td><td align=\"left\">0.017</td></tr></tbody></table></table-wrap>", "<table-wrap id=\"Tab8\"><label>Table 8</label><caption><p>Effect of dietary manganese (Mn) on NBT, BG, total protein, albumin, globulin, A:G ratio, Ig and MPO against multiple stressors in fish.</p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th align=\"left\">Exposure/diets (mg kg⁻¹)</th><th align=\"left\">NBT</th><th align=\"left\">BG</th><th align=\"left\">Total Protein</th><th align=\"left\">Albumin</th><th align=\"left\">Globulin</th><th align=\"left\">A:G ratio</th><th align=\"left\">MPO</th></tr></thead><tbody><tr><td align=\"left\">Ctr/Ctr</td><td char=\".\" align=\"char\">0.56f. ± 0.05</td><td char=\".\" align=\"char\">111.23c ± 5.44</td><td char=\".\" align=\"char\">0.76e ± 0.03</td><td align=\"left\">0.26 ± 0.003</td><td char=\".\" align=\"char\">0.71e ± 0.12</td><td char=\".\" align=\"char\">0.36b ± 0.03</td><td char=\".\" align=\"char\">0.29 ± 0.003</td></tr><tr><td align=\"left\">As/Ctr</td><td char=\".\" align=\"char\">0.21ab ± 0.01</td><td char=\".\" align=\"char\">129.75d ± 3.98</td><td char=\".\" align=\"char\">0.55d ± 0.05</td><td align=\"left\">0.20 ± 0.035</td><td char=\".\" align=\"char\">0.34b ± 0.02</td><td char=\".\" align=\"char\">0.58d ± 0.09</td><td char=\".\" align=\"char\">0.19a ± 0.006</td></tr><tr><td align=\"left\">NH₃/Ctr</td><td char=\".\" align=\"char\">0.24b ± 0.03</td><td char=\".\" align=\"char\">140.86e ± 1.69</td><td char=\".\" align=\"char\">0.50c ± 0.03</td><td align=\"left\">0.16 ± 0.017</td><td char=\".\" align=\"char\">0.35b ± 0.01</td><td char=\".\" align=\"char\">0.45c ± 0.02</td><td char=\".\" align=\"char\">0.18a ± 0.003</td></tr><tr><td align=\"left\">As + NH₃/Ctr</td><td char=\".\" align=\"char\">0.20a ± 0.01</td><td char=\".\" align=\"char\">144.66f. ± 4.18</td><td char=\".\" align=\"char\">0.39b ± 0.03</td><td align=\"left\">0.15 ± 0.012</td><td char=\".\" align=\"char\">0.25a ± 0.02</td><td char=\".\" align=\"char\">0.60e ± 0.06</td><td char=\".\" align=\"char\">0.17a ± 0.007</td></tr><tr><td align=\"left\">NH₃ + T/Ctr</td><td char=\".\" align=\"char\">0.18a ± 0.02</td><td char=\".\" align=\"char\">168.93 g ± 6.17</td><td char=\".\" align=\"char\">0.54d ± 0.04</td><td align=\"left\">0.16 ± 0.006</td><td char=\".\" align=\"char\">0.38b ± 0.03</td><td char=\".\" align=\"char\">0.44c ± 0.03</td><td char=\".\" align=\"char\">0.16a ± 0.001</td></tr><tr><td align=\"left\">As + T + NH₃/Ctr</td><td char=\".\" align=\"char\">0.25b ± 0.03</td><td char=\".\" align=\"char\">146.61f. ± 5.0</td><td char=\".\" align=\"char\">0.32a ± 0.03</td><td align=\"left\">0.11 ± 0.002</td><td char=\".\" align=\"char\">0.21a ± 0.01</td><td char=\".\" align=\"char\">0.54d ± 0.03</td><td char=\".\" align=\"char\">0.16a ± 0.003</td></tr><tr><td align=\"left\">Ctr/Mn-4</td><td char=\".\" align=\"char\">0.48d ± 0.02</td><td char=\".\" align=\"char\">110.37c ± 5.97</td><td char=\".\" align=\"char\">0.81f. ± 0.04</td><td align=\"left\">0.19 ± 0.001</td><td char=\".\" align=\"char\">0.62d ± 0.03</td><td char=\".\" align=\"char\">0.30b ± 0.01</td><td char=\".\" align=\"char\">0.32c ± 0.038</td></tr><tr><td align=\"left\">Ctr/Mn-8</td><td char=\".\" align=\"char\">0.56e ± 0.06</td><td char=\".\" align=\"char\">88.05a ± 2.67</td><td char=\".\" align=\"char\">1.05 g ± 0.04</td><td align=\"left\">0.14 ± 0.009</td><td char=\".\" align=\"char\">0.91 g ± 0.05</td><td char=\".\" align=\"char\">0.15a ± 0.00</td><td char=\".\" align=\"char\">0.38d ± 0.009</td></tr><tr><td align=\"left\">Ctr/Mn-12</td><td char=\".\" align=\"char\">0.32c ± 0.02</td><td char=\".\" align=\"char\">129.75d ± 3.30</td><td char=\".\" align=\"char\">0.46bc ± 0.03</td><td align=\"left\">0.13 ± 0.014</td><td char=\".\" align=\"char\">0.33b ± 0.05</td><td char=\".\" align=\"char\">0.40b ± 0.09</td><td char=\".\" align=\"char\">0.20ab ± 0.006</td></tr><tr><td align=\"left\">As + T + NH_3/Mn-4</td><td char=\".\" align=\"char\">0.44d ± 0.07</td><td char=\".\" align=\"char\">112.13c ± 6.71</td><td char=\".\" align=\"char\">0.76e ± 0.06</td><td align=\"left\">0.23 ± 0.056</td><td char=\".\" align=\"char\">0.53d ± 0.03</td><td char=\".\" align=\"char\">0.47c ± 0.02</td><td char=\".\" align=\"char\">0.27b ± 0.007</td></tr><tr><td align=\"left\">As + T + NH_3/Mn-8</td><td char=\".\" align=\"char\">0.53e ± 0.03</td><td char=\".\" align=\"char\">95.13b ± 2.32</td><td char=\".\" align=\"char\">1.02f. ± 0.08</td><td align=\"left\">0.18 ± 0.014</td><td char=\".\" align=\"char\">0.84f. ± 0.03</td><td char=\".\" align=\"char\">0.21a ± 0.02</td><td char=\".\" align=\"char\">0.38d ± 0.006</td></tr><tr><td align=\"left\">As + T + NH₃/Mn-12</td><td char=\".\" align=\"char\">0.33c ± 0.02</td><td char=\".\" align=\"char\">129.80d ± 4.15</td><td char=\".\" align=\"char\">0.60de ± 0.03</td><td align=\"left\">0.15 ± 0.017</td><td char=\".\" align=\"char\">0.45c ± 0.05</td><td char=\".\" align=\"char\">0.34b ± 0.05</td><td char=\".\" align=\"char\">0.22a ± 0.007</td></tr><tr><td align=\"left\"><italic>P</italic>_Value</td><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td align=\"left\">0.071</td><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/></tr></tbody></table></table-wrap>", "<table-wrap id=\"Tab9\"><label>Table 9</label><caption><p>Effect of dietary manganese (Mn) on ALT, AST, LDH, MDH and AChE enzymatic activities against multiple stressors in fish.</p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th align=\"left\">Exposure/diets (mg kg⁻¹)</th><th align=\"left\">ALT-L</th><th align=\"left\">ALT-G</th><th align=\"left\">AST-L</th><th align=\"left\">AST-G</th><th align=\"left\">LDH-L</th><th align=\"left\">LDH-G</th><th align=\"left\">MDH-L</th><th align=\"left\">MDH-G</th><th align=\"left\">AChE</th></tr></thead><tbody><tr><td align=\"left\">Ctr/Ctr</td><td align=\"left\">11.66c ± 1.11</td><td align=\"left\">5.43b ± 0.49</td><td align=\"left\">11.86 cd ± 0.95</td><td align=\"left\">14.26d ± 1.26</td><td align=\"left\">1.94b ± 0.23</td><td align=\"left\">5.40d ± 0.37</td><td align=\"left\">0.97b ± 0.04</td><td align=\"left\">1.23c ± 0.03</td><td align=\"left\">0.53d ± 0.04</td></tr><tr><td align=\"left\">As/Ctr</td><td align=\"left\">14.49d ± 1.92</td><td align=\"left\">8.04c ± 0.57</td><td align=\"left\">16.14e ± 1.03</td><td align=\"left\">22.81e ± 1.67</td><td align=\"left\">2.93bc ± 0.18</td><td align=\"left\">8.15e ± 0.31</td><td align=\"left\">1.19c ± 0.02</td><td align=\"left\">1.79d ± 0.07</td><td align=\"left\">0.48c ± 0.06</td></tr><tr><td align=\"left\">NH₃/Ctr</td><td align=\"left\">14.87d ± 1.20</td><td align=\"left\">9.57c ± 1.31</td><td align=\"left\">17.54e ± 0.55</td><td align=\"left\">23.38e ± 2.43</td><td align=\"left\">3.31c ± 0.26</td><td align=\"left\">8.57e ± 0.40</td><td align=\"left\">1.55c ± 0.05</td><td align=\"left\">1.69d ± 0.04</td><td align=\"left\">0.43bc ± 0.05</td></tr><tr><td align=\"left\">As + NH₃/Ctr</td><td align=\"left\">16.08e ± 2.27</td><td align=\"left\">10.60d ± 1.11</td><td align=\"left\">19.75f. ± 1.34</td><td align=\"left\">23.44e ± 2.46</td><td align=\"left\">3.25c ± 0.11</td><td align=\"left\">7.82e ± 0.24</td><td align=\"left\">1.16c ± 0.06</td><td align=\"left\">1.52d ± 0.08</td><td align=\"left\">0.38b ± 0.04</td></tr><tr><td align=\"left\">NH₃ + T/Ctr</td><td align=\"left\">16.69e ± 0.95</td><td align=\"left\">14.30e ± 1.62</td><td align=\"left\">19.42f. ± 1.77</td><td align=\"left\">25.21f. ± 2.38</td><td align=\"left\">4.77d ± 0.27</td><td align=\"left\">7.34e ± 0.23</td><td align=\"left\">1.31c ± 0.09</td><td align=\"left\">1.73d ± 0.14</td><td align=\"left\">0.39b ± 0.03</td></tr><tr><td align=\"left\">As + T + NH₃/Ctr</td><td align=\"left\">20.90f. ± 1.71</td><td align=\"left\">18.87f. ± 2.21</td><td align=\"left\">22.58 g ± 2.09</td><td align=\"left\">31.26 g ± 1.70</td><td align=\"left\">5.46e ± 0.18</td><td align=\"left\">11.06f. ± 0.34</td><td align=\"left\">1.81d ± 0.06</td><td align=\"left\">2.05e ± 0.12</td><td align=\"left\">0.33a ± 0.04</td></tr><tr><td align=\"left\">Ctr/Mn-4</td><td align=\"left\">9.07b ± 0.92</td><td align=\"left\">6.47b ± 1.23</td><td align=\"left\">7.91b ± 0.58</td><td align=\"left\">12.19c ± 1.63</td><td align=\"left\">1.86ab ± 0.15</td><td align=\"left\">3.79c ± 0.31</td><td align=\"left\">0.93b ± 0.03</td><td align=\"left\">0.82b ± 0.014</td><td align=\"left\">0.53d ± 0.02</td></tr><tr><td align=\"left\">Ctr/Mn-8</td><td align=\"left\">5.76a ± 0.53</td><td align=\"left\">3.72a ± 0.78</td><td align=\"left\">4.87a ± 0.70</td><td align=\"left\">7.57a ± 0.76</td><td align=\"left\">0.94a ± 0.11</td><td align=\"left\">1.66a ± 0.91</td><td align=\"left\">0.62ab ± 0.15</td><td align=\"left\">0.57a ± 0.04</td><td align=\"left\">0.69e ± 0.07</td></tr><tr><td align=\"left\">Ctr/Mn-12</td><td align=\"left\">14.14d ± 0.93</td><td align=\"left\">14.76e ± 1.14</td><td align=\"left\">13.09d ± 1.46</td><td align=\"left\">12.78c ± 1.15</td><td align=\"left\">2.35bc ± 0.28</td><td align=\"left\">4.86 ± 0.83</td><td align=\"left\">1.29d ± 0.04</td><td align=\"left\">1.28c ± 0.13</td><td align=\"left\">0.39b ± 0.05</td></tr><tr><td align=\"left\">As + T + NH_3/Mn-4</td><td align=\"left\">8.39b ± 0.80</td><td align=\"left\">5.95b ± 0.52</td><td align=\"left\">9.32c ± 1.04</td><td align=\"left\">10.25b ± 0.73</td><td align=\"left\">1.79b ± 0.16</td><td align=\"left\">2.60b ± 0.78</td><td align=\"left\">0.94b ± 0.04</td><td align=\"left\">1.05bc ± 0.12</td><td align=\"left\">0.51d ± 0.05</td></tr><tr><td align=\"left\">As + T + NH_3/Mn-8</td><td align=\"left\">5.62a ± 0.43</td><td align=\"left\">3.07a ± 1.15</td><td align=\"left\">6.09b ± 0.37</td><td align=\"left\">7.88a ± 0.90</td><td align=\"left\">1.27a ± 0.15</td><td align=\"left\">1.54a ± 0.27</td><td align=\"left\">0.42a ± 0.01</td><td align=\"left\">0.68a ± 0.04</td><td align=\"left\">0.72e ± 0.08</td></tr><tr><td align=\"left\">As + T + NH₃/Mn-12</td><td align=\"left\">14.85d ± 1.24</td><td align=\"left\">11.12d ± 1.06</td><td align=\"left\">10.13c ± 0.76</td><td align=\"left\">13.25 cd ± 0.76</td><td align=\"left\">2.78bc ± 0.36</td><td align=\"left\">3.67c ± 0.87</td><td align=\"left\">1.09c ± 0.03</td><td align=\"left\">1.65d ± 0.03</td><td align=\"left\">0.44bc ± 0.06</td></tr><tr><td align=\"left\"><italic>P</italic>_Value</td><td align=\"left\">0.0054</td><td align=\"left\">0.018</td><td align=\"left\">0.0072</td><td align=\"left\">0.0081</td><td align=\"left\">0.003</td><td align=\"left\">0.0063</td><td align=\"left\">0.0029</td><td align=\"left\">0.0015</td><td align=\"left\">0.0039</td></tr></tbody></table></table-wrap>", "<table-wrap id=\"Tab10\"><label>Table 10</label><caption><p>Effect of dietary manganese (Mn) on detoxification of arsenic in different fish tissues reared under control and multiple stressors condition.</p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th align=\"left\">Exposure/diets (mg kg⁻¹)</th><th align=\"left\">Water (µg L⁻¹)</th><th align=\"left\">Liver (mg kg⁻¹)</th><th align=\"left\">Kidney (mg kg⁻¹)</th><th align=\"left\">Gill (mg kg⁻¹)</th><th align=\"left\">Muscle (mg kg⁻¹)</th><th align=\"left\">Brain (mg kg⁻¹)</th><th align=\"left\">Mn-Muscle (mg kg⁻¹)</th></tr></thead><tbody><tr><td align=\"left\">Ctr/Ctr</td><td char=\".\" align=\"char\">0.06 ± 0.001</td><td align=\"left\">0.01 ± 0.001</td><td char=\".\" align=\"char\">0.04 ± 0.0001</td><td char=\".\" align=\"char\">0.04 ± 0.001</td><td align=\"left\">BDL</td><td align=\"left\">BDL</td><td char=\".\" align=\"char\">0.42 ± 0.02</td></tr><tr><td align=\"left\">As/Ctr</td><td char=\".\" align=\"char\">1077.51 ± 33.37</td><td align=\"left\">5.24 ± 0.15</td><td char=\".\" align=\"char\">5.95 ± 0.28</td><td char=\".\" align=\"char\">3.21 ± 0.05</td><td align=\"left\">1.57 ± 0.11</td><td align=\"left\">0.33 ± 0.046</td><td char=\".\" align=\"char\">0.64 ± 0.07</td></tr><tr><td align=\"left\">NH₃/Ctr</td><td char=\".\" align=\"char\">0.04 ± 0.001</td><td align=\"left\">0.03 ± 0.01</td><td char=\".\" align=\"char\">0.11 ± 0.01</td><td char=\".\" align=\"char\">0.02 ± 0.001</td><td align=\"left\">0.04 ± 0.01</td><td align=\"left\">BDL</td><td char=\".\" align=\"char\">0.80 ± 0.12</td></tr><tr><td align=\"left\">As + NH₃/Ctr</td><td char=\".\" align=\"char\">1186.28 ± 21.09</td><td align=\"left\">5.84 ± 0.09</td><td char=\".\" align=\"char\">6.60 ± 0.17</td><td char=\".\" align=\"char\">3.79 ± 0.07</td><td align=\"left\">1.42 ± 0.13</td><td align=\"left\">0.36 ± 0.038</td><td char=\".\" align=\"char\">0.36 ± 0.06</td></tr><tr><td align=\"left\">NH₃ + T/Ctr</td><td char=\".\" align=\"char\">0.09 ± 0.01</td><td align=\"left\">0.04 ± 0.001</td><td char=\".\" align=\"char\">0.07 ± 0.01</td><td char=\".\" align=\"char\">0.08 ± 0.001</td><td align=\"left\">0.02 ± 0.001</td><td align=\"left\">BDL</td><td char=\".\" align=\"char\">0.47 ± 0.04</td></tr><tr><td align=\"left\">As + T + NH₃/Ctr</td><td char=\".\" align=\"char\">1776.47 ± 55.13</td><td align=\"left\">7.40 ± 0.18</td><td char=\".\" align=\"char\">7.08 ± 0.07</td><td char=\".\" align=\"char\">1.76 ± 0.05</td><td align=\"left\">2.08 ± 0.06</td><td align=\"left\">0.49 ± 0.019</td><td char=\".\" align=\"char\">0.33 ± 0.04</td></tr><tr><td align=\"left\">Ctr/Mn-4</td><td char=\".\" align=\"char\">0.03 ± 0.01</td><td align=\"left\">0.01 ± 0.001</td><td char=\".\" align=\"char\">0.07 ± 0.01</td><td char=\".\" align=\"char\">0.09 ± 0.001</td><td align=\"left\">BDL</td><td align=\"left\">BDL</td><td char=\".\" align=\"char\">5.74 ± 0.20</td></tr><tr><td align=\"left\">Ctr/Mn-8</td><td char=\".\" align=\"char\">0.03 ± 0.001</td><td align=\"left\">BDL</td><td char=\".\" align=\"char\">0.05 ± 0.001</td><td char=\".\" align=\"char\">0.02 ± 0.001</td><td align=\"left\">BDL</td><td align=\"left\">BDL</td><td char=\".\" align=\"char\">9.49 ± 0.15</td></tr><tr><td align=\"left\">Ctr/Mn-12</td><td char=\".\" align=\"char\">0.03 ± 0.001</td><td align=\"left\">0.03 ± 0.001</td><td char=\".\" align=\"char\">0.09 ± 0.002</td><td char=\".\" align=\"char\">0.07 ± 0.01</td><td align=\"left\">0.04 ± 0.001</td><td align=\"left\">0.05 ± 0.004</td><td char=\".\" align=\"char\">14.17 ± 0.14</td></tr><tr><td align=\"left\">As + T + NH_3/Mn-4</td><td char=\".\" align=\"char\">1040.62 ± 28.97</td><td align=\"left\">2.99 ± 0.13</td><td char=\".\" align=\"char\">3.54 ± 0.14</td><td char=\".\" align=\"char\">3.03 ± 0.10</td><td align=\"left\">0.74 ± 0.11</td><td align=\"left\">0.73 ± 0.022</td><td char=\".\" align=\"char\">3.22 ± 0.05</td></tr><tr><td align=\"left\">As + T + NH_3/Mn-8</td><td char=\".\" align=\"char\">243.50 ± 6.89</td><td align=\"left\">0.37 ± 0.03</td><td char=\".\" align=\"char\">0.57 ± 0.06</td><td char=\".\" align=\"char\">0.80 ± 0.07</td><td align=\"left\">0.17 ± 0.07</td><td align=\"left\">0.06 ± 0.008</td><td char=\".\" align=\"char\">6.48 ± 0.54</td></tr><tr><td align=\"left\">As + T + NH₃/Mn-12</td><td char=\".\" align=\"char\">1337.87 ± 49.84</td><td align=\"left\">2.85 ± 0.11</td><td char=\".\" align=\"char\">2.17 ± 0.04</td><td char=\".\" align=\"char\">2.89 ± 0.32</td><td align=\"left\">1.67 ± 0.30</td><td align=\"left\">0.81 ± 0.01</td><td char=\".\" align=\"char\">11.18 ± 0.45</td></tr></tbody></table></table-wrap>" ]

[ "<disp-formula id=\"Equa\"><alternatives><tex-math id=\"M1\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\text{FCR}}\\; = \\;{\\text{Total}}\\;{\\text{dry}}\\;{\\text{feed}}\\;{\\text{intake}}\\;\\left( g \\right)/{\\text{Wet}}\\;{\\text{weight}}\\;{\\text{gain}}\\;\\left( g \\right)$$\\end{document}</tex-math><mml:math id=\"M2\" display=\"block\"><mml:mrow><mml:mtext>FCR</mml:mtext><mml:mspace width=\"0.277778em\"/><mml:mo>=</mml:mo><mml:mspace width=\"0.277778em\"/><mml:mtext>Total</mml:mtext><mml:mspace width=\"0.277778em\"/><mml:mtext>dry</mml:mtext><mml:mspace width=\"0.277778em\"/><mml:mtext>feed</mml:mtext><mml:mspace width=\"0.277778em\"/><mml:mtext>intake</mml:mtext><mml:mspace width=\"0.277778em\"/><mml:mfenced close=\")\" open=\"(\"><mml:mi>g</mml:mi></mml:mfenced><mml:mo stretchy=\"false\">/</mml:mo><mml:mtext>Wet</mml:mtext><mml:mspace width=\"0.277778em\"/><mml:mtext>weight</mml:mtext><mml:mspace width=\"0.277778em\"/><mml:mtext>gain</mml:mtext><mml:mspace width=\"0.277778em\"/><mml:mfenced close=\")\" open=\"(\"><mml:mi>g</mml:mi></mml:mfenced></mml:mrow></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equb\"><alternatives><tex-math id=\"M3\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\text{SGR }} = { 1}00 \\, \\left( {{\\text{ln FBW}} - {\\text{ln IBW}}} \\right)/{\\text{ number of days}}$$\\end{document}</tex-math><mml:math id=\"M4\" display=\"block\"><mml:mrow><mml:mrow><mml:mtext>SGR</mml:mtext><mml:mspace width=\"0.333333em\"/></mml:mrow><mml:mo>=</mml:mo><mml:mn>100</mml:mn><mml:mspace width=\"0.166667em\"/><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:mrow><mml:mtext>ln FBW</mml:mtext></mml:mrow><mml:mo>-</mml:mo><mml:mrow><mml:mtext>ln IBW</mml:mtext></mml:mrow></mml:mrow></mml:mfenced><mml:mo stretchy=\"false\">/</mml:mo><mml:mrow><mml:mspace width=\"0.333333em\"/><mml:mtext>number of days</mml:mtext></mml:mrow></mml:mrow></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equc\"><alternatives><tex-math id=\"M5\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\text{Weight gain }}\\left( \\% \\right) \\, = {\\text{ Final body weight }}\\left( {{\\text{FBW}}} \\right) - {\\text{Initial body weight }}\\left( {{\\text{IBW}}} \\right)/{\\text{Initial body weight }}\\left( {{\\text{IBW}}} \\right) \\, \\times {1}00$$\\end{document}</tex-math><mml:math id=\"M6\" display=\"block\"><mml:mrow><mml:mrow><mml:mtext>Weight gain</mml:mtext><mml:mspace width=\"0.333333em\"/></mml:mrow><mml:mfenced close=\")\" open=\"(\"><mml:mo>%</mml:mo></mml:mfenced><mml:mspace width=\"0.166667em\"/><mml:mo>=</mml:mo><mml:mrow><mml:mspace width=\"0.333333em\"/><mml:mtext>Final body weight</mml:mtext><mml:mspace width=\"0.333333em\"/></mml:mrow><mml:mfenced close=\")\" open=\"(\"><mml:mtext>FBW</mml:mtext></mml:mfenced><mml:mo>-</mml:mo><mml:mrow><mml:mtext>Initial body weight</mml:mtext><mml:mspace width=\"0.333333em\"/></mml:mrow><mml:mfenced close=\")\" open=\"(\"><mml:mtext>IBW</mml:mtext></mml:mfenced><mml:mo stretchy=\"false\">/</mml:mo><mml:mrow><mml:mtext>Initial body weight</mml:mtext><mml:mspace width=\"0.333333em\"/></mml:mrow><mml:mfenced close=\")\" open=\"(\"><mml:mtext>IBW</mml:mtext></mml:mfenced><mml:mspace width=\"0.166667em\"/><mml:mo>×</mml:mo><mml:mn>100</mml:mn></mml:mrow></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equd\"><alternatives><tex-math id=\"M7\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\text{Relative feed intake}}, \\, \\left( {{\\text{FI}}} \\right) \\, \\left( {\\% /{\\text{d}}} \\right) \\, = { 1}00 \\, \\times \\, \\left( {{\\text{TFI}}/{\\rm I}{\\text{BW}}} \\right)$$\\end{document}</tex-math><mml:math id=\"M8\" display=\"block\"><mml:mrow><mml:mrow><mml:mtext>Relative feed intake</mml:mtext></mml:mrow><mml:mo>,</mml:mo><mml:mspace width=\"0.166667em\"/><mml:mfenced close=\")\" open=\"(\"><mml:mtext>FI</mml:mtext></mml:mfenced><mml:mspace width=\"0.166667em\"/><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:mo>%</mml:mo><mml:mo stretchy=\"false\">/</mml:mo><mml:mtext>d</mml:mtext></mml:mrow></mml:mfenced><mml:mspace width=\"0.166667em\"/><mml:mo>=</mml:mo><mml:mn>100</mml:mn><mml:mspace width=\"0.166667em\"/><mml:mo>×</mml:mo><mml:mspace width=\"0.166667em\"/><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:mtext>TFI</mml:mtext><mml:mo stretchy=\"false\">/</mml:mo><mml:mi mathvariant=\"normal\">I</mml:mi><mml:mtext>BW</mml:mtext></mml:mrow></mml:mfenced></mml:mrow></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Eque\"><alternatives><tex-math id=\"M9\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\text{PER}} = {\\text{ Total wet weight gain }}\\left( {\\text{g}} \\right)/{\\text{crude protein intake }}\\left( {\\text{g}} \\right)$$\\end{document}</tex-math><mml:math id=\"M10\" display=\"block\"><mml:mrow><mml:mtext>PER</mml:mtext><mml:mo>=</mml:mo><mml:mrow><mml:mspace width=\"0.333333em\"/><mml:mtext>Total wet weight gain</mml:mtext><mml:mspace width=\"0.333333em\"/></mml:mrow><mml:mfenced close=\")\" open=\"(\"><mml:mtext>g</mml:mtext></mml:mfenced><mml:mo stretchy=\"false\">/</mml:mo><mml:mrow><mml:mtext>crude protein intake</mml:mtext><mml:mspace width=\"0.333333em\"/></mml:mrow><mml:mfenced close=\")\" open=\"(\"><mml:mtext>g</mml:mtext></mml:mfenced></mml:mrow></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equf\"><alternatives><tex-math id=\"M11\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\text{Thermal growth coefficient}}, \\, \\left( {{\\text{TGC}}} \\right) \\, = \\, \\left( {{\\text{FBW}}^{{{1}/{3}}} {-}{\\text{ IBW}}^{{{1}/{3}}} } \\right) \\, \\times \\, \\left( {\\Sigma {\\text{D}}0} \\right)^{{ - {1}}} ,{\\text{ where }}\\Sigma {\\text{D}}0{\\text{ is the thermal sum }}\\left( {{\\text{feeding days }} \\times {\\text{ average temperature}}, \\, ^{ \\circ } {\\text{C}}} \\right)$$\\end{document}</tex-math><mml:math id=\"M12\" display=\"block\"><mml:mrow><mml:mrow><mml:mtext>Thermal growth coefficient</mml:mtext></mml:mrow><mml:mo>,</mml:mo><mml:mspace width=\"0.166667em\"/><mml:mfenced close=\")\" open=\"(\"><mml:mtext>TGC</mml:mtext></mml:mfenced><mml:mspace width=\"0.166667em\"/><mml:mo>=</mml:mo><mml:mspace width=\"0.166667em\"/><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:msup><mml:mrow><mml:mtext>FBW</mml:mtext></mml:mrow><mml:mrow><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">/</mml:mo><mml:mn>3</mml:mn></mml:mrow></mml:msup><mml:mo>-</mml:mo><mml:msup><mml:mrow><mml:mspace width=\"0.333333em\"/><mml:mtext>IBW</mml:mtext></mml:mrow><mml:mrow><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">/</mml:mo><mml:mn>3</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:mfenced><mml:mspace width=\"0.166667em\"/><mml:mo>×</mml:mo><mml:mspace width=\"0.166667em\"/><mml:msup><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:mi mathvariant=\"normal\">Σ</mml:mi><mml:mtext>D</mml:mtext><mml:mn>0</mml:mn></mml:mrow></mml:mfenced><mml:mrow><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mo>,</mml:mo><mml:mrow><mml:mspace width=\"0.333333em\"/><mml:mtext>where</mml:mtext><mml:mspace width=\"0.333333em\"/></mml:mrow><mml:mi mathvariant=\"normal\">Σ</mml:mi><mml:mtext>D</mml:mtext><mml:mn>0</mml:mn><mml:mrow><mml:mspace width=\"0.333333em\"/><mml:mtext>is the thermal sum</mml:mtext><mml:mspace width=\"0.333333em\"/></mml:mrow><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:mrow><mml:mtext>feeding days</mml:mtext><mml:mspace width=\"0.333333em\"/></mml:mrow><mml:mo>×</mml:mo><mml:mrow><mml:mspace width=\"0.333333em\"/><mml:mtext>average temperature</mml:mtext></mml:mrow><mml:mo>,</mml:mo><mml:msup><mml:mspace width=\"0.166667em\"/><mml:mo>∘</mml:mo></mml:msup><mml:mtext>C</mml:mtext></mml:mrow></mml:mfenced></mml:mrow></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equg\"><alternatives><tex-math id=\"M13\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\text{Daily growth index}},{\\text{ DGI }}\\left( \\% \\right) \\, = \\, \\left( {{\\text{FBW}}^{{{1}/{3}}} {-}{\\text{ IBW}}^{{{1}/{3}}} } \\right)/{\\text{days }} \\times { 1}00$$\\end{document}</tex-math><mml:math id=\"M14\" display=\"block\"><mml:mrow><mml:mrow><mml:mtext>Daily growth index</mml:mtext></mml:mrow><mml:mo>,</mml:mo><mml:mrow><mml:mspace width=\"0.333333em\"/><mml:mtext>DGI</mml:mtext><mml:mspace width=\"0.333333em\"/></mml:mrow><mml:mfenced close=\")\" open=\"(\"><mml:mo>%</mml:mo></mml:mfenced><mml:mspace width=\"0.166667em\"/><mml:mo>=</mml:mo><mml:mspace width=\"0.166667em\"/><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:msup><mml:mrow><mml:mtext>FBW</mml:mtext></mml:mrow><mml:mrow><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">/</mml:mo><mml:mn>3</mml:mn></mml:mrow></mml:msup><mml:mo>-</mml:mo><mml:msup><mml:mrow><mml:mspace width=\"0.333333em\"/><mml:mtext>IBW</mml:mtext></mml:mrow><mml:mrow><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">/</mml:mo><mml:mn>3</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:mfenced><mml:mo stretchy=\"false\">/</mml:mo><mml:mrow><mml:mtext>days</mml:mtext><mml:mspace width=\"0.333333em\"/></mml:mrow><mml:mo>×</mml:mo><mml:mn>100</mml:mn></mml:mrow></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equh\"><alternatives><tex-math id=\"M15\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\text{Cumulative mortality }}\\left( \\% \\right) \\, =\\frac{{\\text{ Total mortality in each treatment after challenge }}}{{\\text{ Total no}}.{\\text{ of fish challenged for the same treatments}}} \\times {1}00$$\\end{document}</tex-math><mml:math id=\"M16\" display=\"block\"><mml:mrow><mml:mrow><mml:mtext>Cumulative mortality</mml:mtext><mml:mspace width=\"0.333333em\"/></mml:mrow><mml:mfenced close=\")\" open=\"(\"><mml:mo>%</mml:mo></mml:mfenced><mml:mspace width=\"0.166667em\"/><mml:mo>=</mml:mo><mml:mfrac><mml:mrow><mml:mspace width=\"0.333333em\"/><mml:mtext>Total mortality in each treatment after challenge</mml:mtext><mml:mspace width=\"0.333333em\"/></mml:mrow><mml:mrow><mml:mrow><mml:mspace width=\"0.333333em\"/><mml:mtext>Total no</mml:mtext></mml:mrow><mml:mo>.</mml:mo><mml:mrow><mml:mspace width=\"0.333333em\"/><mml:mtext>of fish challenged for the same treatments</mml:mtext></mml:mrow></mml:mrow></mml:mfrac><mml:mo>×</mml:mo><mml:mn>100</mml:mn></mml:mrow></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equi\"><alternatives><tex-math id=\"M17\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\text{Relative }}\\% {\\text{ survival }} = \\frac{{\\text{ Mortality }}\\left( \\% \\right){\\text{ Control }}{-}{\\text{ Mortality }}\\left( \\% \\right){\\text{ Treatment }}}{{\\text{ Mortality }}\\left( \\% \\right){\\text{ Control}}}\\times { 1}00$$\\end{document}</tex-math><mml:math id=\"M18\" display=\"block\"><mml:mrow><mml:mrow><mml:mtext>Relative</mml:mtext><mml:mspace width=\"0.333333em\"/></mml:mrow><mml:mo>%</mml:mo><mml:mrow><mml:mspace width=\"0.333333em\"/><mml:mtext>survival</mml:mtext><mml:mspace width=\"0.333333em\"/></mml:mrow><mml:mo>=</mml:mo><mml:mfrac><mml:mrow><mml:mrow><mml:mspace width=\"0.333333em\"/><mml:mtext>Mortality</mml:mtext><mml:mspace width=\"0.333333em\"/></mml:mrow><mml:mfenced close=\")\" open=\"(\"><mml:mo>%</mml:mo></mml:mfenced><mml:mrow><mml:mspace width=\"0.333333em\"/><mml:mtext>Control</mml:mtext><mml:mspace width=\"0.333333em\"/></mml:mrow><mml:mo>-</mml:mo><mml:mrow><mml:mspace width=\"0.333333em\"/><mml:mtext>Mortality</mml:mtext><mml:mspace width=\"0.333333em\"/></mml:mrow><mml:mfenced close=\")\" open=\"(\"><mml:mo>%</mml:mo></mml:mfenced><mml:mrow><mml:mspace width=\"0.333333em\"/><mml:mtext>Treatment</mml:mtext><mml:mspace width=\"0.333333em\"/></mml:mrow></mml:mrow><mml:mrow><mml:mrow><mml:mspace width=\"0.333333em\"/><mml:mtext>Mortality</mml:mtext><mml:mspace width=\"0.333333em\"/></mml:mrow><mml:mfenced close=\")\" open=\"(\"><mml:mo>%</mml:mo></mml:mfenced><mml:mrow><mml:mspace width=\"0.333333em\"/><mml:mtext>Control</mml:mtext></mml:mrow></mml:mrow></mml:mfrac><mml:mo>×</mml:mo><mml:mn>100</mml:mn></mml:mrow></mml:math></alternatives></disp-formula>" ]

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[ "<table-wrap-foot><p>^aProcured from local market, ^bHimedia Ltd, Himedia Ltd, ^cSD Fine Chemicals Ltd., India.</p><p>*Manual prepared Vitamin mineral mixture; Composition of vitamin mineral mix (quantity/250 g starch powder): vitamin A 55,00,00 IU; vitamin D3 11,00,00 IU; vitamin B1:20 mg; vitamin E 75 mg; vitamin K 1,00 mg; vitamin B12 0.6 mcg; calcium pantothenate 2,50 mg; nicotinamide 1000 mg; pyridoxine: 100 mg; Zn 500 mg; I 1,00 mg; Fe 750 mg; Cu 200 mg; Co 45 mg; Ca 50 g; P 30 g; Se: 2 ppm.</p><p>Digestible energy (DE) (Kcal/100 g)=(%CP×)+(%EE×9)+(TC×4)</p><p>Data expressed as mean ± SE, <italic>n</italic> = 3.</p></table-wrap-foot>", "<table-wrap-foot><p>SOD: Superoxide dismutase; CAT: Catalase; GPx: Glutathione peroxidase; HSP: Heat shock protein; iNOS: Nitric oxide synthase; TNFα: Tumor necrosis factor; TLR: Toll like receptor; Ghr: Growth hormone receptor; IL; Interleukin; Ig: Immunoglobulin; MYST: myostatin SMT; Somatostatin; CYP P450: Cytochrome P450; MT: Metallothionine; Cas 3a and 3b: caspase 3; GH: Growth hormone; IGF1 and 2: Insulin like growth factor.</p></table-wrap-foot>", "<table-wrap-foot><p>Values in the same row with different superscript (a, b, c, d, e, f) differ significantly. Data expressed as Mean ± SE (<italic>n</italic> = 6). CAT, SOD, GST and GPx: Units/mg protein.</p></table-wrap-foot>", "<table-wrap-foot><p>Values in the same row with different superscript (a, b, c, d, e) differ significantly. Data expressed as Mean ± SE (<italic>n</italic> = 3). FCR: feed conversion ratio; SGR: specific growth rate; PER: protein efficiency ratio; DGI: Daily growth index; TGC: Thermal growth coefficient; RFI: relative feed intake.</p></table-wrap-foot>", "<table-wrap-foot><p>Values in the same row with different superscript (a, b, c, d, e, f) differ significantly. Data expressed as Mean ± SE (<italic>n</italic> = 6). LPO: n mole TBARS formed/h/mg protein; Vit C: µg/g of wet tissue; RBC: Number (10⁶ cell/mm³); WBC: Number (10³ Cell/mm³); Hb: gm (dl).</p></table-wrap-foot>", "<table-wrap-foot><p>Values in the same row with different superscript (a, b, c, d, e) differ significantly. Total protein, albumin, globulin: g dL⁻¹Blood glucose: mgdL⁻¹; Data expressed as Mean ± SE (<italic>n</italic> = 3).</p></table-wrap-foot>", "<table-wrap-foot><p>Values in the same row with different superscript (a, b, c, d, e, f) differ significantly. Data expressed as Mean ± SE (<italic>n</italic> = 6). ALT: nmole of sodium pyruvate formed/mg protein/min at 37 °C, AST: nmole oxaloacetate released/min/mg protein at 37 °C. LDH and MDH: units/min/mg protein at 37 °C, AChE: nmole/min/mg protein.</p></table-wrap-foot>", "<fn-group><fn><p><bold>Publisher's note</bold></p><p>Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.</p></fn></fn-group>" ]

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{ "acronym": [], "definition": [] }

[ "<title>Introduction</title>", "<p id=\"Par2\">Ki-67 immunohistochemistry (IHC) serves as a reliable marker of cell proliferation and is widely used to evaluate the aggressiveness and prognosis of human tumors. Notably, Ki-67 has been adopted for prognostication in breast cancer, with elevated Ki-67 expression correlating with poorer prognosis^{##REF##28154782##1##,##REF##20152769##2##}. The Ki-67 proliferation index (PI) in breast cancer is a measure of the percentage of tumor cells with nuclear immunoreactivity relative to the total number of malignant cells assessed^{##REF##16192605##3##}. A meta-analysis of 64,196 patients revealed that higher Ki-67 PI values are associated with worse overall survival in breast cancer, with 25% being a cutoff of strong outcome prognostication^{##REF##26341751##4##}.</p>", "<p id=\"Par3\">The monarchE committee reported that among patients with early-stage HR+, HER2− breast cancer, and nodal involvement, the addition of abemaciclib to hormone therapy significantly improves cancer-specific free survival and decreases the risk of disease recurrence^{##REF##36493792##5##–##REF##32954927##7##}. For tumor stage 1 to 2, nodal stage 0 to 1, ER+/HER2− breast cancer, the International Ki-67 in Breast Cancer Working Group’s (IKWG) consensus in 2021 recommended using Ki-67 to aid in the decision-making of adjuvant chemotherapy only for cases with a very low (&lt; 5%) or very high (&gt; 30%) PI due to substantial inter-rater variability within this range^{##REF##33369635##8##,##REF##21960707##9##}. The panelists of the St. Gallen International Consensus Conference in 2021 generally support this recommendation^{##REF##34242744##10##}. The monarchE phase III clinical trial studied the impact of a high Ki-67 PI on disease recurrence in a cohort of patients with HR+/HER2− node-positive breast cancer with high-risk clinicopathological features (at least 4 positive lymph nodes, or 1 to 3 positive lymph nodes with either tumor size ≥ 5 cm or histological grade 3 disease). Their analyses demonstrated that a Ki-67 PI ≥ 20% in patients treated with endocrine therapy alone was associated with a significantly increased risk of recurrence within three years compared to patients with lower Ki-67 expression^{##REF##34656740##6##,##REF##35384873##11##}. Following this, the American Food and Drug Administration and Health Canada approved the use of abemaciclib (CDK4/6 inhibitor) for patients with HR+/HER2− high-risk early breast cancer and a Ki-67 PI of ≥ 20%^{##UREF##0##12##}. In a recently published landmark study^{##REF##37585627##13##} based on 500 patients, it was demonstrated that a PI score threshold of &lt; 13.25% derived from Ki-67 slides effectively identified women with luminal A breast cancer who could be safely treated without local breast radiation therapy. This underscores the clinical significance of Ki-67 as a marker with significant promise in guiding management decisions for breast cancer patients.</p>", "<p id=\"Par4\">The current gold standard for quantifying Ki-67 PI is to manually evaluate at least 500 malignant cells based on IKWG recommendations^{##REF##33369635##8##,##REF##21960707##9##}. However, this method is labor-intensive, time-consuming, and prone to poor inter-rater reproducibility and errors^{##REF##34503265##14##,##REF##23449362##15##}. As a result, it is hard to standardize and use Ki-67 for clinical assessments. As shown in the recommendations from the IKWG^{##REF##33369635##8##,##REF##21960707##9##}, the assessment by the pathologist is most reliable for PI values below 5% and above 30% (the 5 to 30% range is subject to the most interpretation variability). The Canadian Association of Pathologists recommends that a second pathologist assess PIs in this range, or use a computer assessment tool to improve robustness^{##UREF##0##12##}. Considering this range is critical for treatment decisions, its reliability must be improved. The recent emergence of digital pathology and high-performance AI algorithms offers the possibility that automated PI scoring can overcome these challenges by accurately and efficiently measuring cell count. There have been several AI-based Ki-67 assessment tools developed^{##UREF##1##16##–##REF##31632956##21##}, and the advantages are becoming increasingly evident.</p>", "<p id=\"Par5\">Several comparative studies have reported the role of AI-assisted assessments of Ki-67 PI in breast cancer^{##REF##33590577##19##,##REF##36931740##22##,##UREF##8##32##}. These studies demonstrated that AI-aided assessment of Ki-67 could achieve a lower mean error^{##REF##33840132##23##} and a lower standard error deviation^{##REF##33590577##19##}, however, the impact on inter-rater agreement is less clear. Additionally, while these studies have encompassed broad PI ranges from 0 to 100%, the effect of AI assistance in the clinically crucial 5 to 30% PI interval has not yet been studied.</p>", "<p id=\"Par6\">Herein, we conducted a large-scale, international study that analyzed the effects of AI assistance on key aspects of pathologists' work, including accuracy, inter-rater agreement, and turnaround time (TAT) in the context of Ki-67 scoring for breast cancer. Our focus was on assessing these metrics within the 5 to 30% PI range to better understand the implications and usability of AI-assisted Ki-67 evaluations. Additionally, we gathered insights into pathologists' perspectives, trust levels, and readiness to adopt AI technologies, highlighting the importance of user acceptance. This study provides a strong foundation for understanding the future impact and potential of AI tools for Ki-67 scoring in the daily routine of pathologists.</p>" ]

[ "<title>Materials and methods</title>", "<p id=\"Par7\">Ethics approval for the study was obtained from Toronto Metropolitan University (REB: 2022-154). All experiments were performed in accordance with the Tri-Council policy statement 2 for the ethical conduct of research involving humans.</p>", "<title>Case selection, TMA preparation, and image acquisition</title>", "<p id=\"Par8\">A subset of ten TMAs from the Toronto-British Columbia trial was used for this study^{##REF##25964246##24##}, which was composed of node-negative patients above the age of 50 years with invasive breast cancer^{##REF##15342804##25##}. Tissue microarrays (TMAs) were constructed using a 0.6 mm tumor core procured from formalin-fixed, paraffin-embedded specimens. TMA sections, with a thickness of 0.5 μm, were stained using a 1:500 dilution of SP6 (ThermoFisher Scientific, Waltham, MA, USA)—a Ki-67 antibody—and counterstained with hematoxylin. The study incorporated ten TMAs with high tumor cellularity, averaging 2093 neoplastic cells per TMA and a PI range of 7 to 28%^{##UREF##1##16##}. This range, which poses a challenge for pathologists, encompasses the clinically relevant PI cutoffs identified in prior studies^{##REF##34656740##6##,##REF##35384873##11##,##REF##37585627##13##}.</p>", "<title>AI tool</title>", "<p id=\"Par9\">A deep learning-based AI tool for IHC quantification, UV-Net, developed by Toronto Metropolitan University, was used in the study^{##UREF##3##18##}. This tool detects neoplastic cells in IHC-stained tissue and differentiates Ki-67 positive from Ki-67 negative tumor cells. Its underlying architecture, a modified U-Net, includes additional connections for densely packed nuclei and replaces the standard 3 × 3 convolutional layers with 'V-Blocks'. These V-Block connections maintain high-resolution nuclear features for precise differentiation between nuclei; each V-Block inputs n channels and outputs 2n channels, forming a 'V' shape across four successive stages.</p>", "<p id=\"Par10\">The AI tool was trained using 256 × 256 RGB patches of WSIs from St. Michael's Hospital, and an open-source dataset \"Deepslides\"^{##REF##30359393##26##} from × 20 Aperio AT Turbo and × 40 Aperio ScanScope scanners respectively. Images were annotated with single-pixel centroid markers distinguishing Ki-67 positive and Ki-67 negative tumor nuclei cells^{##REF##33375043##20##}, defining positive nuclei as any brown color above the background, following the IKWG’s recommendations^{##REF##33369635##8##,##REF##21960707##9##}. Single-pixel markers were extended into circular areas using a Gaussian function, this allocated the highest value to the center of the nuclei, incorporated more contextual information, and improved the efficiency of the training process. A Huber loss function was used to regress and predict the centroid of nuclei.</p>", "<p id=\"Par11\">For a given image, the AI tool generates an automated Ki-67 positive and Ki-67 negative overlay (Fig. ##FIG##0##1##), providing an accessible visual interpretation along with the automated PI calculation.</p>", "<p id=\"Par12\">The generalizability of UV-Net was previously validated on multi-institutional datasets from 5 institutions^{##UREF##3##18##}, including WSIs and TMAs from breast cancer images. UV-Net consistently outperformed other architectures across all image variations, registering an average F1-score of 0.83 on expertly annotated data. In comparison, alternative architectures achieved scores between 0.74 and 0.79.</p>", "<p id=\"Par13\">The images on which UV-Net was trained differed from those used in this study, originating from datasets with different scanners and institutions. None of the pathologists involved in this study participated in annotating the training or validation datasets.</p>", "<title>Study design</title>", "<p id=\"Par14\">A cross-sectional study was performed using an anonymous, self-administered, and structured online survey developed using Qualtrics™, which included hyperlinks for viewing digitized TMAs on the cloud through PathcoreFlow™, a browser-based commercial image management solution and viewer for digital pathology^{##UREF##4##27##}. The AI tool for Ki-67 scoring was integrated into PathcoreFlow™ using an Application Programming Interface. The tool provided an overlay of the Ki-67 positive and negative nuclei and calculated PI scores (Fig. ##FIG##0##1##). Participants were presented with a digital invasive breast cancer TMA stained for Ki-67 for each question and were asked to assign a Ki-67 score by entering a percentage value into Qualtrics™. Examples of questions with and without AI assistance are shown in Supplementary Figs. ##SUPPL##0##1## and ##SUPPL##0##2##. Each Ki-67 TMA was reviewed by respondents twice—once without AI assistance and once with AI assistance—resulting in a total of 20 assessment questions. Participants were not explicitly told to use the AI, but rather to observe the AI results and estimate their PI score. They were instructed to compute the Ki-67 PI by counting individual cells with a denomination of 500 cells and to regard any brown staining beyond the background as positive, in line with current guidelines^{##REF##33369635##8##,##REF##21960707##9##}. They were also guided to spend approximately the same time they would during standard procedures with no limit on the time for assessments. Each pathologist used a distinct viewer from a separate workstation. To minimize bias, the order of cases was randomized, ensuring that TMAs with AI assistance were not shown immediately before or after the same TMA without assistance. Additionally, the AI-assisted images were altered in orientation to look different from the unaided images. At the end of the study, participants were requested to provide their demographic information and respond to inquiries regarding their perspectives on AI.</p>", "<title>Study population</title>", "<p id=\"Par15\">Participants were recruited through the professional networks of the authors between September and November 2022. Contact channels included pathology associations, local pathology residency programs, pathologist colleagues, and social media platforms (LinkedIn, Twitter). Eligible participants were trained pathologists with experience in Ki-67 PI scoring. The study included all participants who provided consent and identified themselves as pathology specialists. There were no limitations based on gender, age, or employment status, and only those who finished the study were considered, in total there were 116 completed responses. Spurious responses defined as outliers with large PI errors (more than 20% on a single response) were excluded from the analysis (N = 26 participants). Consequently, the main analysis included 90 respondents, all experienced in using digital pathology. Demographic characteristics are described in Supplemental Table ##SUPPL##0##1##. The participants' median age ranged from 40 to 49 years; however, the most common age group was 30 to 39 years, accounting for 34.4% of the respondents. While the median work experience falls within the 10 to 19 years range, the most prevalent work experience category is 0 to 9 years, representing 26.7% of the total. The majority of respondents are male, with many being retired clinical pathologists from North America. Among those currently working, most practice in academic health sciences centers.</p>", "<title>Ground truth scores</title>", "<p id=\"Par16\">The ground truth Ki-67 PI scores for the 10 TMAs were determined using the gold standard manual counting method, where any brown staining above the background level was deemed positive, following current guidelines^{##REF##33369635##8##,##REF##21960707##9##}. Each TMA was divided into five rows and five columns, creating 400 × 400 pixel tiles, and annotations were made in each region. Nuclei were annotated at the center of each cell, with tumor cells marked as Ki-67 positive if any discernible brown staining above the background was observed and the cell border was visible; otherwise, they were marked as Ki-67 negative. In cases of overlapping tumor cells, each cell was marked individually if its borders were discernible. An anatomical pathology resident (N.N.J.N.) performed the manual annotations, which were verified by a breast pathologist (S.D.). Ground truth PI scores were calculated from these manual annotations. The ground truth PI scores of the ten cases ranged from 7 to 28%.</p>", "<title>Statistical analysis</title>", "<p id=\"Par17\">Statistical analyses were performed to assess the PI scoring error, inter-rater agreement, and TAT among pathologists when using the AI tool, compared to a standard clinical workflow (i.e., without AI). The experiment involved two groups: a control group where pathologists evaluated Ki-67 PI using standard clinical methods, and an experimental group where the same pathologists used the AI tool to assist with Ki-67 PI assessment on the same TMAs. For each participant, two PI estimations and TATs were obtained per TMA, resulting in 900 paired assessments (90 pathologists × 10 cases). For every assessment, several metrics were recorded, including the clinician-estimated raw PI score, the PI error (the absolute difference between the estimated and ground truth PI), and TAT, which denotes the time taken to score the TMA. The paired Wilcoxon signed-rank test^{##UREF##5##28##} was used to compare the differences between the two groups, with significance determined based on the median values of the paired differences. This test was chosen due to the non-normal distribution of the data, as indicated by the Shapiro–Wilk test. All statistical analyses were two-sided, with significance set at <italic>p</italic> &lt; 0.05.</p>", "<p id=\"Par18\">PI scores and PI errors were assessed with and without AI assistance, using continuous and binary values. PI scores and PI errors were first treated as continuous values and summarized by the mean and standard deviation. Box and bar plots were used to visually depict case-based and sub-demographic PI errors, respectively. PI scores and errors were additionally binarized and assessed using low-risk Ki-67 PI &lt; 20%, and high-risk ≥ 20% stratification^{##UREF##0##12##}.</p>", "<p id=\"Par19\">The consistency of scoring among pathologists, with and without AI assistance, was examined using both continuous and binary metrics. For the continuous analysis, the Two-Way Random-Effects Model for single-rater consistency agreement was chosen to assess the inter-rater agreement using the Intraclass Correlation Coefficient (ICC)^{##REF##27330520##29##,##UREF##6##30##}. This model was selected since all cases were evaluated by all raters. The choice of the single-rater model stemmed from the clinical reliance on a singular clinician's decision for Ki-67 scores, rather than averaging scores from multiple clinicians^{##REF##27330520##29##}. The ICC between the pathologists’ PI scores and the ground truth PI was assessed twice: once with and once without AI assistance. Complementary to ICC, Krippendorff's α was calculated to measure inter-rater agreement and chosen for its adaptability in handling continuous data^{##UREF##7##31##}. Bland–Altman and linear regression plots of the PI scores were incorporated to supplement the measure of inter-rater agreement, with parameters such as Pearson’s correlation coefficient, slope, offset, mean, and limits of agreement being considered. Using binarized PI scores (with scores ≥ 20% assigned a 1, and scores &lt; 20% a 0), the percent agreement and Fleiss’ Kappa^{##UREF##8##32##} were calculated for both groups.</p>", "<p id=\"Par20\">The TAT among pathologists was considered the time in seconds to perform the PI score estimation, starting from the moment they began examining the case to the point when the PI score was saved. TATs were summarized by the mean and standard deviation. Box and bar plots were used to visually depict case-based and sub-demographic TATs, respectively. Additionally, the percentage of time reduction computed by the time savings was determined by the formula: (total time saved/total time spent on conventional assessment) X 100%. Statistical analyses were performed using SPSS Version 28 (Armonk, NY, USA).</p>", "<title>Ethics approval</title>", "<p id=\"Par21\">This research study has been reviewed by the Toronto Metropolitan University Research Ethics Board (REB 2022-154). Participants voluntarily consented to participate and to share contact information if they wanted to.</p>" ]

[ "<title>Results</title>", "<title>Scoring accuracy</title>", "<p id=\"Par22\">The respondents' PI scores and PI errors per case and within ranges are shown in Table ##TAB##0##1##. Responses including outliers are shown in Supplementary Table ##SUPPL##0##2##. The overall mean PI error was found to be 2.1 (2.2) using the AI tool, and 5.9 (4.0) without the AI (difference of − 3.8%, 95% CI: −4.10% to −3.51%, <italic>p</italic> &lt; 0.001). The AI tool significantly improved the accuracy of PI scoring. The PI error was plotted per case (Fig. ##FIG##1##2##A) and for each PI interval (Supplementary Fig. ##SUPPL##0##3##). Cases 2 through 10 had significantly less error (<italic>p</italic> &lt; 0.001), and both the &lt; 20% and ≥ 20% PI ranges had statistically significant decreases in error with AI (<italic>p</italic> &lt; 0.001). Furthermore, Fig. ##FIG##1##2##B, C, which display the PI error across various demographics, revealed that AI-aided scoring was superior across all pathologist age ranges and experience levels—indicating that despite variable background and training, AI improved PI accuracy for all groups of pathologists. Supplementary Fig. ##SUPPL##0##4## shows that AI-aided scoring was statistically superior (<italic>p</italic> &lt; 0.001) across all pathologist subdisciplines.</p>", "<p id=\"Par23\">The AI tool demonstrated high accuracy in the study, with a mean PI error rate of 0.6%, which ranged from 0.0 to 6.1%, as shown in Table ##TAB##0##1##.</p>", "<p id=\"Par24\">To quantify the increase of PI estimation accuracy when pathologists used the AI tool, Supplementary Fig. ##SUPPL##0##5## shows the difference in PI error for each case. This difference is calculated as the PI error for the estimated PI score with AI assistance minus the error without AI assistance, highlighting the extent to which the AI tool reduces error rates. Most pathologists experienced increased accuracy with the AI tool, as indicated by positive differences seen in Supplementary Fig. ##SUPPL##0##5##.</p>", "<title>Inter-rater agreement</title>", "<p id=\"Par25\">AI assistance led to a significant improvement in inter-observer reproducibility (with AI assistance: ICC = 0.92 [95% CI 0.85–0.98], Krippendorff’s α = 0.89 [95% CI 0.71–0.92], without AI assistance: ICC = 0.70 [95% CI 0.52–0.89], Krippendorff’s α = 0.65 [95% CI 0.41–0.72]). These statistics are visually depicted in Supplementary Fig. ##SUPPL##0##6##. Bland–Altman analyses (Fig. ##FIG##2##3##B, D) revealed that pathologists with AI assistance exhibited less bias (mean of 0.7 vs. − 2.7) and tighter limits of agreement (6.5 to − 5.1 vs. 10.2 to − 15.6) compared to the ground truth scores. Linear regression models (Fig. ##FIG##2##3##A, C) further support the notion that AI assistance improves inter-rater agreement (with AI assistance: y = 1.06x − 0.46, r = 0.92, SSE = 7792; without AI assistance: y = 0.64x + 3.62, r = 0.58, SSE = 33,992).</p>", "<p id=\"Par26\">After binarizing the pathologists' responses, with scores ≥ 20% assigned as 1 and scores &lt; 20% as 0, the Fleiss’ Kappa values showed better agreement with AI assistance (with AI assistance: 0.86 [95% CI 0.85–0.86]; without AI assistance: 0.40 [95% CI 0.40–0.41]). Table ##TAB##1##2## shows that agreement levels are increased for every case when using AI, with some cases achieving 100% agreement.</p>", "<title>Turnaround time</title>", "<p id=\"Par27\">A visual depiction of TATs for each case is provided in Fig. ##FIG##3##4##A. Table ##TAB##2##3## displays the mean response time, standard deviation, and time saved for each TMA case for PI scoring with and without the AI aid.</p>", "<p id=\"Par28\">Without AI assistance, pathologists required an average of 23.3 s to assess each TMA, with a median time of 7.5 s and an interquartile range (IQR) of 5.5 to 16.2 s. AI assistance led to a statistically significant increase (<italic>p</italic> &lt; 0.001) in efficiency where the average TAT per TMA reduced to 18.6 s, a median time of 6.4 s and a narrower IQR from 4.6 to 12.1 s.</p>", "<p id=\"Par29\">Figure ##FIG##3##4##B illustrates the TAT for each question, showing the progression of TAT across cases as they were presented to the pathologists. Due to initially high response times, likely caused by participants acclimating to the software and study setup, question 1 (Case 2 without aid and Case 7 with aid) was excluded from further analyses.</p>", "<p id=\"Par30\">For evaluations without AI, pathologists averaged 18.3 s per TMA, with a median time of 7.2 s and an IQR of 5.5 to 14.0 s. With AI support, the average TAT per TMA decreased to 16.8 s, the median time was 6.4 s, and the IQR narrowed to 4.7 to 11.6 s. The reduction in TAT was statistically significant among pathologists with experience ranging from 10 to 39 years (Fig. ##FIG##3##4##C) (<italic>p</italic> &lt; 0.001), and for pathology fellows, practicing and retired pathologists (Fig. ##FIG##3##4##D) (<italic>p</italic> &lt; 0.001). Supplemental Fig. ##SUPPL##0##7##shows the mean TAT with and without aid for various disciplines, where roles such as clinical and forensic pathologists were statistically faster (<italic>p</italic> &lt; 0.001).</p>", "<p id=\"Par31\">AI assistance resulted in an average reduction of 1.5 s per TMA [95% CI, −2.4 to −0.6 s, <italic>p</italic> &lt; 0.001]. Supplementary Fig. ##SUPPL##0##8## displays a histogram of the distribution of the total percentage of time saved, calculated using the formula: (total time saved/total time spent on conventional assessment) × 100%. The mean percentage saving was 9.4%, with a median of 11.9%.</p>", "<title>Pathologists’ opinions</title>", "<p id=\"Par32\">Pathologists’ opinions on the use of AI for Ki-67 assessment in breast cancer are summarized in Fig. ##FIG##4##5##. The majority of respondents considered the AI tool's suggestion, found it to be appropriate and agreed that this AI tool could improve accuracy, inter-rater agreement and TAT for Ki-67 assessments (Fig. ##FIG##4##5##A). Many respondents also agreed that they would personally implement and agree with the routine implementation of AI aid for Ki-67 assessments within the next decade (Fig. ##FIG##4##5##B, C).</p>" ]

[ "<title>Discussion</title>", "<p id=\"Par33\">Ki-67 serves as a crucial indicator for predicting cancer recurrence and survival among early-stage high-risk breast cancer patients^{##REF##28154782##1##,##REF##20152769##2##}. It informs decisions regarding adjuvant chemotherapy^{##UREF##0##12##} and radiation therapy opt-out for Luminal A breast cancer patients^{##REF##37585627##13##}. These clinical decisions often rely on PI scores between 5 and 30%; however, this range exhibits significant scoring variability among experts, making standardization and clinical application challenging^{##REF##33369635##8##,##REF##21960707##9##,##UREF##0##12##}. This inconsistency, combined with long assessment times using the current Ki-67 scoring system, has limited the broader clinical application of Ki-67 and resultantly, has not yet been integrated into all clinical workflows^{##UREF##1##16##}. AI technologies are being proposed to improve Ki-67 scoring accuracy, inter-rater agreement, and TAT. This study explores the influence of AI in these three areas by recruiting 90 pathologists to examine ten breast cancer TMAs with PIs in the range of 7 to 28%.</p>", "<p id=\"Par34\">Two previous studies aimed to quantify PI accuracy with and without AI^{##REF##33590577##19##,##REF##33840132##23##}. One study demonstrated that AI-enhanced microscopes improved invasive breast cancer assessment accuracy^{##REF##33840132##23##}. They had 30 pathologists use an AI microscope to evaluate 100 invasive ductal carcinoma IHC-stained whole slide images (WSIs), which provided tumor delineations, and cell annotations. AI use resulted in a mean PI error reduction from 9.60 to 4.53. A similar study was conducted^{##REF##33590577##19##}, where eight pathologists assessed 200 regions of interest using an AI tool. Pathologists identified hotspots on WSIs, after which the AI tool provided cell annotations for the clinician's review. The study found that this method significantly improved the accuracy of Ki-67 PI compared to traditional scoring (14.9 error without AI vs. 6.9 error with AI).</p>", "<p id=\"Par35\">Similarly, this study found that using AI assistance for PI scoring significantly (<italic>p</italic> &lt; 0.001) improved pathologists’ accuracy, reducing both the PI error and its standard deviation across various demographics, including years of experience and specialties. This indicates that AI assistance leads to higher PI accuracy across all levels of pathologists' training, enabling professionals at every career stage to deliver more precise PI scores in the range critical for clinical decision-making. This improvement may help bridge experience gaps and is critical for PI scoring standardization. An underestimation trend, previously reported by^{##REF##25375149##33##}, was also noted in this study, as shown by the PI correlation and Bland–Altman analysis (Fig. ##FIG##2##3##). However, scoring with the support of AI improved PI accuracy for all cases and corrected this underestimation bias. This is exemplified by the scoring near the 20% cutoff, which simulates a clinical decision threshold. In conventional assessments, many pathologists select the incorrect range (≥ 20% or &lt; 20%), particularly for TMAs 7, 8, and 9, with ground truths of 19.8, 23.7, and 28.2, respectively. For instance, TMA 8 had 76.7% of respondents incorrectly estimated the score as &lt; 20%. Errors like these would result in incorrect therapy decisions and poor patient outcomes. Fortunately, with AI assistance, the percentage of pathologists agreeing with the ground truth greatly improved, providing a strong incentive for the clinical use of AI tools in Ki-67 scoring. All cases showed a statistically significant PI error decrease with AI assistance, except for Case 1, with a ground truth PI score of 7.3% (p = 0.133). This exception could be attributed to fewer Ki-67 positive cells requiring counting, which likely simplified the scoring process.</p>", "<p id=\"Par36\">In addition to accuracy, PI scoring agreement is critical to ensure that patients with similar disease phenotypes are delivered the proper therapeutic regimes. However, significant variability in Ki-67 scoring is widely recognized, even in established laboratories. A study led by^{##REF##24203987##34##}, found reproducibility among eight labs was only moderately reliable with contributing factors such as subjective judgements related to PI scoring and tumor region selection. Standardizing scoring methods becomes imperative, as transferring Ki-67 PIs and cutoffs between laboratories would compromise analytical validity. In another study by^{##REF##33803148##35##}, the variability in breast cancer biomarker assessments, including Ki-67, among pathology departments in Sweden was investigated. While positivity rates for HR and HER2 had low variability, there was substantial variation in Ki-67 scoring, where 66% of labs showed significant intra-laboratory variability. This variability could potentially affect the distribution of endocrine and HER2-targeted treatments, emphasizing the need for improved scoring methods to ensure consistent and dependable clinical decision-making. The study by^{##REF##33840132##23##}, aimed to improve Ki67 scoring concordance with their AI-empowered microscope. They found a higher ICC of 0.930 (95% CI: 0.91–0.95) with AI, compared to 0.827 (95% CI: 0.79–0.87) without AI. Similarly^{##REF##36931740##22##}, aimed to quantify the inter-rater agreement for WSIs with AI assistance across various clinical settings. The AI tool evaluated 72 Ki-67 breast cancer slides by annotating Ki-67 cells and providing PI scores. Ten pathologists from eight institutes reviewed the tool and input their potentially differing PI scores. When the scores were categorized using a PI cutoff of 20%, there was an 87.6% agreement between traditional and AI-assisted methods. Results also revealed a Krippendorff's α of 0.69 in conventional eyeballing quantification and 0.72 with AI assistance indicative of increased inter-rater agreement, however, these findings were not significant.</p>", "<p id=\"Par37\">In this study, we evaluated the scoring agreement with and without AI across 90 pathologists, representing one of the largest cohorts analyzed for this task. It was found that over the critical PI range of 7 to 28%, AI improved the inter-rater agreement, with superior ICC, Krippendorff’s α and Fleiss’ Kappa values compared to conventional assessments and higher correlation of PI estimates with the ground truth PI score. Additionally, there was a decrease in offset and variability, as shown in Fig. ##FIG##2##3##. These agreement metrics align with findings from earlier studies^{##REF##36931740##22##,##REF##33840132##23##} and signify that AI tools can standardize Ki-67 scoring, enhance reproducibility and reduce the subjective differences seen with conventional assessments. Therefore, using an AI tool for Ki-67 scoring could lead to more robust assessments and consistent therapeutic decisions.</p>", "<p id=\"Par38\">AI applications have predominantly focused on automating the laborious tasks for pathologists, thereby freeing up time for high-level, critical decision-making, especially those related to more complex disease presentations^{##UREF##1##16##–##REF##33375043##20##,##REF##27423409##36##}. Some research into AI support tools in this field has demonstrated a notable decrease in TAT for pathologists. For instance, a study led by^{##REF##33180129##37##}, which involved 20 pathologists analyzing 240 prostate biopsies, reported that an AI-based assistive tool significantly reduced TAT, with 13.5% less time spent on assisted reviews than on unassisted ones. Similarly, the study by^{##REF##32035484##38##}, demonstrated a statistical improvement (<italic>p</italic> &lt; 0.05) in TATs when 24 raters counted breast mitotic figures in 140 high-power fields, with and without AI support, ultimately achieving a time saving of 27.8%. However, the study by^{##REF##33840132##23##}, reported a longer TAT using an AI-empowered microscope in their study, which involved 100 invasive ductal carcinoma WSIs and 30 pathologists (11.6 s without AI vs. 23.8 s with AI).</p>", "<p id=\"Par39\">Our study found that AI support resulted in faster TATs (18.3 s without AI vs. 16.8 s with AI, <italic>p</italic> &lt; 0.001), equating to a median time saving of 11.9%. Currently, our team only performs Ki-67 testing upon oncologists' requests, as routine Ki-67 assessment is not yet standard practice. This is partly due to the difficulties in standardizing Ki-67, compounded by pathologists' increasing workloads and concerns over burnout^{##UREF##9##39##,##UREF##10##40##}. Pathologists' caseloads have grown in the past decade, from 109 to 116 annually in Canada and 92 to 132 in the U.S.^{##REF##31150073##41##}. With the Canadian Cancer Society expecting 29,400 breast cancer cases in 2023^{##UREF##11##42##}, routine Ki-67 assessments would significantly increase workloads. Therefore, the implementation of AI tools in this context could alleviate workload pressures by offering substantial time savings and supporting the clinical application of this important biomarker.</p>", "<p id=\"Par40\">The gold standard for assessing Ki-67 PI is manual counting^{##REF##33369635##8##,##REF##21960707##9##}; however, due to the labor-intensive nature of this method, many pathologists often resort to rough visual estimations^{##REF##32590453##43##,##REF##23543399##44##}. As indicated in Table ##TAB##2##3## and Fig. ##FIG##3##4##, the shorter TATs suggest that respondents may have relied on visual estimations for Ki-67 scoring. Despite this, the TATs significantly improved (<italic>p</italic> &lt; 0.001) when using AI. This improvement was evident among experienced pathologists; however, some encountered longer TATs after integrating AI, possibly due to unfamiliarity with the AI tool or digital pathology viewing software. Although participants received a brief orientation and two initial examples, the novelty of the tool might have posed a learning curve. Addressing this challenge involves integrating the tool into regular practice and providing comprehensive training before its use.</p>", "<p id=\"Par41\">The perspectives of pathologists highlight a growing enthusiasm towards AI integration for Ki-67 evaluations for breast cancer. A significant 84% of participants agreed the AI’s recommendations were suitable for the task at hand. They recognized AI's ability to improve pathologists' accuracy (76%), enhance inter-rater consistency (82%), and reduce the TAT for Ki-67 evaluations (83%). Additionally, 49% expressed their intent to incorporate AI into their workflow, and 47% anticipated the routine implementation of AI within the next decade. An important observation is that many respondents who were hesitant about personally or routinely implementing AI in clinical practice were retired pathologists. In total, 83% of retired pathologists reported they would not currently implement AI personally or routinely, which is a stark contrast to only 15% of practicing pathologists who expressed the same reluctance. This positive outlook in the pathologist community supports the insights of this study and signals an increasing momentum for the widespread adoption of AI into digital pathology.</p>", "<p id=\"Par42\">The strength of this research is highlighted by the extensive and diverse participation of 90 pathologists, which contributes to the study's generalizability in real-world clinical contexts. Adding to the study's credibility is the focus on Ki-67 values around the critical 20% threshold, which is used for adjuvant therapy decisions. Moreover, the AI nuclei overlay addresses the transparency concerns often associated with AI-generated scores, thus improving clarity and comprehensibility for users. The ongoing discussion around 'explainable AI' highlights the importance of transparency in AI tools' outputs, a crucial factor for their acceptance and adoption^{##UREF##12##45##}. The outcomes of the study emphasize the positive outlook and readiness of pathologists to embrace AI in their workflow and serve to reinforce the growing need for the integration of AI into regular medical practice.</p>", "<p id=\"Par43\">The study has its limitations, one of which includes the potential unintentional inclusion of non-pathologists. The survey required respondents to confirm their status as pathologists through agreement before beginning; however, due to confidentiality limitations, no further verification was possible. In some instances, pathologists' scores deviated from the ground truth by more than 20%, with PI errors reaching up to 50%. Such large errors would render any PI score diagnostically irrelevant, as the variance exceeds the clinical threshold of 20%. These errors might be attributed to input errors or a lack of experience in Ki-67 assessments. Consequently, we used this threshold to filter out potentially erroneous responses. In total, 26 participants who logged responses exceeding the 20% error threshold were subsequently excluded from the study. For completeness, Supplementary Table ##SUPPL##0##2## discloses the PI scores and PI errors of all respondents, including outliers, where the data trends appear similar. The demographics of the study's participants reveal there was limited participation from currently practicing pathologists, representing 14.4% of respondents. This may be attributed to the time constraints faced by practicing pathologists. In future research, efforts will be made to include more practicing pathologists and to evaluate intra-observer variability. Additionally, while the survey provided specific guidelines for calculating the PI and applying Ki-67 positivity criteria, the accuracy and thoroughness of each pathologist's evaluations could not be verified. Lastly, the study deviated from standard practice by using TMAs instead of WSIs for Ki-67 clinical assessments. The rationale behind this choice was the expectation of more precise scoring with TMAs, as this eliminates the need to select high-power fields (a subjective process) and involves a lower number of cells to evaluate, leading to better consistency in visual estimations. Future research should focus on evaluating the accuracy achieved with AI assistance in identifying regions of interest and analyzing WSIs. This should also incorporate a broader range of cases and a wider PI range. Prospective studies involving solely practicing breast pathologists could also yield valuable insights into the real-world application of the AI tool and its impact on clinical decision-making.</p>", "<p id=\"Par44\">In conclusion, this study provides early insights into the potential of an AI tool in improving the accuracy, inter-rater agreement, and workflow efficiency of Ki-67 assessment in breast cancer. As AI tools become more widely adopted, ongoing evaluation and refinement will be essential to fully realize its potential and optimize patient care. Such tools are critical for robustly analyzing large datasets and effectively determining PI thresholds for treatment decisions.</p>" ]

[]

[ "<p id=\"Par1\">The Ki-67 proliferation index (PI) guides treatment decisions in breast cancer but suffers from poor inter-rater reproducibility. Although AI tools have been designed for Ki-67 assessment, their impact on pathologists' work remains understudied. 90 international pathologists were recruited to assess the Ki-67 PI of ten breast cancer tissue microarrays with and without AI. Accuracy, agreement, and turnaround time with and without AI were compared. Pathologists’ perspectives on AI were collected. Using AI led to a significant decrease in PI error (2.1% with AI vs. 5.9% without AI, <italic>p</italic> &lt; 0.001), better inter-rater agreement (ICC: 0.70 vs. 0.92; Krippendorff’s α: 0.63 vs. 0.89; Fleiss’ Kappa: 0.40 vs. 0.86), and an 11.9% overall median reduction in turnaround time. Most pathologists (84%) found the AI reliable. For Ki-67 assessments, 76% of respondents believed AI enhances accuracy, 82% said it improves consistency, and 83% trust it will improve efficiency. This study highlights AI's potential to standardize Ki-67 scoring, especially between 5 and 30% PI—a range with low PI agreement. This could pave the way for a universally accepted PI score to guide treatment decisions, emphasizing the promising role of AI integration into pathologist workflows.</p>", "<title>Subject terms</title>" ]

[ "<title>Supplementary Information</title>", "<p>\n</p>" ]

[ "<title>Supplementary Information</title>", "<p>The online version contains supplementary material available at 10.1038/s41598-024-51723-2.</p>", "<title>Acknowledgements</title>", "<p>This research acknowledges the Canadian Cancer Society and Mitacs.</p>", "<title>Author contributions</title>", "<p>A.D. designed the study, did the statistical analyses, gathered, and interpreted the data, conceived the tables and figures, and drafted the manuscript. N.N.J.N. did the pathologic annotations of the study samples for the ground truth references, contributed to the interpretation of the data, and helped draft the manuscript. J.M. helped design the study. S.D. helped design the study and validated the pathologic annotations of the study samples for the ground truth references. A.K. (principal investigator) designed the study, contributed to the interpretation of the data, helped draft the manuscript, and supervised the project. All authors helped with the recruitment of participants, revised the manuscript, and agreed with the final version of the manuscript.</p>", "<title>Funding</title>", "<p>This publication is funded by the Canadian Cancer Society.</p>", "<title>Data availability</title>", "<p>The datasets used and/or analyzed during the current study are available from the corresponding author upon reasonable request.</p>", "<title>Competing interests</title>", "<p id=\"Par45\">The authors declare no competing interests.</p>" ]

[ "<fig id=\"Fig1\"><label>Figure 1</label><caption><p>Examples of TMA with no AI aid (left) and TMA with AI tool overlay and calculated proliferation index (PI) (right). The TMA shown is case 7.</p></caption></fig>", "<fig id=\"Fig2\"><label>Figure 2</label><caption><p>Graphs of absolute PI error. (<bold>A</bold>) Illustrates the absolute PI error vs. each case. (<bold>B</bold>) Displays the mean absolute PI error vs. years of experience. (<bold>C</bold>) Depicts the mean absolute PI error vs. career stages. Asterisk: statistical significance was found between pathologists and pathologists with AI using the paired Wilcoxon signed-rank test.</p></caption></fig>", "<fig id=\"Fig3\"><label>Figure 3</label><caption><p>(<bold>A</bold>) Linear Regression of pathologists’ scores with AI assistance. (<bold>B</bold>) Bland–Altman of pathologists’ scores with AI assistance. (<bold>C</bold>) Linear regression of pathologists’ scores without AI assistance. (<bold>D</bold>) Bland–Altman of pathologists’ scores without AI assistance.</p></caption></fig>", "<fig id=\"Fig4\"><label>Figure 4</label><caption><p>Graphs of TATs displayed in seconds. (<bold>A</bold>) Illustrates the absolute TAT vs. each case, the average of all cases (All), and the average of all cases excluding Question 1 (All-Q1). (<bold>B</bold>) Represents the TAT vs. the sequential question pairs in the study. (<bold>C</bold>) Displays the mean TAT vs. years of experience. (<bold>D</bold>) Depicts the mean TAT vs. career stages. Asterisk: Statistical significance was found between pathologists and pathologists with AI using the paired Wilcoxon signed-rank test.</p></caption></fig>", "<fig id=\"Fig5\"><label>Figure 5</label><caption><p>(<bold>A</bold>) Pathologists’ opinions on AI for Ki-67 assessments. (<bold>B</bold>) Pathologists’ opinions on their personal implementation timeline of AI into Ki-67 assessments. (<bold>C</bold>) Pathologists’ opinions on the routine implementation timeline of AI into Ki-67 assessments.</p></caption></fig>" ]

[ "<table-wrap id=\"Tab1\"><label>Table 1</label><caption><p>PI scores, PI error and ground truth are shown as mean (SD) per case and within PI ranges. </p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th align=\"left\" rowspan=\"2\">Case</th><th align=\"left\" rowspan=\"2\">Ground truth</th><th align=\"left\" rowspan=\"2\">AI tool</th><th align=\"left\" rowspan=\"2\">AI tool error</th><th align=\"left\" colspan=\"3\">PI scores</th><th align=\"left\" colspan=\"3\">PI error</th></tr><tr><th align=\"left\">No aid</th><th align=\"left\">With aid</th><th align=\"left\"><italic>p</italic> value*</th><th align=\"left\">No aid</th><th align=\"left\">With aid</th><th align=\"left\"><italic>p</italic> value*</th></tr></thead><tbody><tr><td align=\"left\">1</td><td char=\".\" align=\"char\">7.3</td><td char=\".\" align=\"char\">8.2</td><td char=\".\" align=\"char\">0.9</td><td align=\"left\">8.0 (4.4)</td><td align=\"left\">8.6 (1.3)</td><td char=\".\" align=\"char\"><bold> &lt; 0.001</bold></td><td align=\"left\">2.1 (3.9)</td><td align=\"left\">1.5 (1.1)</td><td char=\".\" align=\"char\">0.133</td></tr><tr><td align=\"left\">2</td><td char=\".\" align=\"char\">11.1</td><td char=\".\" align=\"char\">10.5</td><td char=\".\" align=\"char\">0.6</td><td align=\"left\">10.6 (4.3)</td><td align=\"left\">10.8 (1.4)</td><td char=\".\" align=\"char\"><bold> &lt; 0.001</bold></td><td align=\"left\">3.0 (3.1)</td><td align=\"left\">1.0 (1.0)</td><td char=\".\" align=\"char\"><bold> &lt; 0.001</bold></td></tr><tr><td align=\"left\">3</td><td char=\".\" align=\"char\">12.1</td><td char=\".\" align=\"char\">13.3</td><td char=\".\" align=\"char\">1.2</td><td align=\"left\">8.6 (3.9)</td><td align=\"left\">13.8 (1.9)</td><td char=\".\" align=\"char\"><bold> &lt; 0.001</bold></td><td align=\"left\">4.9 (1.8)</td><td align=\"left\">2.0 (1.6)</td><td char=\".\" align=\"char\"><bold> &lt; 0.001</bold></td></tr><tr><td align=\"left\">4</td><td char=\".\" align=\"char\">14.4</td><td char=\".\" align=\"char\">14.8</td><td char=\".\" align=\"char\">0.4</td><td align=\"left\">12.6 (3.5)</td><td align=\"left\">15.0 (2.3)</td><td char=\".\" align=\"char\"><bold> &lt; 0.001</bold></td><td align=\"left\">3.5 (1.8)</td><td align=\"left\">1.1 (2.1)</td><td char=\".\" align=\"char\"><bold> &lt; 0.001</bold></td></tr><tr><td align=\"left\">5</td><td char=\".\" align=\"char\">16.2</td><td char=\".\" align=\"char\">16.2</td><td char=\".\" align=\"char\">0.0</td><td align=\"left\">14.3 (3.6)</td><td align=\"left\">16.6 (1.9)</td><td char=\".\" align=\"char\"><bold> &lt; 0.001</bold></td><td align=\"left\">3.4 (2.9)</td><td align=\"left\">1.1 (1.5)</td><td char=\".\" align=\"char\"><bold> &lt; 0.001</bold></td></tr><tr><td align=\"left\">6</td><td char=\".\" align=\"char\">16.9</td><td char=\".\" align=\"char\">16.3</td><td char=\".\" align=\"char\">0.6</td><td align=\"left\">11.1 (4.0)</td><td align=\"left\">16.8 (1.5)</td><td char=\".\" align=\"char\"><bold> &lt; 0.001</bold></td><td align=\"left\">6.5 (2.5)</td><td align=\"left\">1.0 (1.2)</td><td char=\".\" align=\"char\"><bold> &lt; 0.001</bold></td></tr><tr><td align=\"left\">7</td><td char=\".\" align=\"char\">19.8</td><td char=\".\" align=\"char\">17.8</td><td char=\".\" align=\"char\">2.0</td><td align=\"left\">28.6 (6.0)</td><td align=\"left\">18.0 (3.6)</td><td char=\".\" align=\"char\"><bold> &lt; 0.001</bold></td><td align=\"left\">9.9 (3.9)</td><td align=\"left\">3.0 (2.6)</td><td char=\".\" align=\"char\"><bold> &lt; 0.001</bold></td></tr><tr><td align=\"left\">8</td><td char=\".\" align=\"char\">23.7</td><td char=\".\" align=\"char\">22.0</td><td char=\".\" align=\"char\">1.7</td><td align=\"left\">17.3 (5.5)</td><td align=\"left\">22.0 (2.2)</td><td char=\".\" align=\"char\"><bold> &lt; 0.001</bold></td><td align=\"left\">7.9 (2.9)</td><td align=\"left\">2.0 (1.9)</td><td char=\".\" align=\"char\"><bold> &lt; 0.001</bold></td></tr><tr><td align=\"left\">9</td><td char=\".\" align=\"char\">28.2</td><td char=\".\" align=\"char\">29.7</td><td char=\".\" align=\"char\">1.5</td><td align=\"left\">19.8 (4.7)</td><td align=\"left\">30.3 (2.7)</td><td char=\".\" align=\"char\"><bold> &lt; 0.001</bold></td><td align=\"left\">9.2 (2.9)</td><td align=\"left\">2.5 (2.3)</td><td char=\".\" align=\"char\"><bold> &lt; 0.001</bold></td></tr><tr><td align=\"left\">10</td><td char=\".\" align=\"char\">27.8</td><td char=\".\" align=\"char\">33.9</td><td char=\".\" align=\"char\">6.1</td><td align=\"left\">19.6 (4.1)</td><td align=\"left\">32.3 (3.8)</td><td char=\".\" align=\"char\"><bold> &lt; 0.001</bold></td><td align=\"left\">8.5 (3.4)</td><td align=\"left\">5.6 (1.8)</td><td char=\".\" align=\"char\"><bold> &lt; 0.001</bold></td></tr><tr><td align=\"left\">All</td><td char=\".\" align=\"char\">17.7</td><td char=\".\" align=\"char\">18.3</td><td char=\".\" align=\"char\">0.6</td><td align=\"left\">15.0 (7.5)</td><td align=\"left\">18.4 (7.7)</td><td char=\".\" align=\"char\"><bold> &lt; 0.001</bold></td><td align=\"left\">5.9 (4.0)</td><td align=\"left\">2.1 (2.2)</td><td char=\".\" align=\"char\"><bold> &lt; 0.001</bold></td></tr><tr><td align=\"left\"> &lt; 20%</td><td char=\".\" align=\"char\">14.0</td><td char=\".\" align=\"char\">13.7</td><td char=\".\" align=\"char\">0.3</td><td align=\"left\">19.6 (4.1)</td><td align=\"left\">14.2 (3.8)</td><td char=\".\" align=\"char\"><bold> &lt; 0.001</bold></td><td align=\"left\">4.8 (3.8)</td><td align=\"left\">1.5 (1.8)</td><td char=\".\" align=\"char\"><bold> &lt; 0.001</bold></td></tr><tr><td align=\"left\"> ≥ 20%</td><td char=\".\" align=\"char\">26.6</td><td char=\".\" align=\"char\">28.5</td><td char=\".\" align=\"char\">1.9</td><td align=\"left\">18.9 (4.9)</td><td align=\"left\">28.2 (5.4)</td><td char=\".\" align=\"char\"><bold> &lt; 0.001</bold></td><td align=\"left\">8.5 (3.1)</td><td align=\"left\">3.4 (2.6)</td><td char=\".\" align=\"char\"><bold> &lt; 0.001</bold></td></tr></tbody></table></table-wrap>", "<table-wrap id=\"Tab2\"><label>Table 2</label><caption><p>Average percent agreement for each of the 10 cases and cases above and below a 20% PI threshold. </p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th align=\"left\" colspan=\"5\">Average percent agreement</th></tr><tr><th align=\"left\">Case</th><th align=\"left\">Ground truth (PI)</th><th align=\"left\">Tool (PI)</th><th align=\"left\">No aid (%)</th><th align=\"left\">With aid (%)</th></tr></thead><tbody><tr><td align=\"left\">1</td><td align=\"left\">7.3</td><td char=\".\" align=\"char\">8.2</td><td char=\".\" align=\"char\">95.6</td><td char=\".\" align=\"char\">100.0</td></tr><tr><td align=\"left\">2</td><td align=\"left\">11.1</td><td char=\".\" align=\"char\">10.5</td><td char=\".\" align=\"char\">94.4</td><td char=\".\" align=\"char\">100.0</td></tr><tr><td align=\"left\">3</td><td align=\"left\">12.1</td><td char=\".\" align=\"char\">13.3</td><td char=\".\" align=\"char\">96.7</td><td char=\".\" align=\"char\">98.9</td></tr><tr><td align=\"left\">4</td><td align=\"left\">14.4</td><td char=\".\" align=\"char\">14.8</td><td char=\".\" align=\"char\">92.2</td><td char=\".\" align=\"char\">95.6</td></tr><tr><td align=\"left\">5</td><td align=\"left\">16.2</td><td char=\".\" align=\"char\">16.2</td><td char=\".\" align=\"char\">91.1</td><td char=\".\" align=\"char\">96.7</td></tr><tr><td align=\"left\">6</td><td align=\"left\">16.9</td><td char=\".\" align=\"char\">16.3</td><td char=\".\" align=\"char\">95.6</td><td char=\".\" align=\"char\">94.4</td></tr><tr><td align=\"left\">7</td><td align=\"left\">19.8</td><td char=\".\" align=\"char\">17.8</td><td char=\".\" align=\"char\">88.9*</td><td char=\".\" align=\"char\">86.7</td></tr><tr><td align=\"left\">8</td><td align=\"left\">23.7</td><td char=\".\" align=\"char\">22.0</td><td char=\".\" align=\"char\">76.7*</td><td char=\".\" align=\"char\">96.7</td></tr><tr><td align=\"left\">9</td><td align=\"left\">28.2</td><td char=\".\" align=\"char\">29.7</td><td char=\".\" align=\"char\">67.8*</td><td char=\".\" align=\"char\">100.0</td></tr><tr><td align=\"left\">10</td><td align=\"left\">27.8</td><td char=\".\" align=\"char\">33.9</td><td char=\".\" align=\"char\">56.7</td><td char=\".\" align=\"char\">97.8</td></tr><tr><td align=\"left\">1–7</td><td align=\"left\" colspan=\"2\"> &lt; 20%</td><td char=\".\" align=\"char\">82.4</td><td char=\".\" align=\"char\">96.0</td></tr><tr><td align=\"left\">8–10</td><td align=\"left\" colspan=\"2\"> ≥ 20%</td><td char=\".\" align=\"char\">73.2*</td><td char=\".\" align=\"char\">98.1</td></tr></tbody></table></table-wrap>", "<table-wrap id=\"Tab3\"><label>Table 3</label><caption><p>TAT expressed in seconds as mean (SD). </p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th align=\"left\">Case</th><th align=\"left\">No aid (s)</th><th align=\"left\">With aid (s)</th><th align=\"left\">With aid—no aid (s)</th><th align=\"left\"><italic>p</italic> value*</th></tr></thead><tbody><tr><td align=\"left\">1</td><td align=\"left\">15.5 (34.3)</td><td align=\"left\">12.8 (28.6)</td><td align=\"left\"><bold>−2.7</bold></td><td char=\".\" align=\"char\"><bold>0.02</bold></td></tr><tr><td align=\"left\">2</td><td align=\"left\">71.7 (146.3)</td><td align=\"left\">15.3 (30.1)</td><td align=\"left\"><bold>−56.4</bold></td><td char=\".\" align=\"char\"><bold> &lt; 0.001</bold></td></tr><tr><td align=\"left\">3</td><td align=\"left\">16.6 (25.6)</td><td align=\"left\">17.6 (31.2)</td><td align=\"left\"> + 1.0</td><td char=\".\" align=\"char\">0.08</td></tr><tr><td align=\"left\">4</td><td align=\"left\">22.2 (37.0)</td><td align=\"left\">18.6 (49.5)</td><td align=\"left\"><bold>−3.6</bold></td><td char=\".\" align=\"char\"><bold>0.003</bold></td></tr><tr><td align=\"left\">5</td><td align=\"left\">13.5 (23.1)</td><td align=\"left\">14.2 (23.4)</td><td align=\"left\"> + 0.7</td><td char=\".\" align=\"char\">0.35</td></tr><tr><td align=\"left\">6</td><td align=\"left\">19.2 (44.4)</td><td align=\"left\">25.0 (99.8)</td><td align=\"left\"> <bold>+ 5.8</bold></td><td char=\".\" align=\"char\"><bold>0.004</bold></td></tr><tr><td align=\"left\">7</td><td align=\"left\">14.9 (23.4)</td><td align=\"left\">36.4 (70.8)</td><td align=\"left\"> + 21.5</td><td char=\".\" align=\"char\">0.58</td></tr><tr><td align=\"left\">8</td><td align=\"left\">12.0 (18.8)</td><td align=\"left\">17.0 (34.6)</td><td align=\"left\"> + 5.0</td><td char=\".\" align=\"char\">0.81</td></tr><tr><td align=\"left\">9</td><td align=\"left\">27.4 (57.1)</td><td align=\"left\">12.7 (19.6)</td><td align=\"left\"><bold>−14.7</bold></td><td char=\".\" align=\"char\"><bold> &lt; 0.001</bold></td></tr><tr><td align=\"left\">10</td><td align=\"left\">20.1 (38.4)</td><td align=\"left\">16.2 (50.8)</td><td align=\"left\"><bold>−3.9</bold></td><td char=\".\" align=\"char\"><bold>0.006</bold></td></tr><tr><td align=\"left\">All</td><td align=\"left\">23.3 (59.4)</td><td align=\"left\">18.6 (50.1)</td><td align=\"left\"><bold>−4.6</bold></td><td char=\".\" align=\"char\"><bold> &lt; 0.001</bold></td></tr><tr><td align=\"left\">All—Q1</td><td align=\"left\">18.3 (48.6)</td><td align=\"left\">16.8 (36.9)</td><td align=\"left\"><bold>−1.5</bold></td><td char=\".\" align=\"char\"><bold> &lt; 0.001</bold></td></tr></tbody></table></table-wrap>" ]

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[ "<supplementary-material content-type=\"local-data\" id=\"MOESM1\"></supplementary-material>" ]

[ "<table-wrap-foot><p>*The p values were computed for paired comparisons between pathologists and pathologists with AI with the paired Wilcoxon signed-rank test.</p><p>Significant values are in bold.</p></table-wrap-foot>", "<table-wrap-foot><p>The * highlights situations where the consensus misaligns with the established ground truth—meaning pathologists agree on a PI score that's &gt; 20%, but the actual score was &lt; 20%. For example, in case 7, most raters agreed that the TMA had a PI above 20%, but the ground truth indicates it's below 20% PI.</p></table-wrap-foot>", "<table-wrap-foot><p>*The p values were computed for paired comparisons between pathologists and pathologists with AI with the paired Wilcoxon signed-rank test.</p><p>Significant values are in bold.</p></table-wrap-foot>", "<fn-group><fn><p><bold>Publisher's note</bold></p><p>Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.</p></fn></fn-group>" ]

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[ "<media xlink:href=\"41598_2024_51723_MOESM1_ESM.docx\"><caption><p>Supplementary Information.</p></caption></media>" ]

{ "acronym": [], "definition": [] }

[ "<title>Introduction</title>", "<p id=\"Par2\">In schizophrenia, only ~70% of affected individuals respond to antipsychotic drug treatment, and even fewer go into remission [##REF##26925705##1##, ##REF##22834451##2##]. Pre-treatment prediction of the subsequent response could reduce the time spent on ineffective treatments, shorten patient suffering, and reduce possible mortality [##REF##26842482##3##, ##UREF##0##4##]. The ability to identify brain biomarkers of antipsychotic nonresponders using magnetic resonance imaging may lead to improved prognosis and the detection of malleable central nervous system targets for the development of new treatment strategies.</p>", "<p id=\"Par3\">Second-generation antipsychotics (SGAs) are widely believed to work by decreasing striatal dopamine via dopamine receptor blockage within the mesolimbic pathway to alleviate positive symptoms and increasing cortical dopamine via 5-HT(2A) antagonism in presynaptic neurons within the mesocortial pathway to improve negative symptoms [##REF##28970021##5##, ##REF##32246399##6##]. These neurotransmitters modulate synapses at glutamate (N-methyl-d-aspartate [NMDA]) receptors that are involved in synaptic plasticity and may cause delayed corollary discharges [##REF##17514195##7##]. Animal studies noted that antipsychotic treatment following administration of the copper chelator cuprizone promoted oligodendrocyte development and remyelination [##REF##17684494##8##, ##REF##22555017##9##]. Thus, the integrity of WM may be a sensitive index of the brain pharmacological mechanism of action of SGA.</p>", "<p id=\"Par4\">Diffusion tensor imaging (DTI) is widely used to evaluate the structural integrity of WM, and voxelwise metrics such as fractional anisotropy (FA) and mean diffusivity (MD) are generally considered sensitive measures for axonal/myelin damage [##REF##25993492##10##–##REF##8939209##12##]. A handful of DTI studies have documented the association between FA/MD of WM and response to treatment [##REF##20708198##13##–##REF##17708778##15##]. Further, inconsistencies prior work may be related to differences in clinical variables such as the illness course/chronicity, substance misuse, and previous exposure to antipsychotics. Therefore, investigating patients with drug-naive first-episode psychosis may address to disentangle which brain white matter changes may predict response to treatment.</p>", "<p id=\"Par5\">To date, only three studies have investigated the effects of antipsychotic use on WM tracts in drug-naive first-episode schizophrenia. These studies show that antipsychotic medications appear to alter or improve FA or MD of WM, such as in the bilateral anterior cingulate gyrus (ACG), corticospinal tract (CT), anterior thalamic radiation (ATR), longitudinal fasciculi (ILF), inferior fronto-occipital fasciculi (IFOF), and uncinated fasciculus (UF) structural abnormalities, especially at remission [##REF##23442742##16##–##REF##28826419##18##]. However, while these studies typically report group-level WH tract differences between psychosis before and after antipsychotic administration, whether these findings have predictive value for individualized drug responses is unclear.</p>", "<p id=\"Par6\">WH alterations may occur before the onset of psychosis, and FA/MD of WM changes have been reported in subjects with a high risk of psychosis [##REF##30723287##19##, ##REF##34024906##20##]. The severity of WM alterations such as fronto-temporal, fronto-occipital, and fronto-striatal WH at onset were associated with the severity and persistence of the signs and symptoms of schizophrenia, suggesting that baseline WM alterations may serve as an early marker for differentially characterizing patients with poor or good outcomes [##REF##17949516##21##–##REF##20809999##23##]. Furthermore, the cortex, especially the frontal lobe, is the core component of the dopamine projection system [##REF##16696579##24##, ##UREF##2##25##]. However, it is still unknown whether psychotic symptom-related alterations in FA and MD of WM at the early stage of the disorder may provide aid to individualized prediction of drug response.</p>", "<p id=\"Par7\">In this study, we investigated the above questions using DTI data acquired from first-episode schizophrenia patients with no prior medication. Patients underwent baseline structural MRI scans and were subsequently randomized to receive a single atypical antipsychotic throughout the first 12 weeks. It was hypothesized that the altered FA/MD of WM was related to the severity of psychotic symptoms at baseline, and those changes would show potential as individualized predictive biomarkers of response to SGA.</p>" ]

[ "<title>Methods</title>", "<title>Participants</title>", "<p id=\"Par8\">A total of 68 drug-naive patients between 18 and 45 years old were diagnosed with schizophrenia based on the Structured Clinical Interview for the DSM-IV Axis I disorder and had no previous psychiatric treatment. Patients underwent MRI scans and symptom ratings before assigned to a randomized open-label treatment with risperidone, olanzapine or aripiprazole for up to 1 year (Clinical trials.gov ID: NCT01057849). Clinical symptoms were evaluated using the eight “core symptoms” selected from the Positive and Negative Syndrome Scale (PANSS-8), which has more acceptable internal consistency and comparable sensitivity to early improvement in psychotic symptoms than the PANSS-30 [##REF##29280500##26##]. This analysis included data only from the first 12 weeks of treatment. During this period, patients received a single antipsychotic that started with low dosage and gradually increased to a standard therapeutic range (3–6 mg risperidone, 15–30 mg aripiprazole, or 10–25 mg olanzapine per day) in 2 weeks. Follow-up assessments were conducted at the 4th, 8th, and 12th weeks by trained psychiatrists. To ensure the consistency and reliability of ratings across the study, three psychiatrists with more than 5 years of experience in clinical psychiatry attended a 1-week training workshop on the use of the rating instruments prior to the study. After training, they achieved an interrater reliability of 0.80 for the PANSS-8 score.</p>", "<title>Evaluation of treatment response</title>", "<p id=\"Par9\">Treatment response was operationalized as a reduction in symptom severity to the levels required by the remission criteria of the Schizophrenia Working Group Consensus [##REF##15741458##27##]. According to these criteria, clinical improvement is reached when a simultaneous rating of mild or less (equivalent to 1, 2, or 3) is given in all the following items of the PANSS-8: delusions (P1), conceptual disorganization (P2), hallucinatory behavior (P3), mannerisms and posturing (G5), unusual thought content (G9), blunted affect (N1), social withdrawal (N4), and lack of spontaneity and flow of conversation (N6). The clinical recommendation is that antipsychotic treatment with a specific drug should be continued for 6–8 weeks before switching to a different medication owing to lack of efficacy or adverse effects. Hence, in this study, we defined treatment response as meeting the remission criteria at the 8th or 12th week. Only Fifty patients completed both clinical follow-up assessments at the 8th and 12th weeks and were therefore included in this study as the final sample. The drop-out participants did not differ from the rest of the sample in characteristics and symptom severity (Supplementary Table ##SUPPL##0##S1##).</p>", "<title>MRI data acquisition</title>", "<p id=\"Par10\">The MRI scans were performed before medication using a GE Signa EXCITE 3.0-T scanner (GE Healthcare, Milwaukee, Wisconsin) equipped with an 8-channel phase array head coil. The DTI data were acquired using a bipolar diffusion-weighted spin‒echo planar imaging (EPI) sequence (TR = 10000 ms, TE = 70 ms) with a 128 × 128 matrix over a field of view of 240 × 240 mm and 42 axial slices of 3 mm thickness to cover the whole brain without gap. Each DTI dataset included 20 images of unique diffusion directions (B = 1000) and a nondiffusion image (B = 0). High-resolution T1 data were acquired using a 3D spoiled gradient (3D-SPGR) sequence: TR = 8.5 ms, TE = 3.5 ms, TI = 400 ms, flip angle = 12, 240*240 matrix over a field of view of 240*240 mm, and 156 axial slices of 1 mm thickness. All scans were reviewed by an experienced neuroradiologist to exclude gross brain abnormalities.</p>", "<title>Imaging processing</title>", "<p id=\"Par11\">The routine DTI preprocessing included head motion and eddy current correction, brain extraction, and tensor model fitting was performed using FSL (FMRIB Software Library, <ext-link ext-link-type=\"uri\" xlink:href=\"http://www.fmrib.ox.ac.uk/fsl\">http://www.fmrib.ox.ac.uk/fsl</ext-link>). We used automated fiber quantification software (AFQ) to identify 20 white matter traces in individual subjects. The identification procedure included three primary steps: whole-brain deterministic fiber tractography, waypoint ROI-based tract segmentation, and probability map-based fiber refinement using the 20-tract Johns Hopkins University white matter template. The 20 identified tracts were the left and right ATR, cingulum–cingulate (CC), cingulum–hippocampus pathway, inferior fronto-occipital fasciculus (IFOF), inferior longitudinal fasciculus (ILF), superior longitudinal fasciculus (SLF), uncinated fasciculus and arcuate fasciculus, and the forceps major of the splenium and the forceps minor of the genu of the corpus callosum. After tract identification, we smoothed each tract using a 10-point moving average filter to reduce local variation caused by imaging noise. The diffusion measurements along the tract core, defined as the tract profile, were extracted from each fiber tract, including the FA and MD values. Hence, each tract had two features, and each subject had 40 features to depict their global white matter status.</p>", "<title>Diffusion properties and clinical associations</title>", "<p id=\"Par12\">To confirm the correlation between diffusion properties and symptoms at baseline, partial Pearson correlation was performed to examine relations between 40 white matter features and PANSS-8 scores at baseline, with age, gender, and duration of untreated psychosis as covariates. To adjust the significant values for multiple comparisons, we used the Benjamini–Hochberg false discovery rate (FDR <italic>q</italic> value selected to maintain the false positive error rate &lt;0.05).</p>", "<title>Prediction of treatment outcome with diffusion properties</title>", "<p id=\"Par13\">We next sought to investigate whether baseline FA and MD of WM would be capable of distinguishing antipsychotic responders from nonresponders at the individual level. To this end, we trained a cross-validated generalized LASSO regression model with treatment outcome as the dependent variable and baseline diffusion properties as predictors. To constrain the number of features in the model and meanwhile include all features relevant to the disorder, we preselected the FA and MD measures for model training. Here, only measures significantly associated with baseline symptoms at uncorrected <italic>P</italic> &lt; 0.05 were included in the model as input predictors. We also trained the model with all baseline diffusion properties as predictors as a supplementary analysis (see Supplementary Materials).</p>", "<p id=\"Par14\">The LASSO regression is an L1-norm regularization method that incorporates a shrinkage penalty term λ to avoid model overfitting, which coerces the coefficients of some less important predictors to be shrunken to zero. Specifically, the predictors included in the model were adjusted for age, sex, antipsychotic drug dosage, duration of untreated psychosis, and PANSS-8 scores at baseline. Similar to our prior work [##REF##34525195##28##–##REF##37644811##30##], a repeated nested cross-validation (CV) method (10 outer folds, each with 10 inter folds) was used in which the tuning parameter λ was optimized within the inner cycles and subsequently utilized to predict remaining subjects in the outer cycles. This procedure eventually yielded predicted probabilities of nonresponders for each individual in the main dataset, based on which the classification accuracy was calculated. To ensure the robustness of the results, we repeated the CV 100 times, each time by randomly parcellating the sample. The final classification performance was determined as the average area under curve (AUC) of the receiver operating characteristic (ROC) curves from the 100 runs, and the significance of the performance was determined by 1000 permutations. We also investigated whether the top features selected by the model (at least 8 out of 10 cycles) would be capable of predicting individualized symptom changes in first-episode schizophrenia as a supplementary analysis (see Supplementary Materials).</p>" ]

[ "<title>Results</title>", "<title>12-Week treatment outcome</title>", "<p id=\"Par15\">By the end of the 12th week, 30 patients met the remission criteria as responders, and 20 patients were classified as nonresponders. Demographic and clinical variables at baseline were not significantly different between responders and nonresponders (Table ##TAB##0##1##).</p>", "<title>Diffusion properties and clinical associations at baseline</title>", "<p id=\"Par16\">In partial correlation analysis between PANSS-8 scores and FA/MD of each fiber tract, we found positive correlations between PANSS-8 scores and average MD of the following fiber tracts after FDR correction: left and right IFOF (<italic>r</italic> = 0.564, <italic>q</italic> = 0.002; <italic>r</italic> = 0.456, <italic>q</italic> = 0.013), and left and right ILF (<italic>r</italic> = 0.493, <italic>q</italic> = 0.009; <italic>r</italic> = 0.425, <italic>q</italic> = 0.03) (Fig. ##FIG##0##1##). At a more liberal threshold without FDR correction, PANSS-8 score was significantly correlated with the FAs of the left ILF (<italic>r</italic> = -0.296, <italic>P</italic> = 0.044) and right ILF (<italic>r</italic> = −0.374, <italic>P</italic> = 0.01), as well as MDs of the left ATR (<italic>r</italic> = 0.358, <italic>P</italic> = 0.013), left corticospinal (<italic>r</italic> = 0.362, <italic>P</italic> = 0.012), right corticospinal (<italic>r</italic> = 0.389, <italic>P</italic> = 0.007), genu of corpus callosum (<italic>r</italic> = 0.346, <italic>P</italic> = 0.017), and right SLF (<italic>r</italic> = 0.376, <italic>P</italic> = 0.009). Therefore, these eleven measures (two FA measures and nine MD measures) were subsequently used as predictors in the LASSO regression model.</p>", "<title>Classification performance of treatment response</title>", "<p id=\"Par17\">The average AUC from the 100 repeats of the LASSO regression model was 0.828 (range: 0.81–0.86) (<italic>P</italic> &lt; 0.001, average sensitivity = 0.867 and average specificity = 0.636). Here, the FA of the right ILF and MDs of the left IFOF and the right SLF had nonzero coefficients at least 8 out of 10 cycles during all 100 repeats, and were therefore selected as final features. The post-hoc <italic>t</italic> test revealed significantly higher FA of the right ILF (<italic>t</italic> = 5.69, <italic>P</italic> &lt; 0.001) but lower MDs of the left IFOF and the right SLF in responders compared with nonresponders (<italic>t</italic> = −2.25, <italic>P</italic> = 0.029; <italic>t</italic> = −2.62, <italic>P</italic> = 0.012) (Table ##TAB##1##2## and Fig. ##FIG##1##2##).</p>" ]

[ "<title>Discussion</title>", "<p id=\"Par18\">Here, we provide evidence for a baseline psychotic symptom-related white matter tract biomarker that potentially predicts response to antipsychotic treatment in first-episode schizophrenia. Importantly, the study sample was treatment-naive to ensure that the findings were not confounded by the drug. Two main findings emerged from this study. First, the altered diffusion properties (FA or MD) of fiber tracts in the ATR, corticospinal tract, callosum forceps minor, IFOF, ILF and SLF were related to the severity of symptoms, demonstrating that early microstructural WH changes contribute to the pathophysiology of psychosis. Second, these abnormal fiber tracts, especially the ILF, IFOF, and SLF, significantly predicted the response to antipsychotic treatment at the individual level. This suggests that these symptom-related WH changes could be an outcome marker after the onset of psychosis or even a target for intervention and preventive strategies.</p>", "<p id=\"Par19\">Our findings decreased FA and increased MD of several fiber bundles throughout the brain correlated with core positive and negative symptom severities consistent with a “disconnection” hypothesis of symptoms in schizophrenia [##REF##20855415##31##, ##REF##7583624##32##]. Several meta-analytic studies have investigated the role of WH irregularities in schizophrenia spectrum disorders [##REF##19744536##33##–##REF##19128945##36##]. A meta-analysis of WM alteration in patients with FES indicated widespread abnormalities across white matter tracts, with evidence for reductions in FA in the corpus callosum, the left ILF and IFOF [##REF##23648972##34##]. Even in chronic schizophrenia, the meta-analysis of 15 DTI studies also observed significant FA reductions in the genu and splenium of corpus callosum, the left anterior thalamic radiation, the left IFOF and ILF [##REF##19128945##36##]. Moreover, the relationship between aberrant FA of these WM tracts and psychotic symptoms of schizophrenia were reported among previous studies [##REF##23317110##22##, ##REF##28550466##37##]. Consistent with this study, several studies found the inverse relationship between FA of SLF, ILF, and IFOF and negative symptoms and auditory verbal hallucinations in schizophrenia [##REF##28550466##37##–##UREF##3##39##]. Taken together, these implied that white matter dysintegrity may represent a “trait” marker, related to the underlying pathophysiology in schizophrenia.</p>", "<p id=\"Par20\">Beyond the group level for white matter correlated with symptoms at baseline, our longitudinal follow-up study also provided evidence that these baseline psychotic symptom-related FA and MD of WM may serve as an individualized predictor for antipsychotic treatment response in patients with schizophrenia. Similar to our findings, previous group-level analysis studies found more widespread FA decreases at baseline in FEP patients with a subsequent poorer response, and that baseline global white matter network organization showed greater alterations in FEP patients who subsequently showed a poorer treatment response [##REF##32398721##40##–##REF##28007987##42##]. Taken together, all these studies highlight the usefulness of baseline WM integrity in predicting response to treatment. Furthermore, the LASSO regression model in this study correctly classified 82.8% of patients as responsive, which may represent an important preliminary step to provide clinicians with decision support in selecting the ideal antipsychotic treatment for schizophrenia in a personalized manner. Further studies using larger and independent samples are required to replicate these findings.</p>", "<p id=\"Par21\">We focused on predicting biomarkers of symptom-related FA and MD in WM. Longitudinal studies have observed that longitudinal increases in FA values, especially in the IFOF, ILF, SLF, and anterior thalamic radiation, are significantly correlated with improved symptoms at follow-up [##REF##26852402##17##, ##REF##28826419##18##, ##UREF##4##43##]. Consistent with these findings, the top treatment response predicting features in our study were located in the ILF, IFOF, and SLF. These WM tracts connect frontal, temporal, parietal and occipital areas, which have been implicated in several cognitive functions, such as visuospatial processing, emotional regulation, memory and language, in schizophrenia [##REF##17720147##44##, ##REF##34204171##45##]. In addition, altered FAs in the SLF and IFOF have been found to be a biomarker for auditory hallucinations, and FA in the ILF has been linked to positive symptoms [##UREF##5##46##, ##UREF##6##47##]. Furthermore, these regions are major target of dopamine signaling. Evidence from animal models suggested that the upregulation of D2 receptors in the frontal, parietal, temporal and occipital lobes and the downregulation of D1 receptors in the prefrontal and temporal cortices may be an important component of the therapeutic response to neuroleptic drugs [##REF##8183912##48##]. These receptor blockades have been shown to promote oligodendrocyte repopulation and remyelination of experimentally demyelinated cells in mice [##REF##30476795##49##]. We also found that nonresponders had more severe WM damage as lower FA and higher MD in these fiber bundles at baseline than responders. These findings together suggest that the alterations in the IFOF, ILF and SLF may represent neural markers of the severity and persistence of the signs and symptoms of schizophrenia, and may compromise the potential effects of antipsychotics.</p>", "<p id=\"Par22\">This study has some limitations. First, participants in our study were randomized to receive a single standardized treatment with one of three antipsychotics, but the sample was too small to rule out changes in response patterns based on different medications. Future investigations are encouraged to provide a comparison of the effectiveness of different drugs. Second, the study did not include a placebo control group, so a potential placebo or time effect cannot be excluded. Due to ethical issues, these effects are normally nested in clinical studies and cannot be completely removed. Third, as a machine learning study focusing on individual prediction, the sample size in this study was relatively small, and it lack an independent validation sample. Therefore, these findings should be externally verified with larger samples in the future.</p>", "<p id=\"Par23\">In conclusion, altered FA and MD of fiber tracts in the ATR, corticospinal tract, callosum forceps minor, IFOF, ILF, and SLF were related to the severity of symptoms in first-episode schizophrenia. These effects on WM tracts are not influenced by pharmacotherapy and therefore appear to be disease-related. These baseline psychotic symptom-related WM tracts, especially ILF, IFOF, and SLF, may serve as meaningful individualized predictors of response to SGA. These results may represent an important first step of the translational value of baseline brain structural measures in precision psychiatry.</p>" ]

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[ "<p id=\"Par1\">There is significant heterogeneity in individual responses to antipsychotic drugs, but there is no reliable predictor of antipsychotics response in first-episode psychosis. This study aimed to investigate whether psychotic symptom-related alterations in fractional anisotropy (FA) and mean diffusivity (MD) of white matter (WM) at the early stage of the disorder may aid in the individualized prediction of drug response. Sixty-eight first-episode patients underwent baseline structural MRI scans and were subsequently randomized to receive a single atypical antipsychotic throughout the first 12 weeks. Clinical symptoms were evaluated using the eight “core symptoms” selected from the Positive and Negative Syndrome Scale (PANSS-8). Follow-up assessments were conducted at the 4th, 8th, and 12th weeks by trained psychiatrists. LASSO regression model and cross-validation were conducted to examine the performance of baseline symptom-related alterations FA and MD of WM in the prediction of individualized treatment outcome. Fifty patients completed both clinical follow-up assessments by the 8th and 12th weeks. 30 patients were classified as responders, and 20 patients were classified as nonresponders. At baseline, the altered diffusion properties of fiber tracts in the anterior thalamic radiation, corticospinal tract, callosum forceps minor, longitudinal fasciculi (ILF), inferior frontal-occipital fasciculi (IFOF) and superior longitudinal fasciculus (SLF) were related to the severity of symptoms. These abnormal fiber tracts, especially the ILF, IFOF, and SLF, significantly predicted the response to antipsychotic treatment at the individual level (AUC = 0.828, <italic>P</italic> &lt; 0.001). These findings demonstrate that early microstructural WM changes contribute to the pathophysiology of psychosis and may serve as meaningful individualized predictors of response to antipsychotics.</p>", "<title>Subject terms</title>" ]

[ "<title>Supplementary information</title>", "<p>\n\n</p>" ]

[ "<title>Supplementary information</title>", "<p>The online version contains supplementary material available at 10.1038/s41398-023-02714-w.</p>", "<title>Acknowledgements</title>", "<p>This study was supported by the National Major Project of Scientific and Technical Supporting Programs of China during the 11th Five-year Plan Period (Grant No. 2017BAI17B04), the Key research and development project of Science and Technology, department of Sichuan Province (22ZDFY2064, 2022YFS0179).</p>", "<title>Author contributions</title>", "<p>YC, HC, HD, and XY conceptualized the study. HD, SL, BZ, GZ, ZZ, SL, and HL collected the data. YC and HC performed statistical analysis. YC drafted the manuscript, HC critically reviewed the manuscript. All authors reviewed the manuscript and approved the final version for submission.</p>", "<title>Data availability</title>", "<p>The data that support the findings of this study are available from the corresponding author upon reasonable request.</p>", "<title>Competing interests</title>", "<p id=\"Par24\">The authors declare no competing interests.</p>" ]

[ "<fig id=\"Fig1\"><label>Fig. 1</label><caption><title>Relationship between diffusion properties and the severity of core symptoms at baseline in first-episode schizophrenia.</title><p><bold>A</bold> Fiber tracts significantly associated with symptoms. <bold>B</bold> The red lines indicate MD of fiber tracts were positively correlated with PANSS-8 scores. IFOF inferior fronto-occipital fasciculi, ILF longitudinal fasciculi, MD mean diffusivity.</p></caption></fig>", "<fig id=\"Fig2\"><label>Fig. 2</label><caption><title>Diffusion properties in discriminating nonresponders from responders in first-episode schizophrenia after 12-week antipsychotic monotherapies.</title><p><bold>A</bold> The selected diffusion properties showing highest predictability for nonresponders from cross-validated LASSO regression in first-episode schizophrenia. Red and blue indicated increased and decreased fractional anisotropy, respectively. <bold>B</bold> The receiver operating characteristic (ROC) curve for nonresponders. ILF longitudinal fasciculi, IFOF inferior fronto-occipital fasciculi, SLF superior longitudinal fasciculus.</p></caption></fig>" ]

[ "<table-wrap id=\"Tab1\"><label>Table 1</label><caption><p>Clinical and demographic information for the first-episode schizophrenia patients.</p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th>Characteristic</th><th>Responders, <italic>N</italic> = 30</th><th>Nonresponders, <italic>N</italic> = 20</th><th><italic>t/F</italic></th><th><italic>P</italic></th></tr></thead><tbody><tr><td>Age (years)</td><td>25.13 ± 7.45</td><td>25.65 ± 7.79</td><td>−0.24</td><td>0.815</td></tr><tr><td>Sex (M/F)</td><td>13/17</td><td>7/13</td><td>0.35</td><td>0.556</td></tr><tr><td>Duration of untreated psychosis (months)</td><td>5.73 ± 4.34</td><td>5.65 ± 4.63</td><td>0.07</td><td>0.949</td></tr><tr><td>Baseline PANSS-8</td><td>24.8 ± 5.85</td><td>23.65 ± 6.12</td><td>0.67</td><td>0.507</td></tr><tr><td>Antipsychotic type (risperidone/aripiprazole/olanzapine)</td><td>16/6/8</td><td>6/8/6</td><td>3.33</td><td>0.074</td></tr><tr><td>Mean antipsychotic dosage (olanzapine equivalents, mg/day)</td><td>19.08 ± 4.26</td><td>19.03 ± 4.87</td><td>0.34</td><td>0.564</td></tr></tbody></table></table-wrap>", "<table-wrap id=\"Tab2\"><label>Table 2</label><caption><p>Significant features for discriminating treatment responders and nonresponders.</p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th>Selection times</th><th>Hemisphere</th><th>Label</th><th>Feature type</th><th>Responders</th><th>Nonresponders</th><th><italic>t</italic></th><th><italic>P</italic></th></tr></thead><tbody><tr><td>10</td><td>Right</td><td>ILF</td><td>FA</td><td>0.45 ± 0.2</td><td>0.42 ± 0.02</td><td>5.69</td><td>&lt;0.001</td></tr><tr><td>9</td><td>Left</td><td>IFOF</td><td>MD</td><td>0.75 ± 0.02</td><td>0.77 ± 0.03</td><td>−2.25</td><td>0.029</td></tr><tr><td>8</td><td>Right</td><td>SLF</td><td>MD</td><td>0.68 ± 0.02</td><td>0.7 ± 0.03</td><td>−2.62</td><td>0.012</td></tr></tbody></table></table-wrap>" ]

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[ "<supplementary-material content-type=\"local-data\" id=\"MOESM1\"></supplementary-material>" ]

[ "<table-wrap-foot><p><italic>ILF</italic> longitudinal fasciculi, <italic>IFOF</italic> inferior fronto-occipital fasciculi, <italic>SLF</italic> superior longitudinal fasciculus, <italic>FA</italic> fractional anisotropy, <italic>MD</italic> mean diffusivity.</p></table-wrap-foot>", "<fn-group><fn><p><bold>Publisher’s note</bold> Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.</p></fn></fn-group>" ]

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[ "<media xlink:href=\"41398_2023_2714_MOESM1_ESM.docx\"><caption><p>Supplement materials</p></caption></media>" ]

{ "acronym": [], "definition": [] }

[ "<title>Introduction</title>", "<p id=\"Par2\">Modern science has become increasingly collaborative over the past decades^{##REF##17431139##1##}. Large teams have become almost necessary to tackle complex problems in various disciplines, requiring a large pool of knowledge and skills. On the other hand, small teams may introduce novel paradigms^{##REF##30760923##2##}.</p>", "<p id=\"Par3\">A powerful representation of the collaborative nature of science is given by a collaboration network, in which nodes are authors, and two nodes are connected if they have coauthored at least one paper. With the growing availability of bibliometric data, collaboration networks have been extensively studied, and their structural properties are now well known^{##REF##11149952##3##–##REF##26261301##6##}. Collaboration networks are concrete manifestations of <italic>homophily</italic> between scholars, i.e<italic>.</italic>, of the tendency of individuals to interact with people similar to themselves. People working on the same topic or problem may decide to team up and leverage their respective skills to increase their chances of discovering new results. This is an example of <italic>selection</italic>, where homophily results from the choice of people to engage with similar individuals. On the other hand, collaboration could also induce <italic>social influence</italic>, in that scholars might affect the future behavior of their coauthors. For a thorough discussion on homophily, selection, and social influence, we refer the reader to chapter 4 of the book by Easley and Kleinberg^{##UREF##1##7##}.</p>", "<p id=\"Par4\">Coauthors often expose us to new tools, methods, and theories, even when the latter is not being used for the specific project carried out by the team. The link between diffusion of knowledge and collaboration has been highlighted and explored for some time. For instance, it is known that knowledge flow occurs with a greater probability between scholars who have collaborated in the past^{##UREF##2##8##} and those who are in close proximity in the network^{##UREF##3##9##}.</p>", "<p id=\"Par5\">In particular, once scholars discover new research topics, they may decide to work on them in the future. Switches between research interests have become increasingly frequent over time^{##REF##30602773##10##} and have recently been subjected to investigation^{##UREF##4##11##,##REF##35939670##12##}. The decision to switch may actually be induced by the coauthors in a social contagion process^{##UREF##5##13##–##REF##22972300##17##} where scholar <italic>a</italic>, who spreads the new topic, influences scholar <italic>b</italic> to adopt it. For this reason, epidemic models have been applied to describe the diffusion of ideas^{##REF##14212412##18##–##UREF##7##20##}. In these models, an <italic>infected</italic> individual <italic>a</italic> exposes a <italic>susceptible</italic> individual <italic>b</italic> to a disease with a certain probability of getting infected and continuing the spread. In the case of an idea or a topic, the infection spreads if <italic>b</italic> adopts the new idea or starts working on the new topic. On a macro level, dynamics within collaboration networks like topic switches guide the evolution of disciplines^{##UREF##8##21##,##REF##23323212##22##}.</p>", "<p id=\"Par6\">Here we present an extensive empirical analysis of the relationship between topic switches of scientists and their collaboration patterns. We distinguish active authors, i.e<italic>.</italic>, those who have papers on the new topic, from inactive authors who have never published in that area. For simplicity, we focus only on the first-order neighborhoods in the collaboration network. We find that the probability that the inactive coauthors of an active scholar switch topic grows with the productivity and impact of the latter. The larger the average number of inactive coauthors of active scientists, the smaller the effect. Also, the topic-switch probability for an inactive scholar grows with the number of their active coauthors, with a profile suggesting that the contributions of each coauthor are not independent.</p>" ]

[ "<title>Methods</title>", "<p id=\"Par31\">Data We analyze papers from the February 2023 snapshot of the bibliometric dataset OpenAlex: the successor to Microsoft Academic Graph (MAG). We found incomplete citation coverage for papers published before 1990. So, we only consider papers published between 1990 and 2022 and having at most thirty authors. Papers are tagged with <italic>concepts</italic> (topics) by a classifier trained on the MAG. We use concept tags to construct snapshots for three fields: Physics, Computer Science (CS), and Biology and Medicine (BioMed). Physics contains 19.7M papers, while CS and BioMed each have 27.6M and 43.52M papers, respectively. Within each domain, we select seven, six, and seven topics, respectively.</p>", "<p id=\"Par32\">Within each topic, we consider reference years between 1995 and 2018, where the respective interaction and activation windows contain at least 3000 papers. This threshold ensures a critical mass of papers and authors to conduct the analyses. Each topic we selected has at least ten reference years satisfying the constraint. The statistical tests in the manuscript are aggregated over the different reference years. More information is available in Supplementary Tables ##SUPPL##0##S1–3## online.</p>", "<p id=\"Par33\">Overlap coefficient We use the overlap coefficient to measure the degree of overlap between the different sets of authors picked based on productivity and impact.In our case, the two sets are the same size, so a score of 10% implies that both sets share 10% of the elements.</p>", "<p id=\"Par34\">Author ranking metrics Let <italic>P</italic> be the set of papers published on topic <italic>t</italic> authored by the set of active authors <italic>A</italic> during the interaction window IW. Let <italic>a</italic> be an active author who wrote papers during the IW. We define the following metrics to rank active authors and select the top and bottom 10%.</p>", "<p id=\"Par35\"><italic>Productivity:</italic> the count of papers <italic>a</italic> has authored on topic <italic>t</italic> during the IW. More formally, it is the cardinality of the set .</p>", "<p id=\"Par36\"><italic>Impact:</italic> the average citation count of from the papers in <italic>P</italic>.</p>", "<p id=\"Par37\">We argue that restricting incoming citations from <italic>P</italic> is a good proxy for the impact that <italic>a</italic> has made on that topic. The average number of citations is a better indicator of excellence than the total citation count^{##UREF##14##31##}. Also, considering the average instead of the sum lowers its correlation with productivity, here measured by the <italic>overlap coefficient</italic>, as often the most productive authors are also the most cited ones^{##REF##35939670##12##}. A low correlation lets us safely disregard the confounding effects of the two metrics and allows us to treat them as fairly independent variables. Correlation statistics are reported in Supplementary Tables ##SUPPL##0##S4–6## online. Although citation-based measures are frequently used to quantify research impact, we are aware of the influence of social structures and other hidden biases on scholarly citation behavior^{##UREF##15##32##}. Using more sophisticated measures, however, is beyond the scope of this present work.</p>", "<p id=\"Par38\">Statistical test for difference of samples To test whether two independent samples and are different concerning their means and , we assume the null hypothesis that their means are the same, i.e<italic>.</italic>, . Next, we compute the mean and 95% confidence interval of the distribution of the difference of their means, i.e<italic>.</italic>, , using bootstrapping^{##UREF##16##33##}. We reject the null hypothesis at if the confidence interval of \n<italic>does not</italic> contain 0^{##REF##3082422##34##}. In other words, and are considered statistically different at if the 95% confidence interval of the difference of their respective means does not contain 0. Furthermore, a positive mean of the difference indicates that , while a negative mean indicates .</p>", "<p id=\"Par39\">In our experiments, we aggregate the differences across the reference years for a given topic, and then carry out the procedure described above.</p>", "<p id=\"Par40\">Target activation probability Let <italic>n</italic>(<italic>k</italic>) be the number of inactive authors with exactly <italic>k</italic> contacts during the exposure window, of whom <italic>m</italic>(<italic>k</italic>) become active in the observation window. The <italic>target activation probability</italic>\n<italic>P</italic>(<italic>k</italic>) is the probability of becoming active after having exactly <italic>k</italic> contacts, defined asThe <italic>cumulative target activation probability</italic>\n<italic>C</italic>(<italic>k</italic>) with <italic>k</italic> or more contacts is given by</p>", "<p id=\"Par41\">Simple baseline for membership closure Let <italic>p</italic> represent the probability of activation from a single contact. The probability of activation having <italic>k</italic> contacts, acting independently of each other, is . We compute <italic>p</italic> from the observed data using Eq. (##FORMU##75##1##) as . This is the fraction of inactive authors with <italic>exactly</italic> one contact who became active as . Like before, we calculate the cumulative target activation probability for the baseline with <italic>k</italic> or more contacts asThe denominator is the same as in Eq. (##FORMU##75##1##) and comes from the observed data. The numerator represents the expected number of active authors if the contacts affect the activation independently.</p>", "<p id=\"Par42\">Source activation probability Let be the number of exclusive inactive coauthors of an active author <italic>a</italic> in the IW. Let be the number of those exclusive inactive coauthors who become active in the AW. The <italic>source activation probability</italic> of scholar <italic>a</italic> is thusWe stress that, for the probability to be well-defined, must be greater than zero. Therefore, in our calculations, we focused on active authors with at least one exclusive inactive coauthor.</p>", "<p id=\"Par43\">For any , we compute the fraction of all active authors whose source activation probability is greater than or equal to <italic>f</italic>. is the complementary cumulative probability distribution of the source activation probability . As expected, quickly decreases to 0 with increasing <italic>f</italic>. Because the curves corresponding to two sets of active authors are effectively indistinguishable at the tail, we compare a pair of points at some threshold . We call the <italic>cumulative source activation</italic>.</p>", "<p id=\"Par44\">The choice of the threshold is important. Setting it to 0 or 1 would return the same probability for both sets of authors. It should not also be too small for numerical reasons. For example, if there are only five inactive coauthors, the smallest non-zero fraction cannot be smaller than . Choosing too high a value instead would lead to weaker statistics. So, we fix the value at 0.10 for the results in the main text (Figs. ##FIG##3##4## and ##FIG##4##5##), and at 0.20 in the Supplementary Figs. ##SUPPL##0##S3## and ##SUPPL##0##S4## online.</p>", "<p id=\"Par45\">Chaperoning propensity Let be the number of exclusive inactive coauthors of an active author <italic>a</italic> who become active in the AW, which is the same as the numerator of Eq. (##FORMU##84##4##). Let be the number of those authors who write their first paper on topic <italic>t</italic> with <italic>a</italic> in the AW. The <italic>chaperoning probability</italic> of <italic>a</italic> is defined asWe define the <italic>chaperoning propensity</italic>\n corresponding to a specific threshold as the fraction of all active authors with . We use the aforementioned values of 0.10 (Figs. ##FIG##3##4## and ##FIG##4##5##) and 0.20 (Supplementary Figs. ##SUPPL##0##S3## and ##SUPPL##0##S4## online) for the threshold <italic>f</italic>.</p>" ]

[ "<title>Results</title>", "<p id=\"Par7\">We use the scientific publication dataset OpenAlex^{##UREF##9##23##}. We present the results for twenty topics belonging to three disciplines: Physics, Computer Science, and Biology &amp; Medicine. See “<xref rid=\"Sec6\" ref-type=\"sec\">Methods</xref>” for details.</p>", "<p id=\"Par8\">Our approach is inspired by the pioneering work by Kossinets and Watts on social network evolution^{##REF##16400149##24##}. In it, the authors estimated <italic>triadic closure</italic> of two individuals <italic>a</italic> and <italic>b</italic>, i.e., the probability that <italic>a</italic> and <italic>b</italic> become acquainted as a function of the number of common friends. They took two snapshots of the network at consecutive time ranges: in the earlier snapshot, one keeps track of all pairs of disconnected people, and in the latter, one counts how many of those pairs become connected. A similar approach has been adopted to compute <italic>membership closure</italic>, i.e<italic>.</italic>, the probability that an individual starts participating in an activity having been connected to <italic>k</italic> others who participate in it^{##UREF##10##25##}. We now describe how we adapt this framework to measure how collaborations induce topic switches.</p>", "<p id=\"Par9\">Given a scientific topic <italic>t</italic>, reference year , and window size <italic>T</italic>, we construct two consecutive non-overlapping time ranges spanning years and respectively. We call the first range the <italic>interaction window</italic> (IW), where we track author interactions in the collaboration network, and the latter range, the <italic>activation window</italic> (AW), where we count topic switches. We then identify the set of <italic>active</italic> authors <italic>A</italic> who published papers <italic>P</italic> on topic <italic>t</italic> during the IW. For example, in Fig. ##FIG##0##1##a, . We construct the collaboration network <italic>G</italic> by considering all papers written by authors after <italic>a</italic> becomes active. Note that includes papers outside of <italic>P</italic>, like the ones drawn in gray in Fig. ##FIG##0##1##a. We classify the non-active authors in <italic>G</italic> as <italic>inactive</italic> authors who are the candidates for topic switches in the AW. They turn active when they publish their first paper on topic <italic>t</italic>. In Fig. ##FIG##0##1##b, authors , and are inactive, with and becoming active in the AW. Furthermore, we rank each active author based on two metrics of scientific prominence: <italic>productivity</italic> and <italic>impact</italic>, described in Methods, and calculated at the end of the IW to capture the current perception of <italic>a</italic>’s scholarly output. Finally, for each metric, we identify and mark the authors who rank in the top and the bottom 10%.</p>", "<p id=\"Par10\">Given this general setup, we conduct two complementary experiments that we describe in depth in the following sections. In Experiment I, we measure membership closure among inactive authors to quantitatively assess how past collaborations with active authors manifest in topic switches. In Experiment II, we instead focus on active authors, quantifying the propensity of their inactive coauthors to start working on their topic of expertise. All the measures used in these sections are formally defined in <xref rid=\"Sec6\" ref-type=\"sec\">Methods</xref>.</p>", "<title>Experiment I</title>", "<p id=\"Par11\">Here we investigate membership closure among inactive authors. Specifically, we will answer the following questions:<list list-type=\"bullet\"><list-item><p id=\"Par12\">How is the probability of topic switches related to <italic>k</italic>, the number of contacts with active authors?</p></list-item><list-item><p id=\"Par13\">Does this probability depend on the relative prominence of the active authors?</p></list-item></list>To compute the measure, we first must define what construes as contact with an active author in the IW. We consider two definitions as described below. <list list-type=\"order\"><list-item><p id=\"Par14\">The number of active coauthors, with the same coauthor counted as many times as the number of collaborations. In the collaboration network, this corresponds to the weighted degree when considering only active coauthors.</p></list-item><list-item><p id=\"Par15\">The number of papers written with active coauthors.</p></list-item></list>For example, in Fig. ##FIG##0##1##c, author has five contacts based on the first definition (two each from and and one from ), and three if we use the second (the second, the fifth, and the seventh papers in the IW). We report the findings based on the first definition in the main text. The results from the second definition do not alter the main conclusions and can be found in Supplementary Figs. ##SUPPL##0##S1## and ##SUPPL##0##S2## online.</p>", "<p id=\"Par16\">To address the first question, we compute the cumulative <italic>target activation probability</italic>\n<italic>C</italic>(<italic>k</italic>), i.e<italic>.</italic>, the fraction of inactive authors who become active in the AW as a function of the number of contacts <italic>k</italic>. In Fig. ##FIG##1##2##, we plot <italic>C</italic>(<italic>k</italic>) (in purple) for each of the twenty topics under investigation. Error bars derive from averaging over different time windows for each field. As expected, we see an increasing trend. In particular, the jump from <italic>k</italic> = 0 to <italic>k</italic> = 1 is remarkable, showing that the probability of <italic>spontaneous</italic> activation in the absence of previous contacts (<italic>k</italic> = 0) is much lower than that of activation through collaboration (<italic>k</italic>\n 1). We observe that the higher the number of contacts, the larger the probability. Most of the growth occurs for low values of <italic>k</italic>.</p>", "<p id=\"Par17\">To put these numbers in context, we consider a <italic>simple baseline</italic>\n where we assume each contact has a constant, independent probability of producing a topic switch. Within each topic, we compute the difference between the curves for each value of <italic>k</italic> (see “<xref rid=\"Sec6\" ref-type=\"sec\">Methods</xref>”) over all reference years and plot them below the <italic>x</italic>-axis. Except for the topics of Cluster Analysis, Parallel Computing, and Peptide Sequence, the observed curves deviate from the baseline. This provides some empirical evidence to ascertain that the baseline cannot capture the nuances in the observed data. A positive deviation for the majority of the topics indicates a compounding effect. Fluid Dynamics and Statistical Physics are exceptions, as they undershoot the baseline. This may be because they are broad interdisciplinary fields unlike the others, and having collaborators in different fields may lessen their effect.</p>", "<p id=\"Par18\">Next, we explore the second research question, checking if the contact source’s prominence affects activation chances. Recall that in every IW for a topic, we select active authors in the top 10% and the bottom 10% based on productivity and impact. This separates the most prominent active authors from the least prominent. To mitigate confounding effects, we only consider the subset of inactive authors who are neighbors with strictly one of the two sets of active authors. In Fig. ##FIG##2##3##, we assess the significance of the difference between the cumulative target activation probabilities for inactive authors in contact with active authors in the two bins. Each heatmap row corresponds to a topic, and the color of each cell indicates whether the difference is positive (red), negative (blue), or non-significant (gray). The two panels correspond to prominent authors selected based on productivity (panel <bold>a</bold>) and impact (panel <bold>b</bold>). For productivity, all differences are significant and positive, meaning that contacts with highly productive active authors lead to higher target activation probabilities. For impact, there are a handful of exceptions. Overall, having prominent contacts increases the target activation probability.</p>", "<title>Experiment II</title>", "<p id=\"Par19\">Here we focus on the active authors and their collaborators. For every active author <italic>a</italic>, we consider the subset of their inactive coauthors who have <italic>exclusively</italic> collaborated with <italic>a</italic> in the IW. We call this set the exclusive inactive coauthors of <italic>a</italic>. For example, in Fig. ##FIG##0##1##d, active author has four coauthors , of whom only and exclusively collaborate with in the IW. We do this because effects due to active authors different from <italic>a</italic> would be difficult to disentangle and could confound the analysis and the conclusions. The relevant measure here is the <italic>source activation probability</italic>\n, i.e<italic>.</italic>, the fraction of exclusive inactive coauthors who become active in the AW. The fraction controls for the collaboration neighborhood sizes which could vary widely for different scholars. In Fig. ##FIG##0##1##d, for is = 50%, as only becomes active in the AW.</p>", "<p id=\"Par20\">For a given set of active authors, we obtain , the <italic>complementary cumulative probability distribution</italic> of their source activation probabilities. We select the pools of the most and the least prominent authors as described in Experiment I. The relative effects of the two groups are estimated by comparing the <italic>cumulative source activations</italic>, i.e<italic>.</italic>, points on the respective cumulative distributions at a specific threshold . Results are reported in Fig. ##FIG##3##4##a for a threshold . Our conclusions also hold when considering a threshold , which can be found in Supplementary Fig. ##SUPPL##0##S3## online.</p>", "<p id=\"Par21\">In Fig. ##FIG##3##4##a, each row corresponds to a topic. The different ranges represent the confidence intervals of the mean difference between the cumulative source activations for the two pools of authors for productivity (green) and impact (pink), respectively. For productivity, the difference is significant for all topics but one (Gravitational Wave). The differences are somewhat less pronounced for impact, but are still significant in most cases.</p>", "<p id=\"Par22\">To further corroborate this finding, we specialize the analysis by checking how many exclusive coauthors of <italic>a</italic> also published their first paper on topic <italic>t</italic> in the AW with <italic>a</italic>. This is a way to assess the <italic>chaperoning propensity</italic> of active authors^{##REF##30530676##26##}, and we define the measure in <xref rid=\"Sec6\" ref-type=\"sec\">Methods</xref>. In Fig. ##FIG##3##4##b, we report the confidence intervals of the average difference between the chaperoning propensities for the most prominent and the least prominent active authors for threshold . Similar to Fig. ##FIG##3##4##a, we find that the more productive/impactful an active author is, the more likely their coauthors will start working with them on a new topic. Results for , which confirm this trend, can be found in Supplementary Fig. ##SUPPL##0##S4## online.</p>", "<p id=\"Par23\">While our analysis clearly shows that prominence is a factor, one may wonder if the number of coauthors also plays a role. We posit that, on average, the more collaborators one has, the more tenuous the contact with any of them will be, resulting in lower source activation probabilities. From each group of most prominent authors, we, therefore, pick the top and the bottom 20% based on the average number of coauthors on papers published with exclusive inactive coauthors. By construction, this excludes any paper written on the focal topic. In Fig. ##FIG##4##5##, we perform the same analysis as in Fig. ##FIG##3##4## for the two pools of authors described above. We observe that the confidence intervals of the differences lie to the <italic>left</italic> of zero, i.e<italic>.</italic>, are negative. For productivity, all values are significant. For impact, there are only two topics (Chemotherapy and Radiation Therapy) that are not significant. Overall, inactive coauthors of prominent authors with more collaborators have a lower probability of switching topics. This is consistent with the intuition that the interactions with each coauthor are less frequent/strong in that case and, consequently, less effective at inducing topic switches.</p>" ]

[ "<title>Discussion</title>", "<p id=\"Par24\">Collaboration allows scholars to deepen existing knowledge and be exposed to new ideas. In this paper, we assessed if and how collaboration patterns affect the probability of switching research topics. We determined that the probability for a scholar to start working on a new topic depends on earlier contacts with people already active in that topic. This effect is proportional to the number of contacts, with more contacts resulting in higher probabilities. In most topics, this behavior is distinct from a simple baseline assuming independent effects from the contacts, which likely indicates effects of non-dyadic interactions that prompt further investigation.</p>", "<p id=\"Par25\">Similarly, we measured the probability that inactive coauthors of an active author end up publishing on the new topic, which singles out the effect of the association with that author in the activation process. Specifically, we checked whether the activation probability depends on some features of the active authors. We found that the more prolific and impactful authors have higher chances of inducing coauthors to switch topics and become coauthors in their first paper on the topic.</p>", "<p id=\"Par26\">We stress that, by design, previous interactions between inactive and active authors are limited to works dealing with topics different from the focal topic. Therefore, our analysis suggests that an active author may expose an inactive one to a new topic, even when their interactions do not directly concern that topic. This underlines the social character of scientific interactions, where discussions may deviate from the context that mainly motivates them.</p>", "<p id=\"Par27\">Furthermore, we showed that the larger the number of coauthors of an active author, the lower the chance of a topic switch. This is consistent with a <italic>dilution</italic> of the effect, resulting from the inability to interact strongly with collaborators when their number is large. To the best of our knowledge, we are disclosing this effect for the first time.</p>", "<p id=\"Par28\">A possible explanation of our findings is that topic switches result from a social contagion process, much like the adoption of new products^{##UREF##6##15##,##UREF##11##27##}, or the spreading of political propaganda^{##REF##22972300##17##}. However, we cannot discount selection effects in observational studies like ours^{##REF##22523436##28##}. Having large numbers of active coauthors on a topic may be associated with strong latent homophily between the authors, which may facilitate the future adoption of the topic even without interventions from the active authors. Therefore, the effects we observed may be due to a combination of social contagion and selection.</p>", "<p id=\"Par29\">Our work uses OpenAlex, a valuable open-access bibliometric database. We rely on their author disambiguation and topic classification algorithms to conduct the analyses. These processes are inherently noisy and can introduce implicit biases. In addition, there appears to be incomplete citation coverage which might partly explain why the results for impact are not so robust as those for productivity. Future releases of OpenAlex might mitigate these problems. To counter these issues, we repeated our analysis on multiple topics from three distinct scientific disciplines. While the size of the effects varies with the topic, the paper’s main conclusions hold across topics, with very few exceptions.</p>", "<p id=\"Par30\">In conclusion, our work offers a platform for further investigations on the mechanisms driving topic switches in science. A thorough understanding of these mechanisms requires effective integration of all factors that may play a role. Besides productivity and impact, topic switches may be affected by the institutional affiliations of those involved. On the one hand, it is plausible that people in the same institution have more chances to interact and affect each other’s behavior. On the other hand, collaborations with people from renowned institutions are expected to weigh more in the process. Another discriminating factor could be the number of citations to the collaborator’s papers. The higher the number of citations, the closer the association between collaborators. We could also include the scientific affinity between coauthors through the similarity of their papers. Modern neural language models^{##UREF##12##29##,##UREF##13##30##} allow to embed papers and, consequently, authors in high-dimensional vector spaces, where the distance between two authors is a good proxy of the similarity of their outputs. The analysis we have conducted here can be extended to other sectors of human activity where collaboration plays a key role, like software development and patent design.</p>" ]

[]

[ "<p id=\"Par1\">Collaboration is a key driver of science and innovation. Mainly motivated by the need to leverage different capacities and expertise to solve a scientific problem, collaboration is also an excellent source of information about the future behavior of scholars. In particular, it allows us to infer the likelihood that scientists choose future research directions via the intertwined mechanisms of selection and social influence. Here we thoroughly investigate the interplay between collaboration and topic switches. We find that the probability for a scholar to start working on a new topic increases with the number of previous collaborators, with a pattern showing that the effects of individual collaborators are not independent. The higher the productivity and the impact of authors, the more likely their coworkers will start working on new topics. The average number of coauthors per paper is also inversely related to the topic switch probability, suggesting a dilution of this effect as the number of collaborators increases.</p>", "<title>Subject terms</title>" ]

[ "<title>Supplementary Information</title>", "<p>\n</p>" ]

[ "<title>Supplementary Information</title>", "<p>The online version contains supplementary material available at 10.1038/s41598-024-51606-6.</p>", "<title>Acknowledgements</title>", "<p>We acknowledge the support of the AccelNet-MultiNet program, a project of the National Science Foundation (Award #1927425 and #1927418). This work is also supported by the Air Force Office of Scientific Research under award #FA9550-19-1-0354.This research was also supported in part by Lilly Endowment, Inc., through its support for the Indiana University Pervasive Technology Institute.</p>", "<title>Author contributions</title>", "<p>S.F. designed the research; S.V. and S.S. performed the experiments and data analysis. S.V., S.S., F.R., F.T., and S.F. wrote the manuscript. All authors reviewed the manuscript.</p>", "<title>Data availability</title>", "<p>The datasets generated during and/or analyzed during the current study are available in the <italic>Collaboration-Topic-Switches</italic> repository on <ext-link ext-link-type=\"uri\" xlink:href=\"https://github.com/satyakisikdar/Collaboration-Topic-Switches\">GitHub</ext-link>.</p>", "<title>Competing interests</title>", "<p id=\"Par46\">The authors declare no competing interests.</p>" ]

[ "<fig id=\"Fig1\"><label>Figure 1</label><caption><p>Schematic setup for our analysis. (<bold>a</bold>) Stream of papers across interaction (IW) and activation (AW) windows. Papers tagged with the focal topic <italic>t</italic> are marked in red. (<bold>b</bold>) Author collaboration graph at the end of IW. Authors and are linked by an edge of weight <italic>k</italic> if coauthored <italic>k</italic> papers with within the IW. The authors active in the focal topic by the end of IW are marked in red. (<bold>c</bold>) Focus: inactive authors. Inactive author has five active contacts from three sources {} derived from the collaboration graph in (<bold>b</bold>). (<bold>d</bold>) Focus: active authors. Active author has four coauthors {}, of whom is already active, and also collaborated with in the IW. This leaves the subset of exclusive inactive coauthors . Within this subset, only becomes active in the AW, resulting in ’s source activation probability of . Additionally, writes their first paper with in the AW.</p></caption></fig>", "<fig id=\"Fig2\"><label>Figure 2</label><caption><p>Experiment I. Cumulative target activation probability (in purple) for inactive authors in the AW with shaded 95% confidence intervals. For each <italic>k</italic>, the <italic>y</italic>-value indicates the fraction of inactive authors with at least <italic>k</italic> active contacts in the IW who became active in the AW. The dashed green line with shaded errors represents the baseline described in the text, corresponding to independent effects from the coauthors. The heatmap below the <italic>x</italic>-axis shows the mean difference between the observed and baseline curves for each <italic>k</italic> value. It is gray if the 95% confidence interval contains 0, denoting the <italic>k</italic>-values where the points are statistically indistinguishable at <italic>p</italic>-value 0.05. Positive and negative deviations from the baseline are in red and blue, respectively.</p></caption></fig>", "<fig id=\"Fig3\"><label>Figure 3</label><caption><p>Heatmaps showing the mean difference between the cumulative target activation probabilities of the inactive authors in the AW who had exclusive contacts with the top 10% and bottom 10% of active authors, respectively, selected according to productivity (<bold>a</bold>) and impact (<bold>b</bold>) in the IW. The cells are gray if the 95% confidence interval contains 0. The topic names have been abbreviated to save space. The majority of red cells indicate that the cumulative target activation probabilities for contacts with the top 10% are higher than those with the bottom 10%.</p></caption></fig>", "<fig id=\"Fig4\"><label>Figure 4</label><caption><p>Experiment II. Results for . (<bold>a</bold>) The mean and 95% confidence interval of the means of the difference between the cumulative source activations of active authors in the top 10% and bottom 10% based on productivity (green circles) and impact (pink squares). (<bold>b</bold>) The mean and 95% confidence interval of the means of the difference between the chaperoning propensities of active authors in the top 10% and bottom 10% based on productivity (green circles) and impact (pink squares). The topic names have been abbreviated to save space. A positive difference indicates that the effect is stronger for the top 10% active authors.</p></caption></fig>", "<fig id=\"Fig5\"><label>Figure 5</label><caption><p>Dilution effect. Results for . The mean and 95% confidence interval of the mean of the difference between the cumulative source activations of active authors in the top 20% and bottom 20% bins, based on the average number of coauthors, among the top 10% active authors in productivity (green circles) and impact (pink squares). The topic names have been abbreviated to save space. A negative difference across the topics indicates a <italic>dilution</italic> effect, wherein coauthors of prominent active scholars with fewer collaborators (on average) are more likely to switch topics.</p></caption></fig>" ]

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stretchy=\"false\">}</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq22\"><alternatives><tex-math id=\"M43\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$P^\\prime$$\\end{document}</tex-math><mml:math id=\"M44\"><mml:msup><mml:mi>P</mml:mi><mml:mo>′</mml:mo></mml:msup></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq23\"><alternatives><tex-math id=\"M45\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$a \\in A$$\\end{document}</tex-math><mml:math id=\"M46\"><mml:mrow><mml:mi>a</mml:mi><mml:mo>∈</mml:mo><mml:mi>A</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq24\"><alternatives><tex-math id=\"M47\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$P^\\prime$$\\end{document}</tex-math><mml:math id=\"M48\"><mml:msup><mml:mi>P</mml:mi><mml:mo>′</mml:mo></mml:msup></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq25\"><alternatives><tex-math id=\"M49\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$a_2, a_3$$\\end{document}</tex-math><mml:math id=\"M50\"><mml:mrow><mml:msub><mml:mi>a</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>a</mml:mi><mml:mn>3</mml:mn></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq26\"><alternatives><tex-math id=\"M51\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$a_6$$\\end{document}</tex-math><mml:math id=\"M52\"><mml:msub><mml:mi>a</mml:mi><mml:mn>6</mml:mn></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq27\"><alternatives><tex-math id=\"M53\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$a_2$$\\end{document}</tex-math><mml:math id=\"M54\"><mml:msub><mml:mi>a</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq28\"><alternatives><tex-math id=\"M55\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$a_6$$\\end{document}</tex-math><mml:math id=\"M56\"><mml:msub><mml:mi>a</mml:mi><mml:mn>6</mml:mn></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq29\"><alternatives><tex-math id=\"M57\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$a \\in A$$\\end{document}</tex-math><mml:math id=\"M58\"><mml:mrow><mml:mi>a</mml:mi><mml:mo>∈</mml:mo><mml:mi>A</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq30\"><alternatives><tex-math id=\"M59\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$a_6$$\\end{document}</tex-math><mml:math id=\"M60\"><mml:msub><mml:mi>a</mml:mi><mml:mn>6</mml:mn></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq31\"><alternatives><tex-math id=\"M61\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$a_1$$\\end{document}</tex-math><mml:math id=\"M62\"><mml:msub><mml:mi>a</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq32\"><alternatives><tex-math id=\"M63\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$a_5$$\\end{document}</tex-math><mml:math id=\"M64\"><mml:msub><mml:mi>a</mml:mi><mml:mn>5</mml:mn></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq33\"><alternatives><tex-math id=\"M65\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$a_0$$\\end{document}</tex-math><mml:math id=\"M66\"><mml:msub><mml:mi>a</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq34\"><alternatives><tex-math id=\"M67\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\ge$$\\end{document}</tex-math><mml:math id=\"M68\"><mml:mo>≥</mml:mo></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq35\"><alternatives><tex-math id=\"M69\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$C_\\text {base}(k)$$\\end{document}</tex-math><mml:math id=\"M70\"><mml:mrow><mml:msub><mml:mi>C</mml:mi><mml:mtext>base</mml:mtext></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq36\"><alternatives><tex-math id=\"M71\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$a_0$$\\end{document}</tex-math><mml:math id=\"M72\"><mml:msub><mml:mi>a</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq37\"><alternatives><tex-math id=\"M73\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\{a_1, a_2, a_3, a_6\\}$$\\end{document}</tex-math><mml:math id=\"M74\"><mml:mrow><mml:mo stretchy=\"false\">{</mml:mo><mml:msub><mml:mi>a</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>a</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>a</mml:mi><mml:mn>3</mml:mn></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>a</mml:mi><mml:mn>6</mml:mn></mml:msub><mml:mo stretchy=\"false\">}</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq38\"><alternatives><tex-math id=\"M75\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$a_2$$\\end{document}</tex-math><mml:math id=\"M76\"><mml:msub><mml:mi>a</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq39\"><alternatives><tex-math id=\"M77\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$a_3$$\\end{document}</tex-math><mml:math id=\"M78\"><mml:msub><mml:mi>a</mml:mi><mml:mn>3</mml:mn></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq40\"><alternatives><tex-math id=\"M79\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$a_0$$\\end{document}</tex-math><mml:math id=\"M80\"><mml:msub><mml:mi>a</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq41\"><alternatives><tex-math id=\"M81\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$P_s^a$$\\end{document}</tex-math><mml:math id=\"M82\"><mml:msubsup><mml:mi>P</mml:mi><mml:mi>s</mml:mi><mml:mi>a</mml:mi></mml:msubsup></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq42\"><alternatives><tex-math id=\"M83\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$P_s^a$$\\end{document}</tex-math><mml:math id=\"M84\"><mml:msubsup><mml:mi>P</mml:mi><mml:mi>s</mml:mi><mml:mi>a</mml:mi></mml:msubsup></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq43\"><alternatives><tex-math id=\"M85\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$a_0$$\\end{document}</tex-math><mml:math id=\"M86\"><mml:msub><mml:mi>a</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq44\"><alternatives><tex-math id=\"M87\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\tfrac{1}{2}$$\\end{document}</tex-math><mml:math id=\"M88\"><mml:mstyle displaystyle=\"false\" scriptlevel=\"0\"><mml:mfrac><mml:mn>1</mml:mn><mml:mn>2</mml:mn></mml:mfrac></mml:mstyle></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq45\"><alternatives><tex-math id=\"M89\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$a_2$$\\end{document}</tex-math><mml:math id=\"M90\"><mml:msub><mml:mi>a</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq46\"><alternatives><tex-math id=\"M91\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$C_s$$\\end{document}</tex-math><mml:math id=\"M92\"><mml:msub><mml:mi>C</mml:mi><mml:mi>s</mml:mi></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq47\"><alternatives><tex-math id=\"M93\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$f^*$$\\end{document}</tex-math><mml:math id=\"M94\"><mml:msup><mml:mi>f</mml:mi><mml:mo>∗</mml:mo></mml:msup></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq48\"><alternatives><tex-math id=\"M95\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$f^*= 0.10$$\\end{document}</tex-math><mml:math id=\"M96\"><mml:mrow><mml:msup><mml:mi>f</mml:mi><mml:mo>∗</mml:mo></mml:msup><mml:mo>=</mml:mo><mml:mn>0.10</mml:mn></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq49\"><alternatives><tex-math id=\"M97\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$f^*= 0.20$$\\end{document}</tex-math><mml:math id=\"M98\"><mml:mrow><mml:msup><mml:mi>f</mml:mi><mml:mo>∗</mml:mo></mml:msup><mml:mo>=</mml:mo><mml:mn>0.20</mml:mn></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq50\"><alternatives><tex-math id=\"M99\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$f^*= 0.10$$\\end{document}</tex-math><mml:math id=\"M100\"><mml:mrow><mml:msup><mml:mi>f</mml:mi><mml:mo>∗</mml:mo></mml:msup><mml:mo>=</mml:mo><mml:mn>0.10</mml:mn></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq51\"><alternatives><tex-math id=\"M101\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$95\\%$$\\end{document}</tex-math><mml:math id=\"M102\"><mml:mrow><mml:mn>95</mml:mn><mml:mo>%</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq52\"><alternatives><tex-math id=\"M103\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$95\\%$$\\end{document}</tex-math><mml:math id=\"M104\"><mml:mrow><mml:mn>95</mml:mn><mml:mo>%</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq53\"><alternatives><tex-math id=\"M105\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$f^*= 0.10$$\\end{document}</tex-math><mml:math id=\"M106\"><mml:mrow><mml:msup><mml:mi>f</mml:mi><mml:mo>∗</mml:mo></mml:msup><mml:mo>=</mml:mo><mml:mn>0.10</mml:mn></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq54\"><alternatives><tex-math id=\"M107\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$f^*= 0.20$$\\end{document}</tex-math><mml:math id=\"M108\"><mml:mrow><mml:msup><mml:mi>f</mml:mi><mml:mo>∗</mml:mo></mml:msup><mml:mo>=</mml:mo><mml:mn>0.20</mml:mn></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq55\"><alternatives><tex-math id=\"M109\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$f^*= 0.10$$\\end{document}</tex-math><mml:math id=\"M110\"><mml:mrow><mml:msup><mml:mi>f</mml:mi><mml:mo>∗</mml:mo></mml:msup><mml:mo>=</mml:mo><mml:mn>0.10</mml:mn></mml:mrow></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ6\"><alternatives><tex-math id=\"M111\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} \\text {Overlap}(A, B) = \\frac{|A \\cap B|}{\\min (|A|, |B|)} \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M112\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:mtext>Overlap</mml:mtext><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>A</mml:mi><mml:mo>,</mml:mo><mml:mi>B</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mfrac><mml:mrow><mml:mo stretchy=\"false\">|</mml:mo><mml:mi>A</mml:mi><mml:mo>∩</mml:mo><mml:mi>B</mml:mi><mml:mo stretchy=\"false\">|</mml:mo></mml:mrow><mml:mrow><mml:mo movablelimits=\"true\">min</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mo stretchy=\"false\">|</mml:mo><mml:mi>A</mml:mi><mml:mo stretchy=\"false\">|</mml:mo><mml:mo>,</mml:mo><mml:mo stretchy=\"false\">|</mml:mo><mml:mi>B</mml:mi><mml:mo stretchy=\"false\">|</mml:mo><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mfrac></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq56\"><alternatives><tex-math id=\"M113\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$P_a$$\\end{document}</tex-math><mml:math id=\"M114\"><mml:msub><mml:mi>P</mml:mi><mml:mi>a</mml:mi></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq57\"><alternatives><tex-math id=\"M115\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$P \\cap P_a$$\\end{document}</tex-math><mml:math id=\"M116\"><mml:mrow><mml:mi>P</mml:mi><mml:mo>∩</mml:mo><mml:msub><mml:mi>P</mml:mi><mml:mi>a</mml:mi></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq58\"><alternatives><tex-math id=\"M117\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$P_a$$\\end{document}</tex-math><mml:math id=\"M118\"><mml:msub><mml:mi>P</mml:mi><mml:mi>a</mml:mi></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq59\"><alternatives><tex-math id=\"M119\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$X_1$$\\end{document}</tex-math><mml:math id=\"M120\"><mml:msub><mml:mi>X</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq60\"><alternatives><tex-math id=\"M121\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$X_2$$\\end{document}</tex-math><mml:math id=\"M122\"><mml:msub><mml:mi>X</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq61\"><alternatives><tex-math id=\"M123\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\mu _1$$\\end{document}</tex-math><mml:math id=\"M124\"><mml:msub><mml:mi>μ</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq62\"><alternatives><tex-math id=\"M125\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\mu _2$$\\end{document}</tex-math><mml:math id=\"M126\"><mml:msub><mml:mi>μ</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq63\"><alternatives><tex-math id=\"M127\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$H_0$$\\end{document}</tex-math><mml:math id=\"M128\"><mml:msub><mml:mi>H</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq64\"><alternatives><tex-math id=\"M129\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$H_0: \\mu _1 = \\mu _2$$\\end{document}</tex-math><mml:math id=\"M130\"><mml:mrow><mml:msub><mml:mi>H</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo>:</mml:mo><mml:msub><mml:mi>μ</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>=</mml:mo><mml:msub><mml:mi>μ</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq65\"><alternatives><tex-math id=\"M131\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$(\\mu _1 - \\mu _2)$$\\end{document}</tex-math><mml:math id=\"M132\"><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>μ</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>-</mml:mo><mml:msub><mml:mi>μ</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq66\"><alternatives><tex-math id=\"M133\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$H_0$$\\end{document}</tex-math><mml:math id=\"M134\"><mml:msub><mml:mi>H</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq67\"><alternatives><tex-math id=\"M135\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$p &lt; 0.05$$\\end{document}</tex-math><mml:math id=\"M136\"><mml:mrow><mml:mi>p</mml:mi><mml:mo>&lt;</mml:mo><mml:mn>0.05</mml:mn></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq68\"><alternatives><tex-math id=\"M137\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$(\\mu _1 - \\mu _2)$$\\end{document}</tex-math><mml:math id=\"M138\"><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>μ</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>-</mml:mo><mml:msub><mml:mi>μ</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq69\"><alternatives><tex-math id=\"M139\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$X_1$$\\end{document}</tex-math><mml:math id=\"M140\"><mml:msub><mml:mi>X</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq70\"><alternatives><tex-math id=\"M141\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$X_2$$\\end{document}</tex-math><mml:math id=\"M142\"><mml:msub><mml:mi>X</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq71\"><alternatives><tex-math id=\"M143\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$p &lt; 0.05$$\\end{document}</tex-math><mml:math id=\"M144\"><mml:mrow><mml:mi>p</mml:mi><mml:mo>&lt;</mml:mo><mml:mn>0.05</mml:mn></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq72\"><alternatives><tex-math id=\"M145\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$X_1 &gt; X_2$$\\end{document}</tex-math><mml:math id=\"M146\"><mml:mrow><mml:msub><mml:mi>X</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>&gt;</mml:mo><mml:msub><mml:mi>X</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq73\"><alternatives><tex-math id=\"M147\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$X_1 &lt; X_2$$\\end{document}</tex-math><mml:math id=\"M148\"><mml:mrow><mml:msub><mml:mi>X</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>&lt;</mml:mo><mml:msub><mml:mi>X</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq74\"><alternatives><tex-math id=\"M149\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$X_1 - X_2$$\\end{document}</tex-math><mml:math id=\"M150\"><mml:mrow><mml:msub><mml:mi>X</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>-</mml:mo><mml:msub><mml:mi>X</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ1\"><label>1</label><alternatives><tex-math id=\"M151\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} P(k) = \\frac{m(k)}{n(k)}. \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M152\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:mi>P</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mfrac><mml:mrow><mml:mi>m</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mfrac><mml:mo>.</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ2\"><label>2</label><alternatives><tex-math id=\"M153\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} C(k) = \\tfrac{\\sum _k^\\infty m(k)}{\\sum _k^\\infty n(k)}. \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M154\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:mi>C</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mstyle displaystyle=\"false\" scriptlevel=\"0\"><mml:mfrac><mml:mrow><mml:msubsup><mml:mo>∑</mml:mo><mml:mi>k</mml:mi><mml:mi>∞</mml:mi></mml:msubsup><mml:mi>m</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow><mml:mrow><mml:msubsup><mml:mo>∑</mml:mo><mml:mi>k</mml:mi><mml:mi>∞</mml:mi></mml:msubsup><mml:mi>n</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:mfrac></mml:mstyle><mml:mo>.</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq75\"><alternatives><tex-math id=\"M155\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$P_\\text {base}(k) = 1 - (1 - p)^k$$\\end{document}</tex-math><mml:math id=\"M156\"><mml:mrow><mml:msub><mml:mi>P</mml:mi><mml:mtext>base</mml:mtext></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mn>1</mml:mn><mml:mo>-</mml:mo><mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>1</mml:mn><mml:mo>-</mml:mo><mml:mi>p</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mi>k</mml:mi></mml:msup></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq76\"><alternatives><tex-math id=\"M157\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$p = P(1) = \\tfrac{m(1)}{n(1)}$$\\end{document}</tex-math><mml:math id=\"M158\"><mml:mrow><mml:mi>p</mml:mi><mml:mo>=</mml:mo><mml:mi>P</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mstyle displaystyle=\"false\" scriptlevel=\"0\"><mml:mfrac><mml:mrow><mml:mi>m</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mfrac></mml:mstyle></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq77\"><alternatives><tex-math id=\"M159\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$P_\\text {base}(1) = 1 - (1 - p)^1 = p$$\\end{document}</tex-math><mml:math id=\"M160\"><mml:mrow><mml:msub><mml:mi>P</mml:mi><mml:mtext>base</mml:mtext></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mn>1</mml:mn><mml:mo>-</mml:mo><mml:msup><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>1</mml:mn><mml:mo>-</mml:mo><mml:mi>p</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mn>1</mml:mn></mml:msup><mml:mo>=</mml:mo><mml:mi>p</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq78\"><alternatives><tex-math id=\"M161\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$C_{\\text {base}}(k)$$\\end{document}</tex-math><mml:math id=\"M162\"><mml:mrow><mml:msub><mml:mi>C</mml:mi><mml:mtext>base</mml:mtext></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ3\"><label>3</label><alternatives><tex-math id=\"M163\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} C_{\\text {base}}(k) = \\frac{\\sum _k^\\infty P_\\text {base}(k) \\cdot n(k)}{\\sum _k^\\infty n(k)}. \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M164\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:msub><mml:mi>C</mml:mi><mml:mtext>base</mml:mtext></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mfrac><mml:mrow><mml:msubsup><mml:mo>∑</mml:mo><mml:mi>k</mml:mi><mml:mi>∞</mml:mi></mml:msubsup><mml:msub><mml:mi>P</mml:mi><mml:mtext>base</mml:mtext></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>·</mml:mo><mml:mi>n</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow><mml:mrow><mml:msubsup><mml:mo>∑</mml:mo><mml:mi>k</mml:mi><mml:mi>∞</mml:mi></mml:msubsup><mml:mi>n</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:mfrac><mml:mo>.</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq79\"><alternatives><tex-math id=\"M165\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$n_a$$\\end{document}</tex-math><mml:math id=\"M166\"><mml:msub><mml:mi>n</mml:mi><mml:mi>a</mml:mi></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq80\"><alternatives><tex-math id=\"M167\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$m_a$$\\end{document}</tex-math><mml:math id=\"M168\"><mml:msub><mml:mi>m</mml:mi><mml:mi>a</mml:mi></mml:msub></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ4\"><label>4</label><alternatives><tex-math id=\"M169\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} P_s^a=\\frac{m_a}{n_a}. \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M170\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:msubsup><mml:mi>P</mml:mi><mml:mi>s</mml:mi><mml:mi>a</mml:mi></mml:msubsup><mml:mo>=</mml:mo><mml:mfrac><mml:msub><mml:mi>m</mml:mi><mml:mi>a</mml:mi></mml:msub><mml:msub><mml:mi>n</mml:mi><mml:mi>a</mml:mi></mml:msub></mml:mfrac><mml:mo>.</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq81\"><alternatives><tex-math id=\"M171\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$n_a$$\\end{document}</tex-math><mml:math id=\"M172\"><mml:msub><mml:mi>n</mml:mi><mml:mi>a</mml:mi></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq82\"><alternatives><tex-math id=\"M173\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$0 \\le f \\le 1$$\\end{document}</tex-math><mml:math id=\"M174\"><mml:mrow><mml:mn>0</mml:mn><mml:mo>≤</mml:mo><mml:mi>f</mml:mi><mml:mo>≤</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq83\"><alternatives><tex-math id=\"M175\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$C_s(f)$$\\end{document}</tex-math><mml:math id=\"M176\"><mml:mrow><mml:msub><mml:mi>C</mml:mi><mml:mi>s</mml:mi></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>f</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq84\"><alternatives><tex-math id=\"M177\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$C_s(f)$$\\end{document}</tex-math><mml:math id=\"M178\"><mml:mrow><mml:msub><mml:mi>C</mml:mi><mml:mi>s</mml:mi></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>f</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq85\"><alternatives><tex-math id=\"M179\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$P_s^a$$\\end{document}</tex-math><mml:math id=\"M180\"><mml:msubsup><mml:mi>P</mml:mi><mml:mi>s</mml:mi><mml:mi>a</mml:mi></mml:msubsup></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq86\"><alternatives><tex-math id=\"M181\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$C_s(f)$$\\end{document}</tex-math><mml:math id=\"M182\"><mml:mrow><mml:msub><mml:mi>C</mml:mi><mml:mi>s</mml:mi></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>f</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq87\"><alternatives><tex-math id=\"M183\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$f^*$$\\end{document}</tex-math><mml:math id=\"M184\"><mml:msup><mml:mi>f</mml:mi><mml:mo>∗</mml:mo></mml:msup></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq88\"><alternatives><tex-math id=\"M185\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$C_s(f^*)$$\\end{document}</tex-math><mml:math id=\"M186\"><mml:mrow><mml:msub><mml:mi>C</mml:mi><mml:mi>s</mml:mi></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msup><mml:mi>f</mml:mi><mml:mo>∗</mml:mo></mml:msup><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq89\"><alternatives><tex-math id=\"M187\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$f^*$$\\end{document}</tex-math><mml:math id=\"M188\"><mml:msup><mml:mi>f</mml:mi><mml:mo>∗</mml:mo></mml:msup></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq90\"><alternatives><tex-math id=\"M189\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$1/5=0.20$$\\end{document}</tex-math><mml:math id=\"M190\"><mml:mrow><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">/</mml:mo><mml:mn>5</mml:mn><mml:mo>=</mml:mo><mml:mn>0.20</mml:mn></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq91\"><alternatives><tex-math id=\"M191\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$m_a$$\\end{document}</tex-math><mml:math id=\"M192\"><mml:msub><mml:mi>m</mml:mi><mml:mi>a</mml:mi></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula 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id=\"M198\"><mml:mrow><mml:msub><mml:mi>P</mml:mi><mml:mi>c</mml:mi></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>f</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq94\"><alternatives><tex-math id=\"M199\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$f \\in [0, 1]$$\\end{document}</tex-math><mml:math id=\"M200\"><mml:mrow><mml:mi>f</mml:mi><mml:mo>∈</mml:mo><mml:mo stretchy=\"false\">[</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">]</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq95\"><alternatives><tex-math id=\"M201\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$P_c^a \\ge f$$\\end{document}</tex-math><mml:math id=\"M202\"><mml:mrow><mml:msubsup><mml:mi>P</mml:mi><mml:mi>c</mml:mi><mml:mi>a</mml:mi></mml:msubsup><mml:mo>≥</mml:mo><mml:mi>f</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>" ]

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[ "<supplementary-material content-type=\"local-data\" id=\"MOESM1\"></supplementary-material>" ]

[ "<fn-group><fn><p><bold>Publisher's note</bold></p><p>Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.</p></fn><fn><p>These authors contributed equally: Sara Venturini and Satyaki Sikdar.</p></fn></fn-group>" ]

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[ "<media xlink:href=\"41598_2024_51606_MOESM1_ESM.pdf\"><caption><p>Supplementary Information.</p></caption></media>" ]

{ "acronym": [], "definition": [] }

[ "<title>Introduction</title>", "<p id=\"Par2\">Coronary artery disease (CAD) is the leading cause of mortality around the globe. In developed and developing countries, CAD is the foremost cause of death. According to a previous study, people in the Middle East demonstrate heart disease at younger ages^{##REF##33605111##1##}. In Iran, 50% of annual deaths are the consequence of CAD^{##REF##31038733##2##}.</p>", "<p id=\"Par3\">CAD usually affects people aged above 50 years. Nonetheless, in some cases, women under 55 and men under 45 years develop CAD, termed “premature CAD”^{##REF##31601367##3##}. No universally accepted age threshold exists for premature CAD. A prior investigation assigned the age of 49 as the cutoff for premature CAD in males based on several autopsy reports^{##REF##29730481##4##}. Here, we determined the age of 49 as a cutoff for assigning premature CAD in males.</p>", "<p id=\"Par4\">Patients with premature CAD tend to develop consequent ischemic events at a higher rate than other age groups^{##REF##33287625##5##}.CAD is the consequence of atherosclerosis, an inflammatory disease contributing to the accumulation of fatty deposits within the arterial wall which leads to atherosclerotic plaque formation^{##UREF##0##6##}.</p>", "<p id=\"Par5\">The results of a recent genome-wide association study (GWAS) ascribed many loci as risk loci for premature CAD. Among these, the 9p21.3 locus is the most striking genetic risk factor. The 58-kb region of the 9p21 locus is assigned as the CAD risk interval region. No protein-coding gene resides in this segment. The risk haplotype in this region includes interlinked noncoding single-nucleotide polymorphisms (SNPs)^{##REF##30735646##7##}.</p>", "<p id=\"Par6\"><italic>CDKN2B-AS1</italic>, also known as “<italic>ANRIL</italic> (antisense noncoding RNA in the <italic>INK4A</italic> locus)”, is a long noncoding RNA (lncRNA) overlapping the <italic>CDKN2B</italic> gene. Further, rs10757274 (NC_000009.12:g.22096056A&gt;G), rs2383206 (g.22115027A&gt;G), rs2383207 (g.22115960A&gt;G), rs496892 (g.22024352C&gt;T), rs10757278 (g.22124478A&gt;G), and rs10738605 (g.22049131C&gt;G) reside in the 9p21 locus. While rs10757274, rs2383206, rs2383207, and rs496892 are in the intronic regions of the <italic>ANRIL</italic> sequence, rs10738605 resides in exon six of <italic>ANRIL</italic> (isoform. 1). On the other hand, rs10757278 is in the intergenic region downstream of the <italic>ANRIL</italic> gene. Extensive GWAS analyses have been conducted on the associations between the abovementioned SNPs and CAD^{##UREF##1##8##,##REF##28107200##9##}. In our study, however, we narrowed down the CAD-associated GWAS variants to SNPs that appeared in publications most frequently as CAD-associated variants in different populations.</p>", "<p id=\"Par7\">Of the 14 splicing isoforms of <italic>ANRIL</italic> annotated on Ref Seq, five isoforms are short and nine are long. The expression pattern of the two groups differs noticeably. The expression of short isoforms is higher than that of long ones. Specific genetic variants are associated with increased levels of short variants and a reduced expression of long isoforms suggesting the contribution of genetic variants to <italic>ANRIL</italic> expression^{##REF##28107200##9##,##REF##28653984##10##}.</p>", "<p id=\"Par8\">A prior study discovered that the relatively lower <italic>ANRIL</italic> expression could be increased following treatment^{##REF##31957852##11##}. The interplay between the genotypes of 9p21 risk SNPs and <italic>ANRIL</italic> expression has been extensively explained, with research having validated differential exon expression levels of <italic>ANRIL</italic>, along with various circular and linear isoforms^{##REF##28653984##10##,##REF##30460243##12##–##REF##21151960##14##}.</p>", "<p id=\"Par9\">In vascular smooth muscle cells and mononuclear cells, <italic>ANRIL</italic> promotes proliferation and atherosclerosis progression^{##REF##30460243##12##}. It has been demonstrated that <italic>ANRIL</italic> is upregulated in serum samples from patients with multiple malignancies, including glioma and breast cancer^{##REF##33388059##15##,##UREF##2##16##}. The higher expression of circulating <italic>ANRIL</italic> in the peripheral blood and serum of patients with diabetes and ischemic stroke has been reported^{##REF##36129598##17##,##UREF##3##18##}. As demonstrated by Holdt and Teupser^{##REF##30460243##12##}, atherosclerosis progression is associated with decreased expression of circular <italic>ANRIL</italic> and increased expression of linear <italic>ANRIL</italic>.</p>", "<p id=\"Par10\">Circulating transcripts are enriched in serum compared to whole blood sharing 80% of RNAs with other tissues^{##UREF##4##19##}. Serum is more sensitive than plasma in biomarker discoveries^{##REF##26878386##20##} and being free of EDTA, which induces platelet activation, provides a more realistic picture of circulating RNA's physiological pathways^{##UREF##5##21##}.</p>", "<p id=\"Par11\">As proposed by Lawford, from the therapeutic intervention perspective, the earlier the diagnosis is established, the better the outcome will be^{##UREF##0##6##}. Early diagnosis of CAD is crucial in Iran due to the increasing incidence of premature CAD in recent years. Sedentary lifestyles, familial CAD, type 2 diabetes (T2D), hyperlipidemia, and smoking are the most frequently correlated risk factors, although the precise involvement of genetic susceptibility factors should not be underestimated in population-based studies^{##REF##33732287##22##,##REF##35941419##23##}.</p>", "<p id=\"Par12\">In the present study, we aimed to investigate the associations between the genotypes of six SNPs within (rs10757274, rs2383206, rs2383207, rs496892, and rs10738605) and proximal to (rs10757278) the <italic>ANRIL</italic> gene sequence, in addition to <italic>ANRIL</italic> expression, and PCAD solely and in combination. According to similar studies in other populations, we hypothesized that circulating <italic>ANRIL</italic> expression differs between premature CAD and non-CAD groups. We also assumed that in premature CAD patients of the Iranian population, a correlation exists between the genetic profile of <italic>ANRIL</italic> SNPs, rs10757278 and <italic>ANRIL</italic> expression, since it has been proposed that genetic factors are more important than environmental factors in the development of CAD among young adults. The growing number of premature CAD occurrences in Iran prompted us to investigate the reasons.</p>" ]

[ "<title>Methods</title>", "<title>Sample collection</title>", "<p id=\"Par39\">The present study recruited 93 premature CAD and 87 non-CAD age-matched subjects hospitalized at Rajaie Cardiovascular Medical and Research Center, Tehran, Iran, between 2017 and 2019. All patients were diagnosed by angiography. A questionnaire and a formal consent form were completed and signed by all the subjects. The study protocol was approved by the Ethics Committee of Rajaie Cardiovascular Medical and Research Center (RHC.AC.IR.REC.1396.62), and was conducted in accordance with the Helsinki Declaration. The informed consent was obtained from all subjects and/or their legal guardian(s). Patients (women &lt; 55 y/o and men &lt; 49 y/o) diagnosed with premature CAD by coronary angiography were included, while patients with malignant tumors, severe liver disease, severe kidney dysfunction, infectious diseases, immune diseases, communication dysfunction, or cognitive dysfunction were excluded. The non-CAD group included women &lt; 55 y/o and men &lt; 49 y/o with no evidence of coronary artery disease. The exclusion criteria was as well as the one applied for CAD subjects. T2D patients with and without PCAD were also included in the study. All patients in Rajaie Cardiovascular Medical and Research Center were evaluated by the medical team for their diabetes status before being included in the current study. Clinical considerations regarding diabetes have been taken into account by clinicians prior to any treatment or intervention.</p>", "<title>DNA extraction</title>", "<p id=\"Par40\">DNA was extracted from peripheral blood using the DNeasy Blood &amp; Tissue Kit (QIAGEN, Germany). The quantity of the extracted DNAs was assessed using NanoDrop 2000/2000c Spectrophotometers (Thermo Fisher Scientific, USA).</p>", "<title>Serum isolation, RNA extraction, and cDNA synthesis</title>", "<p id=\"Par41\">Following blood collection, serum was isolated by incubation at room temperature for 30 min. Afterward, clots were removed by centrifuge at 3500×<italic>g</italic> for 20 min. Subsequently, the serum was isolated in a ribonuclease (RNase)-free tube and kept at − 80 ℃ for longer preservation.</p>", "<p id=\"Par42\">Total RNAs were isolated from serum using a plasma/ serum RNA purification kit (NORGEN BIOTEK, Canada). The RNAs were then concentrated by the RNA Clean-up and Concentration Kit (NORGEN BIOTEK, Canada). Subsequently, reverse transcription was performed with a PrimeScript First-Strand cDNA Synthesis Kit (Takara, Japan).</p>", "<title>Real-time polymerase chain reaction (PCR)</title>", "<p id=\"Par43\">Specific PCR primers were designed using Gene Runner (version 6.5.51) and Primer3, synthesized by Macrogen (Korea) (##SUPPL##0##S1## Table). The primers were designed for different exons of <italic>ANRIL</italic> (RefSeq NR_003529.3): one pair on exon one, one on exon five-six, and one on exon 19 (Fig. ##FIG##0##1##). The real-time PCR analysis was performed using the BioFACT 2X Real-Time PCR Master Mix (High ROX), including SYBR Green (BioFACT, South Korea), in an Applied Biosystems StepOne Plus instrument (Applied Biosystems, USA). Next, cDNA was added to 10 μL of SYBR Green and 0.5 μM of each primer in a 20 μL reaction. All the reactions were repeated in duplicates, and mean threshold cycles were used for further analyses. The real-time thermal program was as follows: 95 °C for 30 s and 62 °C for 35 s, for which <italic>5srRNA</italic> was utilized as an internal control. The expression level was calculated by 2^−ΔCt.</p>", "<title>PCR and sequencing</title>", "<p id=\"Par44\">For SNP genotyping, primers were designed to detect a 300–700 base pair (bp) product. The amplicon was then sequenced. The sequence and characteristics of the primers used in this study are listed in ##SUPPL##0##S1## Table. Afterward, 100 ng DNA was added to 10 μL of Taq DNA Polymerase Master Mix RED (Ampliqon, Denmark) for PCR reactions, along with 0.5 μM of each primer in each 20 μL reaction. PCR was performed for 30 cycles in the following thermal program: 95 °C for 5 min, 30 cycles at 95 °C for 30 s, 61 °C for 30 s, 72 °C for 30 s, and a final extension at 72 °C for 5 min. The PCR products were electrophoresed on a 1–2% agarose gel, stained with FluoroDye (SMOBIO, Taiwan), and visualized under ultraviolet light. Subsequently, Sanger sequencing was performed, and the sequencing files were analyzed using CodonCode Aligner 10.0.2 for Windows (<ext-link ext-link-type=\"uri\" xlink:href=\"http://www.codoncode.com\">www.codoncode.com</ext-link>).</p>", "<title>Linkage disequilibrium (LD) analysis</title>", "<p id=\"Par45\">Haploview (Haploview Software, USA, <ext-link ext-link-type=\"uri\" xlink:href=\"http://www.broadinstitute.org/haploview\">www.broadinstitute.org/haploview</ext-link>) was employed to analyze LD^{##REF##15297300##42##}. Haploview generated D’, logarithm of the odds (LOD), and r² factors. The results are presented in ##SUPPL##0##S2## Table.</p>", "<title>In silico analysis</title>", "<title>Finding transcription-factor binding sites in the SNPs’ regions</title>", "<p id=\"Par46\">Transcription-factor (TF) binding sites in <italic>ANRIL</italic> introns were discovered through the application of related databases in the UCSC Genome Browser track data hubs (<ext-link ext-link-type=\"uri\" xlink:href=\"https://genome.ucsc.edu/\">https://genome.ucsc.edu/</ext-link>)^{##REF##24227676##43##}. The list of the databases is as follows: TF ChIP-seq Clusters from ENCODE3, eCLIP (by biosample) (ENCODE), eCLIP (by target) (ENCODE), ENCODE TFs, ENC RNA Binding, ENC TF Binding, TFBS Conserved, JASPAR CORE 2022-Predicted TF Binding Site, TOBIAS footprint prediction, human P53 Binding And Expression Resource (BAER) hub, ReMap 2022 Regulatory Atlas, and UniBind 2021 Hub.</p>", "<title>Statistical analysis</title>", "<p id=\"Par47\">The mean and the standard deviation were calculated in GraphPad Prism 9 for Windows (GraphPad Software, San Diego, California USA, <ext-link ext-link-type=\"uri\" xlink:href=\"http://www.graphpad.com\">www.graphpad.com</ext-link>) for each parameter between the premature CAD and non-CAD subjects. Risk-factor probabilities were analyzed with the aid of the χ² test for qualitative variables and the <italic>t</italic> test for quantitative variables. The D'Agostino–Pearson omnibus normality test was also performed. Additionally, a nonparametric <italic>t</italic> test (Mann–Whitney) was applied for values not fitting in a normal distribution. One-way ANOVA was utilized to analyze the values of more than two groups, and the Spearman correlation was applied to generate a correlation matrix for all variables. The Receiver Operating Characteristic (ROC) analysis was applied to assess the diagnostic potential of <italic>ANRIL</italic> expression. ROC curves were drawn using GraphPad Prism 9.</p>", "<title>Ethics approval</title>", "<p id=\"Par48\">The study protocol was approved by the Ethics Committee of Rajaie Cardiovascular Medical and Research Center (RHC.AC.IR.REC.1396.62), and the study was conducted in accordance with the Helsinki Declaration. The informed consent was obtained from all subjects and/or their legal guardian(s).</p>" ]

[ "<title>Results</title>", "<title>Clinical and demographic features represented an association with premature CAD</title>", "<p id=\"Par13\">The results of clinical and demographic characteristic analysis are included in Table ##TAB##0##1##. The levels of triglyceride, total cholesterol, low-density lipoprotein, and fasting blood sugar were significantly associated with premature CAD. Furthermore, T2D, stroke history, and familial CAD were significantly more frequent in the premature CAD group than in the non-CAD group. The distribution of age in CAD and non-CAD groups respective to and irrespective of sex is depicted in ##SUPPL##0##S1## Figure.</p>", "<title><italic>ANRIL</italic> was expressed differentially in the premature CAD group compared with the non-CAD group</title>", "<p id=\"Par14\"><italic>ANRIL</italic> featured 14 splicing variants in the Ref Seq database and encompassed many regulatory elements along its sequence (Fig. ##FIG##0##1##). <italic>ANRIL</italic> was significantly downregulated in the serum samples of the premature CAD group compared with the non-CAD control group (<italic>P</italic> &lt; 0.05) (Fig. ##FIG##1##2##). The detection of <italic>ANRIL</italic> yielded diverse results through different primer sets. The expression analysis of <italic>ANRIL</italic> was performed using three different sets of primers amplifying exons 1, 5–6, and 19. The detection of middle exons (exons five-six) showed a nonsignificant decrease in <italic>ANRIL</italic> expression in the premature CAD group compared with the non-CAD group (Fig. ##FIG##1##2##D), while the detection of terminal exons manifested a significant downregulation (Fig. ##FIG##1##2##A,G).</p>", "<title>Diabetes correlated with higher <italic>ANRIL</italic> expression levels</title>", "<p id=\"Par15\">Patients with both premature CAD and T2D showed lower expression of <italic>ANRIL</italic> than those categorized as non-CAD. Variations in <italic>ANRIL</italic> expression using different primers were adopted here as well (Fig. ##FIG##1##2##B,E,H). Diabetic patients with no symptoms of premature CAD indicated higher levels of <italic>ANRIL</italic> than those who did not have T2D or had premature CAD. Nevertheless, in the patients with premature CAD and T2D, the expression of <italic>ANRIL</italic> remained low altogether.</p>", "<title>The history of familial CAD was in line with lower <italic>ANRIL</italic> expression levels</title>", "<p id=\"Par16\">A history of familial CAD per se lacked contribution to significantly lower <italic>ANRIL</italic> expression. However, along with premature CAD, it preserved the low expression of <italic>ANRIL</italic> altogether.</p>", "<title>Genetic variants within and proximal to the <italic>ANRIL</italic> sequence displayed different genotype and allele frequencies but not a significant association with premature CAD</title>", "<p id=\"Par17\">Associations between premature CAD status and the genotypes of six SNPs were investigated. No significant associations were found between premature CAD and rs10757274, rs2383206, rs2383207, rs496892, rs10757278, and rs10738605. An overview of the studied variants is summarized in ##SUPPL##0##S3## Table. The odds ratios and <italic>P</italic> values for the different genetic models are listed in Table ##TAB##1##2##. The distributions of the genotypes for all 6 SNPs were consistent with the Hardy Weinberg law at a significance level of 0.05 (Table ##TAB##1##2##). The frequencies of risk genotypes for all the SNPs were higher in the premature CAD group than in the non-CAD group but the differences lacked statistical significance.</p>", "<title>An association existed between <italic>ANRIL</italic> expression and the genotypes of rs10757274, rs2383206, rs2383207, rs496892, rs10757278, and rs10738605</title>", "<p id=\"Par18\">While rs10738605 resided in exon six of <italic>ANRIL</italic>, the other SNPs were in noncoding regions. Among these, rs10757274 and rs496892 are included in the sequences of long interspersed elements (LINEs). Multiple binding sites for transcription factors (TFs) were included in the regions containing the studied SNPs (##SUPPL##0##S2## Fig.).</p>", "<p id=\"Par19\">The expression of <italic>ANRIL</italic> in the premature CAD subjects carrying the risk genotypes was lower than that in the control group (Fig. ##FIG##2##3##). For rs10757274, rs2383206, rs2383207, and rs10757278, the expression of exon 19 of <italic>ANRIL</italic> (a representative of long isoforms) was significantly downregulated in PCAD carriers of risk genotypes, however, detection of the first exon of <italic>ANRIL</italic> was significantly reduced in PCAD carriers of risk genotypes by rs496892 and rs10738605. For rs10757274, rs2383206, rs2383207, rs10757278, and rs10738605, the risk genotypes were GG compared to TT for rs496892.</p>", "<title>The Receiver Operating Characteristic (ROC) analysis revealed that <italic>ANRIL</italic> expression could differentiate premature CAD from non-CAD regarding rs10738605 and rs496892</title>", "<p id=\"Par20\">The ROC analysis was performed to investigate whether subjects with the risk genotypes of the studied SNPs could be differentiated into premature CAD and non-CAD groups regarding <italic>ANRIL</italic> expression. The ROC curves (Fig. ##FIG##2##3##G,H) revealed that in the subjects carrying the risk genotype of rs10738605, the expression of <italic>ANRIL</italic> was capable of separating premature CAD from non-CAD subjects with a sensitivity of 83%. Further, in the subjects carrying the risk genotypes for a combination of rs10738605 and rs496892, <italic>ANRIL</italic> expression predicted the patients correctly with 84% sensitivity (Fig. ##FIG##2##3##H).</p>", "<title>The LD analysis ascertained two distinct LD blocks</title>", "<p id=\"Par21\">Haploview revealed strong LD between rs10757274, rs2383206, rs2383207, and rs10757278 and between rs496892 and rs10738605. Haploview demonstrated two distinct blocks: one for rs496894 and rs10738605 and another block for rs10757274, rs2383206, rs2383207, and rs10757278 (F##FIG##2##i##g. ##FIG##2##3##I).</p>", "<title>The correlation analysis demonstrated strong links between some variables</title>", "<p id=\"Par22\">The correlation matrix is depicted as a heat map in Fig. ##FIG##2##3##J. A negative correlation was detected between the status of premature CAD and age, triglyceride, fasting blood sugar, stroke history, T2D, and familial CAD. Positive correlations were observed between the expression of <italic>ANRIL</italic> using different primers (E1, E5-6, and E19), and between the status of the rs10757274, rs2383206, rs2383207, and rs10757278 genotypes as well. Some weaker positive correlations were discovered between the expression of <italic>ANRIL</italic> and the level of high-density lipoprotein and between familial CAD and familial myocardial infarction. Positive correlations were also indicated between the levels of triglyceride, cholesterol, fasting blood sugar, stroke history, and T2D (Fig. ##FIG##2##3##J).</p>" ]

[ "<title>Discussion</title>", "<p id=\"Par23\">Atherosclerosis normally progresses in the form of late plaques in the fourth decade of life and involves older people. Nonetheless, in some cases, women aged below 55 and men younger than 45 experience CAD symptoms^{##REF##31601367##3##}. The principal cause of such incidence remains indefinite, although previous studies have reported the contribution of genetic factors in promoting atherosclerosis at younger ages^{##UREF##0##6##,##UREF##6##24##}.</p>", "<p id=\"Par24\">The gold-standard diagnosis of CAD is angiography, which is an invasive technique. Developing noninvasive and low-cost screening methods to predict premature CAD occurrence in young adults is crucial since CAD has various socioeconomic effects on populations.</p>", "<p id=\"Par25\">According to a previous study, <italic>ANRIL</italic> expression exhibited post-treatment recovery. A negative correlation also existed between the number of diseased vessels and the expression of <italic>ANRIL</italic>^{##REF##31957852##11##}. The detection of terminal exons (E1 and E19) manifested promising potential to differentiate premature CAD from non-CAD, while the identification of medial exons (E5-6) lacked such a capability. This might be due to the differential expression of linear and circular isoforms in serum since internal exons are mostly enriched in circular variants^{##REF##28653984##10##}.</p>", "<p id=\"Par26\">Diabetes is the most common risk factor for premature CAD in the Iranian population^{##REF##31038733##2##}. It is associated with higher levels of <italic>ANRIL</italic> expression. Our CAD patients who suffered from T2D demonstrated lower expression of <italic>ANRIL</italic>, denoting the dominant effect of CAD over T2D. In contrast, our diabetic patients with no premature CAD symptoms expressed <italic>ANRIL</italic> to significantly higher extents. Our data chime in with the findings of a previous study^{##UREF##7##25##}. A history of familial CAD was in keeping with the influence of CAD itself on the expression of <italic>ANRIL</italic> and demonstrated no independent association with CAD.</p>", "<p id=\"Par27\">A substantial body of evidence indicates that lncRNAs function through their distinct secondary structure, resulting in the dynamic accessibility of exons during cellular functions. The results of a study demonstrated a relationship between the presence of Alu elements in exons and their differential expression^{##REF##28653984##10##}.</p>", "<p id=\"Par28\">There are shreds of evidence that noncoding variants can affect expression as well as exonic ones. Intronic SNPs exert their functional role through their existence in exon–intron junctions, splicing branch points and enhancers, or through the creation of cryptic splice sites. Although most functional SNPs reside within a 30-bp distance from splice sites, SNPs in the middle of introns should not be underestimated^{##REF##30906297##26##}.</p>", "<p id=\"Par29\">While some of our investigated SNPs reside in the binding sites of transcription factors, some are included in LINEs, explaining the regulatory effect of these variants on <italic>ANRIL</italic> expression. LINEs can regulate gene expression by mediating chromatin remodeling, regulating transcription, and altering the stability of transcripts. They can function as promoters or enhancers as well^{##REF##26912865##27##}.</p>", "<p id=\"Par30\">Previous studies on the Iranian population have introduced rs10757274 and rs2383206 as SNPs associated with CAD^{##REF##28868267##28##–##UREF##9##30##}. Only a few investigations are available on premature CAD in the Iranian population^{##REF##31038733##2##}.</p>", "<p id=\"Par31\">There are few studies on <italic>ANRIL</italic> expression in the Iranian population. Previous studies in the Iranian population measured <italic>ANRIL</italic> expression in peripheral blood samples from patients with (aged) CAD^{##UREF##7##25##,##REF##30234067##31##}, whereas this study examined <italic>ANRIL</italic> expression in sera of premature CAD patients.</p>", "<p id=\"Par32\">Serum provides a higher sensitivity for biomarker detection and contains stable RNAs mostly enriched in extracellular vesicles^{##REF##26878386##20##}. As of yet, there are no serum expression data available for the Iranian population. Yari et al.^{##REF##30234067##31##} demonstrated no significant difference in the expression of NR_003529 transcript between CAD patients and the control group. However, we revealed a significant association between lower expression of <italic>ANRIL</italic> (NR_003529), the longest <italic>ANRIL</italic> transcript, and premature CAD in the Iranian population. Despite this discrepancy, their expression analysis of EU741058 transcript was consistent with ours regarding the first exon primer. Rahimi et al.^{##UREF##7##25##} reported an upregulation in <italic>ANRIL</italic> expression in diabetic patients with CAD compared to non-CAD diabetic subjects. Here, we investigated <italic>ANRIL</italic> expression in premature CAD patients with and without diabetes. Serum expression of <italic>ANRIL</italic> showed a different pattern compared to peripheral blood mononuclear cells' expression.</p>", "<p id=\"Par33\">The correlation between <italic>ANRIL</italic> expression and the status of SNPs in its sequence has not yet been reported in Iranian patients with premature CAD. Here, we demonstrated that PCAD carriers of risk genotypes for rs496892 and rs10738605 showed lower expression of first exon of <italic>ANRIL</italic>. rs496892 and rs10738605, both are proximal to the first exons of <italic>ANRIL</italic> and revealed a strong linkage disequilibrium in our population. On the contrary, rs10757274, rs2383206, rs2383207, and rs10757278 reside in the 3' of <italic>ANRIL</italic> and correlate with lower expression of exon 19. These four SNPs share a haplotype block together supporting our findings on the effect of these SNPs on expression of terminal exons of <italic>ANRIL</italic>.</p>", "<p id=\"Par34\">Linking DNA variants to function is one of the major challenges in deciphering the influence of genetics on CAD. Unlike monogenic diseases, multifactorial disorders result from many small contributing factors. Therefore, large-scale studies are needed to validate the association between these variants and diseases^{##REF##30901535##32##}.</p>", "<p id=\"Par35\">Our ROC analysis demonstrated that <italic>ANRIL</italic> lacked predictive value as a biomarker for premature CAD per se (data not shown). Nevertheless, <italic>ANRIL</italic> expression could distinguish between premature CAD and non-CAD in carriers of rs496892 and rs10738605 risk genotypes (Fig. ##FIG##2##3##G,H). In this group, the expression of <italic>ANRIL</italic> could accurately determine the status of premature CAD by 83% and 84% sensitivity (respectively for rs10738605 and a combination of rs10738605 and rs496892); still, the specificity values were slight (54% and 55%, respectively). Further research requires larger sample sizes.</p>", "<p id=\"Par36\">GWASs are incapable of discovering disease-causing SNPs solely. They only narrow down genetic variants to the most associated SNPs with the disease. Due to the population stratification in large-scale GWASs, the confirmation of the associations by replication studies in smaller and more homogenous populations is crucial^{##REF##27939749##33##,##UREF##10##34##}. Our findings did not replicate the association between the studied SNPs and premature CAD significantly. It might be a consequence of the small sample size, which was limited because of the exclusive age group of the included subjects. As proposed by the 1000 Genome Project, genetic variants are categorized into four groups: population-specific, continental area-specific, continental areas-shared, and all continents-shared^{##UREF##11##35##}. The structure of the population also contributes to differences in the frequency of genotypes between sub-populations even in a shared geographic location^{##REF##18817904##36##}. It is essential to investigate the association between genetic variants and diseases in populations of various ancestries. Different expression and LD profiling among populations were convincing evidence to discover the expression, LD, and genetic association screening of the <italic>ANRIL</italic> region in the Iranian population. The interplay of SNP and expression has been reported to vary between populations as well^{##REF##18313023##37##–##REF##20811451##39##}. The association between genetic variants and diseases in 1 population does not simply generalize to other populations. Besides ethnicity, age and other environmental factors affect the variation^{##REF##31001318##40##}.</p>", "<p id=\"Par37\">DNA variants and RNA expression lack predictive value when used solely, while a combination of both approaches is worthwhile. The last decade has witnessed a considerable fall in the costs of sequencing and expression analyses, raising the hopes of using genetics as a helping arm in clinics. Our principal notion is that a combination of DNA and RNA variants is capable of predicting premature CAD in young adults. Premature CAD is considered a burden on the health systems of countries: not only does it negatively impact a young workforce but also it contributes to societal challenges. Traditional screening methods lack the strength to precisely predict premature CAD^{##UREF##12##41##}. What could substantially strengthen the armamentarium is the recruitment of artificial intelligence and machine learning.</p>" ]

[]

[ "<p id=\"Par1\">Coronary artery disease (CAD) is the major cause of mortality in the world. Premature development of CAD can be attributed to women under 55 and men under 45. Many genetic factors play a part in premature CAD. Among them, <italic>ANRIL</italic>, a long noncoding RNA is located at the 9p21 risk locus, and its expression seems to be correlated with CAD. In the current study, premature CAD and control blood samples, with and without Type 2 Diabetes (T2D), were genotyped for six SNPs at the 9p21 locus. Additionally, <italic>ANRIL</italic> serum expression was assessed in both groups using real-time PCR. It was performed using different primers targeting exons 1, 5–6, and 19. The χ² test for association, along with t-tests and ANOVA, was employed for statistical analysis. In this study, we did not find any significant correlation between premature coronary artery disease and rs10757274, rs2383206, rs2383207, rs496892, rs10757278 and rs10738605. However, a lower <italic>ANRIL</italic> expression was correlated with each SNP risk genotype. Despite the correlation between lower <italic>ANRIL</italic> expression and CAD, Type 2 diabetes was associated with higher <italic>ANRIL</italic> expression. Altogether, the correlation between <italic>ANRIL</italic> expression and the genotypes of the studied SNPs indicated that genetic variants, even those in intronic regions, affect long noncoding RNA expression levels. In conclusion, we recommend combining genetic variants with expression analysis when developing screening strategies for families with premature CAD. To prevent the devastating outcomes of CAD in young adults, it is crucial to discover noninvasive genetic-based screening tests.</p>", "<title>Subject terms</title>" ]

[ "<title>Limitations of this study</title>", "<p id=\"Par38\">It is necessary to consider the limitations of the current study when interpreting its outcomes. Firstly, our small sample size precludes the generalizability of our results concerning the association of rs10757274, rs2383206, rs2383207, rs496892, rs10757278, and rs10738605 in Iranian patients with premature CAD. Secondly, our age threshold limited the number of participants in this study. Therefore, we strongly recommend replicating this study with a larger sample size of the Iranian population. To gain a better understanding of the independent effect of premature CAD occurrence on <italic>ANRIL</italic> expression, parallel studies on older individuals with CAD are strongly recommended.</p>", "<title>Supplementary Information</title>", "<p>\n</p>" ]

[ "<title>Supplementary Information</title>", "<p>The online version contains supplementary material available at 10.1038/s41598-024-51715-2.</p>", "<title>Author contributions</title>", "<p>E.T.B.: experiment design, lab work, data production, data interpretation, writing first draft of manuscript; A.Z., F.S., &amp; M.M.: cardiovascular counselling, angiography results’ interpretation, data interpretation; H.B.: statistical analysis, data interpretation; S.J.M.: project design, data interpretation, manuscript edit; M.M.: project design, experiment design, data interpretation, manuscript final edit. All authors reviewed and confirmed the final draft.</p>", "<title>Funding</title>", "<p>This research was supported in part by a research grant awarded to Dr. Mahshid Malakootian from NIMAD (National Institute for Medical Research Development) [no. 943104] and Research Deputyship of Rajaie Cardiovascular Medical and Research Center [no. 96060].</p>", "<title>Data availability</title>", "<p>The datasets generated during and/or analysed during the current study are available in the GenBank repository, sequences are accessible via OQ744705–OQ745198 accession numbers for CAD patients and OQ745199–OQ745577 for non-CAD subjects. The detailed list of accession numbers is available from the corresponding author.</p>", "<title>Competing interests</title>", "<p id=\"Par49\">The authors declare no competing interests.</p>" ]

[ "<fig id=\"Fig1\"><label>Figure 1</label><caption><p>Genomic location of <italic>ANRIL</italic> (<italic>CDKN2B-AS1</italic>) and its splicing variants. <italic>ANRIL</italic> resides in 9p21 locus and spans approximately 126 kb of genomic DNA, comprising 20 exons in total (black rectangles) and 14 splicing isoforms annotated in RefSeq records. The main regulatory elements are depicted below the scale bar. The discussed SNPs are indicated above the <italic>ANRIL</italic> exons. The orientations of primers are displayed by arrows above the exons. <italic>ANRIL</italic>, antisense noncoding RNA in the <italic>INK4A</italic> locus. <italic>CTCF</italic> CCCTC-binding factor.</p></caption></fig>", "<fig id=\"Fig2\"><label>Figure 2</label><caption><p>The expression of <italic>ANRIL</italic> regarding CAD, T2D, and the history of familial CAD. (<bold>A</bold>–<bold>C</bold>) for exon one, (<bold>D</bold>–<bold>F</bold>) for exons five-six, and G, H, and I for exon 19 detection. The expression of <italic>ANRIL</italic> using E1 and E19 primers was significantly lower in CAD vs non-CAD (<bold>A</bold>, <bold>G</bold>) while E5-6 primers revealed a non-significant decrease (<bold>D</bold>). (<bold>B</bold>, <bold>E</bold>, <bold>H</bold>) The expression of <italic>ANRIL</italic> in the presence and absence of T2D and CAD. T2D contributes to higher expression of <italic>ANRIL</italic>. E5-6 detection revealed a considerable downregulation of <italic>ANRIL</italic> in CAD patients compared to non-CAD ones and displayed a significantly higher expression of <italic>ANRIL</italic> in subjects with diabetes I. (<bold>C</bold>, <bold>F</bold>, <bold>I</bold>) The expression of <italic>ANRIL</italic> regarding familial CAD. E19 primers detected significant downregulation of <italic>ANRIL</italic> in patients with CAD as well as in those having a history of familial CAD, compared to the control group. One-way ANOVA was performed to compare expression between different categories (the respective <italic>P</italic>-values are indicated below each graph). Multiple comparison tests were also applied by comparing the mean of each column with the mean of every other column and were corrected by Turkey test (asterisks represent for multiple comparison tests). The presence and absence of each disease is depicted by + and − signs in front of each status in x axis (<bold>B</bold>, <bold>C</bold>, <bold>E</bold>, <bold>F</bold>, <bold>H</bold>, <bold>I</bold>). <italic>P</italic> values ≤ 0.0001 are given ****, while <italic>P</italic> ≤ 0.001, <italic>P</italic> ≤ 0.01, and <italic>P</italic> ≤ 0.05 are given ***, **, and * respectively. <italic>ANRIL</italic>, antisense noncoding RNA in the <italic>INK4A</italic> locus; CAD, Coronary Artery Disease; T2D, Type 2 Diabetes.</p></caption></fig>", "<fig id=\"Fig3\"><label>Figure 3</label><caption><p>The correlation between SNPs within and proximal to the <italic>ANRIL</italic> sequence and the serum expression of <italic>ANRIL</italic> (both exon one and exon 19) in premature CAD compared to the non-CAD group. The risk genotype for rs10757274: A&gt;G, rs2383206: A&gt;G, rs2383207: A&gt;G, rs10757278: A&gt;G and rs10738605: C&gt;G is GG, and the risk genotype for rs496892: C&gt;T is TT (<bold>A</bold>–<bold>F</bold>). The P- values are regarding one-way ANOVA analyses between all groups and asterisks represent the P-values &lt; 0.05 in multiple comparison analyses. (<bold>G</bold>, <bold>H</bold>) ROC curve analysis for rs10738605, rs496892, and the expression of <italic>ANRIL</italic>. (<bold>I</bold>) Linkage disequilibrium analysis for the selected SNPs. The name of studied SNPs are at the top of the figure and D’ factor (%) is displayed in the squares below. The empty red squares indicate Dʹ &gt; 0.9. rs10757274, rs2383206, rs2383207 and rs10757278 are in one block and rs496892 and rs10738605 share a block together. (<bold>J</bold>) The Spearman correlation between all variables. The graph is generated using the Spearman correlation coefficients (r). The scale bar on the right demonstrates the range of the r coefficient in color. The r values close to 1 (blue squares) demonstrate an increasing monotonic pattern between variables and values close to − 1 (red squares) correspond to decreasing monotonic trends between variables. P values ≤ 0.0001 are given ****, while P ≤ 0.001, P ≤ 0.01, and P ≤ 0.05 are given ***, **, and * respectively. <italic>ANRIL</italic> antisense noncoding RNA in the <italic>INK4A</italic> locus, <italic>CAD</italic> Coronary Artery Disease, <italic>ROC</italic> Receiver Operating Characteristic, <italic>BMI</italic> body mass index, <italic>DBP</italic> diastolic blood pressure, <italic>FBS</italic> fasting blood sugar, <italic>HDL-C</italic> high-density lipoprotein-cholesterol, <italic>LDL-C</italic> low-density lipoprotein-cholesterol, <italic>SBP</italic> systolic blood pressure, <italic>T2D</italic> Type 2 diabetes, <italic>TC</italic> total cholesterol.</p></caption></fig>" ]

[ "<table-wrap id=\"Tab1\"><label>Table 1</label><caption><p>Clinical and demographic characteristics of the included patients.</p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th align=\"left\">Parameters</th><th align=\"left\">CAD (n = 93)</th><th align=\"left\">Non-CAD (n = 87)</th><th align=\"left\"><italic>P</italic>-value</th><th align=\"left\"><italic>P</italic>-value summary</th></tr></thead><tbody><tr><td align=\"left\" colspan=\"5\">Age (years), (mean ± SD)</td></tr><tr><td align=\"left\"> Female</td><td align=\"left\">45.07 ± 3.738</td><td align=\"left\">41.82 ± 7.920</td><td align=\"left\">0.0655†</td><td align=\"left\">ns</td></tr><tr><td align=\"left\"> Male</td><td align=\"left\">39.98 ± 5.034</td><td align=\"left\">36.93 ± 5.497</td><td align=\"left\">0.0569<bold>†</bold></td><td align=\"left\">ns</td></tr><tr><td align=\"left\">BMI (kg/m²), (mean ± SD)</td><td align=\"left\">28.52 ± 4.525</td><td align=\"left\">27.93 ± 5.999</td><td align=\"left\">0.2177</td><td align=\"left\">ns</td></tr><tr><td align=\"left\">SBP (mean ± SD)</td><td align=\"left\">127.0 ± 13.71</td><td align=\"left\">124.9 ± 11.95</td><td align=\"left\">0.3890</td><td align=\"left\">ns</td></tr><tr><td align=\"left\">DBP (mean ± SD)</td><td align=\"left\">78.64 ± 6.372</td><td align=\"left\">79.22 ± 5.520</td><td align=\"left\">0.8449</td><td align=\"left\">ns</td></tr><tr><td align=\"left\">Triglyceride (mg/dL), (mean ± SD)</td><td align=\"left\">157.3 ± 93.87</td><td align=\"left\">116.6 ± 54.86</td><td align=\"left\">0.0033</td><td align=\"left\">** (P &lt; 0.05)</td></tr><tr><td align=\"left\">TC (mg/dL), (mean ± SD)</td><td align=\"left\">148.4 ± 47.31</td><td align=\"left\">132.5 ± 30.18</td><td align=\"left\">0.0153</td><td align=\"left\">* (P &lt; 0.05)</td></tr><tr><td align=\"left\">HDL-C (mg/dL), (mean ± SD)</td><td align=\"left\">36.10 ± 7.177</td><td align=\"left\">38.19 ± 7.776</td><td align=\"left\">0.753</td><td align=\"left\">ns</td></tr><tr><td align=\"left\">LDL-C (mg/dL), (mean ± SD)</td><td align=\"left\">83.77 ± 37.07</td><td align=\"left\">72.39 ± 21.76</td><td align=\"left\">0.0207</td><td align=\"left\">* (P &lt; 0.05)</td></tr><tr><td align=\"left\">FBS (mg/dL), (mean ± SD)</td><td align=\"left\">136.3 ± 52.58</td><td align=\"left\">101.5 ± 30.11</td><td align=\"left\">&lt; 0.0001</td><td align=\"left\">**** (P &lt; 0.05)</td></tr><tr><td align=\"left\">Stroke history (% frequency)</td><td align=\"left\">28.26%</td><td align=\"left\">7.93%</td><td align=\"left\">0.0019</td><td align=\"left\">** (P &lt; 0.05)</td></tr><tr><td align=\"left\">T2D (% frequency)</td><td align=\"left\">34.78%</td><td align=\"left\">12.69%</td><td align=\"left\">0.0025</td><td align=\"left\">** (P &lt; 0.05)</td></tr><tr><td align=\"left\">Familial CAD (% frequency)</td><td align=\"left\">78.49%</td><td align=\"left\">62.5%</td><td align=\"left\">0.0316</td><td align=\"left\">* (P &lt; 0.05)</td></tr><tr><td align=\"left\">Familial BP (% frequency)</td><td align=\"left\">68.47%</td><td align=\"left\">73.77%</td><td align=\"left\">0.5872</td><td align=\"left\">ns</td></tr><tr><td align=\"left\">Familial T2D (% frequency)</td><td align=\"left\">48.38%</td><td align=\"left\">40.32%</td><td align=\"left\">0.4103</td><td align=\"left\">ns</td></tr><tr><td align=\"left\">Familial high cholesterol (% frequency)</td><td align=\"left\">47.82%</td><td align=\"left\">37.09%</td><td align=\"left\">0.2460</td><td align=\"left\">ns</td></tr><tr><td align=\"left\">Familial MI (% frequency)</td><td align=\"left\">59.13%</td><td align=\"left\">49.18%</td><td align=\"left\">0.2489</td><td align=\"left\">ns</td></tr><tr><td align=\"left\">Smoking (% frequency)</td><td align=\"left\">36.55%</td><td align=\"left\">24.28%</td><td align=\"left\">0.1244</td><td align=\"left\">ns</td></tr><tr><td align=\"left\">Alcohol drinking (% frequency)</td><td align=\"left\">19.56%</td><td align=\"left\">12.90%</td><td align=\"left\">0.3808</td><td align=\"left\">ns</td></tr><tr><td align=\"left\">Stress (% frequency)</td><td align=\"left\">75.26%</td><td align=\"left\">83.07%</td><td align=\"left\">0.3254</td><td align=\"left\">ns</td></tr></tbody></table></table-wrap>", "<table-wrap id=\"Tab2\"><label>Table 2</label><caption><p>The distribution of alleles and genotypes of variants.</p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th align=\"left\"/><th align=\"left\">Non-CAD (N/%)</th><th align=\"left\">CAD (N/%)</th><th align=\"left\">OR (95% CI)</th><th align=\"left\"><italic>P</italic>-value</th></tr></thead><tbody><tr><td align=\"left\" colspan=\"5\">rs10757274 (A&gt;G)</td></tr><tr><td align=\"left\" colspan=\"5\"> Allele frequency</td></tr><tr><td align=\"left\">  A</td><td align=\"left\">62 (40.79)</td><td align=\"left\">72 (39.13)</td><td align=\"left\">1.00</td><td align=\"left\" rowspan=\"2\">0.75</td></tr><tr><td align=\"left\">  G</td><td align=\"left\">90 (59.21)</td><td align=\"left\">112 (60.78)</td><td align=\"left\">0.93 (0.60–1.43)</td></tr><tr><td align=\"left\" colspan=\"5\"> Genotypes (Codominant)</td></tr><tr><td align=\"left\">  A/A</td><td align=\"left\">15 (19.74)</td><td align=\"left\">18 (19.57)</td><td align=\"left\">1.00</td><td align=\"left\"/></tr><tr><td align=\"left\">  A/G</td><td align=\"left\">32 (42.10)</td><td align=\"left\">36 (39.13)</td><td align=\"left\">0.93 (0.40–2.11)</td><td align=\"left\">0.87</td></tr><tr><td align=\"left\">  G/G</td><td align=\"left\">29 (38.16)</td><td align=\"left\">38 (41.30)</td><td align=\"left\">1.09 (0.46–2.48)</td><td align=\"left\">0.83</td></tr><tr><td align=\"left\" colspan=\"5\"> Genotypes (Dominant)</td></tr><tr><td align=\"left\">  A/A</td><td align=\"left\">15 (19.74)</td><td align=\"left\">18 (19.57)</td><td align=\"left\">1.00</td><td align=\"left\"/></tr><tr><td align=\"left\">  A/G–G/G</td><td align=\"left\">61 (80.26)</td><td align=\"left\">74 (80.43)</td><td align=\"left\">1.01 (0.47–2.16)</td><td align=\"left\">0.97</td></tr><tr><td align=\"left\" colspan=\"5\"> Genotypes (Recessive)</td></tr><tr><td align=\"left\">  A/A–A/G</td><td align=\"left\">47 (61.84)</td><td align=\"left\">54 (58.70)</td><td align=\"left\">1.00</td><td align=\"left\"/></tr><tr><td align=\"left\">  G/G</td><td align=\"left\">29 (38.16)</td><td align=\"left\">38 (41.30)</td><td align=\"left\">1.14 (0.62–2.10)</td><td align=\"left\">0.67</td></tr><tr><td align=\"left\" colspan=\"5\"> Genotypes (Over-dominant)</td></tr><tr><td align=\"left\">  A/A–G/G</td><td align=\"left\">44 (57.89)</td><td align=\"left\">56 (60.87)</td><td align=\"left\">1.00</td><td align=\"left\"/></tr><tr><td align=\"left\">  A/G</td><td align=\"left\">32 (42.11)</td><td align=\"left\">36 (39.13)</td><td align=\"left\">0.88 (0.48–1.61)</td><td align=\"left\">0.70</td></tr><tr><td align=\"left\">  HWE†</td><td align=\"left\"><p><italic>X</italic>^{<italic>2</italic>} = 1.25</p><p><italic>P</italic> = 0.53</p></td><td align=\"left\"><p><italic>X</italic>^{<italic>2</italic>} = 2.93</p><p><italic>P</italic> = 0.23</p></td><td align=\"left\"/><td align=\"left\"/></tr><tr><td align=\"left\" colspan=\"5\">rs2383206 (A&gt;G)</td></tr><tr><td align=\"left\" colspan=\"5\"> Allele frequency</td></tr><tr><td align=\"left\">  A</td><td align=\"left\">52 (34.21)</td><td align=\"left\">66 (35.87)</td><td align=\"left\">1.00</td><td align=\"left\" rowspan=\"2\">0.75</td></tr><tr><td align=\"left\">  G</td><td align=\"left\">100 (65.72)</td><td align=\"left\">118 (64.13)</td><td align=\"left\">0.93 (0.59–1.45)</td></tr><tr><td align=\"left\" colspan=\"5\"> Genotypes (Codominant)</td></tr><tr><td align=\"left\">  A/A</td><td align=\"left\">9 (11.84)</td><td align=\"left\">13 (14.13)</td><td align=\"left\">1.00</td><td align=\"left\"/></tr><tr><td align=\"left\">  A/G</td><td align=\"left\">34 (44.73)</td><td align=\"left\">40 (43.47)</td><td align=\"left\">0.81 (0.32–2.11)</td><td align=\"left\">0.68</td></tr><tr><td align=\"left\">  G/G</td><td align=\"left\">33 (43.42)</td><td align=\"left\">39 (42.39)</td><td align=\"left\">0.82 (0.32–2.13)</td><td align=\"left\">0.68</td></tr><tr><td align=\"left\" colspan=\"5\"> Genotypes (Dominant)</td></tr><tr><td align=\"left\">  A/A</td><td align=\"left\">9 (11.84)</td><td align=\"left\">13 (14.13)</td><td align=\"left\">1.00</td><td align=\"left\"/></tr><tr><td align=\"left\">  A/G–G/G</td><td align=\"left\">67 (88.16)</td><td align=\"left\">79 (85.87)</td><td align=\"left\">0.82 (0.34–1.93)</td><td align=\"left\">0.66</td></tr><tr><td align=\"left\" colspan=\"5\"> Genotypes (Recessive)</td></tr><tr><td align=\"left\">  A/A–A/G</td><td align=\"left\">43 (56.58)</td><td align=\"left\">53 (57.61)</td><td align=\"left\">1.00</td><td align=\"left\"/></tr><tr><td align=\"left\">  G/G</td><td align=\"left\">33 (43.42)</td><td align=\"left\">39 (42.39)</td><td align=\"left\">0.96 (0.53–1.73)</td><td align=\"left\">0.89</td></tr><tr><td align=\"left\" colspan=\"5\"> Genotypes (Over-dominant)</td></tr><tr><td align=\"left\">  A/A–G/G</td><td align=\"left\">42 (55.26)</td><td align=\"left\">52 (56.52)</td><td align=\"left\">1.00</td><td align=\"left\"/></tr><tr><td align=\"left\">  A/G</td><td align=\"left\">34 (44.73)</td><td align=\"left\">40 (43.47)</td><td align=\"left\">0.95 (0.52–1.71)</td><td align=\"left\">0.87</td></tr><tr><td align=\"left\">  HWE</td><td align=\"left\"><p><italic>X</italic>^{<italic>2</italic>} = 0.002</p><p><italic>P</italic> = 0.99</p></td><td align=\"left\"><p><italic>X</italic>^{<italic>2</italic>} = 0.28</p><p><italic>P</italic> = 0.87</p></td><td align=\"left\"/><td align=\"left\"/></tr><tr><td align=\"left\" colspan=\"5\">rs2383207 (A&gt;G)</td></tr><tr><td align=\"left\" colspan=\"5\"> Allele frequency</td></tr><tr><td align=\"left\">  A</td><td align=\"left\">51 (34)</td><td align=\"left\">63 (34.24)</td><td align=\"left\">1.00</td><td align=\"left\" rowspan=\"2\">0.96</td></tr><tr><td align=\"left\">  G</td><td align=\"left\">99 (66)</td><td align=\"left\">121 (65.76)</td><td align=\"left\">0.98 (0.62–1.56)</td></tr><tr><td align=\"left\" colspan=\"5\"> Genotypes (Codominant)</td></tr><tr><td align=\"left\">  A/A</td><td align=\"left\">8 (10.66)</td><td align=\"left\">12 (13.04)</td><td align=\"left\">1.00</td><td align=\"left\"/></tr><tr><td align=\"left\">  A/G</td><td align=\"left\">35 (46.66)</td><td align=\"left\">39 (42.39)</td><td align=\"left\">0.74 (0.26–2.07)</td><td align=\"left\">0.56</td></tr><tr><td align=\"left\">  G/G</td><td align=\"left\">32 (42.66)</td><td align=\"left\">41 (44.56)</td><td align=\"left\">0.85 (0.30–2.40)</td><td align=\"left\">0.76</td></tr><tr><td align=\"left\" colspan=\"5\"> Genotypes (Dominant)</td></tr><tr><td align=\"left\">  A/A</td><td align=\"left\">8 (10.66)</td><td align=\"left\">12 (13.04)</td><td align=\"left\">1.00</td><td align=\"left\"/></tr><tr><td align=\"left\">  A/G–G/G</td><td align=\"left\">67 (89.33)</td><td align=\"left\">80 (86.96)</td><td align=\"left\">0.79 (0.31–2.00)</td><td align=\"left\">0.64</td></tr><tr><td align=\"left\" colspan=\"5\"> Genotypes (Recessive)</td></tr><tr><td align=\"left\">  A/A–A/G</td><td align=\"left\">43 (57.33)</td><td align=\"left\">51 (55.43)</td><td align=\"left\">1.00</td><td align=\"left\"/></tr><tr><td align=\"left\"> G/G</td><td align=\"left\">32 (42.66)</td><td align=\"left\">41 (44.56)</td><td align=\"left\">1.08 (0.59–1.96)</td><td align=\"left\">0.80</td></tr><tr><td align=\"left\" colspan=\"5\"> Genotypes (Over-dominant)</td></tr><tr><td align=\"left\">  A/A–G/G</td><td align=\"left\">40 (53.33)</td><td align=\"left\">53 (57.61)</td><td align=\"left\">1.00</td><td align=\"left\"/></tr><tr><td align=\"left\">  A/G</td><td align=\"left\">35 (46.67)</td><td align=\"left\">39 (42.39)</td><td align=\"left\">0.84 (0.46–1.52)</td><td align=\"left\">0.58</td></tr><tr><td align=\"left\">  HWE</td><td align=\"left\"><p><italic>X</italic>^{<italic>2</italic>} = 0.11</p><p><italic>P</italic> = 0.94</p></td><td align=\"left\"><p><italic>X</italic>^{<italic>2</italic>} = 0.31</p><p><italic>P</italic> = 0.85</p></td><td align=\"left\"/><td align=\"left\"/></tr><tr><td align=\"left\" colspan=\"5\">rs496892 (C&gt;T)</td></tr><tr><td align=\"left\" colspan=\"5\"> Allele frequency</td></tr><tr><td align=\"left\">  C</td><td align=\"left\">108 (71.05)</td><td align=\"left\">124 (68.89)</td><td align=\"left\">1.00</td><td align=\"left\" rowspan=\"2\">0.66</td></tr><tr><td align=\"left\">  T</td><td align=\"left\">44 (28.95)</td><td align=\"left\">56 (31.11)</td><td align=\"left\">1.10 (0.69–1.77)</td></tr><tr><td align=\"left\" colspan=\"5\"> Genotypes (Codominant)</td></tr><tr><td align=\"left\">  C/C</td><td align=\"left\">38 (50)</td><td align=\"left\">45 (50)</td><td align=\"left\">1.00</td><td align=\"left\"/></tr><tr><td align=\"left\">  C/T</td><td align=\"left\">32 (42.10)</td><td align=\"left\">34 (37.77)</td><td align=\"left\">0.89 (0.46–1.73)</td><td align=\"left\">0.74</td></tr><tr><td align=\"left\">  T/T</td><td align=\"left\">6 (7.89)</td><td align=\"left\">11 (12.22)</td><td align=\"left\">1.54 (0.56–4.69)</td><td align=\"left\">0.42</td></tr><tr><td align=\"left\" colspan=\"5\"> Genotypes (Dominant)</td></tr><tr><td align=\"left\">  C/C</td><td align=\"left\">38 (50)</td><td align=\"left\">45 (50)</td><td align=\"left\">1.00</td><td align=\"left\"/></tr><tr><td align=\"left\">  C/T–T/T</td><td align=\"left\">38 (50)</td><td align=\"left\">45 (50)</td><td align=\"left\">1.00 (0.54–1.82)</td><td align=\"left\">&gt; 0.99</td></tr><tr><td align=\"left\" colspan=\"5\"> Genotypes (Recessive)</td></tr><tr><td align=\"left\">  C/C–C/T</td><td align=\"left\">70 (92.11)</td><td align=\"left\">79 (87.78)</td><td align=\"left\">1.00</td><td align=\"left\"/></tr><tr><td align=\"left\">  T/T</td><td align=\"left\">6 (7.89)</td><td align=\"left\">11 (12.22)</td><td align=\"left\">1.62 (0.55–4.62)</td><td align=\"left\">0.35</td></tr><tr><td align=\"left\" colspan=\"5\"> Genotypes (Over-dominant)</td></tr><tr><td align=\"left\">  C/C–T/T</td><td align=\"left\">44 (57.89)</td><td align=\"left\">56 (62.22)</td><td align=\"left\">1.00</td><td align=\"left\"/></tr><tr><td align=\"left\">  C/T</td><td align=\"left\">32 (42.10)</td><td align=\"left\">34 (37.77)</td><td align=\"left\">0.83 (0.45–1.53)</td><td align=\"left\">0.57</td></tr><tr><td align=\"left\">  HWE</td><td align=\"left\"><p><italic>X</italic>^{<italic>2</italic>} = 0.04</p><p><italic>P</italic> = 0.97</p></td><td align=\"left\"><p><italic>X</italic>^{<italic>2</italic>} = 1.26</p><p><italic>P</italic> = 0.53</p></td><td align=\"left\"/><td align=\"left\"/></tr><tr><td align=\"left\" colspan=\"5\">rs10757278 (A&gt;G)</td></tr><tr><td align=\"left\" colspan=\"5\"> Allele frequency</td></tr><tr><td align=\"left\">  A</td><td align=\"left\">67 (45.27)</td><td align=\"left\">79 (43.41)</td><td align=\"left\">1.00</td><td align=\"left\" rowspan=\"2\">0.73</td></tr><tr><td align=\"left\">  G</td><td align=\"left\">81 (54.73)</td><td align=\"left\">103 (56.59)</td><td align=\"left\">1.07 (0.68–1.68)</td></tr><tr><td align=\"left\" colspan=\"5\"> Genotypes (Codominant)</td></tr><tr><td align=\"left\">  A/A</td><td align=\"left\">17 (22.97)</td><td align=\"left\">21 (23.07)</td><td align=\"left\">1.00</td><td align=\"left\"/></tr><tr><td align=\"left\">  A/G</td><td align=\"left\">33 (44.59)</td><td align=\"left\">37 (40.65)</td><td align=\"left\">0.90 (0.42–2.05)</td><td align=\"left\">0.81</td></tr><tr><td align=\"left\">  G/G</td><td align=\"left\">24 (32.43)</td><td align=\"left\">33 (36.26)</td><td align=\"left\">1.11 (0.49–2.48)</td><td align=\"left\">0.79</td></tr><tr><td align=\"left\" colspan=\"5\"> Genotypes (Dominant)</td></tr><tr><td align=\"left\">  A/A</td><td align=\"left\">17 (22.97)</td><td align=\"left\">21 (23.07)</td><td align=\"left\">1.00</td><td align=\"left\"/></tr><tr><td align=\"left\">  A/G–G/G</td><td align=\"left\">57 (77.03)</td><td align=\"left\">70 (76.92)</td><td align=\"left\">0.99 (0.48–2.02)</td><td align=\"left\">0.98</td></tr><tr><td align=\"left\" colspan=\"5\"> Genotypes (Recessive)</td></tr><tr><td align=\"left\">  A/A–A/G</td><td align=\"left\">50 (67.57)</td><td align=\"left\">58 (63.74)</td><td align=\"left\">1.00</td><td align=\"left\"/></tr><tr><td align=\"left\">  G/G</td><td align=\"left\">24 (32.43)</td><td align=\"left\">33 (36.26)</td><td align=\"left\">1.18 (0.62–2.23)</td><td align=\"left\">0.60</td></tr><tr><td align=\"left\" colspan=\"5\"> Genotypes (Over-dominant)</td></tr><tr><td align=\"left\">  A/A–G/G</td><td align=\"left\">41 (55.41)</td><td align=\"left\">54 (59.34)</td><td align=\"left\">1.00</td><td align=\"left\"/></tr><tr><td align=\"left\">  A/G</td><td align=\"left\">33 (44.59)</td><td align=\"left\">37 (40.65)</td><td align=\"left\">0.85 (0.46–1.55)</td><td align=\"left\">0.61</td></tr><tr><td align=\"left\"> HWE</td><td align=\"left\"><p><italic>X</italic>^{<italic>2</italic>} = 0.74</p><p><italic>P</italic> = 0.69</p></td><td align=\"left\"><p><italic>X</italic>^{<italic>2</italic>} = 2.70</p><p><italic>P</italic> = 0.25</p></td><td align=\"left\"/><td align=\"left\"/></tr><tr><td align=\"left\" colspan=\"5\">rs10738605 (C&gt;G)</td></tr><tr><td align=\"left\" colspan=\"5\"> Allele frequency</td></tr><tr><td align=\"left\">  C</td><td align=\"left\">46 (31.08)</td><td align=\"left\">59 (33.15)</td><td align=\"left\">1.00</td><td align=\"left\" rowspan=\"2\">0.69</td></tr><tr><td align=\"left\">  G</td><td align=\"left\">102 (68.92)</td><td align=\"left\">119 (66.85)</td><td align=\"left\">0.90 (0.57–1.47)</td></tr><tr><td align=\"left\" colspan=\"5\"> Genotypes (Codominant)</td></tr><tr><td align=\"left\">  C/C</td><td align=\"left\">6 (8.10)</td><td align=\"left\">13 (14.60)</td><td align=\"left\">1.00</td><td align=\"left\"/></tr><tr><td align=\"left\">  C/G</td><td align=\"left\">34 (45.94)</td><td align=\"left\">33 (37.07)</td><td align=\"left\">0.44 (0.14–1.36)</td><td align=\"left\">0.13</td></tr><tr><td align=\"left\">  G/G</td><td align=\"left\">34 (45.94)</td><td align=\"left\">43 (48.31)</td><td align=\"left\">0.58 (0.19–1.73)</td><td align=\"left\">0.31</td></tr><tr><td align=\"left\" colspan=\"5\"> Genotypes (Dominant)</td></tr><tr><td align=\"left\">  C/C</td><td align=\"left\">6 (8.10)</td><td align=\"left\">13 (14.60)</td><td align=\"left\">1.00</td><td align=\"left\"/></tr><tr><td align=\"left\">  C/G–G/G</td><td align=\"left\">68 (91.89)</td><td align=\"left\">76 (85.39)</td><td align=\"left\">0.51 (0.18–1.38)</td><td align=\"left\">0.19</td></tr><tr><td align=\"left\" colspan=\"5\"> Genotypes (Recessive)</td></tr><tr><td align=\"left\">  C/C–C/G</td><td align=\"left\">40 (54.05)</td><td align=\"left\">46 (51.69)</td><td align=\"left\">1.00</td><td align=\"left\"/></tr><tr><td align=\"left\">  G/G</td><td align=\"left\">34 (45.94)</td><td align=\"left\">43 (48.31)</td><td align=\"left\">1.10 (0.60–2.00)</td><td align=\"left\">0.76</td></tr><tr><td align=\"left\" colspan=\"5\"> Genotypes (Over-dominant)</td></tr><tr><td align=\"left\">  C/C–G/G</td><td align=\"left\">40 (54.05)</td><td align=\"left\">56 (62.92)</td><td align=\"left\">1.00</td><td align=\"left\"/></tr><tr><td align=\"left\">  C/G</td><td align=\"left\">34 (45.94)</td><td align=\"left\">33 (37.07)</td><td align=\"left\">0.69 (0.37–1.28)</td><td align=\"left\">0.25</td></tr><tr><td align=\"left\">  HWE</td><td align=\"left\"><p><italic>X</italic>^{<italic>2</italic>} = 0.38</p><p><italic>P</italic> = 0.82</p></td><td align=\"left\"><p><italic>X</italic>^{<italic>2</italic>} = 2.37</p><p><italic>P</italic> = 0.30</p></td><td align=\"left\"/><td align=\"left\"/></tr></tbody></table></table-wrap>" ]

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[ "<supplementary-material content-type=\"local-data\" id=\"MOESM1\"></supplementary-material>" ]

[ "<table-wrap-foot><p><italic>BMI</italic> body mass index, <italic>CAD</italic> Coronary Artery Disease, <italic>DBP</italic> diastolic blood pressure, <italic>FBS</italic> fasting blood sugar, <italic>HDL-C</italic> high-density lipoprotein-cholesterol, <italic>LDL-C</italic> low-density lipoprotein-cholesterol, <italic>ns</italic> not significant, <italic>SBP</italic> systolic blood pressure, <italic>T2D</italic> Type 2 diabetes, <italic>TC</italic> total cholesterol.</p><p>^a†<italic>P</italic>-value for the comparison of age between CAD and Non-CAD irrespective of sex = 0.0745 (ns).</p></table-wrap-foot>", "<table-wrap-foot><p>The frequency of alleles and genotypes of rs10757274, rs2383206, rs2383207, rs496892, rs10757278, and rs10738605 in CAD and non-CAD groups are listed. The odd ratios and <italic>P-</italic>values are outputs of χ² tests. For Hardy–Weinberg Equilibrium, χ² values are also included in the table.</p><p><bold>†</bold>Hardy–Weinberg Equilibrium.</p></table-wrap-foot>", "<fn-group><fn><p><bold>Publisher's note</bold></p><p>Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.</p></fn></fn-group>" ]

[ "<graphic xlink:href=\"41598_2024_51715_Fig1_HTML\" id=\"MO1\"/>", "<graphic xlink:href=\"41598_2024_51715_Fig2_HTML\" id=\"MO2\"/>", "<graphic xlink:href=\"41598_2024_51715_Fig3_HTML\" id=\"MO3\"/>" ]

[ "<media xlink:href=\"41598_2024_51715_MOESM1_ESM.pdf\"><caption><p>Supplementary Information.</p></caption></media>" ]

{ "acronym": [], "definition": [] }

[ "<title>Introduction</title>", "<p id=\"Par2\">The presence of cell-in-cell (CIC) structures has been observed in a wide range of human cancers and is mainly associated with poor prognosis [##REF##26504802##1##–##REF##34178655##7##]. CIC formation involves a dynamic interaction between an outer and an inner cell. It can be the result of different types of mechanisms involving either homotypic or heterotypic interactions and initiated by either the outer (endocytic CIC) or the inner (invasive CIC) cell [##REF##34685548##8##]. Entosis is defined as a homotypic invasion occurring between tumor cells of the same type where one living cell is internalized inside another one. After internalization by the outer cell, the inner cell is found in the entotic vacuole [##REF##28970591##9##], where it will be degraded generally by autophagy, a non-apoptotic cell death involving the lysosome fusion. Entosis was discovered in vitro in breast cancer cells (MCF-7) cultured under different stress factors such as extracellular matrix detachment, glucose starvation, or ultraviolet radiation [##REF##18045538##10##–##REF##34401679##12##]. Entosis was also described in cells with prolonged aberrant mitosis to eliminate the aneuploid progenies by engulfment, maintaining thus the genome integrity [##UREF##0##13##]. Mechanistically, entosis was found dependent on cell-cell adhesion molecules such as cadherins and driven by imbalances in actomyosin contractility which is mainly under the control of the RhoA/ROCK pathway [##REF##18045538##10##, ##REF##25342560##14##].</p>", "<p id=\"Par3\">Rnd3, also known as RhoE, is an atypical RhoGTPase protein and a negative regulator of the Rho/ROCK pathway. Others and we previously described Rnd3 as a tumor suppressor in liver cancer [##REF##27555595##15##, ##UREF##1##16##]. Indeed, <italic>RND3</italic> expression is downregulated in human hepatocellular carcinoma (HCC) samples compared to non-tumoral liver tissues and correlated with poor survival rates [##REF##22234932##17##, ##REF##22829315##18##]. It is implicated in cancer development through the regulation of cell growth and invasion [##REF##22234932##17##, ##REF##35256752##19##]. While working on the role of Rnd3 in HCC, we observed CIC structures, prompting us to study this phenomenon in liver cancer cells. Here, we reported that HCC cells are susceptible to form entosis after Rnd3 downregulation in cultured liver cancer cell lines and in xenografts in mice. Entosis is highly dependent on the RhoA/ROCK pathway, but not on E-cadherin. We also found that Rnd3 loss leads to Lamp-1 up-regulation, required for the internalization and degradation of the entotic cell. Finally, we related that entosis is rare in human HCC tissues, but associated with poor prognosis. Our results suggest that entotic engulfment induced by the loss of Rnd3 in HCC promotes liver tumor progression.</p>" ]

[ "<title>Materials/subjects and methods</title>", "<title>Cell culture</title>", "<p id=\"Par18\">The liver cancer cell lines (Hep3B, Huh7, HepG2, Huh6) and the breast adenocarcinoma cell line MCF-7 were cultured in Dulbecco modified Eagle’s medium (DMEM 1x glutamax, Fisher Scientific) supplemented with 10% heat-inactivated fetal bovine serum and incubated at 37 °C in a humidified 5% CO2 atmosphere. Cell line authentication was performed using short tandem repeat analysis, and the absence of mycoplasma contamination in cell culture media was tested every week. The nutrient deprivation was performed by culturing the cells in DMEM 1x glutamax with a very low concentration of glucose (Fisher Scientific) and supplemented with 10% of dialyzed heat-inactivated FBS for 10–13 h. For induction of cell suspension, the adherent cancer cells were trypsinized and cultured in non-treated plastic Petri dishes for 13 h. Stable cell lines with fluorescent nuclei or fluorescent F-actin were generated through transduction with lentiviruses expressing H2B-GFP (Addgene #25999), H2B-RFP (Addgene #26001) or LifeAct-mRuby. Transient knockdowns were done by transfection of small interfering RNAs (siRNAs) into cells using the lipofectamine RNAi max (Invitrogen) according to its manufacturer’s protocol. siRNAs targeting Rnd3 (SiRnd3 #1, SiRnd3 #2, SiRnd3 #3), p190RhoGAP-A (Sip190RhoGAP-A #1, Sip190RhoGAP-A #2), RhoA and E-cadherin (siE-cadherin #1) were purchased from Eurofins Genomics and the sequences are presented in Supplemental Table ##SUPPL##5##5##. siE-cadherin #2 were purchased from Thermo Fisher Scientific (CDH1, cat#: 4427037, ID: s2768). Control siRNA corresponds to AllStars Negative control from Qiagen. Transient transfection of plasmids was performed using Lipofectamine^TM 3000 reagent according to the manufacturer’s protocol (Thermo Fisher Scientific). The E-Cadherin-GFP construct was a generous gift from Dr. Peter Coopman (Montpellier). To induce a stable suppression of endogenous Rnd3 expression, we used Hep3B-shRnd3 and Huh7-shRnd3 cell lines with conditional, doxycycline-dependent, expression of shRnd3 as described previously [##REF##29940177##30##]. Hep3B-shCtrl and Huh7-shCtrl cell lines, conditionally expressing the control shRNA targeting the firefly luciferase, were used as controls. Stable cell lines were cultured as described above and shRNA expressions were induced by 50 mg/mL of doxycycline. For ROCK inhibition, cells were treated with Y-27632 (Sigma Aldrich) at 5 µM for 24 h. Sorafenib (Bay 43–9006, Enzo Life Science) was used at two doses 8 or 10 µm to treat Huh7 cell lines.</p>", "<title>Antibodies, immunoblot analysis</title>", "<p id=\"Par19\">Cells were lysed in RIPA lysis Buffer (Sigma) supplemented with protease inhibitor cocktail (Roche Diagnostics) and protein concentration was determined using Bio-Rad Protein Assay (Lowry). 60 μg of proteins from each sample were separated on 10% polyacrylamide gel (Bio-Rad) and blotted onto nitrocellulose membranes (0.2 μm nitrocellulose, Bio-Rad) using Trans-Blot Turbo Transfer System (Bio-Rad). The membranes were blocked in Odyssey blocking buffer for 30 min and then incubated with each of the following specific primary antibodies: Mouse anti-RhoE/Rnd3 (1:1000, clone 4, Cell Signaling, #3664 S), Rabbit anti-actin (1:2000, Sigma Aldrich, #A2066), Mouse anti-HSP90 (1:1000, Santa Cruz, #sc-69703), Rabbit anti-Mypt-1 (1:500, Millipore, #07-672-I), Rabbit anti-phospho-Mypt-1 (Thr696) (1:500, Millipore, #ABS45), Mouse anti-RhoA (1:1000, Santa Cruz, #sc-418), Mouse anti-p190A (1:1000, BD Biosciences, #610149), Mouse anti-E-cadherin (1:1000; BD Biosciences, #610182), Mouse anti-HSP90 (1:1000, Santa Cruz, #sc-69703), Mouse anti-LAMP1 (1:1000, Santa Cruz, #sc-20011), Rabbit anti-GFP (1/1000, Abcam, #ab290) either 1 h at room temperature or overnight at 4 °C. After incubation with the specific secondary antibodies, all blots were analyzed with the Bio-Rad Chemidoc system.</p>", "<title>Immunofluorescence staining</title>", "<p id=\"Par20\">The following primary antibodies were used for immunofluorescence (IF): Mouse anti-beta-catenin (1:400; BD Biosciences, #610154), Mouse anti-E-cadherin (1:50; BD Biosciences, #610182), Mouse anti-LAMP1 (1:100, Santa Cruz, #sc-20011). IF was performed on adherent or suspended cells after cytospin using the Shandon Cytospin 2 centrifuge at 110 rpm for 5 min. Cells were fixed and permeabilized in 4% paraformaldehyde PFA (Electron Microscopy Science, #15710) and 1% Triton X-100 respectively for 10 min at RT, followed by PBS washing and blocking in PBS-5%BSA for 20 min. Cells were then incubated with primary antibodies for 45 min followed, after PBS washing, by a incubation with the secondary antibodies (Interchim) specific for primary antibodies. F-actin was stained using fluorescent phalloidin (Molecular Probes). Finally, the cells were counterstained with DAPI (Sigma, #D9542) and the coverslip mounting on the glass slide was done using Fluoromount-G medium (Interchim #FP-483331).</p>", "<title>Time-lapse microscopy</title>", "<p id=\"Par21\">Hep3B cells expressing H2B-GFP and Lifeact-mRuby were cultured on glass-bottom dishes 35 mm high (Ibidi) after Rnd3 silencing, and time-lapse microscopy was performed in 37 °C and 5% CO₂ live-cell incubation chambers. The fluorescence was acquired every 2 h for 48 h using the spinning-disk LiveSR confocal microscope. The image analysis and the video reconstitution were done using ImageJ program.</p>", "<title>Correlative light-electron microscopy (CLEM)</title>", "<p id=\"Par22\">Hep3B cells expressing H2B-GFP were transfected with siRNA targeting Rnd3 (SiRnd3 #1) and cultured on glass coverslips gridded and numbered (Delta microscopy, #72265-12). After selection of the area of interest containing the entotic events using fluorescent microscopy, cells were fixed with 2.5% glutaraldehyde (Electron Microscopy Science, #15960) for 30 min at RT followed by incubation at 4° for 1 h in the dark. After washing with Sorensen’s phosphate buffer 0.2 M, pH 7.4 (Electron Microscopy Science, #11601-10), cells were dehydrated in graded series of ethanol solutions, including 50%, 70%, 90% for 5 min and 100% for 2 times, 5 min at RT in the dark. Finally, the cells were dried with CO₂ and observed using a Zeiss GeminiSEM300 microscope.</p>", "<title>Entosis quantification</title>", "<p id=\"Par23\">The entotic event is defined by 1) the presence of a cell inside another one with a deformed nucleus and 2) the high concentration of actin staining around the inner cell. The entotic cells were observed with epifluorescence microscopy (Zeiss). Quantification was done by counting 300–400 of total cells for each condition. The percentage of entotic events was calculated by dividing the number of entotic events by the total.</p>", "<title>SILAC labeling, laser capture microdissection, and proteomic analysis</title>", "<p id=\"Par24\">To discriminate laser-captured proteins from undesirable exogenous contaminating proteins, Hep3B-H2B-GFP cells were first metabolically labeled using stable isotope labeling with amino acids in cell culture (SILAC) method, as previously published [##REF##29795195##22##]. For that, Hep3B-H2B-GFP cells were cultured in DMEM medium (Dulbecco’s modified Eagle’s medium, Invitrogen) supplemented with 10% dialyzed fetal bovine serum, 200 mg/L L-proline, and 84 mg/L <sc>l</sc>-Arginine and Lysine. The incorporation of labeled amino acids was done after six cycles of cellular doubling. After transfection with siRnd3 #1, cells were seeded for 12–24 h on LamPen coated with collagen matrix allowing cells to adhere. Cells were fixed with PFA and the laser capture microdissection was performed using PALM type 4 micro-beam (Zeiss). SILAC-labeled cells were transfected with Rnd3-targeting siRNAs and entotic cells with crescent-shaped nuclei were microdissected. To obtain enough material for proteomic analysis, we manually collected 2000 cells in triplicate. With 2000 isolated entotic cells, we have identified a mean of 2880 ¹³C peptides corresponding to 406 proteins with at least 2 specific peptides. Given the small quantity of material analyzed, the first step guarantees the specificity of identifications by distinguishing labeled proteins from microdissected cells from unlabeled environmental contaminants (keratins, skin and hair proteins, etc…). In parallel, a standard range (500 ng to 5 ng) of protein quantity was done on cells labeled with SILAC and transfected with control siRNA (siCtrl) in order to compare the same quantity of entotic cells microdissected and obtained after transfection with siRnd3 to cells transfected with siCtrl. Three independent biological replicates on total protein extracts from SILAC-labeled cells were compared by label-free protein quantification. Proteins were loaded on a 10% acrylamide SDS-PAGE gel and proteins were visualized by Colloidal Blue staining. Migration was stopped when samples had just entered the resolving gel and the unresolved region of the gel was cut into only one segment. The steps of sample preparation and protein digestion by the trypsin were performed as previously described [##REF##28646562##31##]. NanoLC-MS/MS analysis was performed using an Ultimate 3000 RSLC Nano-UPHLC system (Thermo Scientific, USA) coupled to a nanospray Orbitrap Fusion™ Lumos™ Tribrid™ Mass Spectrometer (Thermo Fisher Scientific, California, USA). Each peptide extracts were loaded on a 300 µm ID x 5 mm PepMap C₁₈ precolumn (Thermo Scientific, USA) at a flow rate of 10 µL/min. After a 3 min desalting step, peptides were separated on a 50 cm EasySpray column (75 µm ID, 2 µm C₁₈ beads, 100 Å pore size, ES903, Thermo Fisher Scientific) with a 4–40% linear gradient of solvent B (0.1% formic acid in 80% ACN) in 57 min. The separation flow rate was set at 300 nL/min. The mass spectrometer operated in positive ion mode at a 2.0 kV needle voltage. Data was acquired using Xcalibur 4.4 software in a data-dependent mode. MS scans (<italic>m</italic>/<italic>z</italic> 375–1500) were recorded at a resolution of R = 120000 (@ <italic>m</italic>/<italic>z</italic> 200), a standard AGC target, and an injection time in automatic mode, followed by a top speed duty cycle of up to 3 s for MS/MS acquisition. Precursor ions (2 to 7 charge states) were isolated in the quadrupole with a mass window of 1.6 Th and fragmented with HCD@28% normalized collision energy. MS/MS data was acquired in the ion trap with rapid scan mode, a 20% normalized AGC target, and a maximum injection time in dynamic mode. Selected precursors were excluded for 60 s. Protein identification was done in Proteome Discoverer 2.5. Mascot 2.5 algorithm was used for protein identification in batch mode by searching against a UniProt <italic>Homo sapiens</italic> protein database (75796 entries, released September 3, 2020; <ext-link ext-link-type=\"uri\" xlink:href=\"https://www.uniprot.org/\">https://www.uniprot.org/</ext-link> website). Two missed enzyme cleavages were allowed for the trypsin. Mass tolerances in MS and MS/MS were set to 10 ppm and 0.6 Da. Oxidation (M), acetylation (K), SILAC modifications (K, R) were searched as dynamic modifications, and carbamidomethylation (C) as static modifications. Raw LC-MS/MS data were imported into Proline Studio for feature detection, alignment, and quantification. Protein identification was accepted only with at least 2 specific peptides with a pretty rank=1 and with a protein FDR value less than 1.0% calculated using the “decoy” option in Mascot. Label-free quantification of MS1 level by extracted ion chromatograms (XIC) was carried out with parameters indicated previously [##REF##28646562##31##]. The normalization was carried out on the median of ratios. The inference of missing values was applied with 5% of the background noise. The mass spectrometry proteomics data have been deposited to the ProteomeXchange Consortium via the PRIDE partner repository [##REF##36370099##32##] with the dataset identifier PXD043640.</p>", "<title>Immunohistochemistry on HCC samples</title>", "<p id=\"Par25\">Mouse HCC samples were from Hep3B xenografts previously described [##REF##35256752##19##]. This model consisted of subcutaneous inoculation of Hep3B-shCtrl or Hep3B-shRnd3 cells in immunodeficient NOG mice treated with doxycycline to induce or not Rnd3 knockdown. The analysis of entosis was performed on tumor sections stained with hematoxylin to mark the cytoplasm and nuclei of cells. Human samples came from resected or explanted livers of HCC patients treated in Bordeaux from 1992–2005. The characteristics of HCCs used for the IHC analysis (10 HCCs) are indicated in Table ##TAB##0##1##. Formalin-fixed, paraffin-embedded sections of human or mouse HCC samples were used for Hematoxylin-Eosin or immunohistochemical staining. Staining of Rnd3 and pan-keratin/beta-catenin was performed as previously published [##REF##22234932##17##, ##REF##30979717##33##].</p>", "<title>Statistical analysis</title>", "<p id=\"Par26\">The frequency of entotic cells was represented as a percentage (mean ± SD) of the total counted cells for at least three experiments. Data were analyzed using GraphPad Prism 10. For all experiments, significance was determined with the Mann–Whitney <italic>U</italic>-test, *<italic>p</italic> &lt; 0.05, **<italic>p</italic> &lt; 0.01<italic>,</italic> ***<italic>p</italic> <italic>&lt;</italic> 0.001.</p>" ]

[ "<title>Results</title>", "<title>Liver cancer cells are prone to perform entosis</title>", "<p id=\"Par4\">In order to investigate entosis in liver cancer, cultured cells (Hep3B, Huh7, Huh6, or HepG2) were subjected to stress conditions known to favor entosis in breast cancer cells such as nutrient deprivation [##REF##28683313##11##] or matrix detachment [##REF##18045538##10##]. MCF-7 human breast cancer cells were used as a positive control. Cell internalization was analyzed and quantified by confocal microscopy after immunostaining of nuclei, F-actin, and β-catenin to visualize nucleus deformation, shape, and cell periphery respectively. The outer cell shows a crescent-shaped nucleus, whereas the inner cell is rounded and retains its plasma membrane (Fig. ##FIG##0##1A##). In normal conditions, entotic structures can be observed in about 1–3% of total adherent liver cancer cells. After nutrient deprivation, a significant increase in the percentage of entotic cells is observed in all four cell lines, with up to 10% of entotic cells in HepG2 and Huh6 cells, when compared to normal culture conditions (Fig. ##FIG##0##1B–C##). Matrix detachment is also described as an inducer of entosis in MCF-7 cells [##REF##18045538##10##]. In order to test this stimulus, cells were cultured in non-adherent conditions and deposited on a slide by cytospin for immunostaining. The results show an increase of entotic cells after matrix detachment in all liver cancer cells, even if the percentage of entotic cells remains lower in liver cancer cells (6–14%) when compared to MCF-7 cells (Fig. ##FIG##0##1D##).</p>", "<p id=\"Par5\">As entosis is favored under stress conditions, we attempted to address whether hypoxia or drug treatment may trigger this mechanism in liver cancer cells. Hypoxic conditions were monitored by the increase in the mRNA expression of <italic>GLUT1</italic> and <italic>VEGF</italic>, two target genes of Hypoxia-Inducible Factor (Hif1α) [##REF##18259200##20##] (Supplementary Fig. ##SUPPL##0##1A##). The quantification of entotic Hep3B and Huh7 cells under hypoxia did not show any significant change compared to basal levels (Supplementary Fig. ##SUPPL##0##1B##). We also evaluated whether HCC treatments like Sorafenib could be an inducer of entosis. While Sorafenib treatment altered pERK/ERK ratio as expected, no significant difference was observed in the percentage of entosis in treated versus untreated HCC cells (Supplementary Fig. ##SUPPL##0##1C##). Altogether, our results demonstrate that liver cancer cells are prone to perform entosis upon proper stimuli such as nutrient deprivation or matrix detachment.</p>", "<title>CIC formation is driven by the loss of Rnd3 expression</title>", "<p id=\"Par6\">While working on the role of Rnd3 in HCC progression [##REF##22234932##17##, ##REF##35256752##19##], we noticed that, upon Rnd3 silencing, some cells appeared to be inside other cells, reminiscent of entosis. To examine whether the silencing of Rnd3 may trigger this mechanism, we first analyzed Rnd3 expression in liver cancer and MCF-7 cells. While Hep3B and Huh7 cells strongly express Rnd3 and Huh6 cells slightly less, Rnd3 is not expressed in HepG2 or MCF-7 cells (Supplementary Fig. ##SUPPL##0##2A##). We thus choose to silence Rnd3 in Hep3B and Huh7 cells using two different approaches: transient transfection of siRNAs using three different siRNAs targeting Rnd3 (SiRnd3#1, #2, #3), or inducible Rnd3 knockdown (KD) in cell lines stably expressing shRNA [##REF##35256752##19##]. We found that Rnd3 KD led to a significant increase of CIC events in Hep3B and Huh7 cells regardless of the approach used (Fig. ##FIG##1##2A##). Neighboring cells engulfed each other, with about 10% of adherent cells containing engulfed neighbors. CIC events induced by the loss of Rnd3 show similar characteristics to stress-induced entotic cells, such as crescent-shaped nuclei and round-engulfed cells (Fig. ##FIG##1##2B##). More complex cell structures were observed, with three or more cells involved in sequential engulfments (Fig. ##FIG##1##2C##). Thus, Rnd3 loss-mediated CIC events share common features with entosis observed in nutrient-depleted or matrix-detached conditions. Moreover, the combination of stimuli, i.e., silencing of Rnd3 and starvation did not increase the entosis mechanism in Hep3B and Huh7 cells, suggesting the involvement of the same pathways to mediate cell engulfment (Supplementary Fig. ##SUPPL##0##2B, C##). The loss of Rnd3 expression, and therefore function, appears to be a trigger of cell engulfment in HCC cells. We next aimed to identify entosis in vivo in tumor tissues using the xenograft mouse model described previously [##REF##35256752##19##]. We noticed a significant increase in entosis percentage in tumors generated from Rnd3-KD cells when compared to control Hep3B cells (Fig. ##FIG##1##2D##). Thus, Rnd3 loss favors HCC CIC events both in vitro and in vivo.</p>", "<title>Characteristics of entosis stages after silencing of Rnd3</title>", "<p id=\"Par7\">To investigate the CIC events induced by the loss of Rnd3, Hep3B cells expressing GFP-H2B and LifeAct-mRuby were treated with siRnd3 #1 and analyzed by time-lapse microscopy (Video ##SUPPL##6##1##). Entosis was initiated by the contact between the two cells, and then the internalization was marked by a high concentration of actin at the border between the two cells. The nucleus of the outer cell acquired a change in its shape due to the nucleus of the inner cell that pushes and deforms it. The final stage was characterized by the degradation of the inner cell, visible by the disappearance of the inner cell nuclei (Fig. ##FIG##2##3A##). Using correlative light-scanning electron microscopy (CLEM) allowing analysis of the same entotic cell by fluorescence microscopy and scanning EM (Fig. ##FIG##2##3B##), we found that the internalized cell has a rounded shape and seemed to be denser than the outer cell, with apparent cytoskeletal filaments. The inner cell appeared to be inside a large vacuole, the so-called entotic vacuole. The nucleus of the host cell acquired a crescent-like shape and was pushed to the cell periphery. The analysis of CLEM demonstrated that CIC structures mediated by the loss of Rnd3 are similar to entosis, showing a complete internalization of one cell inside another.</p>", "<p id=\"Par8\">To further examine whether the loss of Rnd3 was required in inner cells, outer cells, or both, we generated Hep3B cells with green and red nuclei by expression of GFP-H2B (G) and mCherry-H2B (R), respectively (Supplementary Fig. ##SUPPL##0##3A##). Hep3B-mcherry-H2B were then transfected with siRnd3#1 and co-cultured with untreated Hep3B-GFP-H2B (Fig. ##FIG##2##3C##; Supplementary Fig. ##SUPPL##0##3A, B##). 48 h after mixing, entotic cells were analyzed by fluorescence microscopy and counted for each combination (G in G, G in R, R in G, or R in R) (Fig. ##FIG##2##3C##). Among all entotic cells, the percentage of events between wild-type cells (G in G) is about 10%, and it reaches 50% between those transfected with siRnd3#1 (R in R). This result demonstrated that the decrease of Rnd3 expression is important in both the inner and outer cells for internalization. We noted that the percentage of the combination R in G is higher than that G in R combination suggesting that the loss of Rnd3 in the inner cell could be sufficient to induce its internalization in the wild-type outer cell.</p>", "<title>Entosis mediated by the loss of Rnd3 is dependent on an active Rho/ROCK pathway and a decrease of E-cadherin expression</title>", "<p id=\"Par9\">The Rho/ROCK pathway was implicated in the entosis induced after matrix detachment and starvation. As Rnd3 is a known antagonist of this signaling pathway, we analyzed whether the RhoA/ROCK pathway was required for cell engulfment after Rnd3 silencing. We first combined RhoA and Rnd3 knockdowns in HCC cells to assess RhoA involvement (Supplementary Fig. ##SUPPL##0##4A##). We demonstrated that the inhibition of RhoA reverses the effect of Rnd3 silencing by decreasing the percentage of entosis in Hep3B cells (Fig. ##FIG##3##4A##, left panel). We next used the Y-27632 compound to inhibit RhoA downstream effector, ROCK. We confirmed that Y-27632 treatment was able to inhibit MYPT1 phosphorylation induced upon Rnd3 KD in Hep3B cells (Supplementary Fig. ##SUPPL##0##4B##). Moreover, similar to RhoA KD, we found that this treatment decreases the percentage of entosis induced by the silencing of Rnd3 (Fig. ##FIG##3##4A##, right panel). Like Rnd3, p190RhoGAP (p190A) is a negative regulator of the Rho/ROCK pathway. HCC cells were transfected with siRNA targeting p190A (Supplementary Fig. ##SUPPL##0##4C##). Consistently the silencing of p190A increased the percentage of entotic cells in Hep3B and Huh7 cell lines, similar results to those obtained after Rnd3 inhibition (Fig. ##FIG##3##4B##). The inhibition of p190A expression in Hep3B and Huh7 cells in the presence of siRNA targeting Rnd3 did not modify the percentage of entotic cells compared to the condition where Rnd3 expression was inhibited alone, confirming that Rnd3 and p190A act in the same pathway for the induction of entosis (Supplementary Fig. ##SUPPL##0##5##). Altogether, these data demonstrate that entosis mediated by the silencing of Rnd3 is dependent on the RhoA/ROCK pathway in HCC cells. Therefore, deregulation of the Rho/ROCK pathway alters the ability of HCC cells to perform entosis.</p>", "<p id=\"Par10\">Entosis was described to be dependent on E-cadherin-based cell-cell junctions [##REF##25342560##14##]. However, we have previously found a decrease in E-cadherin expression upon silencing of Rnd3 in HCC cells [##REF##22234932##17##], which raises the question of the involvement of E-cadherin in the entosis observed here. We thus used E-cadherin-targeting siRNAs and analyzed their impact on entosis induced by the silencing of Rnd3 in HCC cells (Fig. ##FIG##3##4C##; Supplementary Fig. ##SUPPL##0##6A##). As previously published, E-cadherin expression decreased upon Rnd3 silencing in both cell types (Fig. ##FIG##3##4C##). Our results showed that inhibition of E-cadherin in Hep3B and Huh7 cells did not alter the percentage of entotic cells compared to Rnd3 knockdown alone (Fig. ##FIG##3##4C##), demonstrating that entosis mediated by Rnd3 loss is independent on E-cadherin in HCC cells. These data prompted us to explore the impact of E-cadherin silencing alone on entosis in HCC cells. Surprisingly, in contrast to breast cancer cells [##REF##25342558##21##], we found that the decrease of E-cadherin expression promoted entosis in HCC cells (Supplementary Fig. ##SUPPL##0##6B##). We then asked whether overexpression of E-cadherin could rescue the suppression effects of Rnd3 on entosis. To do so, we overexpressed a GFP tagged-version of E-Cadherin in Rnd3-silenced Hep3B cells (Supplementary Fig. ##SUPPL##0##7A##). Whereas overexpression of GFP alone did not alter Rnd3 KD-induced entosis, we observed that overexpression of E-Cadherin-GFP significantly decreased the percentage of entotic cells suggesting that both Rnd3 and E-cadherin down-expression are required to promote entosis in Hep3B cells (Supplementary Fig. ##SUPPL##0##7B##).</p>", "<title>Identification of LAMP1 as an effector of Rnd3 loss-mediated entosis</title>", "<p id=\"Par11\">In order to better characterize the entosis mediated by Rnd3 loss, we sought to apply a global approach. However, isolation of entotic cells is not an easy task, and we failed to do so using flow cytometry. We, therefore, applied a pipeline combining isolation of entotic cells by laser microdissection in a Hep3B-H2B-GFP cell population transfected with siRnd3 #1 (Fig. ##FIG##4##5A##, left panel) and mass spectrometry-based proteomic analysis according to our published protocol [##REF##29795195##22##]. We obtained 139 proteins that were underrepresented in the entotic proteome with an entosis/total proteome abundance ratio ≤0.5 and 55 enriched proteins with an entosis/total proteome abundance ratio ≥2 (Supplemental Tables ##SUPPL##1##1## and ##SUPPL##2##2##). We used Gene Ontology (GO) to classify the identified proteins (Supplemental Tables ##SUPPL##3##3## and ##SUPPL##4##4##). We highlighted that the underrepresented proteins were mainly involved in cellular metabolic processes with 63% (<italic>n</italic> = 88) of proteins associated with these biological processes. Proteins associated with intracellular anatomical structures and organelles were also found significantly decreased in entotic samples, suggesting a disassembly of cellular elements. Notably, we found a decrease in the abundance of Lamin-B1 and Lamin-B2, proteins of the nuclear envelope (Supplemental Table ##SUPPL##2##2##). In parallel, we found an enrichment of RNA binding proteins (42%, 23 proteins) including ribosomal and/or translation proteins (Supplemental Table ##SUPPL##3##3##). Network analysis using the Ingenuity Pathways Analysis platform (IPA, Qiagen) of the upregulated proteins further showed an implication of these candidates mainly in cell death/survival (Fig. ##FIG##4##5A##, right panel). We indeed found an overrepresentation of ER-associated proteins involved in stress response and/or folding of proteins such as Prdx2, HSPA5 (Bip), or protein disulfide-isomerases (P4HB, PDIA6), and PPIA, suggesting the recognition of cellular intrusion as a stressful cellular event (Supplemental Table ##SUPPL##1##1##). Network analysis using IPA also revealed a significantly altered functional network linked to the LAMP1 protein in entotic cells, where proteasome, chaperone, and ER stress proteins were also present (Supplemental Fig. ##SUPPL##0##8##A). LAMP1 is known to be implicated in the degradation of the inner cells by autophagy during the entotic mechanism. We confirmed the high expression of LAMP1 in the inner cells in the early stage of degradation (Fig. ##FIG##4##5B##). To validate the involvement of LAMP1 in entosis induced by Rnd3 silencing, we inhibited Rnd3 in the presence or absence of siRNA targeting LAMP1 in Hep3B and Huh7 cells (Fig. ##FIG##4##5C##). Whereas, as previously shown, the silencing of Rnd3 increased the percentage of entosis in both cell lines, this percentage returned to control levels when Rnd3 and LAMP1 were co-silenced (Fig. ##FIG##4##5D##). These data demonstrated the involvement of LAMP1 not only in the final stage of inner cell degradation but also in the overall mechanism of entosis induced by silencing of Rnd3. Interestingly, using RNAseq analysis of Hep3B cells (data not shown), <italic>LAMP1</italic> gene expression was found upregulated upon Rnd3 silencing (Supplemental Fig. ##SUPPL##0##8##B). These data were confirmed at the protein level in both Hep3B and Huh7 cells (Fig. ##FIG##4##5C##, bottom panel). This up-regulation of LAMP1 upon Rnd3 silencing appears to be dependent on RhoA as RhoA-KD rescued LAMP1 protein level (Supplemental Fig. ##SUPPL##0##8C##), suggesting that LAMP1 and RhoA pathway are related factors. We further analyzed whether the overexpression of LAMP1 is sufficient to promote entosis. The ectopic expression of LAMP1 in Hep3B cells only slightly increased the percentage of entotic cells, which remains very low compared to that obtained with Rnd3-silencing alone. However, the combination of overexpression of LAMP1 and Rnd3 inhibition significantly increased the entosis percentage compared to Rnd3-silencing alone (Fig. ##FIG##4##5E##). All these results suggest that Rnd3 favors entosis through LAMP1 expression and function. Thus, LAMP1 is an important factor to be considered all over the entotic mechanism.</p>", "<title>Entosis in patient tumors correlates with invasive features</title>", "<p id=\"Par12\">We next investigated the presence of entotic cells in HCC patient tissues. Using membrane (pan-keratin/β-catenin) and nucleus staining, we found few entotic cells in tumor samples (Fig. ##FIG##5##6A##). These events were rare, but visible in the tumor part, using an immunofluorescence approach on tissues. We then performed Rnd3 staining on HCC sections and looked for entotic cells in negative and positive Rnd3 areas. We noticed that the entotic cells are present in tumor sections with low expression of Rnd3 (Fig. ##FIG##5##6B##) supporting our in vitro data concerning the implication of Rnd3 downregulation in the entosis process. Using a cohort of 10 patient samples (Table ##TAB##0##1##), we then correlated the number of entotic cells with the characteristics of patient tumors. Similar to the low level of Rnd3 expression [##REF##22234932##17##], we found that the number of entotic cells was significantly higher in tumors with satellite nodules or vascular invasion, which are indicative of local invasion of HCC (Fig. ##FIG##5##6C##). All these results suggest the association between the entotic events, loss of Rnd3 and tumor progression.</p>" ]

[ "<title>Discussion</title>", "<p id=\"Par13\">In this study, we characterized entosis in liver cancer cells, which resembles to that largely described in the literature in breast cancer cells [##REF##27048820##23##]. We demonstrated that entosis could be induced by matrix detachment and nutrient deprivation in HCC cells. In addition, we identified that entosis can be efficiently triggered in HCC cells through the loss of <italic>RND3</italic> expression. Others and we previously described the downregulation of this gene expression in human HCC [##REF##22234932##17##, ##REF##22829315##18##, ##REF##22213123##24##]. Moreover, Rnd3 loss was associated with liver cancer cell proliferation, invasion, chemoresistance, and senescence [##REF##22234932##17##–##REF##35256752##19##, ##REF##27213590##25##]. We revealed here that Rnd3 loss also favors entosis in HCC tumors. Recently, Rnd3 has been involved in the p53-dependent entotic mechanism driven by transient mitotic arrest in breast cancer cells [##REF##33110215##26##]. However, in contrast to our results, it was found that Rnd3 expression under the regulation of p53 was essential to promote entosis in breast cancer cells. Thus, the Rnd3 role appears to be different in breast and liver cancers. In this line, Rnd3 expression does not seem to be regulated by p53 in HCC cells as no correlation between <italic>RND3</italic> expression and TP53 mutations could be established in HCC human samples [##REF##22234932##17##].</p>", "<p id=\"Par14\">Rnd3 is a negative regulator of the RhoA/ROCK pathway. Herein, we confirmed the strong dependency of the entotic mechanism on this pathway. We found that entosis induced by the loss of <italic>RND3</italic> is reduced after the inhibition of the Rho/ROCK pathway. Mechanically, entosis depends on actomyosin contractility regulated by the RhoA GTPase activity in the invading cell. It was described that outer cells have lower actomyosin contractility, consistent with a more deformable status compared to invading cells. In our co-culture experiments, the percentage of wild-type cells inside Rnd3-negative cells was very low, suggesting that the penetrating cells require an increase in contractility. However, in liver cancer cells, the loss of Rnd3 is important in both the inner and outer cells, as a maximum of entosis was found when Rnd3 expression is decreased in both the inner and the outer cells. This result is consistent with the existence of a required threshold for Rnd3 expression to favor entosis. Thus, a knockdown recreating an imbalance in contractility, in agreement with the literature, the more rigid cell could be internalized by a more deformable cell [##REF##25342558##21##]. Consistently, we observed a high concentration of actin and the presence of fibers in the inner cells by electronic microscopy.</p>", "<p id=\"Par15\">Several studies have shown the importance of E-cadherin in entosis induced by matrix detachment and starvation [##REF##18045538##10##, ##REF##28683313##11##, ##REF##25342558##21##]. Interestingly, we showed that entosis induced by the silencing of Rnd3 occurs independently of E-cadherin, and is even altered upon E-Cadherin re-expression. Consistent with our published results showing the decrease of E-cadherin expression upon silencing of Rnd3 in HCC cells [##REF##22234932##17##], we even found that loss of E-cadherin favors entosis in Hep3B and Huh7 cells. Thus, entosis induced by the loss of Rnd3 could be dependent on other transmembrane proteins that may trigger cell-cell interaction.</p>", "<p id=\"Par16\">A proteomic analysis of laser-captured entotic cells allowed us to identify an overrepresentation of LAMP1 and ER-associated proteins involved in stress response and/or folding of proteins. Statistical analyses by Gene Set Enrichment Analysis did not reveal significant enrichment of canonical pathways of cellular degradation or stress. However, a functional environment of the LAMP1 protein is significantly modified with key proteins of these pathways. It is possible that entosis pathways involve only certain proteins known to be implicated in degradation and stress and mobilize a specific protein set. Our data confirm that entotic cells express highly LAMP1, a protein described in the literature as implicated in the final stage of entosis, i.e., degradation of inner cells. Indeed, the membrane of the entotic vacuole surrounding the internalized cells recruits LAMP1 and LC3 independently of the autophagosome formation. The fusion between the entotic vacuole and the lysosome of outer cells allows to degrade the inner cells [##REF##22002674##27##]. Herein, we highlighted that <italic>LAMP1</italic> expression is upregulated at the mRNA and protein levels upon Rnd3 silencing, in a RhoA-dependent manner. Our results are consistent with an involvement of LAMP1 in inner cell degradation after internalization induced by the loss of Rnd3. However, we found that LAMP1 is involved not only in degradation but also in the whole mechanism of entosis. LAMP1 could be among the different signals necessary for internalization. Although LAMP1 primarily resides at the lysosomal membrane, its localization to cell surface expression was described to mediate cell-cell adhesion and favor melanoma cell invasion [##REF##25614122##28##].</p>", "<p id=\"Par17\">The degradation of entotic cells could serve to feed cells, supporting the survival of the outer tumor cells in stress conditions. Entosis may also contribute to tumorigenesis by inducing aneuploidy [##REF##21336303##29##]. Others and we previously described Rnd3 as a potential metastasis suppressor as its down-expression was associated with poor prognosis in HCC patients. The results presented here support the hypothesis that loss of Rnd3 could participate in HCC progression through the promotion of entosis. Accordingly, entosis in HCC patient tissues correlates with the presence of satellite nodules and vascular invasion. Thus, targeting the entotic mechanism may be valuable as a novel therapeutic avenue to impair HCC progression.</p>" ]

[]

[ "<p id=\"Par1\">Entosis is a process that leads to the formation of cell-in-cell structures commonly found in cancers. Here, we identified entosis in hepatocellular carcinoma and the loss of Rnd3 (also known as RhoE) as an efficient inducer of this mechanism. We characterized the different stages and the molecular regulators of entosis induced after Rnd3 silencing. We demonstrated that this process depends on the RhoA/ROCK pathway, but not on E-cadherin. The proteomic profiling of entotic cells allowed us to identify LAMP1 as a protein upregulated by Rnd3 silencing and implicated not only in the degradation final stage of entosis, but also in the full mechanism. Moreover, we found a positive correlation between the presence of entotic cells and the metastatic potential of tumors in human patient samples. Altogether, these data suggest the involvement of entosis in liver tumor progression and highlight a new perspective for entosis analysis in medicine research as a novel therapeutic target.</p>", "<title>Subject terms</title>" ]

[ "<title>Supplementary information</title>", "<p>\n\n\n\n\n\n\n\n\n\n</p>" ]

[ "<title>Supplementary information</title>", "<p>The online version contains supplementary material available at 10.1038/s41419-024-06420-3.</p>", "<title>Acknowledgements</title>", "<p>We thank N. Dugot-Senant from the Histopathology platform (TBMCore, UMS005, Bordeaux). Flow cytometry, hypoxia, and qRT-PCR experiments were respectively performed at the FACSility, CellOxia, and OneCell facilities (TBMCore, UMS005, Bordeaux). Electron and photonic microscopies were done in the Bordeaux Imaging Center, a service unit of the CNRS-INSERM and Bordeaux University, a member of the national infrastructure France Bio-Imaging, with the help of Dr. Etienne Gontier and Isabelle Svahn. We thank Dr. Peter Coopman (Montpellier) for providing us with the E-Cadherin-GFP construct. We express our gratitude to Paul Divet, Cyril Dourthe, and Pr. Clotilde Billottet (BRIC, Bordeaux) for their help, with, respectively, evaluation of entotic cells, bubble plot representation of proteomic data, and collecting patient tumor data. SB was supported by Bordeaux University and a postdoctoral fellowship from the Fondation ARC/Région Nouvelle-Aquitaine. This work was supported by grants from the Fondation ARC, La Ligue contre le Cancer (comité regional, Gironde), and from Institut National du Cancer (PLBIO-INCa2014-182) to VM. VM’s team was supported by La Fondation pour la Recherche Médicale “Equipe labellisée 2018”.</p>", "<title>Author contributions</title>", "<p>Study design: SB, VM. Generation of experimental data: SB, LD, C Dantzer, SDT, JWD. Analysis and interpretation of data: SB, LD, AAR. Providing biological samples from HCC patients: PBS, C Desdouets. Writing of the manuscript: SB, VM. Critical reading of the manuscript: LD, JWD, AAR, FS. Supervision of the project: VM.</p>", "<title>Data availability</title>", "<p>The mass spectrometry proteomic data have been deposited to the ProteomeXchange consortium with the dataset identifier PXD043640. Additional data are available from the corresponding author upon reasonable request.</p>", "<title>Competing interests</title>", "<p id=\"Par27\">The authors declare no competing interests.</p>", "<title>Ethics approval and consent to participate</title>", "<p id=\"Par28\">All animal procedures were approved by the ethical committee of the University of Bordeaux according to the French government regulations. To minimize animal use, we retrospectively analyze samples from a previous study [##REF##35256752##19##]. Human samples were acquired from resected livers of patients with HCC treated in Bordeaux from 1992 to 2005. Written informed consent was obtained from all patients according to French legislation.</p>" ]

[ "<fig id=\"Fig1\"><label>Fig. 1</label><caption><title>Liver cancer cells perform entosis upon starvation and matrix detachment.</title><p><bold>A</bold> Representative Hep3B entotic cell characterized using confocal microscopy by the nucleus deformation showed with the DAPI staining (blue) and high concentration of actin with the phalloidin (green). Beta-catenin staining (red) was used to confirm that the inner cell is inside the outer cell. Scale bar, 15 µm. <bold>B</bold> Nutrient starvation induces entosis as determined by immunofluorescence in Hep3B and Huh7 cells stained for nucleus (blue), actin (green), and beta-catenin (red). Scale bar, 15 µm. <bold>C</bold> Graphs show the quantification of entosis induced by starvation in Hep3B, Huh7, Huh6, and HepG2 cells. <bold>D</bold> Matrix detachment favors entosis in liver cancer cells. Hep3B, Huh7, Huh6, and HepG2 cells were cultured in non-adherent conditions and analyzed for their ability to form entosis. Graphs show the quantification of entosis induced by suspension culture conditions. <bold>B</bold>–<bold>D</bold> MCF-7 breast cancer cells were used as a positive control. Error bars: SD of three or more independent experiments. Significance was determined with the Mann–Whitney <italic>U</italic>-test.</p></caption></fig>", "<fig id=\"Fig2\"><label>Fig. 2</label><caption><title>Rnd3 silencing promotes entosis in hepatocellular carcinoma cells.</title><p><bold>A</bold> The inhibition of Rnd3 expression was performed in Hep3B (upper panel) and in Huh7 (lower panel) cells using either a shRNA targeting Rnd3 (ShRnd3) inducible with doxycycline treatment and compared to a control shRNA (ShCtrl) or three independent siRNA targeting Rnd3 SiRnd3 #1, SiRnd3 #2, SiRnd3 #3 and SiCtrl as control. Rnd3 knock-down was assessed by Western blot, β-actin is the loading control. Graphs show the quantification of entotic cells upon Rnd3 KD. Error bars: SD of three or more independent experiments. Significance was determined with the Mann–Whitney <italic>U</italic>-test. <bold>B</bold>, <bold>C</bold> Representative of a single (<bold>B</bold>) or a double (<bold>C</bold>) event of entosis after silencing of Rnd3 expression in Hep3B cells with siRnd3 (<bold>B</bold>, upper panel) or with shRnd3 (<bold>B</bold>, lower panel; and <bold>C</bold>). Cells were labeled with DAPI, phalloidin, and beta-catenin antibody to visualize the nucleus, actin, and membrane, respectively. Scale bar, 15 µm. <bold>D</bold> Entosis was visualized in fixed tumor tissues from mice subcutaneously inoculated with Hep3B-shCtrl (<italic>n</italic> = 3) or Hep3B-shRnd3 (<italic>n</italic> = 5) cells. Tumor sections were stained with hematoxylin-eosin (HE) and the percentage of entotic cells was quantified. Scale bar, 50 µm. Error bars: SD of three or more independent samples. Significance was determined with the Mann–Whitney <italic>U</italic>-test.</p></caption></fig>", "<fig id=\"Fig3\"><label>Fig. 3</label><caption><title>Characterization of entosis mediated by the loss of Rnd3 expression.</title><p><bold>A</bold> Dynamics of entosis upon Rnd3 silencing. Hep3B cell lines were transduced with H2B-GFP (yellow) and LifeAct-mRuby (cyan) to mark the nucleus and the actin, respectively. Cells were transfected with siRNA targeting Rnd3 and spinning-disk microscopy analysis was done over 48 h. The gallery corresponds to Video ##SUPPL##6##1##. Time is in hours. Three stages were defined: (i) cell contact, (ii) internalization, arrows show the internalization of the inner binuclear cell 21 h after contact between cells and (iii) degradation of the inner cell; the inner cell degradation starts about 2 h after internalization and the cell degradation takes almost 10 h to be completed. Scale bar, 28 µm. <bold>B</bold> Analysis of entosis by correlative light-scanning electron microscopy. Entotic cells characterized by the nucleus (H2B-GFP, green) deformation and high concentration of actin (LifeAct-mRuby, red) in the inner cell were chosen using immunofluorescence microscopy (scale bar, 100 µm) and then the same event was analyzed by scanning electron microscopy. Entotic cells were taken with two magnifications 1000x (scale bar, 5 µm) and 2000x (scale bar, 10 µm). <bold>C</bold> Two populations of Hep3B cells were mixed, one population transduced with H2B-GFP, and another one transduced with H2B-RFP and transfected with siRNA targeting Rnd3. 48 h after mixing, the quantification of entotic cells was performed by evaluating the percentage of different combinations, Green in Green (wild-type in wild-type cells), Green in Red (Wild-type in siRnd3-transfected cells), Red in Green (siRnd3-transfected in wild-type) and Red in Red (siRnd3-transfected in siRnd3-transfected cells). Error bars: SD of three or more independent experiments. Significance was determined with the Mann–Whitney <italic>U</italic>-test. Examples of events counted after mixing of Rnd3-silencing RFP cells and wild-type GFP cells; G in G = Green in Green, G in R = Green in Red, R in G = Red in Green, R in R = Red in Red. Scale bar, 15 µm.</p></caption></fig>", "<fig id=\"Fig4\"><label>Fig. 4</label><caption><title>Entosis induced by Rnd3 silencing is dependent on Rho/ROCK pathway and occurs independently of E-cadherin expression.</title><p><bold>A</bold> Hep3B cells were transfected with siRNA targeting Rnd3 (siRnd3 #1) in the presence or absence of siRNA targeting RhoA (siRhoA) (left panel), or after treatment or not with Y-27632 (right panel) and the percentage of entosis was evaluated. <bold>B</bold> Hep3B or Huh7 cell lines were treated with siRNA targeting p190RhoGAP-A (siP190A #1 and siP190A #2). The Western blots show the inhibition of p190A and the graphs indicate the percentage of entotic cells. <bold>C</bold> Hep3B or Huh7 cells were transfected with siRNA targeting Rnd3 in the presence or absence of siRNA targeting E-cadherin (siEcad #1 or siEcad #2). The inhibition of Rnd3 and E-cadherin was confirmed by Western blot and the percentage of entotic cells was evaluated. Error bars: SD of three or more independent experiments. Significance was determined with the Mann–Whitney <italic>U-</italic>test.</p></caption></fig>", "<fig id=\"Fig5\"><label>Fig. 5</label><caption><title>Proteomic analysis and identification of LAMP1 as an effector of Rnd3 loss-mediated entosis.</title><p><bold>A</bold> Hep3B-H2B-GFP cells were labeled with SILAC and transfected with siRNA targeting Rnd3 (siRnd3 #1) and the entotic cells were microdissected and analyzed using mass spectrometry. Upregulated proteins were analyzed using ingenuity pathway analysis (IPA) and represented in a bubble plot showing pathways implicated. The bubble size represents the number of upregulated proteins involved in a specific pathway. <bold>B</bold> Hep3B entotic cells stained to visualize nuclei (Blue), actin (Green), and LAMP1 (Red). Scale bar, 15 µm. <bold>C</bold> Hep3B or Huh7 cells were transfected with siRNA targeting Rnd3 in the presence or absence of siRNA targeting LAMP1 (siLAMP1 #1). The inhibition was confirmed by Western blot. The graphs show the quantification of Rnd3 and LAMP1 knock-downs in Huh7 and Hep3B cells. <bold>D</bold> Evaluation of entotic cells in Hep3B and Huh7 cell lines after Rnd3 and LAMP1 silencing. <bold>E</bold> Overexpression of LAMP1-mcherry in Hep3B cells after silencing of Rnd3. pcDNA is the control vector. Representative images of the overexpression showed by the mcherry staining and the quantification of entotic cells are shown in the right panel of (<bold>E</bold>). Scale bar, 15 µm. Error bars: SD of three or more independent experiments. Significance was determined with the Mann-Whitney <italic>U</italic>-test.</p></caption></fig>", "<fig id=\"Fig6\"><label>Fig. 6</label><caption><title>Entosis in patient tumors correlates with invasive features.</title><p><bold>A</bold> Representative entotic cells in HCC tumor tissues stained with DAPI (Blue) and pan-keratin/bate-catenin (Green). Scale bar, 50 µm. <bold>B</bold> HCC tissues stained using hematoxylin staining, and Rnd3 antibody. Scale bar: 2.5 mm. Entotic cells were evaluated in the positive (red) or negative (gray) Rnd3 areas. Scale bar: 50 µm. <bold>C</bold> Correlation between the number of entotic cells and the presence (Yes) or not (No) of satellite nodules or vascular invasion. <italic>N</italic> = 10 patient samples, significance was determined with the Mann–Whitney <italic>U-</italic>test.</p></caption></fig>" ]

[ "<table-wrap id=\"Tab1\"><label>Table 1</label><caption><p>Clinical and tumoral characteristics of patients.</p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th>Clinical characteristics</th><th>Value</th></tr></thead><tbody><tr><td>Age (mean ± standard deviation)</td><td>69.5 ± 9.1 years</td></tr><tr><td>  Gender (male)</td><td>100%</td></tr><tr><td>  HBV infection</td><td>0%</td></tr><tr><td>  HCV infection</td><td>44.4%</td></tr><tr><td>  Cirrhosis</td><td>44.4%</td></tr><tr><td><bold>Tumor characteristics</bold></td><td><bold>Value</bold></td></tr><tr><td> Diameter (mean ± standard deviation)</td><td>56.63 ± 39.76</td></tr><tr><td>  Satellite nodule</td><td>50%</td></tr><tr><td>  Vascular invasion</td><td>70%</td></tr></tbody></table></table-wrap>" ]

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[ "<supplementary-material content-type=\"local-data\" id=\"MOESM1\"></supplementary-material>", "<supplementary-material content-type=\"local-data\" id=\"MOESM2\"></supplementary-material>", "<supplementary-material content-type=\"local-data\" id=\"MOESM3\"></supplementary-material>", "<supplementary-material content-type=\"local-data\" id=\"MOESM4\"></supplementary-material>", "<supplementary-material content-type=\"local-data\" id=\"MOESM5\"></supplementary-material>", "<supplementary-material content-type=\"local-data\" id=\"MOESM6\"></supplementary-material>", "<supplementary-material content-type=\"local-data\" id=\"MOESM7\"></supplementary-material>", "<supplementary-material content-type=\"local-data\" id=\"MOESM8\"></supplementary-material>", "<supplementary-material content-type=\"local-data\" id=\"MOESM9\"></supplementary-material>" ]

[ "<fn-group><fn><p>Edited by Professor Stephen Tait</p></fn><fn><p><bold>Publisher’s note</bold> Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.</p></fn></fn-group>" ]

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[ "<media xlink:href=\"41419_2024_6420_MOESM1_ESM.pdf\"><caption><p>Supplemental Figures</p></caption></media>", "<media xlink:href=\"41419_2024_6420_MOESM2_ESM.xlsx\"><caption><p>Supplemental Table 1</p></caption></media>", "<media xlink:href=\"41419_2024_6420_MOESM3_ESM.xlsx\"><caption><p>Supplemental Table 2</p></caption></media>", "<media xlink:href=\"41419_2024_6420_MOESM4_ESM.xlsx\"><caption><p>Supplemental Table 3</p></caption></media>", "<media xlink:href=\"41419_2024_6420_MOESM5_ESM.xlsx\"><caption><p>Supplemental Table 4</p></caption></media>", "<media xlink:href=\"41419_2024_6420_MOESM6_ESM.xlsx\"><caption><p>Supplemental Table 5</p></caption></media>", "<media xlink:href=\"41419_2024_6420_MOESM7_ESM.avi\"><caption><p>The video shows the entosis mechanism upon Rnd3 silencing in Hep3B cells</p></caption></media>", "<media xlink:href=\"41419_2024_6420_MOESM8_ESM.docx\"><caption><p>Supplemental legends</p></caption></media>", "<media xlink:href=\"41419_2024_6420_MOESM9_ESM.pdf\"><caption><p>Original Data File</p></caption></media>" ]

{ "acronym": [], "definition": [] }

[ "<title>Introduction</title>", "<p id=\"Par3\">Ferroelectricity, characterized by a remanent and switchable polarization, just experiences the 100th anniversary of its first discovery in Rochelle salt in 1920^{##UREF##0##1##,##UREF##1##2##}. The research and application of ferroelectrics were first limited to bulk ceramics, such as in the domains of actuators, piezoelectric transducers, and pyroelectric detectors. Since 1989^{##REF##17755995##3##}, with the development of thin-films, the switchable polarization by an electric field greater than the coercive field has attracted amounts of effort for the military and commercial applications of ferroelectric films in non-volatile memories^{##REF##17303745##4##–##REF##31894148##15##}. The high switching speed at sub-nanosecond and the excellent endurance of over 10¹² make the ferroelectric memory a competitive candidate for replacing the current flash memory products which suffers from slow speed of ~10⁻³s and limited endurance of ~10⁴ cycles^{##UREF##6##16##}. Furthermore, the programmability of polarization renders the ferroelectric memory viable for in-memory computing^{##REF##37770677##17##–##REF##30698976##24##}, which offers huge advantages in terms of computing power, latency, and energy efficiency by performing parallel multiply-accumulate (MAC) calculations directly with Ohm’s law for multiplication and Kirchhoff’s law for accumulation^{##REF##31996818##25##–##UREF##13##28##}. Recently, several breakthroughs in materials technology, such as Si-compatible hafnia-based ferroelectrics with 3D integration^{##UREF##14##29##–##REF##33858991##33##}, low-weight molecular ferroelectrics^{##REF##28729511##34##–##REF##30872522##36##} and polar topology in ferroelectrics^{##REF##25883317##37##–##REF##33674493##40##}, have further generated unprecedented enthusiasm for the development of ferroelectric memories.</p>", "<p id=\"Par4\">Currently, there are three basic ferroelectric memory structures namely capacitors^{##REF##17755995##3##,##REF##35653464##41##}, tunnel junctions^{##UREF##15##42##,##REF##27143121##43##} and field-effect transistors^{##REF##33397907##18##}. The capacitor-type ferroelectric random access memory (FeRAM) shows outstanding endurance of over 10¹⁵ and is now commercially available in a cell structure of one transistor and one capacitor (1T1C)^{##REF##35653464##41##} (Fig. ##FIG##0##1a##). However, the charge reading of polarization reversal in the capacitor-type FeRAM not only asks for a large area for detectable charges but is also itself destructive and requires a rewrite process after each polarization-reversal reading operation, which hinders the direct analog MAC calculations for in-memory computing. To overcome this destructive readout, the ferroelectric tunnel junction (FTJ) varies its conductance through modulation of the tunnel barrier height by polarization reversal^{##REF##25056141##44##,##UREF##16##45##}. The simple structure of metal/ferroelectric/metal in FTJ allows a high memory density, but the few-nanometers-thick ferroelectric films for direct tunneling suffer from poor endurance (Fig. ##FIG##0##1b##)^{##UREF##16##45##}. The alternative three-terminal ferroelectric field-effect transistor (FeFET), where the gate dielectric is replaced by a ferroelectric layer in a standard metal-oxide-semiconductor field-effect transistor (MOSFET), encodes its memory states by modulating the conductance of the channel in the semiconductor by the gate polarization^{##UREF##17##46##}. However, the lack of an epitaxial template results in mesoscopically disordered and polycrystalline ferroelectric on the semiconductor channel which intrinsically leads to uncontrolled device-to-device variation in nanoscale FeFET devices^{##UREF##17##46##} (Fig. ##FIG##0##1c##). This variation issue makes the commercialization of FeFET challenging. On the other hand, a much simple two-terminal memristor is preferred for a high-density hardware-level in-memory computing^{##UREF##18##47##,##REF##30894760##48##}. Therefore, it is extremely appealing to develop a novel memory technology that not only exploits all merits of fast switching speed, good endurance, non-volatile states, and low operation energy consumption of ferroelectrics, but also combines a simple structure and non-destructive readout mode for high-density memory and computing applications.</p>", "<p id=\"Par5\">In this work, we propose a two-terminal ferroelectric fin diode (FFD) in which a ferroelectric capacitor and a fin-like semiconductor channel are combined to share both top and bottom electrodes (Fig. ##FIG##0##1d##). This FDD memory absorbs merits of both non-destructive conductance read mode as in FTJs and long endurance as in FeRAM while it allows ferroelectric directly on electrode other than semiconductor, eliminating the intrinsic source of device-to-device variation in a traditional FeFET. Technology computer-aided design (TCAD) simulations demonstrate that the Schottky barrier at the semiconductor/electrode interface twists the electric field distribution, adding a transverse electric field to the vertical semiconductor channel of the FFD. The remanent polarization plastically-reversible by the distorted electric field permits the FFD both digital and analog memory functionalities. Such a FFD operates using different ferroelectric materials (namely the organic P(VDF-TrFE) polymer and inorganic industrially used PZT compounds) which emphasizes its universal character. Compared to the vast family of the current nonvolatile memories, this FFD shows exhaustive properties of prior memories with high performances such as an endurance of over 10¹⁰ cycles, an ON/OFF ratio of ~10², a feature size of 30 nm, an operating energy of ~ 20 fJ and an operation speed of 100 ns. Benefiting from the simplicity for fabricating this FFD and its self-rectification ratio of ~ 10⁴, a passive crossbar array with 1.6 k units is constituted and used to demonstrate the in-memory computing of a simple pattern classification task. The high device-to-device uniformity is reflected by a small σ/μ value of ~0.023 for positive coercive voltage and ~0.019 for negative coercive voltage using a Gaussian distribution. This work opens a new avenue for efficient memories and emerging-computing architectures for big data and artificial intelligence applications.</p>" ]

[ "<title>Methods</title>", "<title>P(VDF-TrFE) based FFD</title>", "<p id=\"Par25\">The structure and fabrication process of the designed FFD are orderly presented in Fig. ##FIG##1##2a##. First of all, the striped 100-nm Pt films sputtered by DC magnetron sputtering instrument on the cleaned SiO₂/Si were used as the gate and drain electrodes. The working power, chamber pressure, and Ar flow rate were 250 W, 0.3 Pa, and 50 sccm, respectively (step 1); Secondly, the P(VDF-TrFE) (70:30 mol%) ferroelectric polymer was dissolved in the diethyl carbonate with 2.5 wt%. The P(VDF-TrFE) is formed by spin coating the polymers in an initially homogeneous evaporative solution, followed by a thermal anneal at 135 °C for 4 h to facilitate the ferroelectric <italic>β</italic>-phase growth as shown in step 2; Thirdly, the ~20-nm Al was deposited on the 120-nm ferroelectric layer by thermal evaporation (the deposition rate is 10 Å/s); The crossbar structure between top electrodes and bottom electrodes are accomplished by switching the metal mask before depositing the electrodes. Next, the excess organic polymer (P(VDF-TrFE)) was removed by oxygen ion etching, and the P(VDF-TrFE) films covered by Al was retained. Finally, assisted by a metal mask, the 30-nm ZnO was sputtered onto the crossbar network using RF magnetron sputtering instrument at a chamber pressure of 0.8 pa, Ar flow rate of 50 sccm and working power of 80 W.</p>", "<title>PZT based FFD</title>", "<p id=\"Par26\">The structure and fabrication process of the designed FFD based on PZT are orderly presented in Fig. S##SUPPL##0##5##. Assisted by a metal mask, the ~300-nm Al electrodes with a periodic striped structure were deposited on the purchased PZT (200 nm)/Pt (50 nm)/SiO₂/Si substrate by thermal evaporation (the deposition rate is 10 Å/s). The excess PZT was removed by argon ion etching, and the PZT covered by Al electrodes was retained. Finally, assisted by a metal mask, the 30-nm ZnO was sputtered to cover the electrodes boundary using RF magnetron sputtering instrument with a chamber pressure of 0.8 pa, Ar flow rate of 50 sccm, working power of 80 W.</p>", "<title>The reversed FFD</title>", "<p id=\"Par27\">The fabrication process of the designed reversed FFD based on P(VDF-TrFE) are orderly presented in Fig. S##SUPPL##0##11##. Firstly, assisted by lithography, the 100 nm-thick Al electrodes were deposited on the purchased SiO₂/Si substrate by thermal evaporation (the deposition rate is 10 Å/s) (step1). Secondly, the P(VDF-TrFE) (70:30 mol%) ferroelectric polymer was dissolved in the diethyl carbonate with 2.5 wt%. The P(VDF-TrFE) ( ~ 360 nm, 6 layers) is formed by spin coating the polymers in an initially homogeneous evaporative solution, followed by a thermal anneal at 135 °C for 4 h to facilitate the ferroelectric β-phase growth as shown in step 2. The Pt electrode was prepared by means of photolithographic alignment as shown in Fig. S##SUPPL##0##11b–d## and step 3. The Pt films are sputtered by DC magnetron sputtering instrument, and the working power, chamber pressure, and Ar flow rate were 250 W, 0.3 Pa, and 50 sccm, respectively. Thirdly, the excess P(VDF-TrFE) was removed by oxygen ion etching, and the P(VDF-TrFE) covered by Pt electrodes was retained (step 4). Finally, assisted by a metal mask, the 30-nm ZnO was sputtered to cover the electrodes boundary using RF magnetron sputtering instrument with a chamber pressure of 0.8 pa, Ar flow rate of 50 sccm, working power of 80 W (step 5).</p>", "<title>A FFD nano device</title>", "<p id=\"Par28\">The fabrication process of the designed FFD nano device based on P(VDF-TrFE) are orderly presented in Fig. S##SUPPL##0##12##. First of all, the striped 100-nm Pt films sputtered by DC magnetron sputtering instrument on the cleaned SiO₂/Si were used as the gate and drain electrodes (step 1). The working power, chamber pressure, and Ar flow rate were 250 W, 0.3 Pa, and 50 sccm, respectively; Secondly, the P(VDF-TrFE) (70:30 mol%) ferroelectric polymer was dissolved in the diethyl carbonate with 2.5 wt%. The 60 nm-thick P(VDF-TrFE) (~60 nm, 1 layer) is formed by spin coating polymers in an initially homogeneous evaporative solution, followed by a thermal anneal at 135 °C for 4 h to facilitate the ferroelectric <italic>β</italic>-phase growth as shown in step 2; Thirdly, assisted by lithography, the ~1-µm photoresist (PMMA) with a periodic striped structure were prepared on the P(VDF-TrFE) (60 nm)/Pt (50 nm)/SiO₂/Si substrate by spin coating and a thermal anneal (200 °C for 2 h) (step 3). Next, assisted by a metal mask, the 30-nm Al was deposited onto the crossbar network using thermal evaporation instrument (the deposition rate is 10 Å/s) (step 4). The excess Al was removed by argon ion etching, and the Al upright and close to the PMMA was retained as shown in step 5. Next, the remaining P(VDF-TrFE) was cleaned up by oxygen ion etching (step 6). Finally, assisted by a metal mask, the 30-nm ZnO was sputtered onto the crossbar network using RF magnetron sputtering instrument at a chamber pressure of 0.8 pa, Ar flow rate of 50 sccm and working power of 80 W (step 7).</p>", "<title>A 40 × 40 passive crossbar array</title>", "<p id=\"Par29\">Both the bottom electrode and the top electrode are assisted by striped mask plates arranged with 40 columns. The detailed preparation process is the same as that in the P(VDF-TrFE)-based FFD devices.</p>", "<title>STEM measurements</title>", "<p id=\"Par30\">The Pt/P(VDF-TrFE)/Al/ZnO STEM sample was fabricated using a focused ion beam machine (Hellios G4 UX). Structural characterization was conducted using a JEM- ARM200 scanning transmission electron microscope equipped with an ASCOR probe corrector operating at an accelerating voltage of 300 kV.</p>", "<title>Electrical measurements</title>", "<p id=\"Par31\">The <italic>I–V</italic> curves, programing speed and retention property of the FFD were measured, unless noted otherwise, under ambient conditions using a Keithley 4200A-SCS parameter analyzer with remote preamplifiers. The read voltage of 5 V (0.3 V) is used to obtain the retention characteristic of the FFD based on P(VDF-TrFE) (PZT). A HP4194A impedance analyzer with an ac amplitude of 0.04 V is used to measure the capacitance versus frequency (<italic>C</italic>-<italic>f</italic>) at various temperatures. The <italic>P</italic>–<italic>V</italic> hysteresis loops and the fatigue characteristic of the FFD were investigated using ferroelectric analyzer (TF Analyzer 3000). In fatigue measurements, the fatigue pulses with an amplitude of 25 V (10 V) and a frequency of 10 kHz (2 MHz) were used to reverse the ferroelectric domains in P(VDF-TrFE) (PZT) based devices. The checked <italic>P</italic>–<italic>V</italic> curves were measured using a triangular wave with the same amplitude and frequency. A quasi static <italic>I</italic>–<italic>V</italic> sweeping was performed after ferroelectric fatigue texts with different cycles to obtain the cycles-dependent <italic>I</italic>–<italic>V</italic> curves. For all temperature-dependent measurements, the temperature was changed at a rate of 1 K/min. Each temperature was maintained for 2 mins using a computer-controlled cryostat (MMR Tech, Inc.) before the following electrical measurements. The electrical performance of the FFD array was achieved by a multi-channel array test system (PXIe-4631) controlled by a labview program.</p>", "<title>TCAD simulation</title>", "<p id=\"Par32\">Silvaco TCAD simulator has been used to simulate the device we proposed. The structure is created by a 2D process simulation editor (Athena), where the size parameters of device modeling are consistent with the experiments. The contact between Pt and ZnO was defined as a Schottky contact with a barrier height of 1.02 eV, and the contact between Al and ZnO was set as an ohmic contact by default. A doping concentration in the ZnO is set as 10²⁰cm^-3. The dielectric constant and the band gap of P(VDF-TrFE) are set as 10.7 and 6 eV, respectively. Physical models including mobility, recombination, and fermi-Dirac are employed in the simulation.</p>" ]

[ "<title>Results</title>", "<title>Structure of two-terminal FFD</title>", "<p id=\"Par6\">Ferroelectric thin films sandwiched between a top electrode and a bottom electrode are used to support a sidewall semiconductor, forming a vertical fin-like structure where the channel length defined by the thickness of the ferroelectric can be easily controlled at the nanoscale. As will be discussed below, by inducing a Schottky contact, for example between the bottom electrode and the fin-like semiconductor channel/bridge, the bottom electrode then plays a dual role i.e.; both the drain and the gate of a typical FeFET. While the fin-like semiconductor channel contributes most currents, the ferroelectric domain switching rearranges the electric field configuration at ferroelectric/semiconductor interface and results in a resistive switching. This combination structure of sandwiched ferroelectric and fin-like Schottky sidewall is morphologically called as ferroelectric fin diode (FFD).</p>", "<p id=\"Par7\">The FFD device can be easily fabricated using a common photolithography technique. Figure ##FIG##1##2a## presents the fabrication process of a FFD with the ferroelectric P(VDF-TrFE) and a n-type ZnO channel (please refer to the Methods section for details). The final heterostructure is examined by cross-sectional scanning transmission electron microscopy (STEM) (Fig. ##FIG##1##2b, c##) that shows a clear stair-like device structure. According to the images of the energy dispersive X-ray spectrometry (EDS) mapping of Si, Pt, F, Al, and Zn elements, the expected FFD device schematized in Fig. ##FIG##1##2a5## is indeed confirmed. The thickness of Pt/P(VDF-TrFE)/Al/ZnO layers is 150/100/20/30 nm, respectively. Note that the oblique ZnO channel with a deflection angle of 30° to the vertical may arise from the nonlinear etching process. Since the ZnO semiconductor and the P(VDF-TrFE) ferroelectric layers play an important role in this novel device, their topography and crystal structure are checked by atomic force microscope (AFM) and X-ray diffraction (XRD), respectively. Deposited on Pt electrode using magnetron sputtering method, the ZnO films show AFM images (inset of Fig. ##FIG##1##2d##) with no macroscopic defects, pinholes or visible cracks and a root mean square roughness of 0.52 nm, which suggests that uniform and smooth ZnO films are obtained. The P(VDF-TrFE) films on Pt electrode have also a good quality and uniformity with a root mean square roughness of 2.01 nm (inset of Fig. ##FIG##1##2f##). The XRD patterns show a Bragg peak at 34.1° for ZnO (Fig. ##FIG##1##2d##) and 19.6° for P(VDF-TrFE) films (Fig. ##FIG##1##2f##), which are specific to the wurtzite hexagonal phase (Fig. ##FIG##1##2e##) and ferroelectric β phase^{##UREF##19##49##}, respectively. The ferroelectricity of P(VDF-TrFE) films is further confirmed by the butterfly-shaped loop of the piezoelectric force microscope (PFM) amplitude and 180°-reversed loop of the PFM phase (Fig. ##FIG##1##2g##). The reproducible ferroelectric domain switching is attested by the out-of-plane PFM images showing alternate of up and down polarization domains. Indeed, in Fig. ##FIG##1##2h, i##, stripe domains characteristics of ferroelectricity are written by alternately applying a +20 V and −20 V bias to the Pt electrode while the PFM tip is grounded. A clear 180° phase shift is observed between adjacent domains (Fig. ##FIG##1##2i##) with depressed amplitude value at domain walls (Fig. ##FIG##1##2h##).</p>", "<title>Polarization-driven resistive switching</title>", "<p id=\"Par8\">By quasi-statically sweeping a voltage from −20 V to +20 V on the bottom Pt electrode, the semi-logarithmic current versus voltage (<italic>I</italic>–<italic>V</italic>) curve shows an anticlockwise hysteresis with an ON/OFF ratio of ~10² (Fig. ##FIG##2##3a##). Moreover, the <italic>I–V</italic> curve of the FFD further demonstrates that the structure of this new device can significantly reduce the reverse (<italic>V</italic> &lt; 0) current (Fig. S##SUPPL##0##1a##). To demonstrate whether the observed resistive switching is dominated by the polarization reversal, ferroelectric-phase-dependent resistive switching experiments are performed. As shown in the temperature-dependent dielectric permittivity of our P(VDF-TrFE) films (Fig. ##FIG##2##3b##), the ferroelectric-to-paraelectric phase transition occurs at a Curie point <italic>T</italic>_c of 380 K on heating while it drops to <italic>T</italic>_c = 355 K on cooling, which agrees well with previous reports^{##UREF##20##50##}. This thermal hysteresis of 25 K indicates that the ferroelectric transition of this copolymer is of first order. This phase transition is further attested by directly measuring the polarization versus voltage (<italic>P</italic>–<italic>V</italic>) loops. As shown in top panels in Fig. ##FIG##2##3c##, <italic>P</italic>–<italic>V</italic> loops, measured at 100 Hz, exhibit clear hysteresis with a remanent polarization (<italic>P</italic>_r) of ~7.5 μC/cm² at room temperature. As the temperature increases, the <italic>P</italic>–<italic>V</italic> loop is altered at 373 K (both <italic>P</italic>_r and the coercive voltage <italic>V</italic>_c reduce) and disappears at 393 K i.e., above <italic>T</italic>_c = 380 K evidenced on heating with temperature dependent dielectric permittivity, resulting in a linear behavior characteristic of a paraelectric state. When cooling back to 300 K, the <italic>P</italic>–<italic>V</italic> hysteresis loop recovers. Likely, the resistive switching (bottom panels in Fig. ##FIG##2##3c##) behaves concomitantly with the <italic>P</italic>–<italic>V</italic> loop and vanishes when the P(VDF-TrFE) films are in the paraelectric phase. Note that the operating voltage can be significantly decreased by lowering the thickness of the ferroelectric layer and/or selecting ferroelectric materials with small coercive fields (Fig. S##SUPPL##0##1b–f##) to meet the CMOS technology requirements. Note that a tradeoff issue between operation voltage and conduction current, that is, a thinner film for lower operation voltage results in higher current, is usually suffered in 2-terminal resistive devices. In this view, it will be more energy efficient to decrease the operation voltage by using ferroelectric materials that possesses a much small coercive field. The direct correlation between ferroelectricity and resistive switching strongly supports that the resistive switching in the FFD device is dominated by the polarization reversal.</p>", "<p id=\"Par9\">To eliminate the influence of other possible mechanism such as defects or charging effect, we fabricate a referenced device sample in which the ferroelectric layer is replaced by a non-ferroelectric aluminum oxide (Al₂O₃) (Fig. S##SUPPL##0##2a##). The thickness of Al₂O₃ and ZnO layer in the referenced device is 27 nm and 30 nm respectively (Fig. S##SUPPL##0##3##). The resistive switching in the FFD devices shows counterclockwise <italic>I</italic>–<italic>V</italic> curves in the first quadrant. However, when ferroelectric layer is replaced by dielectric of Al₂O₃, a small clockwise <italic>I</italic>–<italic>V</italic> curve is observed (Fig. S##SUPPL##0##2b##). This clockwise hysteresis can be understanded by the charge trapping effect^{##UREF##21##51##} or the presence of impurities in the ZnO or interface. The opposite hysteresis in ferroelectric device with that in Al₂O₃ dielectric confirms that the counterclockwise hysteresis in our ferroelectric device is not dominated by the charge trapping effect or impurities effect. A FeFET device is then fabricated and shows that the n-type ZnO channel can be effectively tuned by ferroelectric polarization (Fig. S##SUPPL##0##4##), implying that the charge trapping or impurities effect are much weak and negligible compared with the field effect by ferroelectric polarization.</p>", "<p id=\"Par10\">It is worth mentioning that the <italic>I</italic>–<italic>V</italic> curves are asymmetric with a self-rectifying ratio of ~ 10⁴ (Fig. ##FIG##2##3a##). This self-rectifying characteristic results from the Schottky contact between the Pt electrode and n-type ZnO films. The work function of Al, Cu, Au and Pt metals is obtained from ultraviolet photoelectron spectroscopy (UPS) to be 3.80 eV, 4.63 eV, 4.86 eV and 5.04 eV respectively, while an affinity of 4.02 eV is obtained in ZnO semiconductor (Fig. S##SUPPL##0##5##). With a clean interface contact, a higher work function than the affinity enables a Schottky contact at metal/ZnO interface, while a lower work function results in an ohmic one. The Pt (5.04 eV)/ZnO (4.02 eV) Schottky barrier and Al (3.80 eV)/ZnO (4.02 eV) ohmic contact are confirmed by their rectified and liner <italic>I</italic>–<italic>V</italic> curves respectively (Fig. S##SUPPL##0##6##). The Schottky barrier of the Pt/ZnO interface is further confirmed by the linear relationship between ln(<italic>I</italic>/<italic>T</italic>²) and <italic>V</italic>^1/2 (Fig. ##FIG##2##3d##). By measuring the temperature-dependent slope of this linear relationship, a barrier height value of ~ 0.85 eV is obtained and agrees well with the energy band alignment when considering the energy level of the ZnO channel and Pt and Al electrodes (Fig. ##FIG##2##3e##).</p>", "<p id=\"Par11\">To better understand how the polarization affects the electronic transport, a TCAD simulation is performed. Interestingly, because of the existence of Pt/ZnO Schottky barrier, there is lateral electric field pointing forward (backward) to the vertical ZnO channel when a voltage of +20 V (-20 V) is applied to the Pt electrode (Fig. ##FIG##2##3##g and j). For example, when a voltage of -20 V is applied to the Pt electrode, the negatively-biased Schottky barrier suffers most voltage and results in curved electric potential in the ferroelectric layer (Fig. ##FIG##2##3j##). The electric field, defined by minus the gradient of the electric potential, is represented with black arrows in Fig. ##FIG##2##3##g and j. The lateral electric field, estimated to be of ~ 140 MV/m (near the Al/ZnO interface) or -200 MV/m (at most ZnO channel region) under the +20 V or -20 V bias, is high enough to reverse the polarization in ferroelectric copolymers^{##UREF##8##21##,##UREF##9##22##}. Figure ##FIG##2##3##h, k show the phenomenological model of polarization state after removing the voltage bias. By sweeping a negative voltage that is larger than the coercive voltage (-<italic>V</italic>_c ~ -15 V) on the Pt electrode, the lateral electric field makes the polarization obliquely backward to the vertical ZnO channel. The remanent polarization with net negative bound charges at the ferroelectric/channel interface electrically weakens the carrier density of the n-type channel^{##UREF##3##12##,##REF##37770677##17##} and leads to a thicker Schottky barrier (Fig. ##FIG##2##3i##), resulting in the low conductance state (LCS). These bound charges can be switched away by aligning the polarization upward, even forward to the vertical ZnO channel near the Al/ZnO interface, when applying a positive voltage higher than +<italic>V</italic>_c (Fig. ##FIG##2##3h##), which permits the Schottky barrier to recover its original state (Fig. ##FIG##2##3f##) and gives the high conductance state (HCS). This model explains well the observed asymmetric resistive switching (Fig. ##FIG##2##3a##) because of the non-uniform and amplitude-dependent lateral field in the ferroelectric layer (Fig. ##FIG##2##3##g and j) due to the vertical ZnO channel in FFD devices. This contrasts with the symmetric behavior in more traditional FeFET (Fig. S##SUPPL##0##4##) with expectedly vertical uniform field.</p>", "<title>Robustness performance and universality of FFD</title>", "<p id=\"Par12\">The fatigue behavior is a very important parameter, which is related to the service life of memory devices and the prospect of their potential industrialization. The ferroelectric and electrical fatigue characteristics of FFDs have been studied systematically. Figure ##FIG##3##4a, b## (up panels) and S##SUPPL##0##7a–d## demonstrate the performance of a typical P(VDF-TrFE)-based FFD. The remanent polarization (<italic>P</italic>_r⁺ and <italic>P</italic>_r^-) and the corresponding <italic>P</italic>–<italic>V</italic> loops as a function of measured cycles are presented in Fig. ##FIG##3##4a## (up panel) and Fig. S##SUPPL##0##7c## respectively. Interestingly, accompanying with the robust polarization reversal of up to 10⁷ cycles, the device shows stable resistive switching throughout (Fig. ##FIG##3##4b## (up panel) and Fig. S##SUPPL##0##7d##). It is reasonable that the conductance sustains a better endurance than <italic>P</italic>_r. The contribution to the resistive switching significantly involves the lateral region around the ferroelectric/semiconductor interface where the up-left switching (Fig. ##FIG##2##3##g, h and ##FIG##2##j, k##) is more robust as it does not involve nucleation like in the up-down switching in the parallel capacitor region between bottom Pt electrode and top Al electrode, where <italic>P</italic>_r is measured.</p>", "<p id=\"Par13\">To highlight the universal aspect of both these robust performances and the device structure itself, an industrialized inorganic ferroelectric, i.e. lead zirconate titanate (PZT) on a 4-inch Si wafer (Fig. S##SUPPL##0##8##), is used to fabricate the FFD (Fig. S##SUPPL##0##9##) and put in parallel to the one made with the organic P(VDF-TrFE) ferroelectric. As expected, an endurance of up to 10¹⁰ cycles in both polarization reversal and resistive switching is obtained in a typical PZT-based FFD (Fig. ##FIG##3##4a, b## (bottom panels) and S##SUPPL##0##7e, f##). The same self-rectifying and counterclockwise characteristics of the resistive hysteresis in such devices based on both inorganic PZT and organic P(VDF-TrFE) indicates that the as-designed FFDs are already at high technology readiness level.</p>", "<p id=\"Par14\">Figure ##FIG##3##4c## shows the retention property of FFDs. Benefiting from the nonvolatility of ferroelectrics, both HCS and LCS show slight degradation, if any, in a period of ~10⁴s in either P(VDF-TrFE) (top panel) or PZT (bottom panel) based devices. In addition, the programming speed of FFDs is studied (Fig. S##SUPPL##0##10a, b##). The device is firstly written to LCS by applying a -28 V (-8 V) pulse with a width as long as 100 ms to the P(VDF-TrFE) (PZT) device. Then +35 V ( + 10 V) pulses with an increased width from 1 μs to 1 ms (100 ns to 100 μs) are used to write the P(VDF-TrFE) (PZT) device to HCS. Each programing voltage is followed by a read voltage pulse of +3.0 V (+0.2 V) with a width of 100 ms to check the conductance state. The achieved speed of P(VDF-TrFE)- and PZT-based devices is found to be 1 μs and 100 ns, respectively.</p>", "<p id=\"Par15\">The essential width of Al top electrode needed for the resistive properties is the limit of feature size in our devices. A reversed structure is used to verify the scalability of our memory devices (Fig. S##SUPPL##0##11##). This reversed structure allows the design of across (I in Fig. S##SUPPL##0##11d, e##), just touch (inset of Fig. ##FIG##3##4d and II## in Fig. S##SUPPL##0##11d, e##) and separate (III in Fig. S##SUPPL##0##11d, e##) between vertical projection of Al electrode and Pt electrode where the just touching case is equivalent to infinitely reducing the width of Al. It is found that the resistive switching exists well when Al electrode and Pt electrode have across (Fig. S##SUPPL##0##11f##) and even just “zero” overlap (Fig. ##FIG##3##4d##) within our photolithography error, implying a nanoscale scalability in our memory devices. Inspired by this, we further fabricated nano devices where the width of Al top electrode is only 30 nm (Fig. ##FIG##3##4e## and Fig. S##SUPPL##0##12##). As shown in Fig. ##FIG##3##4f##, the resistive switching survives well in the 30-nm nano device.</p>", "<p id=\"Par16\">The energy consumption for each write operation of the ferroelectric fin diode is evaluated using the formula: <italic>E</italic> = UIt. For example, when an operating voltage with an amplitude of 20 V is used for a reversed FFD based on P(VDF-TrFE) (Fig. S##SUPPL##0##11f##), the energy consumption is estimated to be ~1 fJ and ~ 20 fJ for a reset operation and a set operation respectively.</p>", "<title>Device uniformity and analog storage for in-memory computing</title>", "<p id=\"Par17\">In a traditional ferroelectric transistor (Fig. ##FIG##0##1c##), the ferroelectric layer has to be deposited on the semiconductor layer other than a seed electrode. The lack of an epitaxial template results in poor ferroelectric quality that is usually mesoscopically disordered and polycrystalline^{##UREF##17##46##}. This leads to uncontrolled device-to-device variation in nanoscale devices. On the contrary, in a FFD (Fig. ##FIG##0##1d##), the ferroelectric layer is directly deposited on a seed electrode (for example, Pt electrode for PZT films) and covered by a top electrode. This sandwiching metal/ferroelectric/metal structure, indeed the same as that of a commercial ferroelectric capacitor, possesses a high ferroelectric quality and good uniformity even in nanoscale. Note that during the following semiconductor deposition, the ferroelectric layer is protected by the top electrode. Thus, a good uniformity is expected in ferroelectric fin diodes.</p>", "<p id=\"Par18\">A 4 × 4 passive array with P(VDF-TrFE)-based fin diodes (Fig. ##FIG##4##5a##) is fabricated on a Si/SiO₂ (300 nm thick) wafer. The current cross-talk issue in this passive crossbar array can be effectively eliminated by the self-rectifying characteristic in each device unit^{##REF##30894760##48##}. The device uniformity is checked by performing <italic>I</italic>–<italic>V</italic> curves in all 16 device units (Fig. ##FIG##4##5b##) which shows that the dispersion of the response from one device unit to another is very small. The device-to-device variation is evaluated using the ratio of σ/μ in a Gaussian distribution function, where μ and σ are the mean value and standard deviation of the current, respectively. A good uniformity is found with a σ/μ value of ~0.18 for HCS and ~0.08 for LCS, respectively (Fig. ##FIG##4##5c##).</p>", "<p id=\"Par19\">The electronic transport of a typical device unit is comprehensively measured. Five successive <italic>I</italic>–<italic>V</italic> curves between -20 V and +20 V are collected and show little deviation (Fig. S##SUPPL##0##13##). Ferroelectric is well known for its nonvolatility and analog programing characteristic^{##UREF##22##52##}. Figure ##FIG##4##5d## demonstrates the analog switching of polarization with multiple <italic>P</italic>_r in amplitude-variant <italic>P</italic>–<italic>V</italic> curves of the FFD based on P(VDF-TrFE). Accordingly, by changing the sweeping amplitude of the positive voltage from 8 V to 20 V with a step of 1 V, clearly separated and nested hysteresis loops are observed (Fig. ##FIG##4##5e##), implying multiple intermediate conductance states in the FFD. This is caused by the multiple intermediate polarization states in the ferroelectric layer as shown in Fig. ##FIG##4##5d##. The nonvolatility of these intermediate conductance states can be reflected by these open hysteresis loops. Six distinguishing states are also chosen to measure their retention characteristic and no degeneration is observed in a time scale of &gt;100 s for all six states (Fig. ##FIG##4##5f##). These persistent multiple conductance states provide the essential ingredients to emulate the so-called synaptic plasticity, plastic changes of synaptic weights, that is at the heart of the learning processes. The quasi-linear conductance potentiation (strengthening) and depression (weakening) with 25 discrete states can be achieved (Fig. ##FIG##4##5g##) by repeatedly applying a voltage pulse sequence of 25 positive voltage pulses (+25 V/10 μs) followed by 25 negative voltage pulses (-18 V/10 μs), respectively (Fig. S##SUPPL##0##14##). This analog behavior of conductance resembles the long-term potentiation/depression (LTP/D) process in synaptic devices.</p>", "<p id=\"Par20\">An artificial neural network constituted by crossbar array based on FFDs for digits recognition is simulated using the experimentally measured conductance states. Both the small image version (8 × 8 pixels) of handwritten digits from the “Optical Recognition of Handwritten Digits” (ORHD) dataset and the large image version (28 × 28 pixels) of hand-written digits from the “Modified National Institute of Standards and Technology” (MNIST) dataset are used to perform the back-propagation simulation. For the large (small) image version, a multilayer perceptron (MLP) neural network with 784 (64) input neurons, 300 (30) hidden neurons, and 10 output neurons is utilized by using Cross-Sim simulator (Fig. ##FIG##4##5h##)^{##UREF##23##53##}. In the simulation, the crossbar, regarded as part of a “neural core”, performs vector-matrix multiplication and outer-product update operations. Generally, the performance of a neural network is greatly influenced by the controllability (e.g., nonlinearity and write noise) of synaptic devices, which can be quantitatively analyzed using the probability distribution of the conductance change (Δ<italic>G</italic>) induced by a write operation. The plots of Δ<italic>G</italic> versus initial Δ<italic>G</italic>_{<italic>0</italic>}, derived from the cyclic conductance potentiation and depression, are presented in insets of Fig. ##FIG##4##5i## and Fig. S##SUPPL##0##15##, respectively. For small digits, the classification accuracy approaches 82.6% within the second training epoch and approaches 92.6% after 16 training epochs, which is close to 96.4% the theoretical limit of an ideal numeric training for small digits (Fig. ##FIG##4##5i##). For large digits, our simulations show a classification accuracy of 82.9% (Fig. S##SUPPL##0##15##). These results demonstrate that the analog characteristic of FFDs has the potential for in-memory computing applications.</p>", "<p id=\"Par21\">To further confirm the in-memory computing application of our FFD devices, we have fabricated a passive crossbar array with 1.6 k units (Fig. ##FIG##5##6a, b##). The ferroelectricity is checked by transient <italic>I</italic>–<italic>V</italic> curves in 200 random-selected devices where transient current peaks correspond to ferroelectric coercive voltages (Fig. S##SUPPL##0##16##). A uniformity with a σ/μ value of ~0.023 for positive coercive voltage and ~0.019 for negative coercive voltage in a Gaussian distribution is obtained (Fig. ##FIG##5##6c##). The coercive voltage and diode characteristic together enable the intended programing in the FFD passive crossbar array (see supplementary note ##SUPPL##0##1##, Fig. S##SUPPL##0##17##). The resistive switching in 400 devices at cross points of alternate rows and alternate columns is carefully checked one by one. Stirringly, all units show resistive switching (Fig. S##SUPPL##0##18##). These ON and OFF conductance states can be distinguished clearly (Fig. S##SUPPL##0##19##) and the ON/OFF ratio at the read voltage of 3 V fluctuates around 10 (Fig. ##FIG##5##6d##). We believe that the decay of ON/OFF ratio results from the sneak path issue while the fluctuation of electrical performances is due to the immature fabrication techniques.</p>", "<p id=\"Par22\">A simple pattern recognition task based the 1.6 k passive crossbar array is then demonstrated. Figure ##FIG##5##6e–g## display the schematic diagram for clarifying three 4 × 4-pixel images using a hardware-based artificial neural network (ANN). The images of “L”, “u” and “n” are trained using the Manhattan update rule (Fig. S##SUPPL##0##20##) to obtain weights guiding to program the hardware ANN for recognizing the three letters. A region with 16 × 6 units are chosen to demonstrate the pattern recognition task where each weight is encoded by conductance difference between neighboring pair units (Fig. ##FIG##5##6f, g##). Fig. S##SUPPL##0##21## shows the programming process and final conductance distribution in the 16 × 6 hardware ANN. During the inference progress, the content pixels are encoded into a voltage pulse with a width of 10 ms and vacant pixels into a voltage pulse with a width of 0 ms, that is absence of a voltage pulse. The patterns recognition is implemented successfully by collecting these column currents where the current difference (<italic>I</italic>_i⁺-<italic>I</italic>_i^-) standing for the target image show much large value than others (Fig. ##FIG##5##6h–j##). Images with noise are also tested to confirm the tolerance of our hardware ANN (Fig. S##SUPPL##0##22##). As summarized in Fig. ##FIG##5##6k##, when zero, one, and two pixels are randomly flipped, the recognition accuracy is 100%, 50%, and 37.5%, respectively. The pattern recognition task mentioned above was automatically measured using multi-channel array test system as shown in Fig. S##SUPPL##0##23##.</p>", "<title>Comparison with state-of-the-art non-volatile memory devices</title>", "<p id=\"Par23\">Key parameters of state-of-of-the-art non-volatile memories including Not And logic gates (NAND Flash), phase change memory (PCM), FeRAM, resistive RAM (RRAM) and Magnetic RAM (MRAM) have been collected in recent review reports^{##REF##35653464##41##}. A comparison of the performance with other memories is summarized in Table ##SUPPL##0##S1##. Among the vast family of nonvolatile memories, our FFD memory cumulatively demonstrates very high performances with an endurance of over 10¹⁰ cycles, a self-rectification ratio of ~10⁴, an operation speed of 100 ns, a feature size of 30 nm and cell size of 4 F², and an ultralow energy consumption of ~20 fJ. The simple two-terminal structure and the high self-rectification ratio of ~10⁴ permit to efficiently design passive crossbar arrays for high-density memories as well as emerging in-memory computing application.</p>" ]

[ "<title>Discussion</title>", "<p id=\"Par24\">A robust ferroelectric-based non-volatile memory with a novel FFD structure is proposed as a new building block for future electronic circuit architectures. The device absorbs merits of non-destructive read mode with resistive switching as in FTJs and long endurance as in FeRAM while it allows ferroelectric directly on electrode other than semiconductor, eliminating the intrinsic source of device-to-device variation in a traditional FeFET. Both digital and analog memory functionalities can be achieved in such device. It can operate with different ferroelectric materials illustrating its universal character. It demonstrates superior performances when compared to state-of-the-art nonvolatile memories with an endurance of over 10¹⁰ cycles, an ON/OFF ratio of ~10², a feature size of 30 nm and cell size of 4 F², an operating energy of ~20 fJ and an operation speed of 100 ns. Analog storage using multiple conductance states is demonstrated showing such a device is suitable for synaptic learning plasticity. The simple two-terminal structure and its self-rectifying ratio of ~ 10⁴ permit a passive crossbar array with 1.6 k units in which in-memory computing of a simple pattern classification task is accomplished. The high device-to-device uniformity is reflected by a small σ/μ value of ~0.023 for positive coercive voltage and ~0.019 for negative coercive voltage using a Gaussian distribution. This work paves a way to use this new electronic unit for designing passive crossbar arrays for either memories or in-memory computing applications.</p>" ]

[]

[ "<p id=\"Par1\">Among today’s nonvolatile memories, ferroelectric-based capacitors, tunnel junctions and field-effect transistors (FET) are already industrially integrated and/or intensively investigated to improve their performances. Concurrently, because of the tremendous development of artificial intelligence and big-data issues, there is an urgent need to realize high-density crossbar arrays, a prerequisite for the future of memories and emerging computing algorithms. Here, a two-terminal ferroelectric fin diode (FFD) in which a ferroelectric capacitor and a fin-like semiconductor channel are combined to share both top and bottom electrodes is designed. Such a device not only shows both digital and analog memory functionalities but is also robust and universal as it works using two very different ferroelectric materials. When compared to all current nonvolatile memories, it cumulatively demonstrates an endurance up to 10¹⁰ cycles, an ON/OFF ratio of ~10², a feature size of 30 nm, an operating energy of ~20 fJ and an operation speed of 100 ns. Beyond these superior performances, the simple two-terminal structure and their self-rectifying ratio of ~ 10⁴ permit to consider them as new electronic building blocks for designing passive crossbar arrays which are crucial for the future in-memory computing.</p>", "<p id=\"Par2\">Designing efficient high-density crossbar arrays are nowadays highly demanded for many artificial intelligence applications. Here, the authors propose a two-terminal ferroelectric fin diode non-volatile memory in which a ferroelectric capacitor and a fin-like semiconductor channel are combined to share both top and bottom electrodes with high performance and easy fabrication process</p>", "<title>Subject terms</title>" ]

[ "<title>Supplementary information</title>", "<p>\n\n\n</p>" ]

[ "<title>Supplementary information</title>", "<p>The online version contains supplementary material available at 10.1038/s41467-024-44759-5.</p>", "<title>Acknowledgements</title>", "<p>B.T. would like to thank fundings of National Key Research and Development Program of China (No. 2021YFA1200700), National Natural Science Foundation of China (No. T2222025, 62174053 and 61804055), Open Research Projects of Zhejiang Lab (2021MD0AB03), Shanghai Science and Technology Innovation Action Plan (No. 19JC1416700, 21JC1402000 and 21520714100) and the Fundamental Research Funds for the Central Universities.</p>", "<title>Author contributions</title>", "<p>B.T. conceived the concept. B.T., Q.Z. and C.D. supervised the research. G.F. fabricated the devices. G.F., L.C. and B.T. performed the electrical and piezoelectric force microscope measurements. G.F., S.X. and Q.B. performed the ultraviolet photoelectron spectroscopy. B.T., G.F., X.Z., J.L., K.S. and J.J. performed the simulations and experimental classification tasks. Z.Y. and R.H. performed the STEM. X.L., B.T. and W.T. performed the TCAD. Q.Z., F.Y., H.P., X.T., X.G., J.W., A.J., B.D., J.C. and C.D. advised on the experiments and data analysis. G.F., B.T. and B.D. co-write the manuscript. All authors discussed the results and revised the manuscript.</p>", "<title>Peer review</title>", "<title>Peer review information</title>", "<p id=\"Par33\"><italic>Nature Communications</italic> thanks the anonymous reviewers for their contribution to the peer review of this work. A peer review file is available.</p>", "<title>Data availability</title>", "<p>The data that support the findings of this study are available from the corresponding author upon reasonable request.</p>", "<title>Code availability</title>", "<p>The codes that support the findings of this study are available from the corresponding author upon reasonable request.</p>", "<title>Competing interests</title>", "<p id=\"Par34\">The authors declare no competing interests.</p>" ]

[ "<fig id=\"Fig1\"><label>Fig. 1</label><caption><title>Comparison of ferroelectric memory performances.</title><p><bold>a</bold>–<bold>d</bold> Memory performances in the ferroelectric random access memory capacitor (FeRAM) (<bold>a</bold>), ferroelectric tunnel junctions (FTJ) (<bold>b</bold>), ferroelectric field-effect transistor (FeFET) (<bold>c</bold>) and the proposed ferroelectric fin diode (FFD) (<bold>d</bold>). The parameters in FeRAM, FTJ and FeFET devices are obtained from refs. ^{##REF##35653464##41##,##UREF##16##45##,##UREF##17##46##} respectively.</p></caption></fig>", "<fig id=\"Fig2\"><label>Fig. 2</label><caption><title>Structure of an FFD device.</title><p><bold>a</bold> The fabrication process of the FFD based on P(VDF-TrFE). <bold>b</bold> Scanning transmission electron microscopy (STEM) imaging of the device with Pt/ P(VDF-TrFE)/Al/ZnO layers. <bold>c</bold> Energy dispersive X-ray spectroscopy (EDS) mappings of Si, Pt, F, Al, and Zn elements in the device cross profile. <bold>d</bold> The X-Ray Diffraction (XRD) pattern of ZnO films. Inset: The Atomic Force Microscope (AFM) topography image of ZnO films on Pt electrode. Scale bar: 500 nm (<bold>e</bold>) The schematic of ZnO structure with wurtzite hexagonal phase. <bold>f</bold> The XRD pattern of P(VDF-TrFE) films on Pt electrodes. Inset: the AFM topography of P(VDF-TrEF). <bold>g</bold> The butterfly-shaped piezoelectric force microscope (PFM) amplitude loop and 180°-reversed PFM phase loop of P(VDF-TrFE) films on Pt electrodes. <bold>h</bold>, <bold>i</bold> The PFM amplitude image (<bold>h</bold>) and phase image (<bold>i</bold>) after scanning the grounded PFM tip on the P(VDF-TrFE) films with a + 20 V and −20 V bias alternately applied to the Pt electrode. Scale bar in (<bold>f</bold>, <bold>h</bold>, <bold>i</bold>) has the same value of 3 μm.</p></caption></fig>", "<fig id=\"Fig3\"><label>Fig. 3</label><caption><title>Polarization-driven resistive switching.</title><p><bold>a</bold> The quasi-static current versus voltage (<italic>I–V</italic>) curve in a typical FFD. <bold>b</bold> The temperature-dependence of the dielectric permittivity. <bold>c</bold> The polarization versus voltage <italic>(P-V)</italic> curves at a frequency of 100 Hz and quasi-static <italic>I–V</italic> curves of the FFD at the same temperatures of 300 K, 333 K, 353 K, 373 K, 393 K and back to 300 K, respectively. <bold>d</bold> ln(J/T²) versus 1/T plots at the voltage bias of 5 V. The linear fit gives a Schottky barrier height of ~0.85 eV. Inset: ln(J/T²) versus V^1/2 plots at temperature of 333 K, 343 K, 353 K, 363 K, 373 K, respectively. <bold>e</bold> The schematic diagram of the electronic band structure of Pt electrode, ZnO semiconductor and Al electrode. <bold>f</bold>–<bold>k</bold> The TCAD-obtained energy band alignment of Pt/ZnO/Al structure after poling by a 20 V bias (<bold>f</bold>) and −20 V bias (<bold>i</bold>). The TCAD-obtained distribution of electric potential in the FFD under poling of 20 V bias (<bold>g</bold>) and −20 V bias (<bold>j</bold>). The scheme of polarization vectors alignment in the FFD after poling by 20 V bias (<bold>h</bold>) and −20 V bias (<bold>k</bold>).</p></caption></fig>", "<fig id=\"Fig4\"><label>Fig. 4</label><caption><title>Robust memory cycling and universality.</title><p><bold>a</bold>, <bold>b</bold> The evolution of the remanent polarization (<italic>P</italic>_r) (<bold>a</bold>) and conductance (<bold>b</bold>) with endurance cycles for a typical FFD based on P(VDF-TrFE) (top panels) and PZT (bottom panels). <bold>c</bold> The retention characteristics for a typical FFD based on P(VDF-TrFE) (top panel) and PZT (bottom panel). Insets give the scheme of the device structure. <bold>d</bold> The quasi-static <italic>I</italic>–<italic>V</italic> curves of a reversed FFD device with just “zero” overlapped electrode pairs of Al and Pt. During the electrical measurements, the Al electrode is grounded as in other devices. <bold>e</bold> STEM imaging of a FFD nano device where the width of Al top electrode is only 30 nm and the EDS mappings of Pt, Al, Zn and F elements in the device cross profile. Inset shows the schematic diagram of the FFD nano device. <bold>f</bold> The quasi-static <italic>I</italic>–<italic>V</italic> curves of a typical FFD nano device.</p></caption></fig>", "<fig id=\"Fig5\"><label>Fig. 5</label><caption><title>The uniformity and analog storage for in-memory computing.</title><p><bold>a</bold> The schematic diagram of a 4 × 4 passive crossbar array constituted by FFDs on a Si/SiO₂ substrate. Insets show the schemes (left) and the optical image (right) of a unit device. <bold>b</bold> The overlay plots of quasi-static <italic>I</italic>–<italic>V</italic> curves of all 16 devices. <bold>c</bold> The distribution of LCS and HCS conductance states for all 16 devices. The red lines indicate the fit by gaussian distribution. <bold>d</bold> The multiple <italic>P</italic>–<italic>V</italic> curves of the FFD based on P(VDF-TrFE) with a constant sweeping frequency of 100 Hz and a continuously increased amplitude from 13 V to 25 V with a step of 1 V. <bold>e</bold> The multiple quasi-static <italic>I</italic>–<italic>V</italic> curves in a logarithmic coordinate obtained by gradually changing the sweeping amplitude from 8 V to 20 V with a step of 1 V. Inset gives the <italic>I</italic>–<italic>V</italic> curves in a linear coordinate. <bold>f</bold> The retention characteristics of six intermediate conductance states. <bold>g</bold> The evolution of conductance potentiation and depression (i.e. LTP and LTD) during the poling with a repeated voltage pulse sequence made by 25 positive voltage pulses (+25 V/10 μs) followed by 25 negative voltage pulses (−18 V/10 μs). A read voltage pulse (+3 V/10 μs) follows each poling voltage pulse to obtain the non-volatile conductance. <bold>h</bold> A schematic diagram of a three-layer artificial neural network. <bold>i</bold> Evolution of the accuracy with training epochs achieved by simulating the FFDs-based artificial neural network for recognizing handwritten digits with 8 × 8 pixels. Insets show the probability distributions of the change in conductance (Δ<italic>G</italic>) induced by a write operation versus initial conductance at potentiation process.</p></caption></fig>", "<fig id=\"Fig6\"><label>Fig. 6</label><caption><title>In-memory computing of a pattern classification task within a passive crossbar array.</title><p><bold>a</bold>, <bold>b</bold> The optical image (<bold>a</bold>) and schematic diagram (<bold>b</bold>) of a 40 × 40 passive crossbar array based on FFD devices. <bold>c</bold> The distribution of negative coercive voltage and positive coercive voltage obtained from transient <italic>I</italic>–<italic>V</italic> curves in Fig. S##SUPPL##0##16##. The red lines indicate the fit by gaussian distribution. <bold>d</bold> Statistics of resistive switching ratio at a read voltage of 3 V in the 40 × 40 passive crossbar array. <bold>e</bold> A schematic diagram of an artificial neural network (ANN) for clarifying three 4 × 4-pixel images. <bold>f</bold> The schematic diagram indicating how the hardware to implement the ANN in (<bold>e</bold>). <bold>g</bold> The schematic diagram indicating how the image are encoded to input into the hardware ANN. Each weight is encoded by conductance difference between neighboring pair units. <bold>h</bold>–<bold>j</bold> The current difference (<italic>I</italic>_i = <italic>I</italic>_i⁺-<italic>I</italic>_i^-) collected from columns when images of “L” (<bold>h</bold>) “u” (<bold>i</bold>) and “n” (<bold>j</bold>) are inputted into the 16 × 6 hardware ANN. The patterns recognition is implemented with the value of <italic>I</italic>_i standing for the target <italic>i</italic> image being the biggest. <bold>k</bold> When zero, one, and two pixels are randomly flipped, and recognition accuracy is 100%, 50%, and 37.5%, respectively.</p></caption></fig>" ]

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[ "<inline-formula id=\"IEq1\"><alternatives><tex-math id=\"M1\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\,{{{{{\\rm{f}}}}}}\\left({{{{{\\rm{G}}}}}}\\right)=\\frac{1}{\\sqrt{2\\pi }\\sigma }{e}^{-\\frac{{(G-\\mu )}^{2}}{2{\\sigma }^{2}}}$$\\end{document}</tex-math><mml:math id=\"M2\"><mml:mspace width=\"0.25em\"/><mml:mi mathvariant=\"normal\">f</mml:mi><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:mi mathvariant=\"normal\">G</mml:mi></mml:mrow></mml:mfenced><mml:mo>=</mml:mo><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:msqrt><mml:mrow><mml:mn>2</mml:mn><mml:mi>π</mml:mi></mml:mrow></mml:msqrt><mml:mi>σ</mml:mi></mml:mrow></mml:mfrac><mml:msup><mml:mrow><mml:mi>e</mml:mi></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mfrac><mml:mrow><mml:msup><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>G</mml:mi><mml:mo>−</mml:mo><mml:mi>μ</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:msup><mml:mrow><mml:mi>σ</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:mfrac></mml:mrow></mml:msup></mml:math></alternatives></inline-formula>" ]

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[ "<supplementary-material content-type=\"local-data\" id=\"MOESM1\"></supplementary-material>", "<supplementary-material content-type=\"local-data\" id=\"MOESM2\"></supplementary-material>" ]

[ "<fn-group><fn><p><bold>Publisher’s note</bold> Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.</p></fn><fn><p>These authors contributed equally: Guangdi Feng, Qiuxiang Zhu.</p></fn></fn-group>" ]

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[ "<media xlink:href=\"41467_2024_44759_MOESM1_ESM.pdf\"><caption><p>Supplementary Information</p></caption></media>", "<media xlink:href=\"41467_2024_44759_MOESM2_ESM.pdf\"><caption><p>Peer Review File</p></caption></media>" ]

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[ "<title>Introduction</title>", "<p id=\"Par2\">Industries that work with dyestuff, textiles, leather, paper, plastics, etc., typically drain synthetic dyes along with their effluent. The synthetic dye wastewater is mainly composed of organic components, which are complex and easy to show color in water, and usually slow or difficult to degrade in the natural environment^{##REF##30118202##1##,##REF##35136666##2##}. Among the synthetic dyes, anionic azo dyes account for half of the dye synthesis and industrial application^{##UREF##0##3##}. Due to the low coloring rate on natural fibers, anionic dyes account for a large proportion of the dye wastewater discharged by printing and dyeing factories. Methyl orange [(MO) dimethylaminoazobenzenesulfonate] is a common and typical azo anionic dye. This water-soluble organic synthetic dye has very high colorability and presents a bright orange color when dissolved in water. Azo dyes such as methyl orange contain aromatic and –N=N– groups in their molecules, which are highly toxic, carcinogenic and teratogenic^{##REF##31806438##4##,##REF##33424369##5##}, and are harmful to the environment and organisms^{##UREF##1##6##,##UREF##2##7##}. In addition, the dyes in the wastewater can lead to the deterioration of water quality^{##UREF##3##8##}, so the wastewater containing dyes must be treated innocuously and the dye components need to be removed in order to discharge to the natural water environment or carry out secondary use^{##REF##33097347##9##,##UREF##4##10##}. The adsorption process is considered a valuable technique for the removal of dyes from wastewater, alongside many other physical and chemical processes^{##UREF##5##11##–##UREF##6##13##}. Numerous research endeavours have been conducted with the aim of identifying cost-effective and efficient adsorbents for the purpose of reducing dye concentrations in aqueous solutions. The researchers incorporated several materials, such as activated carbon, peat, chitin, silica, among others^{##REF##11150798##14##}. Chitosan or its derivatives have been demonstrated to be an efficacious substance for the removal of anionic dyes^{##REF##19716652##15##–##UREF##9##22##} through its amino groups which have the ability to undergo cationization, resulting in a strong electrostatic interaction with anionic dyes when exposed to acidic conditions.</p>", "<p id=\"Par3\">In order to enhance the adsorption efficiency, previous studies have indicated that the adsorbent material underwent modification by the introduction of high chelating coordination of sulphur (S) and nitrogen (N) functional groups^{##REF##28027491##23##–##UREF##12##26##}. Nevertheless, the publications outlined a multitude of reaction steps required for polyamine design. Aminated Chitosan derivatives were synthesized by many authors using different approaches, however these studies used two or even three steps to achieve their goal^{##REF##28027491##23##–##UREF##17##31##}. In a recent study^{##REF##31218044##32##}, aminated Chitosan materials have been developed using a one-pot method. Further enhancement of the adsorption capabilities of Chitosan derivatives has been tried through crosslinking process using various cross-linking reagents which have dual purpose namely, stabilizing Chitosan in acid solutions, rendering it insoluble, and augmenting its mechanical qualities.</p>", "<p id=\"Par4\">The novelty of the current study focused on presents a newly developed one-step amination technique for Chitosan through coupling of the Chitosan (CS) with 2-chloroethylamine (ENH2) using click chemistry. The obtained Chitosan derivatives with an increased content of amine groups (CS-ENH2) have subsequently crosslinked using Glutaraldehyde, leading to the formation of amino-ethyl Chitosan Schiff bases. The CS-ENH2 Schiff bases were used as an adsorbent to remove Methyl Orange (MO) dye from aqueous solutions in batch mode process, and its performance was compared with the native Chitosan Schiff base. The impact of several factors, including adsorption time, initial dye concentration, adsorption temperature, adsorption pH, agitation speed, and adsorbent dosage, on the adsorption of Mo dye have been studied. These Schiff bases are further characterized using advanced analytical techniques such as Fourier Transform Infrared spectroscopy (FTIR), Thermal (TGA and DSC) analysis, and Scanning Electron Microscopy (SEM).</p>" ]

[ "<title>Materials and methods</title>", "<title>Materials</title>", "<p id=\"Par5\">Chitosan medium molecular weight (≥ 75% DD) and methyl orange dye (MO; 85%) were procured from Sigma-Aldrich (Germany). The compound 2-Chloro ethyl amine hydrochloride (99%) was acquired from Sigma-Aldrich (Germany). The other chemicals used in this study were obtained from El-Nasr Pharmaceutical Co for Chemicals (Egypt) included sulfuric acid (98%), sodium hydroxide (99%), and phenolphthalein (98%). The Glutaraldehyde (GA) used in this study was a 25.0 wt% aqueous solution acquired from ACROS Organics.</p>", "<title>Methods</title>", "<title>Preparation of amino-ethyl chitosan hydrogel</title>", "<p id=\"Par6\">A solution containing 2 grammes of CS was prepared by dissolving it in 40 millilitres of acetic acid with a concentration of 2%. Subsequently, different quantities of Cl-ENH2 (0, 6.4, 12.7, 25.5, and 51 mM) were added individually to the Chitosan solution and the obtained adsorbents were coded as CS, CS-ENH2-1, CS-ENH2-2, CS-ENH2-3, and CS-ENH2-4, respectively. The temperature was then increased to 70 degrees Celsius and the mixture was stirred for duration of 2 h. Subsequently, a volume of 0.5 ml of Glutaraldehyde solution with a concentration of 25% was introduced into the mixture while maintaining a consistent stirring motion for duration of one hour. The CS-ENH2 hydrogel was subjected to drying at a temperature of 60 °C for duration of one night. The dried samples underwent a milling process and were subsequently subjected to multiple washes using hot distilled water in order to eliminate any remaining unreacted components. Subsequently, the samples were subjected to a drying process and subsequently stored within desiccators to facilitate subsequent analysis and adsorption tests.</p>", "<title>Structural and morphological Characterization</title>", "<title>Infrared Spectrophotometric (FTIR)</title>", "<p id=\"Par7\">To confirm the change and establish the structure of the hydrogel, Fourier transform infrared spectroscopy was conducted using a Shimadzu FTIR-8400 S spectrophotometer from Japan.</p>", "<title>Thermal gravimetric analysis (TGA)</title>", "<p id=\"Par8\">The analysis of materials was conducted using a thermo gravimetric analyzer (Shimadzu TGA –50, Japan) with a Nitrogen flow rate of 30 ml/min. This analysis aimed to observe any structural changes resulting from the modification. The measurement of weight loss in the samples commenced at room temperature and continued up to 600 °C, with a heating rate of 10 °C per minute.</p>", "<title>Differential scanning calorimeter (DSC)</title>", "<p id=\"Par9\">Differential scanning calorimetry was performed on the samples using a Shimadzu DSC-60A apparatus from Japan. The analysis was conducted over a temperature range from ambient to 350 °C, with a heating rate of 10 °C/min and under a nitrogen flow of 30 ml/min.</p>", "<title>Scanning electron microscopic analysis (SEM)</title>", "<p id=\"Par10\">Prior to examination by scanning electron microscopy, the samples were subjected to a vacuum environment and coated with a thin layer of gold. The morphological changes on the surface of the samples were monitored using a secondary electron detector of a scanning electron microscope (SEM) model Joel Jsm 6360LA, manufactured in Japan.</p>", "<title>Physicochemical characterization</title>", "<title>Water uptake (%)</title>", "<p id=\"Par11\">The Water uptake behaviour of the prepared hydrogel was investigated using distilled water (pH 5.4). Accurately weighed amounts of hydrogels were immersed in water and allowed to swell for 24 h at R.T. The swollen hydrogel was periodically separated, and the moisture adhered to the surface of hydrogel was removed by blotting them gently in between two filter papers, immediately followed by weighing. The swelling degree of samples was determined according to the following formula^{##UREF##18##33##}:where M_t is the weight of the swollen hydrogel, and M0 is the initial dry weight.</p>", "<title>The ion exchange capacity</title>", "<p id=\"Par12\">A known weight of chitosan or schiff base hydrogels were added to the known volume of 0.1 M H₂SO₄ solution, and the mixture was kept under shaking for three h. The mixture was filtered, and an aliquot was titrated against a standard solution of sodium hydroxide. Similarly, control titration without the addition of Chitosan was also run. From the difference in the volume of NaOH required for neutralization, the ionic capacity of chitosan samples was calculated using the following equation:where V₂ and V₁ are the volumes of NaOH required for complete neutralization of H₂SO₄ in the absence and presence of chitosan membrane, respectively, A is the normality of NaOH and W is the weight of sample taken for analysis^{##UREF##19##34##}.</p>", "<title>Batch equilibrium studies</title>", "<p id=\"Par13\">A stock solution of methyl orange (MO) dye with a concentration of 1g/L (1000 ppm) was made in distilled water. Subsequently, the desired concentrations, 10, 20, 25, 50, and 100 ppm, were achieved by diluting of 1, 2, 2.5, 5, and 10 mL of the stock solution with distilled water up to 100 mL using 100 mL flasks. The adsorption tests were performed using 100 mL flasks. A certain quantity of adsorbent was added to 25 mL dye solution with varying dye concentrations and pH values. The mixture was then agitated in an orbital shaker for a predetermined duration. The concentrations of MO in the initial and final aqueous solutions were determined by employing a UV–Vis spectrophotometer set to a wavelength of 465 nm. The quantity of dye adsorbed was determined by subtracting the initial concentration from the equilibrium concentration. The calculation of the percentage elimination value was determined using the following mathematical relationship:where; C₀ is the initial dye concentration and C_t is the final dye concentration in supernatant.</p>" ]

[ "<title>Results and discussion</title>", "<title>Characterization</title>", "<title>Physicochemical characterization</title>", "<p id=\"Par14\">The confirmation of the immobilisation of excess amine into the chitosan backbone is achieved through the measurement of ion exchange capacity. Figure ##FIG##0##1## illustrates a nearly linear augmentation in the ion exchange capacity as the quantity of reacted 2-chloroethyl amine in the modification procedure is elevated. This finding substantiates the observed elevation in the concentration of free amines inside the hydrogels that were synthesised.</p>", "<p id=\"Par15\">The water absorption capacity of the hydrogels that were created was assessed and is illustrated in Fig. ##FIG##0##1##. A linear rise in water absorption is seen, providing confirmation of the immobilisation of more free amino groups. Furthermore, the incorporation of immobilised ethyl amine groups as grafted chains onto the Chitosan backbone serves several purposes. Firstly, it enhances the internal space within the hydrogel. Secondly, it reduces the crystallinity of the hydrogel. Lastly, it increases the size of the three-dimensional network, thereby promoting the diffusion of water molecules and consequently augmenting the amount of water uptake.</p>", "<title>Infrared spectrophotometric</title>", "<p id=\"Par16\">The characteristics’ spectrum of CS displays a strong absorption band at 3437 cm⁻¹ due to OH and amine N–H symmetrical stretching vibration. A peak at 2921 cm⁻¹ was due to symmetrical C-H stretching vibration attributed to pyranose ring. The sharp peak at 1383 cm⁻¹ was assigned to CH3 in amide group. The broad peak at 1095 cm⁻¹ was indicated C–O–C stretching vibration in CS^{##UREF##16##30##}, peaks at 1649 and 1425 cm⁻¹ were due to C=O stretching (amide I) and N–H stretching (amide II). The absorption band at 1153 cm^-1 was assigned to the anti-symmetric stretching of C–O–C bridge and 1095 cm⁻¹, 1010 cm⁻¹ were assigned to the skeletal vibration involving the C–O stretching. In the other hand, the infrared spectrum of ethylamine, wave numbers ~ 1500 to 400 cm⁻¹ is considered the fingerprint region for the identification of ethylamine and most organic compounds. It is due to a unique set of complex overlapping vibrations of the atoms of the molecule of ethylamine. The most prominent infrared absorption lines of ethylamine at wavenumbers ~ 3500 to 3300 cm⁻¹ is a broad band for N–H bond stretching vibrations, characteristic of amines. The hydrogen bonding interferes with the N–H stretching vibrations producing the broad band peaking at around ~ 3400 cm⁻¹. There are also characteristic bands due to N–H vibrations at wave numbers 1650–1580 cm⁻¹. Vibrations characteristic of C-N bonds in aliphatic amines like ethylamine occur at 1220–1020 cm⁻¹. Around 3000–2800 cm⁻¹ are absorptions due to C–H stretching vibrations—they overlap with the N–H stretching vibrations. The similarity between most of the characteristic peaks of the Chitosan and 2-Chlorethyl amine, makes overlapping of both parent materials peaks. Figure ##FIG##1##2## depict the Fourier-transform infrared (FT-IR) spectroscopic analysis of Chitosan and amino-ethyl Chitosan hydrogels. The provided charts depict the characteristic bands of functional groups in polysaccharides, including prominent band within the range of 3400 cm⁻¹, which correspond to the vibrational modes of hydroxyl and amine groups involved in starching. Chitosan shows band at a wave number of 3433.4 cm⁻¹. A shift of the wave number to 3435.33, 3439.2, 3437.26, and 3338.9 cm⁻¹ of the CS-ENH2-1, CS-ENH2-2, CS-ENH2-3, and CS-ENH2-4, respectively, has been recognized with varied absorption intestines. The presence of bands at a wave number of 2938 cm⁻¹ is indicative of the presence of aliphatic C–H bonds. A shift of the wave number to 2924, 2930, 2922, and 3014.84 cm⁻¹ of the CS-ENH2-1, CS-ENH2-2, CS-ENH2-3, and CS-ENH2-4, respectively, has been recognized with varied absorption intestines. The amide groups of Chitosan were seen at a wave number of 1641 cm⁻¹. A shift of the wave number to 1637.62, 1653, and 1614.47 cm⁻¹ of the CS-ENH2-2, CS-ENH2-3, and CS-ENH2-4, respectively, has been recognized with varied absorption intestines. The spectral bands observed within the range of 1000–1300 cm⁻¹ are associated with the C–O stretching vibrations occurring in the glucose ring of Chitosan. Chitosan shows band at a wave number of 1068.6 cm⁻¹. A shift of the wave number to 1076.3, 1078.24, 1070.53, and 1074.39 cm⁻¹ of the CS-ENH2-1, CS-ENH2-2, CS-ENH2-3, and CS-ENH2-4, respectively, has been recognized with varied absorption intestines.</p>", "<title>Thermal gravimetric analysis (TGA)</title>", "<p id=\"Par17\">Thermogravimetric analysis (TGA) was conducted on the hydrogel derivatives of CS and CS-ENH2, and the results are illustrated in Fig. ##FIG##2##3##. The presented chart displays data pertaining to the thermal degradation process occurring in the presence of a Nitrogen atmosphere. The observed trend indicates a gradual decline in the measured weight samples values, commencing from the initial temperature of the surrounding environment and continuing until about 150 °C. The recorded samples weight loss percent values fall within the range of 10.87–12.79%, which can potentially be attributed to the reduction in moisture content inside the polymers, as suggested by previous studies. The occurrence of a secondary samples weight loss percent at an elevated temperature, ranging from 17.48 to 27.25%, can perhaps be attributed to the oxidative degradation of the pyranose ring within the chitosan backbone. During this phase, the occurrence of samples weight loss can be attributed to the decomposition of the amine groups inside the pyranose ring, leading to the formation of novel crosslinked fragments. The residue that was generated underwent a gradual decomposition process by the application of increased temperature within the range^{##UREF##20##35##–##UREF##22##37##}.</p>", "<title>Differential scanning calorimetry (DSC)</title>", "<p id=\"Par18\">The differential scanning calorimetry (DSC) investigation was performed on the hydrogel derivatives of CS and CS-ENH2, as depicted in Fig. ##FIG##3##4##. The initial endothermic peak observed in all studied materials, occurring between 50 and 120 °C, can be attributed to the increase in moisture content. The inclusion of hydrophilic functional groups, such as hydroxyl and amine groups, along the polymer chain imparts a strong attraction to water molecules. This property allows the polymer to effectively absorb and retain water from the surrounding atmosphere or during the production process. The second thermal event observed in the chart can perhaps be attributed to the disintegration of the glucose amine (GlcN) units inside the Chitosan hydrogels. This decomposition process is characterised by an exothermic peak occurring at a temperature of 210 °C^{##UREF##23##38##}.</p>", "<title>Scanning electron microscope (SEM) and EDAX analysis</title>", "<p id=\"Par19\">The morphological study of Chitosan and amino-ethyl Chitosan hydrogel was conducted utilising a scanning electron microscope, as depicted in Fig. ##FIG##4##5##a–e. The images demonstrate a marginal elevation in surface roughness and pore structure as the amine content of chitosan is augmented. The observed phenomenon can be attributed to the favourable interaction between the extended side chains containing terminal amine groups and the molecular structure of Chitosan. These chains offer an alternative active site for the process of crosslinking, as opposed to the original amine groups, perhaps resulting in increased space between polymer chains. The verification of the structural modifications of Chitosan to amino-ethyl Chitosan by introducing ethyl amine groups has been conducted using EDAX analysis (Fig. ##FIG##4##5##f). The study reveals an increase in the mass percentage of carbon (C) and nitrogen (N) in comparison to the oxygen (O) mass percentage observed in the Chitosan blank sample. Figure ##FIG##4##5##g, h present empirical support for the adsorption of MO on the Chitosan adsorbent, as indicated by the presence of S and Na elements and an increase in the mass percentage of C and N, accompanied by a corresponding decrease in the mass percentage of O. On the contrary, the adsorption of MO on the CS-ENH2-4 adsorbent (Fig. ##FIG##4##5##j) exhibits an increase in the presence of S and Na elements compared to the CS-ENH2-4 adsorbent (Fig. ##FIG##4##5##i). Additionally, there is a slight increase in the C mass percent, while the N and O mass percents experience a slight decrease. It is noteworthy to mention that the elements S and Na exhibit a higher mass percentage in comparison to Chitosan-MO, as depicted in Fig. ##FIG##4##5##h. This observation provides evidence for the enhanced performance of the CS-ENH2-4 adsorbent in comparison to Chitosan.</p>", "<title>Adsorption process</title>", "<p id=\"Par20\">This study explores the sorption characteristics of methyl orange (MO) using Chitosan and amino-ethyl Chitosan hydrogels in the presence of artificially contaminated water solution. A series of experiments were conducted to examine the adsorption characteristics of hydrogels towards MO under various adsorption settings, including adsorption time, adsorption temperature, adsorption pH, agitation rate, adsorbent dosage, and starting dye concentration.</p>", "<title>Effect of the adsorption time</title>", "<p id=\"Par21\">Figure ##FIG##5##6##a depicts the impact of adsorption time on the efficacy of MO removal percentage (%) using different Chitosan adsorbents. From the Figure, It is acknowledged that the percentage of dye removal exhibits a linear rapid progress as a first initial period for all the Chitosan adsorbents. That period varied with the amine content of the adsorbent. For Chitosan, it is recognized to be 90 min, which after almost no noticeable MO removal was noticed suggesting the consumption of all the amine active adsorption sites and attainment of equilibrium. In the other hand, it can be noticed that all the derivatives of amino-ethyl Chitosan exhibit a higher linear increase in the percentage of MO removal, compared to Chitosan, in the following order CS-ENH2-1, CS-ENH2-2, CS-ENH2-3, and CS-ENH2-4 correlated to the increment of the amine content adsorption sites. At this stage, all the active adsorption sites are free and accessable to MO molecules, in addition to the existence of high MO concentration gradient between the adsorbents solid phase and the MO liquid phase. These driving force leads to a high rate of MO removal compared to Chitosan counterpart. Subsequently, with a progress of adsorption time, that driving force was reduced as a result of consuming most of the adsorption active sites and reduced of the concentration gradient. A slower rate of increase is observed until reaching saturation at 240 min where the adsorption is controlled by the diffusion of MO molecules to the interior pores of the adsorbents. The performance of Chitosan is influenced by secondary key factor namely hydrophobic interactions between the hydrophobic aliphatic moieties of the adsorbents and the aromatic ones of the MO molecules. The hydrophobic interactions arise from the presence of the methyl group in the acetamide moiety (inside partial acetyl amine groups) and the –CH and –CH2 groups in the glucose ring. The adsorption process of MO, an anionic dye, onto Chitosan adsorbents was conducted using a combination of physical sorption behavior and chemisorptions, which occurred due to the electrostatic interaction between opposite charges in accordance with previously published data where Chitosan Schiff bases have used in the removal of MO dye from aqueous solution^{##UREF##24##39##–##UREF##26##41##}. The Chitosan structure as a linear cationic polymer was modifie to improve its behaviour via preparing two different crosslinked Chitosan Schiff bases hydrogels using a glutaraldehyde crosslinker. The Chitosan was coupled with succinimide (Ch/Su) and 1-methyl-2-pyrrolidinone (Ch/Mp)^{##UREF##24##39##}. Cross-linked Chitosan derivate Schiff bases obtained from the coupling of Chitosan with 1-vinyl 2- pyrrolidone [Schiff base (I)] and 4-amino acetanilide (Schiff base (II))^{##UREF##25##40##}. Other Chitosan Schiff base derivative was developed by reaction of Chitosan with 4-methoxybenzaldehyde to have Chitosan Schiff base (Cs/MeB) in the presence of Glutaraldehyde as a crosslinker^{##UREF##26##41##}.</p>", "<p id=\"Par22\">In the case of the existing adsorbents, augmenting the degree of ethylamine substitution leads to heightened hydrophobic interactions and more advantageous active sites comprising amine groups. In order to investigate the effects of incorporating additional amine groups by fictionalization with 2-chloroethyl amine, the cationic exchange capacity of the amino-ethyl Chitosan derivatives that were synthesized was determined. This measurement was then associated with the percentage of MO removal, as shown in Fig. ##FIG##5##6##b. The figure illustrates a notable linear correlation between the removal percentage of MO in CS and the concentration of CS-ENH2-4 in the barrel, as seen by the nearly twofold rise in removal percentage from 43.3 to 84.2% when the ion exchange capacity (IEC) of CS and CS-ENH2-4 increased from 7.4 to 12.8 meq/g. This accomplishment demonstrates the efficacy of using more amine groups to boost the removal effectiveness of MO, on one hand. On the contrary, this provides an indication of the prevailing electrostatic interaction between opposing charges, specifically the positive charge on CS-ENH2 derivatives and the negative charge on the MO molecules.</p>", "<title>Effect of temperature</title>", "<p id=\"Par23\">The study investigated the impact of variations in environmental temperature on the percentage of MO removal utilising cross-linked Chitosan and amino-ethyl Chitosan hydrogels. The temperature range examined spanned from 25 to 60 degrees Celsius, as depicted in Fig. ##FIG##6##7##. Based on the data presented in the Figure, it is evident that there is a consistent linear increase in the percentage of MO removal by all of the adsorbents utilised, with a similar rate of growth observed. The elevation of the medium temperature amplifies the mobility of the big dye ions, so expediting their stochastic motion within the solution. Consequently, this augmentation leads to an increase in the frequency of collisions between the dye ions and the adsorbent surface. Furthermore, it induces an increase in volume inside the internal framework of the adsorbent, hence facilitating deeper penetration of the larger dye molecules^{##UREF##27##42##}. Similar finding where Cross-linked Chitosan derivate Schiff bases obtained from the coupling of Chitosan with 1-vinyl 2- pyrrolidone [Schiff base (I)] and 4-amino acetanilide (Schiff base (II)) were used in the removal of MO dye^{##UREF##25##40##}. The authors explained the effect of the temperature increment would increase the mobility of the large dye ions as well as produce a swelling effect on the internal structure of the Chitosan. That consequently facilitates the diffusion of the large dye molecules^{##UREF##25##40##}. It is noteworthy to notice that the amino-ethyl Chitosan derivatives exhibited a larger boost in the percentage of MO removal at the lowest temperature (25 °C), with the CS-ENH2-4 sample achieving a 100% removal rate. On the contrary, it was observed that the increase in the elimination percentage of MO was only 48.5% at the greatest temperature (60 °C). Hence, the adsorption capacity is predominantly influenced by the chemical interaction occurring between the functional groups present on the internal surface of the adsorbent (namely, the surface of its pores) and the adsorbate. This capacity is expected to augment with increasing temperature.</p>", "<title>Effect of pH</title>", "<p id=\"Par24\">Figure ##FIG##7##8## illustrates the impact of the pH value of the initial MO solutions on the efficiency of adsorption. The absorption of MO dye is significantly higher in acidic solutions compared to alkaline settings. Under acidic conditions, the presence of free amine groups along the chitosan backbone leads to their protonation, resulting in the formation of a positive charge on the surface of the hydrogel. This positive charge facilitates electrostatic interactions with the negatively charged sulfonate group of MO. Comparable findings have reported by other authors^{##UREF##28##43##,##UREF##29##44##}. The adsorption capabilities of chitosan hydrogels are observed to decrease at alkaline pH levels. At the given pH, the surface charges of chitosan exhibited a negative polarity, hence impeding the adsorption process due to the electrostatic repulsion between the negatively charged dye molecules and the adsorbent (chitosan hydrogel). One intriguing finding in this study is the inverse relationship between the removal percentages of MO and the cation exchange capacity of amino-ethyl Chitosan samples. Notably, the CS-ENH2-4 sample exhibited the lowest reduction rate, with a rapid decrease in MO removal % found after reaching a pH of 7.0. On the contrary, it was observed that all the adsorbents exhibited similar percentages of MO removal in pH 10.0, ranging from 35 to 40%. This observation shows that the primary factor influencing the adsorption process is the hydrophobic-hydrophobic interaction^{##UREF##30##45##}. Similar finding results where joint steady adsorption pH range from 6.0 to 8.0 of the Chitosan Schiff bases adsorbents can be explained by two reasons. The first is the reduced number of the free amine groups’ numbers affected by the deprotonation. The second is to increase the physical adsorption role of the chitosan Schiff bases adsorbents via hydrophobic-hydrophobic interaction, which is expected to be higher in the Ch/Mp Schiff base hydrogel containing heterocyclic ring with an attached methyl group^{##UREF##24##39##}. On the other hand, the dye removal % by Cs/MeB was slightly affected by the increase of pH were decreased from 95% at pH 4.0 to 86% at pH 9.0. This behaviour confirmed the dominated hydrophobic-hydrophobic physical adsorption between the benzene rings and the methyl hydrophobic groups of the Cs/MeB adsorbent and the MO dye molecules in a wide range of pH; from 4.0 to 9.0, while the elimination of the cationic charge of the last free amine groups at pH 10.0 leads to the collapse of the Cs/MeB hydrogel structure with loss of its water content leading to reduce the pores volume and so the internal pores surface area. The high and almost constant MO removal % by Cs/MeB hydrogel in a wide pH range is a great advantage for its application in the treatment of industrial effluents contaminated with MO dye^{##UREF##26##41##}.</p>", "<title>Effect of agitation rate</title>", "<p id=\"Par25\">The impact of agitation rate on the adsorption of MO was investigated by conducting experiments at various agitation rates ranging from 50 to 250 rpm, while keeping the kinetic parameters constant. Figure ##FIG##8##9## presents an overview of the findings derived from the study. The adsorption of the MO is observed to exhibit a notable enhancement as the agitation rate is raised within the range of 50–200 rpm, after which it reaches a plateau until 250 rpm. These findings can be attributed to the observation that higher agitation speeds enhance the diffusion of MO towards the surface of the adsorbents. The relationship between intraparticle diffusivity and adsorption capacity, as well as the surface characteristics of adsorbents, has been suggested to be of significant importance. Increasing the agitation rate has the potential to surpass the thickness of the liquid layer and the resistance to mass transfer on the surfaces of the adsorbents being examined. Therefore, it can be inferred that a shaking rate of 200 rpm is adequate to facilitate the accessibility of all surface binding sites for the uptake of methyl orange^{##UREF##25##40##,##UREF##31##46##,##UREF##32##47##}.</p>", "<title>Effect of the initial dye concentration</title>", "<p id=\"Par26\">The impact of the initial concentration of methyl orange on the process of adsorption was investigated over a range of concentrations (10, 20, 25, 50, and 100 ppm). The findings of this investigation are presented in Fig. ##FIG##9##10##. The figure yields two primary observations. The initial observation pertains to the complete elimination of the lowest concentration (10 ppm) of MO by all hydrogel samples employed. This outcome can be attributed to the presence of an ample number of active sites on the Chitosan sample, thereby accommodating all MO molecules. Consequently, the aminated chitosan samples, which possess a greater number of active sites, do not exhibit any discernible impact due to the limited availability of MO. On the other hand, a notable increase in the percentage of removal of MO has been seen as the initial concentration of MO is increased up to 100 ppm. Specifically, the CS-ENH2-4 sample exhibits a four-fold higher removal percentage of MO compared to the CS sample. The second primary observation pertained to the decline in the percentages of MO removal as the starting MO concentration increased. The process of reduction consists of two distinct steps. The initial acute stage was noticed when the concentration of MO was increased from 10 to 25 ppm. This resulted in a decrease in the reduction percentage, in conjunction with the degree of amination of the aminated chitosan samples. Eventually, a nearly linear reduction rate was achieved with the CS-NH2-4 sample. The second stage of reduction, characterised by a nearly identical decrease in the rate, was observed for all samples of adsorbents, with a recognition threshold of up to 100 ppm. A similar finding was observed where the removal percentage linearly reduced with an increase in initial MO concentration using 4-dimethylamino benzaldehyde chitosan Schiff base and benzophenone chitosan Schiff base. This trend is due to the electrostatic repulsion between the dye molecules with increasing concentration, resulting in a competition between the dye molecules for the limited active sites in the adsorbent^{##UREF##33##48##}.</p>", "<title>Effect of the absorbent dose</title>", "<p id=\"Par27\">Figure ##FIG##10##11## illustrates the impact of varying doses of Chitosan adsorbents on the percentage of MO removal, while keeping the kinetic parameters constant. In general, it was observed that an escalation in the dosage resulted in a proportional rise in the elimination % of MO for both CS and CS-NH2-1 samples, exhibiting a nearly linear relationship. The clearance percentages of MO exhibited a gradual increase when lower rates were applied to the other aminated samples. Ultimately, the adsorption efficiency achieved with 0.3 g of adsorbents exhibits minimal variation, falling within the range of 90–100%. The CS-NH2-4 sample has a plateau effect, commencing at a dosage of 0.2 g, whereby it achieves a clearance percentage of 95%. This phenomenon may be attributed to the observation that, when the starting dye concentration remains constant, increases in the quantity of adsorbent material results in a larger surface area and a greater number of sorption sites^{##UREF##34##49##,##UREF##35##50##}. Similar quite tendency have been reported using other sorbents reported in the previous work^{##REF##30599158##51##}.</p>", "<title>Reusability</title>", "<p id=\"Par28\">The sorption–desorption cycle, as depicted in Fig. ##FIG##11##12##, can be utilised to estimate the recovery of MO absorbed from aqueous solution. The reusability of the CS-NH2-4 adsorbent for removing MO from aqueous solutions was investigated by analysing the sorption–desorption cycles. The cycle was repeated ten times using a sodium hydroxide solution. The figure clearly demonstrates a continuous, nearly linear reduction in MO removal. However, the decrease in MO removal efficiency was not significant; the percentage of MO removal reached 66% in the tenth cycle, compared to 80.1% in the first cycle. Only 18% of the MO removal efficiency was lost after ten cycles of sorption–desorption processes. The CS-NH2-4 adsorbent exhibits favourable sorption–desorption performance and can be confidently used without a noticeable decrease in its sorption capacity for MO removal.</p>", "<title>Comparative adsorption capacity study</title>", "<p id=\"Par29\">Table ##TAB##0##1## presents a comparison of the highest adsorption capacity for MO on the CS-NH2-4 adsorbent in relation to other adsorbents documented in the literature^{##UREF##24##39##–##UREF##26##41##,##UREF##33##48##,##REF##25843838##52##–##UREF##37##55##}. Based on the tabular data, it can be observed that the adsorption capacity of the hydrogels composed of alginate and alginate/poly aspartate is comparatively lower than that of the hydrogels being analysed^{##UREF##37##55##}. The example experienced a reversal upon the utilisation of Calcium alginate MWNTs^{##REF##24751058##53##}, resulting in a six-fold increase in adsorption capacity. In contrast, the adsorption capacity of CS-NH2-4 is comparatively lower when compared to other Chitosan and Chitosan derivatives^{##UREF##24##39##–##UREF##26##41##,##REF##25843838##52##}. Magnetic multi-walled carbon nanotubes (MWCNTs) and bottom ash have also demonstrated elevated adsorption capacity^{##REF##24751058##53##,##REF##17379402##56##}. The modest adsorption capacity of the CS-NH2-4 adsorbent can be attributed to several factors, which can be summarised as follows:<list list-type=\"order\"><list-item><p id=\"Par30\">The scarcity of amine active sites that are protonated for the purpose of adsorption at a pH level of 7.0.</p></list-item><list-item><p id=\"Par31\">The limited expansion of the material at a pH level of 7.0, resulting in a diffusion constraint for the MO (material of interest).</p></list-item><list-item><p id=\"Par32\">The utilisation of the initially available active amine group sites during chemical crosslinking procedures involving Glutaraldehyde. It is advisable to conduct a more comprehensive investigation into the specific parameters of the crosslinking process in order to enhance the adsorption capacity.</p></list-item></list></p>" ]

[ "<title>Results and discussion</title>", "<title>Characterization</title>", "<title>Physicochemical characterization</title>", "<p id=\"Par14\">The confirmation of the immobilisation of excess amine into the chitosan backbone is achieved through the measurement of ion exchange capacity. Figure ##FIG##0##1## illustrates a nearly linear augmentation in the ion exchange capacity as the quantity of reacted 2-chloroethyl amine in the modification procedure is elevated. This finding substantiates the observed elevation in the concentration of free amines inside the hydrogels that were synthesised.</p>", "<p id=\"Par15\">The water absorption capacity of the hydrogels that were created was assessed and is illustrated in Fig. ##FIG##0##1##. A linear rise in water absorption is seen, providing confirmation of the immobilisation of more free amino groups. Furthermore, the incorporation of immobilised ethyl amine groups as grafted chains onto the Chitosan backbone serves several purposes. Firstly, it enhances the internal space within the hydrogel. Secondly, it reduces the crystallinity of the hydrogel. Lastly, it increases the size of the three-dimensional network, thereby promoting the diffusion of water molecules and consequently augmenting the amount of water uptake.</p>", "<title>Infrared spectrophotometric</title>", "<p id=\"Par16\">The characteristics’ spectrum of CS displays a strong absorption band at 3437 cm⁻¹ due to OH and amine N–H symmetrical stretching vibration. A peak at 2921 cm⁻¹ was due to symmetrical C-H stretching vibration attributed to pyranose ring. The sharp peak at 1383 cm⁻¹ was assigned to CH3 in amide group. The broad peak at 1095 cm⁻¹ was indicated C–O–C stretching vibration in CS^{##UREF##16##30##}, peaks at 1649 and 1425 cm⁻¹ were due to C=O stretching (amide I) and N–H stretching (amide II). The absorption band at 1153 cm^-1 was assigned to the anti-symmetric stretching of C–O–C bridge and 1095 cm⁻¹, 1010 cm⁻¹ were assigned to the skeletal vibration involving the C–O stretching. In the other hand, the infrared spectrum of ethylamine, wave numbers ~ 1500 to 400 cm⁻¹ is considered the fingerprint region for the identification of ethylamine and most organic compounds. It is due to a unique set of complex overlapping vibrations of the atoms of the molecule of ethylamine. The most prominent infrared absorption lines of ethylamine at wavenumbers ~ 3500 to 3300 cm⁻¹ is a broad band for N–H bond stretching vibrations, characteristic of amines. The hydrogen bonding interferes with the N–H stretching vibrations producing the broad band peaking at around ~ 3400 cm⁻¹. There are also characteristic bands due to N–H vibrations at wave numbers 1650–1580 cm⁻¹. Vibrations characteristic of C-N bonds in aliphatic amines like ethylamine occur at 1220–1020 cm⁻¹. Around 3000–2800 cm⁻¹ are absorptions due to C–H stretching vibrations—they overlap with the N–H stretching vibrations. The similarity between most of the characteristic peaks of the Chitosan and 2-Chlorethyl amine, makes overlapping of both parent materials peaks. Figure ##FIG##1##2## depict the Fourier-transform infrared (FT-IR) spectroscopic analysis of Chitosan and amino-ethyl Chitosan hydrogels. The provided charts depict the characteristic bands of functional groups in polysaccharides, including prominent band within the range of 3400 cm⁻¹, which correspond to the vibrational modes of hydroxyl and amine groups involved in starching. Chitosan shows band at a wave number of 3433.4 cm⁻¹. A shift of the wave number to 3435.33, 3439.2, 3437.26, and 3338.9 cm⁻¹ of the CS-ENH2-1, CS-ENH2-2, CS-ENH2-3, and CS-ENH2-4, respectively, has been recognized with varied absorption intestines. The presence of bands at a wave number of 2938 cm⁻¹ is indicative of the presence of aliphatic C–H bonds. A shift of the wave number to 2924, 2930, 2922, and 3014.84 cm⁻¹ of the CS-ENH2-1, CS-ENH2-2, CS-ENH2-3, and CS-ENH2-4, respectively, has been recognized with varied absorption intestines. The amide groups of Chitosan were seen at a wave number of 1641 cm⁻¹. A shift of the wave number to 1637.62, 1653, and 1614.47 cm⁻¹ of the CS-ENH2-2, CS-ENH2-3, and CS-ENH2-4, respectively, has been recognized with varied absorption intestines. The spectral bands observed within the range of 1000–1300 cm⁻¹ are associated with the C–O stretching vibrations occurring in the glucose ring of Chitosan. Chitosan shows band at a wave number of 1068.6 cm⁻¹. A shift of the wave number to 1076.3, 1078.24, 1070.53, and 1074.39 cm⁻¹ of the CS-ENH2-1, CS-ENH2-2, CS-ENH2-3, and CS-ENH2-4, respectively, has been recognized with varied absorption intestines.</p>", "<title>Thermal gravimetric analysis (TGA)</title>", "<p id=\"Par17\">Thermogravimetric analysis (TGA) was conducted on the hydrogel derivatives of CS and CS-ENH2, and the results are illustrated in Fig. ##FIG##2##3##. The presented chart displays data pertaining to the thermal degradation process occurring in the presence of a Nitrogen atmosphere. The observed trend indicates a gradual decline in the measured weight samples values, commencing from the initial temperature of the surrounding environment and continuing until about 150 °C. The recorded samples weight loss percent values fall within the range of 10.87–12.79%, which can potentially be attributed to the reduction in moisture content inside the polymers, as suggested by previous studies. The occurrence of a secondary samples weight loss percent at an elevated temperature, ranging from 17.48 to 27.25%, can perhaps be attributed to the oxidative degradation of the pyranose ring within the chitosan backbone. During this phase, the occurrence of samples weight loss can be attributed to the decomposition of the amine groups inside the pyranose ring, leading to the formation of novel crosslinked fragments. The residue that was generated underwent a gradual decomposition process by the application of increased temperature within the range^{##UREF##20##35##–##UREF##22##37##}.</p>", "<title>Differential scanning calorimetry (DSC)</title>", "<p id=\"Par18\">The differential scanning calorimetry (DSC) investigation was performed on the hydrogel derivatives of CS and CS-ENH2, as depicted in Fig. ##FIG##3##4##. The initial endothermic peak observed in all studied materials, occurring between 50 and 120 °C, can be attributed to the increase in moisture content. The inclusion of hydrophilic functional groups, such as hydroxyl and amine groups, along the polymer chain imparts a strong attraction to water molecules. This property allows the polymer to effectively absorb and retain water from the surrounding atmosphere or during the production process. The second thermal event observed in the chart can perhaps be attributed to the disintegration of the glucose amine (GlcN) units inside the Chitosan hydrogels. This decomposition process is characterised by an exothermic peak occurring at a temperature of 210 °C^{##UREF##23##38##}.</p>", "<title>Scanning electron microscope (SEM) and EDAX analysis</title>", "<p id=\"Par19\">The morphological study of Chitosan and amino-ethyl Chitosan hydrogel was conducted utilising a scanning electron microscope, as depicted in Fig. ##FIG##4##5##a–e. The images demonstrate a marginal elevation in surface roughness and pore structure as the amine content of chitosan is augmented. The observed phenomenon can be attributed to the favourable interaction between the extended side chains containing terminal amine groups and the molecular structure of Chitosan. These chains offer an alternative active site for the process of crosslinking, as opposed to the original amine groups, perhaps resulting in increased space between polymer chains. The verification of the structural modifications of Chitosan to amino-ethyl Chitosan by introducing ethyl amine groups has been conducted using EDAX analysis (Fig. ##FIG##4##5##f). The study reveals an increase in the mass percentage of carbon (C) and nitrogen (N) in comparison to the oxygen (O) mass percentage observed in the Chitosan blank sample. Figure ##FIG##4##5##g, h present empirical support for the adsorption of MO on the Chitosan adsorbent, as indicated by the presence of S and Na elements and an increase in the mass percentage of C and N, accompanied by a corresponding decrease in the mass percentage of O. On the contrary, the adsorption of MO on the CS-ENH2-4 adsorbent (Fig. ##FIG##4##5##j) exhibits an increase in the presence of S and Na elements compared to the CS-ENH2-4 adsorbent (Fig. ##FIG##4##5##i). Additionally, there is a slight increase in the C mass percent, while the N and O mass percents experience a slight decrease. It is noteworthy to mention that the elements S and Na exhibit a higher mass percentage in comparison to Chitosan-MO, as depicted in Fig. ##FIG##4##5##h. This observation provides evidence for the enhanced performance of the CS-ENH2-4 adsorbent in comparison to Chitosan.</p>", "<title>Adsorption process</title>", "<p id=\"Par20\">This study explores the sorption characteristics of methyl orange (MO) using Chitosan and amino-ethyl Chitosan hydrogels in the presence of artificially contaminated water solution. A series of experiments were conducted to examine the adsorption characteristics of hydrogels towards MO under various adsorption settings, including adsorption time, adsorption temperature, adsorption pH, agitation rate, adsorbent dosage, and starting dye concentration.</p>", "<title>Effect of the adsorption time</title>", "<p id=\"Par21\">Figure ##FIG##5##6##a depicts the impact of adsorption time on the efficacy of MO removal percentage (%) using different Chitosan adsorbents. From the Figure, It is acknowledged that the percentage of dye removal exhibits a linear rapid progress as a first initial period for all the Chitosan adsorbents. That period varied with the amine content of the adsorbent. For Chitosan, it is recognized to be 90 min, which after almost no noticeable MO removal was noticed suggesting the consumption of all the amine active adsorption sites and attainment of equilibrium. In the other hand, it can be noticed that all the derivatives of amino-ethyl Chitosan exhibit a higher linear increase in the percentage of MO removal, compared to Chitosan, in the following order CS-ENH2-1, CS-ENH2-2, CS-ENH2-3, and CS-ENH2-4 correlated to the increment of the amine content adsorption sites. At this stage, all the active adsorption sites are free and accessable to MO molecules, in addition to the existence of high MO concentration gradient between the adsorbents solid phase and the MO liquid phase. These driving force leads to a high rate of MO removal compared to Chitosan counterpart. Subsequently, with a progress of adsorption time, that driving force was reduced as a result of consuming most of the adsorption active sites and reduced of the concentration gradient. A slower rate of increase is observed until reaching saturation at 240 min where the adsorption is controlled by the diffusion of MO molecules to the interior pores of the adsorbents. The performance of Chitosan is influenced by secondary key factor namely hydrophobic interactions between the hydrophobic aliphatic moieties of the adsorbents and the aromatic ones of the MO molecules. The hydrophobic interactions arise from the presence of the methyl group in the acetamide moiety (inside partial acetyl amine groups) and the –CH and –CH2 groups in the glucose ring. The adsorption process of MO, an anionic dye, onto Chitosan adsorbents was conducted using a combination of physical sorption behavior and chemisorptions, which occurred due to the electrostatic interaction between opposite charges in accordance with previously published data where Chitosan Schiff bases have used in the removal of MO dye from aqueous solution^{##UREF##24##39##–##UREF##26##41##}. The Chitosan structure as a linear cationic polymer was modifie to improve its behaviour via preparing two different crosslinked Chitosan Schiff bases hydrogels using a glutaraldehyde crosslinker. The Chitosan was coupled with succinimide (Ch/Su) and 1-methyl-2-pyrrolidinone (Ch/Mp)^{##UREF##24##39##}. Cross-linked Chitosan derivate Schiff bases obtained from the coupling of Chitosan with 1-vinyl 2- pyrrolidone [Schiff base (I)] and 4-amino acetanilide (Schiff base (II))^{##UREF##25##40##}. Other Chitosan Schiff base derivative was developed by reaction of Chitosan with 4-methoxybenzaldehyde to have Chitosan Schiff base (Cs/MeB) in the presence of Glutaraldehyde as a crosslinker^{##UREF##26##41##}.</p>", "<p id=\"Par22\">In the case of the existing adsorbents, augmenting the degree of ethylamine substitution leads to heightened hydrophobic interactions and more advantageous active sites comprising amine groups. In order to investigate the effects of incorporating additional amine groups by fictionalization with 2-chloroethyl amine, the cationic exchange capacity of the amino-ethyl Chitosan derivatives that were synthesized was determined. This measurement was then associated with the percentage of MO removal, as shown in Fig. ##FIG##5##6##b. The figure illustrates a notable linear correlation between the removal percentage of MO in CS and the concentration of CS-ENH2-4 in the barrel, as seen by the nearly twofold rise in removal percentage from 43.3 to 84.2% when the ion exchange capacity (IEC) of CS and CS-ENH2-4 increased from 7.4 to 12.8 meq/g. This accomplishment demonstrates the efficacy of using more amine groups to boost the removal effectiveness of MO, on one hand. On the contrary, this provides an indication of the prevailing electrostatic interaction between opposing charges, specifically the positive charge on CS-ENH2 derivatives and the negative charge on the MO molecules.</p>", "<title>Effect of temperature</title>", "<p id=\"Par23\">The study investigated the impact of variations in environmental temperature on the percentage of MO removal utilising cross-linked Chitosan and amino-ethyl Chitosan hydrogels. The temperature range examined spanned from 25 to 60 degrees Celsius, as depicted in Fig. ##FIG##6##7##. Based on the data presented in the Figure, it is evident that there is a consistent linear increase in the percentage of MO removal by all of the adsorbents utilised, with a similar rate of growth observed. The elevation of the medium temperature amplifies the mobility of the big dye ions, so expediting their stochastic motion within the solution. Consequently, this augmentation leads to an increase in the frequency of collisions between the dye ions and the adsorbent surface. Furthermore, it induces an increase in volume inside the internal framework of the adsorbent, hence facilitating deeper penetration of the larger dye molecules^{##UREF##27##42##}. Similar finding where Cross-linked Chitosan derivate Schiff bases obtained from the coupling of Chitosan with 1-vinyl 2- pyrrolidone [Schiff base (I)] and 4-amino acetanilide (Schiff base (II)) were used in the removal of MO dye^{##UREF##25##40##}. The authors explained the effect of the temperature increment would increase the mobility of the large dye ions as well as produce a swelling effect on the internal structure of the Chitosan. That consequently facilitates the diffusion of the large dye molecules^{##UREF##25##40##}. It is noteworthy to notice that the amino-ethyl Chitosan derivatives exhibited a larger boost in the percentage of MO removal at the lowest temperature (25 °C), with the CS-ENH2-4 sample achieving a 100% removal rate. On the contrary, it was observed that the increase in the elimination percentage of MO was only 48.5% at the greatest temperature (60 °C). Hence, the adsorption capacity is predominantly influenced by the chemical interaction occurring between the functional groups present on the internal surface of the adsorbent (namely, the surface of its pores) and the adsorbate. This capacity is expected to augment with increasing temperature.</p>", "<title>Effect of pH</title>", "<p id=\"Par24\">Figure ##FIG##7##8## illustrates the impact of the pH value of the initial MO solutions on the efficiency of adsorption. The absorption of MO dye is significantly higher in acidic solutions compared to alkaline settings. Under acidic conditions, the presence of free amine groups along the chitosan backbone leads to their protonation, resulting in the formation of a positive charge on the surface of the hydrogel. This positive charge facilitates electrostatic interactions with the negatively charged sulfonate group of MO. Comparable findings have reported by other authors^{##UREF##28##43##,##UREF##29##44##}. The adsorption capabilities of chitosan hydrogels are observed to decrease at alkaline pH levels. At the given pH, the surface charges of chitosan exhibited a negative polarity, hence impeding the adsorption process due to the electrostatic repulsion between the negatively charged dye molecules and the adsorbent (chitosan hydrogel). One intriguing finding in this study is the inverse relationship between the removal percentages of MO and the cation exchange capacity of amino-ethyl Chitosan samples. Notably, the CS-ENH2-4 sample exhibited the lowest reduction rate, with a rapid decrease in MO removal % found after reaching a pH of 7.0. On the contrary, it was observed that all the adsorbents exhibited similar percentages of MO removal in pH 10.0, ranging from 35 to 40%. This observation shows that the primary factor influencing the adsorption process is the hydrophobic-hydrophobic interaction^{##UREF##30##45##}. Similar finding results where joint steady adsorption pH range from 6.0 to 8.0 of the Chitosan Schiff bases adsorbents can be explained by two reasons. The first is the reduced number of the free amine groups’ numbers affected by the deprotonation. The second is to increase the physical adsorption role of the chitosan Schiff bases adsorbents via hydrophobic-hydrophobic interaction, which is expected to be higher in the Ch/Mp Schiff base hydrogel containing heterocyclic ring with an attached methyl group^{##UREF##24##39##}. On the other hand, the dye removal % by Cs/MeB was slightly affected by the increase of pH were decreased from 95% at pH 4.0 to 86% at pH 9.0. This behaviour confirmed the dominated hydrophobic-hydrophobic physical adsorption between the benzene rings and the methyl hydrophobic groups of the Cs/MeB adsorbent and the MO dye molecules in a wide range of pH; from 4.0 to 9.0, while the elimination of the cationic charge of the last free amine groups at pH 10.0 leads to the collapse of the Cs/MeB hydrogel structure with loss of its water content leading to reduce the pores volume and so the internal pores surface area. The high and almost constant MO removal % by Cs/MeB hydrogel in a wide pH range is a great advantage for its application in the treatment of industrial effluents contaminated with MO dye^{##UREF##26##41##}.</p>", "<title>Effect of agitation rate</title>", "<p id=\"Par25\">The impact of agitation rate on the adsorption of MO was investigated by conducting experiments at various agitation rates ranging from 50 to 250 rpm, while keeping the kinetic parameters constant. Figure ##FIG##8##9## presents an overview of the findings derived from the study. The adsorption of the MO is observed to exhibit a notable enhancement as the agitation rate is raised within the range of 50–200 rpm, after which it reaches a plateau until 250 rpm. These findings can be attributed to the observation that higher agitation speeds enhance the diffusion of MO towards the surface of the adsorbents. The relationship between intraparticle diffusivity and adsorption capacity, as well as the surface characteristics of adsorbents, has been suggested to be of significant importance. Increasing the agitation rate has the potential to surpass the thickness of the liquid layer and the resistance to mass transfer on the surfaces of the adsorbents being examined. Therefore, it can be inferred that a shaking rate of 200 rpm is adequate to facilitate the accessibility of all surface binding sites for the uptake of methyl orange^{##UREF##25##40##,##UREF##31##46##,##UREF##32##47##}.</p>", "<title>Effect of the initial dye concentration</title>", "<p id=\"Par26\">The impact of the initial concentration of methyl orange on the process of adsorption was investigated over a range of concentrations (10, 20, 25, 50, and 100 ppm). The findings of this investigation are presented in Fig. ##FIG##9##10##. The figure yields two primary observations. The initial observation pertains to the complete elimination of the lowest concentration (10 ppm) of MO by all hydrogel samples employed. This outcome can be attributed to the presence of an ample number of active sites on the Chitosan sample, thereby accommodating all MO molecules. Consequently, the aminated chitosan samples, which possess a greater number of active sites, do not exhibit any discernible impact due to the limited availability of MO. On the other hand, a notable increase in the percentage of removal of MO has been seen as the initial concentration of MO is increased up to 100 ppm. Specifically, the CS-ENH2-4 sample exhibits a four-fold higher removal percentage of MO compared to the CS sample. The second primary observation pertained to the decline in the percentages of MO removal as the starting MO concentration increased. The process of reduction consists of two distinct steps. The initial acute stage was noticed when the concentration of MO was increased from 10 to 25 ppm. This resulted in a decrease in the reduction percentage, in conjunction with the degree of amination of the aminated chitosan samples. Eventually, a nearly linear reduction rate was achieved with the CS-NH2-4 sample. The second stage of reduction, characterised by a nearly identical decrease in the rate, was observed for all samples of adsorbents, with a recognition threshold of up to 100 ppm. A similar finding was observed where the removal percentage linearly reduced with an increase in initial MO concentration using 4-dimethylamino benzaldehyde chitosan Schiff base and benzophenone chitosan Schiff base. This trend is due to the electrostatic repulsion between the dye molecules with increasing concentration, resulting in a competition between the dye molecules for the limited active sites in the adsorbent^{##UREF##33##48##}.</p>", "<title>Effect of the absorbent dose</title>", "<p id=\"Par27\">Figure ##FIG##10##11## illustrates the impact of varying doses of Chitosan adsorbents on the percentage of MO removal, while keeping the kinetic parameters constant. In general, it was observed that an escalation in the dosage resulted in a proportional rise in the elimination % of MO for both CS and CS-NH2-1 samples, exhibiting a nearly linear relationship. The clearance percentages of MO exhibited a gradual increase when lower rates were applied to the other aminated samples. Ultimately, the adsorption efficiency achieved with 0.3 g of adsorbents exhibits minimal variation, falling within the range of 90–100%. The CS-NH2-4 sample has a plateau effect, commencing at a dosage of 0.2 g, whereby it achieves a clearance percentage of 95%. This phenomenon may be attributed to the observation that, when the starting dye concentration remains constant, increases in the quantity of adsorbent material results in a larger surface area and a greater number of sorption sites^{##UREF##34##49##,##UREF##35##50##}. Similar quite tendency have been reported using other sorbents reported in the previous work^{##REF##30599158##51##}.</p>", "<title>Reusability</title>", "<p id=\"Par28\">The sorption–desorption cycle, as depicted in Fig. ##FIG##11##12##, can be utilised to estimate the recovery of MO absorbed from aqueous solution. The reusability of the CS-NH2-4 adsorbent for removing MO from aqueous solutions was investigated by analysing the sorption–desorption cycles. The cycle was repeated ten times using a sodium hydroxide solution. The figure clearly demonstrates a continuous, nearly linear reduction in MO removal. However, the decrease in MO removal efficiency was not significant; the percentage of MO removal reached 66% in the tenth cycle, compared to 80.1% in the first cycle. Only 18% of the MO removal efficiency was lost after ten cycles of sorption–desorption processes. The CS-NH2-4 adsorbent exhibits favourable sorption–desorption performance and can be confidently used without a noticeable decrease in its sorption capacity for MO removal.</p>", "<title>Comparative adsorption capacity study</title>", "<p id=\"Par29\">Table ##TAB##0##1## presents a comparison of the highest adsorption capacity for MO on the CS-NH2-4 adsorbent in relation to other adsorbents documented in the literature^{##UREF##24##39##–##UREF##26##41##,##UREF##33##48##,##REF##25843838##52##–##UREF##37##55##}. Based on the tabular data, it can be observed that the adsorption capacity of the hydrogels composed of alginate and alginate/poly aspartate is comparatively lower than that of the hydrogels being analysed^{##UREF##37##55##}. The example experienced a reversal upon the utilisation of Calcium alginate MWNTs^{##REF##24751058##53##}, resulting in a six-fold increase in adsorption capacity. In contrast, the adsorption capacity of CS-NH2-4 is comparatively lower when compared to other Chitosan and Chitosan derivatives^{##UREF##24##39##–##UREF##26##41##,##REF##25843838##52##}. Magnetic multi-walled carbon nanotubes (MWCNTs) and bottom ash have also demonstrated elevated adsorption capacity^{##REF##24751058##53##,##REF##17379402##56##}. The modest adsorption capacity of the CS-NH2-4 adsorbent can be attributed to several factors, which can be summarised as follows:<list list-type=\"order\"><list-item><p id=\"Par30\">The scarcity of amine active sites that are protonated for the purpose of adsorption at a pH level of 7.0.</p></list-item><list-item><p id=\"Par31\">The limited expansion of the material at a pH level of 7.0, resulting in a diffusion constraint for the MO (material of interest).</p></list-item><list-item><p id=\"Par32\">The utilisation of the initially available active amine group sites during chemical crosslinking procedures involving Glutaraldehyde. It is advisable to conduct a more comprehensive investigation into the specific parameters of the crosslinking process in order to enhance the adsorption capacity.</p></list-item></list></p>" ]

[ "<title>Conclusion</title>", "<p id=\"Par33\">Novel amino-ethyl Chitosan derivatives (CS-ENH2) with varied and higher amine group content than Chitosan have been developed using one step click chemistry reaction of Chitosan (CS) with 2-chloroethylamine (ENH2) amination reagent, then crosslinked using Glutaraldehyde to form new amino-ethyl Chitosan Schiff bases. The introduced amine groups increased the amine content of Chitosan by 70% which corresponding to increase the cationic exchange capacity of Chitosan (CS) from 7.4 to 12.8 meq/g of CS-ENH2-4 sample. The developed CS-ENH2-4 adsorbent shown 300% adsorption capacity of Methyl Orange (MO) dye solution, 100 ppm, compared to native Chitosan one. The study of adsorption time show fast and linear rate in the first 60 min, then lower rate of adsorption was observed until equilibrium starts to reach at 90 min. The CS-ENH2-4 adsorbent shows an almost constant MO removal percentage over a pH range from 2.0 to 7.0 compared with linear decline of the Chitosan counterpart. Both of the adsorption temperature and agitation speed have the same trend which the MO removal percentage increased along with. The experimental findings indicated that the highest percentage of MO dye removal was achieved under the conditions of pH 2, a temperature of 60 °C, agitation speed of 250 rpm, and adsorption duration of 90 min. Furthermore, the CS-ENH2-4 adsorbent exhibits a favorable potential for reusability, as it only experienced a reduction of 18% of its adsorption effectiveness after undergoing 10 cycles of adsorption–desorption. The structural, morphological, and physiochemical characterization of the developed amino-ethyl Chitosan derivatives underwent by Fourier Transform Infrared spectroscopy (FTIR), Thermal analysis (TGA and DSC), Scanning Electron Microscopy (SEM), and Energy Dispersive X-ray Analysis (EDAX). The later proved the amination process and the adsorption of MO through the increase of the N% by about 70%, in accordance with the results of the ion exchange capacity; 73%, and appearance of new S and Na elements in the analysis of MO-CS-ENH2-4, respectively.</p>" ]

[ "<p id=\"Par1\">The present study introduces a new and straightforward method for the amination of Chitosan. This method involves coupling Chitosan (CS) with 2-chloroethylamine (ENH2) in a single step to produce an amino-ethyl Chitosan derivatives with increased amine group content (CS-ENH2) using click chemistry. The resulting derivatives were then crosslinked using Glutaraldehyde to form amino-ethyl Chitosan Schiff bases. The novel amino-ethyl Chitosan Schiff bases were subsequently utilized as adsorbents for the removal of Methyl Orange (MO) dye from aqueous solutions using a batch technique, and the performance of the produced Schiff bases was compared with that of the native Chitosan Schiff base. The CS-ENH2 adsorbents show improved adsorption capacity up to 300% of the native Chitosan Schiff base with almost double removal rate. The adsorption temperature has a positive impact in general while almost 100% of MO removed at 60 °C using CS-ENH2 adsorbents compared with 66% of the native Chitosan Schiff base adsorbent. The adsorption pH shows a negative impact on the MO removal percent. That effect reduced sharply using the CS-ENH2 adsorbents with higher amination degree while the MO removal percent almost being constant over a wide range of pH; 2.0–7.0. The agitation speed has the same positive effect over all the adsorbents. However, the rate of MO removal percent decreased with increase the agitation speed up to 250 rpm. The experimental findings demonstrated that the highest percentage of MO dye removal was achieved under the conditions of pH 2.0, a temperature of 60 °C, agitation speed of 250 rpm, and adsorption duration of 90 min. These Schiff bases were subsequently characterized using advanced analytical techniques including Fourier Transform Infrared spectroscopy, Thermal analysis (TGA and DSC), and Scanning Electron Microscopy.</p>", "<title>Subject terms</title>", "<p>Open access funding provided by The Science, Technology &amp; Innovation Funding Authority (STDF) in cooperation with The Egyptian Knowledge Bank (EKB).</p>" ]

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[ "<title>Author contributions</title>", "<p>Prof. Dr. A.M.O. and Prof. Dr. T.M.T. have proposed the point of research, design the experimental work, and follow the executive of the work. Prof. Dr. W.A.S. and Prof. Dr. R.A. follow the executive of the work, and supervise the writing of the first draft of the manuscript. Chem. M.M.A.-E. has executed the experiments, drawing the figures and tabulating the data, writes the first draft of the manuscript. Prof. Dr. M.S.M.-E. revises the final version of the manuscript and submits to the journal.</p>", "<title>Funding</title>", "<p>Open access funding provided by The Science, Technology &amp; Innovation Funding Authority (STDF) in cooperation with The Egyptian Knowledge Bank (EKB).</p>", "<title>Data availability</title>", "<p>The datasets used and/or analysed during the current study available from the corresponding author on reasonable request.</p>", "<title>Competing interests</title>", "<p id=\"Par34\">The authors declare no competing interests.</p>" ]

[ "<fig id=\"Fig1\"><label>Figure 1</label><caption><p>Water uptake and ion exchange capacity of Chitosan and amino-ethyl Chitosan hydrogels.</p></caption></fig>", "<fig id=\"Fig2\"><label>Figure 2</label><caption><p>FTIR of Chitosan and amino-ethyl Chitosan hydrogels.</p></caption></fig>", "<fig id=\"Fig3\"><label>Figure 3</label><caption><p>TGA of Chitosan and amino-ethyl Chitosan hydrogels.</p></caption></fig>", "<fig id=\"Fig4\"><label>Figure 4</label><caption><p>DSC of Chitosan and amino-ethyl Chitosan hydrogels.</p></caption></fig>", "<fig id=\"Fig5\"><label>Figure 5</label><caption><p>SEM pictures of Chitosan and amino-ethyl Chitosan hydrogels (<bold>a</bold>–<bold>e</bold>), and EDAX analysis of (<bold>f</bold>) Chitosan and amino-ethyl chitosan derivatives, (<bold>g</bold>) CS, (<bold>h</bold>) CS-MO, (<bold>i</bold>) CS-ENH2-4, and (<bold>j</bold>) CS-ENH2-4-MO.</p></caption></fig>", "<fig id=\"Fig6\"><label>Figure 6</label><caption><p>(<bold>A</bold>) Effect of contact time of adsorption behavior of MO onto Chitosan and amino-ethyl Chitosan hydrogels. (<bold>B</bold>) Correlation between ions exchange capacity of Chitosan and amino-ethyl Chitosan hydrogels and MO removing (%).</p></caption></fig>", "<fig id=\"Fig7\"><label>Figure 7</label><caption><p>Effect of the adsorption temperature on the MO removing (%) using Chitosan and amino-ethyl Chitosan hydrogels.</p></caption></fig>", "<fig id=\"Fig8\"><label>Figure 8</label><caption><p>Effect of the dye pH on the adsorption of MO onto Chitosan and amino-ethyl Chitosan hydrogels.</p></caption></fig>", "<fig id=\"Fig9\"><label>Figure 9</label><caption><p>Effect of the agitation rate on the adsorption of MO onto Chitosan and amino-ethyl Chitosan hydrogels.</p></caption></fig>", "<fig id=\"Fig10\"><label>Figure 10</label><caption><p>Effect of the initial MO dye concentration on the adsorption of MO using Chitosan and amino-ethyl Chitosan hydrogels.</p></caption></fig>", "<fig id=\"Fig11\"><label>Figure 11</label><caption><p>Effect of the adsorbent dose on the adsorption of the MO using Chitosan and amino-ethyl Chitosan hydrogels.</p></caption></fig>", "<fig id=\"Fig12\"><label>Figure 12</label><caption><p>Reusability of the CS-NH2-4 adsorbent in removal of MO dye; [Adsorption conditions; 25 ppm MO, 60 min, pH 7.0, 25 °C, and 0.1 g adsorbent. Desorption conditions; 0.1N NaOH, 25 °C, 60 min].</p></caption></fig>" ]

[ "<table-wrap id=\"Tab1\"><label>Table 1</label><caption><p>Comparison of the Adsorption Capacity of CS-NH2-4 adsorbent with Other Adsorbents.</p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th align=\"left\">Adsorbent</th><th align=\"left\">Adsorption capacity (mg/g)</th><th align=\"left\">References</th></tr></thead><tbody><tr><td align=\"left\">Chitosan—1-vinyl 2-pyrrolidone Schiff base</td><td align=\"left\">20</td><td align=\"left\">^{##UREF##25##40##}</td></tr><tr><td align=\"left\">Chitosan/organic rectorite</td><td align=\"left\">5.56</td><td align=\"left\">^{##REF##25843838##52##}</td></tr><tr><td align=\"left\">Calcium alginate MWNTs</td><td align=\"left\">12.5</td><td align=\"left\">^{##REF##24751058##53##}</td></tr><tr><td align=\"left\">Magnetic MWCNTs</td><td align=\"left\">10.89</td><td align=\"left\">^{##REF##24751058##53##}</td></tr><tr><td align=\"left\">Alginate/polyaspartate hydrogels</td><td align=\"left\">0.22–0.28</td><td align=\"left\">^{##UREF##37##55##}</td></tr><tr><td align=\"left\">Alginate</td><td align=\"left\">0.08–0.28</td><td align=\"left\">^{##UREF##37##55##}</td></tr><tr><td align=\"left\">Bottom ash</td><td align=\"left\">3.618</td><td align=\"left\">^{##REF##17379402##56##}</td></tr><tr><td align=\"left\">Chitosan</td><td align=\"left\">5.54</td><td align=\"left\" rowspan=\"2\">^{##UREF##26##41##}</td></tr><tr><td align=\"left\">Chitosan/4-methoxybenzaldehyde Schiff base</td><td align=\"left\">7.73</td></tr><tr><td align=\"left\">Chitosan</td><td align=\"left\">8.867</td><td align=\"left\" rowspan=\"3\">^{##UREF##24##39##}</td></tr><tr><td align=\"left\">Chitosan/succinimide Schiff base</td><td align=\"left\">10.0</td></tr><tr><td align=\"left\">Chitosan/1-methyl-2-pyrrolidinone Schiff base</td><td align=\"left\">7.2</td></tr><tr><td align=\"left\">Chitosan</td><td align=\"left\">19.92</td><td align=\"left\" rowspan=\"3\">^{##UREF##33##48##}</td></tr><tr><td align=\"left\">Chitosan/4-dimethylamino benzaldehyde Schiff base</td><td align=\"left\">19.34</td></tr><tr><td align=\"left\">Chitosan/benzophenone Schiff base</td><td align=\"left\">22.37</td></tr><tr><td align=\"left\">Chitosan</td><td align=\"left\">2.368</td><td align=\"left\" rowspan=\"2\">This study</td></tr><tr><td align=\"left\">Amino Ethyl Chitosan Hydogel (CS-NH2-4)</td><td align=\"left\">5.0</td></tr></tbody></table></table-wrap>" ]

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{ "acronym": [], "definition": [] }

[ "<title>Introduction</title>", "<p id=\"Par2\">Atrial fibrillation (AF) is the most common sustained arrhythmia and it is a major public health problem worldwide associated with substantial morbidity and mortality^{##REF##8114238##1##,##REF##27057292##2##}. Risk factors for atrial fibrillation include age, hypertension, coronary artery disease, diabetes mellitus (DM), chronic obstructive pulmonary disease (COPD), obstructive sleep apnea (OSA), valvular heart disease, and congestive heart failure^{##REF##12781903##3##–##REF##9337224##6##}.</p>", "<p id=\"Par3\">Among these risk factors, hypertension is the most common cause of AF because hypertension is the most prevalent disease among these conditions^{##REF##27057292##2##,##REF##29285490##7##–##REF##33834045##9##}. In some studies, up to 90% of AF patients are observed to be hypertensive^{##UREF##0##10##}. However, the mechanisms underlying the increased susceptibility to atrial fibrillation in hypertensive patients are not completely understood. Nonetheless, studies that illustrated the development of AF in various risk factors indicated the occurrence of pathophysiological changes in the atrium at a cellular and molecular level, leading to interstitial fibrosis in the atrial tissue. The remodeling of the atrial structure increases the propensity to re-entry atrial tachyarrhythmia^{##REF##20589977##11##–##REF##29759598##15##}.</p>", "<p id=\"Par4\">The renin–angiotensin–aldosterone system (RAAS) may play a role in the occurrence of AF. Angiotensin-II has been shown to regulate cardiac cell proliferation and modulating myocyte ion channels^{##REF##29348255##8##}. Thus, the use of angiotensin converting enzyme (ACE) inhibitors and Angiotensin-II receptor blockers (ARBs) may be effective in the prevention of AF in patients with heart failure^{##REF##29348255##8##,##REF##18456194##16##,##REF##33959029##17##}. However, it is difficult to determine if these agents are effective in the prevention of AF in patients with hypertension. This study aims to investigate the relationship of LA size with the occurrence of AF in hypertensive patients.</p>" ]

[ "<title>Methods</title>", "<title>Study design</title>", "<p id=\"Par5\">This was a retrospective observational cross – sectional study that reviewed records of patients diagnosed with hypertension and admitted to the cardiology department of King Abdullah university hospital (KAUH) in Irbid, Jordan from 1/1/2021 to 31/12/2021. The diagnosis of AF and associated risk factors were obtained from patient’s electronic records. Records included the Electrocardiographic reports (ECG’s), laboratory data, echocardiographic reports and clinical data (history and progress notes). Left atrium (LA) size was measured by transthoracic echocardiography machine (Hp-Sonos 5500, USA), measuring anteroposterior dimensions by M-mode directed by two dimensional (2-D) (real-time) echo in long axis parasternal view measured from LA posterior wall leading edge to leading edge at the level of aortic sinuses.</p>", "<title>Study participants</title>", "<p id=\"Par6\">All patients with hypertension who were admitted to the cardiac units (CCU and IMCU) of KAUH during January to December of 2021 were included. Diagnosis of hypertension was documented from electronic charts based on history of hypertension and being on antihypertensive treatment. Presence of AF was documented from the electrocardiograms and retrieved from patients’ electronic charts and the progress notes of the patients during hospitalization or during the follow up in the outpatient clinics.</p>", "<title>Inclusion and exclusion criteria</title>", "<p id=\"Par7\">Only AF in setting of hypertension was included in this study (144 cases). AF due to rheumatic valvular heart disease (3 cases), pre-excitation (2 cases), AF associated with thyrotoxicosis (1 case), AF associated with chronic Cor pulmonale (2 cases) and AF associated with obstructive sleep apnea (OSA; 1 case) were excluded from the study population, because these cases were not having hypertension.</p>", "<title>Study variables</title>", "<p id=\"Par8\">Patients’ gender, age, and smoking status were reported. Further, patients’ records of average systolic blood pressure, diastolic blood pressure, left atrial size, left ventricular ejection fraction (%), left ventricular dimensions in diastole, left ventricular dimensions in systole, and serum creatinine (µmol/L) were included.</p>", "<p id=\"Par9\">Risk factors for AF (age, sex, diabetes mellitus, coronary artery disease, dilated cardiomyopathies (DCM), congenital heart disease, valvular heart disease, Cor-pulmonale and congestive cardiac failure) were also retrieved from electronic charts of the patients and registered on an excel sheet.</p>", "<title>Analysis</title>", "<p id=\"Par10\">The baseline characteristics of the patients with atrial fibrillation and those without were compared using students <italic>t</italic> test for continuous variables, and chi-square test (X²) for categorical variables. Furthermore, binary logistic regression was used to predict the occurrence of atrial fibrillation based on age, left atrial size, coronary artery disease, left ventricular ejection fraction, left ventricular dimensions in systole and diastole, and heart failure with the occurrence of atrial fibrillation after controlling for gender, smoking, and diabetes. Analysis was performed using the statistical package (“SPSS” software version 23), and a p-value less than 0.05 was considered the level of significance.</p>", "<title>Ethics approval and consent to participate</title>", "<p id=\"Par11\">This study was approved by the institutional review board of Jordan University of Science and Technology (number 124-2022). The requirement for informed consent from the study subjects was waived by the IRB of (Jordan University of Science and Technology/Research committee) due to the retrospective study design. Only patients’ file number were extracted with the data and no names or identifiable information was included. In addition, the committee ensured that all methods used in this research was performed in accordance with relevant guidelines/regulations.</p>" ]

[ "<title>Results</title>", "<p id=\"Par12\">There was a total of 958 hypertensive patients included in the study. The mean age of study patients was 61.40 (± 11.46) years, most of them were males (65.4%), and 40.3% were current smokers. There was a high prevalence of coronary artery disease (CAD) and diabetes mellitus (DM) among participating patients (51.3% and 59.2%, respectively) and about 1 out of each 5 patients had HF (Table ##TAB##0##1##).</p>", "<p id=\"Par13\">Patients with AF represented 15% of the sample (n = 144). Among AF patients, there were 6 patients (4.2%) with valvular health diseases, 10 (6.9%) with Cor pulmonale or COPD, and 6 patients (4.2%) with obstructive sleep apnea (OSA). Those patients had only one of these condition, except 3 patients who had both Cor pulmonale and OSA.</p>", "<p id=\"Par14\">There was a significant statistical difference between patients with AF and patients without AF for the means of the following measured variables: LA size, LVEF%, LV dimensions in diastole and systole (P-values &lt; 0.05). Serum creatinine mean levels were not statistically different between patients with AF and those without AF (Table ##TAB##0##1##).</p>", "<p id=\"Par15\">The binary logistic regression model demonstrated a significant relationship of age, LA size, CAD, and HF with the occurrence of AF after controlling for gender, smoking, and diabetes. For each year increase in age, the probability of AF occurrence increased by 4.7% [OR = 1.047, 95% CI   1.026–1.069, p = value &lt; 0.0001]. In addition, the risk of AF increased 3.2 times more among patients with increased LA size [OR = 3.204, 95% CI   1.749–5.870, p = value &lt; 0.0001]. There was an increased risk of AF by 16% for each 1 cm increase in LA size, as noticed in Table ##TAB##1##2##.</p>", "<p id=\"Par16\">The reasons of admission to cardiac units (CCU and IMCU) during the year (January 1st to December 31st of 2021) are illustrated in Table ##TAB##2##3##.</p>" ]

[ "<title>Discussion</title>", "<p id=\"Par17\">Atrial fibrillation is associated with several risk factors and predisposing conditions that increases its occurrence, including hypertension that accelerates the development and progression of AF^{##REF##18060995##18##}. The prevalence of AF in this current study was 15%. This rate is similar to a study that showed prevalence of AF in hypertensive patients was equal to 14.9%. In that study, there were 9474 hypertensive patients followed for a median period of 24.1 years and a total of 1414 cases of AF (14.9%) was identified during the follow-up period^{##REF##33834045##9##}.</p>", "<p id=\"Par18\">Most of the studies has identified hypertension as a major contributor to the development of AF, and therefore, the study population of this study were all hypertensive patients in order to identify other intrinsic cardiac conditions that could affect the occurrence of AF. The American Heart Association (AHA) listed factors that are associated with AF development, including advanced age, a prolonged uncontrolled hypertension, underlying heart disease such as valve problems or hypertrophic cardiomyopathy, family history, and other chronic conditions like diabetes or sleep apnea^{##UREF##1##19##}. In the current study, age, heart disease, CAD, and LA size were significantly associated with a higher likelihood of AF occurrence among hypertensive patients. Similarly, the Framingham heart study reported that the incidence of AF increased with age, heart failure, coronary heart disease, among other factors. AF was almost doubled every 10-year increment in age. The significant relationship of age with AF was shown in other studies as well^{##REF##16527828##20##,##REF##28592961##21##}. In addition, the relationship of coronary heart disease and heart failure with AF was well established in previous Framingham heart studies and other studies, which was significant even with adjustment for age and gender^{##REF##18060995##18##}.</p>", "<p id=\"Par19\">There are few studies that investigated the relationship of atrial or ventricular size with the occurrence or the progression of AF. However, Kannel et al. in 2 scoping review studies illustrated that echocardiographic predictors of AF included left atrial enlargement, left ventricular fractional shortening, left ventricular wall thickness, and mitral annular calcification^{##REF##9809895##5##,##REF##18060995##18##}. Left atrial size was an independent predictor for persistent AF, even when factors like age and gender were controlled^{##REF##29285490##7##,##REF##33875748##22##}, which is similar to the findings of the current study. Further, Gerdts et al. had demonstrated that left ventricular hypertrophy and LA enlargement attributed to higher rate of AF in hypertensive patients^{##REF##11897755##23##}.</p>", "<p id=\"Par20\">The occurrence of AF seems to be complex in nature due to the several risk factors that could play a role and impose a multifactorial effect on the disease. Moreover, the structural intrinsic changes of the heart that is likely to progress with the advancement of age contribute to the development of AF. While a patient with chronic conditions, such as diabetes or hypertension, advances in age, progressive changes manifest in LA anatomy and function, may promote AF^{##REF##29348255##8##}.</p>", "<p id=\"Par21\">Various inflammatory markers and mediators such as C-reactive protein (CRP), tumor necrosis factor (TNF-a), interleukin -2 (IL-2), interleukin – 6 (IL-6), interleukin – 8 (IL-8), and monocyte chemoattractant protein—I (MCP – I) have been linked to the development and outcome of AF^{##REF##29759598##15##,##REF##11739301##24##–##REF##20455973##28##}.</p>", "<p id=\"Par22\">Animal models and human studies revealed a strong relationship between atrial myopathy and incidence of AF. Atrial myopathy as characterized by atrial fibrotic remodeling leading to electrical and autonomic changes which facilitate the development of AF^{##REF##31768479##29##}. Atrial myopathy leads to structural and electrophysiological changes of the left atrium (i.e. inflammation, oxidative stress, stretch, and fibrosis) which in turn progress into electrical and autonomic remodeling and prothrombotic state^{##REF##31768479##29##}.</p>", "<p id=\"Par23\">Diagnosis of atrial myopathy entails the usage of several tools, such as electrical (ECG), echocardiography, laboratory tests (inflammatory biomarkers), tissue biopsy, and 4 – dimensional magnetic resonance imaging (4-D MRI). The clinical implications of diagnosing atrial myopathy by these means may assist clinicians to consider anticoagulation therapy, even before the onset of AF, for selected subgroups of patients with low risk of strokes. Therefore, Determining the LA size, echocardiographically, can be considered a clinical risk identifier in preclinical AF which should be included in routine comprehensive echocardiography evaluation^{##REF##23194937##27##}.</p>", "<title>Study limitations</title>", "<p id=\"Par24\">Since this study is a retrospective study, we could not identify paroxysmal AF from chronic permanent AF. Another limitation is that we were not able to include the left ventricular mass index because the diastolic posterior wall thickness was not measured in the majority of patients.</p>" ]

[ "<title>Conclusion</title>", "<p id=\"Par25\">Findings suggest that LA enlargement in hypertensive patients play a major role in the development of AF through structural and electrophysiological changes to the atrium and other possible mechanisms^{##REF##28592961##21##}. Although there are several risk factors associated with the development of AF, chronic conditions and advancement of age remains the most critical detrimental factors in the onset and progression of AF. Moreover, large scale randomized studies are needed to establish a definitive role of antihypertensive therapy in reducing AF incidence.</p>" ]

[ "<p id=\"Par1\">Atrial fibrillation (AF) is the most common sustained arrhythmia and it is a major public health problem worldwide. Hypertension is one of the major risk factors for the development of AF. This study is carried out to determine the prevalence and independent risk factors for atrial fibrillation (AF) in hypertensive patients and to evaluate the relationship of AF with left atrial size. This is a retrospective observational cross – sectional study that used a retrospective electronic chart review of all admitted patients to cardiology department at King Abdullah university hospital (KAUH) in Irbid, Jordan, with a diagnosis of hypertension along with various acute cardiac admissions, including AF during 1-year period (January 1st to December 31 of 2021). Risk factors for AF (age, sex, DM, coronary artery disease, valvular heart disease, Cor-pulmonale, obstructive sleep apnea, and congestive cardiac failure) were retrieved from electronic charts of the patients. A total of 958 patients were admitted to the coronary care unit (CCU) and intermediate care unit (IMCU) during a 1-year period. Among them, 276 had 2 or 3 admissions. The main reason of admission was acute coronary syndrome (n = 491), heart failure (n = 180), and AF (n = 144), indicating AF prevalence of 15%. However, there were 40 patients with combined causes. All patients in the study (n = 958) were diagnosed with hypertension, including patients with atrial fibrillation (n = 144). The mean age of patients was 61.4 (± 11.46) years, and approximately two thirds of them were males (65.4%). The binary logistic regression model demonstrated a significant statistical relationship of age, left atrial size, coronary artery disease, left ventricular ejection fraction, left ventricular dimensions in systole and diastole, and heart failure with the occurrence of AF after controlling for gender, smoking, and diabetes. Findings indicate that left atrial size plays a significant role in the development of AF in patients with hypertension. However, the prevalence of AF significantly increased with advancing age in both sexes because of increased left ventricular hypertrophy, which leads to increased left atrial size.</p>", "<title>Subject terms</title>" ]

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[ "<title>Author contributions</title>", "<p>A.S. and R.S. conceptualization and study design. A.S. and B.J. methodology and supervised data collection. T.Z., S.G. and T.A. data collection and validation. R.S. analysis. A.S. and R.S. writing the manuscript.</p>", "<title>Data availability</title>", "<p>The datasets used and/or analyzed during the current study available from the corresponding author on reasonable request.</p>", "<title>Competing interests</title>", "<p id=\"Par26\">The authors declare no competing interests.</p>" ]

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[ "<table-wrap id=\"Tab1\"><label>Table 1</label><caption><p>Demographic and other characteristics of patients.</p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th align=\"left\">Characteristics</th><th align=\"left\">Atrial fibrillation</th><th align=\"left\">No atrial fibrillation</th><th align=\"left\">Total*</th><th align=\"left\">P-value</th></tr></thead><tbody><tr><td align=\"left\">Female</td><td align=\"left\">68 (20.5%)</td><td align=\"left\">263 (79.5%)</td><td align=\"left\">331 (34.6%)</td><td align=\"left\" rowspan=\"2\"><bold>0.001</bold></td></tr><tr><td align=\"left\">Male</td><td align=\"left\">76 (12.1%)</td><td align=\"left\">551 (87.9%)</td><td align=\"left\">627 (65.4%)</td></tr><tr><td align=\"left\">Total</td><td align=\"left\">144</td><td align=\"left\">814</td><td align=\"left\">958</td><td align=\"left\"/></tr><tr><td align=\"left\">Age</td><td align=\"left\">(67.97 ± 10.68)</td><td align=\"left\">(60.24 ± 11.21)</td><td align=\"left\">(61.40 ± 11.46)</td><td align=\"left\"><bold> &lt; 0.001</bold></td></tr><tr><td align=\"left\">Smoking (yes)</td><td align=\"left\">56 (14.5%)</td><td align=\"left\">330 (85.5%)</td><td align=\"left\">386 (40.3%)</td><td align=\"left\" rowspan=\"2\">0.71</td></tr><tr><td align=\"left\">Smoking (no)</td><td align=\"left\">88 (15.4%)</td><td align=\"left\">484 (84.6%)</td><td align=\"left\">572 (59.7%)</td></tr><tr><td align=\"left\">Total</td><td align=\"left\">144</td><td align=\"left\">814</td><td align=\"left\">958</td><td align=\"left\"/></tr><tr><td align=\"left\">Coronary artery disease (yes)</td><td align=\"left\">97 (19.8%)</td><td align=\"left\">394 (80.2%)</td><td align=\"left\">491 (51.3%)</td><td align=\"left\" rowspan=\"2\"><bold> &lt; 0.001</bold></td></tr><tr><td align=\"left\">Coronary artery disease (no)</td><td align=\"left\">47 (10.1%)</td><td align=\"left\">420 (89.9%)</td><td align=\"left\">467 (48.7%)</td></tr><tr><td align=\"left\">Total</td><td align=\"left\">144</td><td align=\"left\">814</td><td align=\"left\">958</td><td align=\"left\"/></tr><tr><td align=\"left\">Heart failure (yes)</td><td align=\"left\">74 (41.1%)</td><td align=\"left\">106 (58.9%)</td><td align=\"left\">180 (18.8%)</td><td align=\"left\" rowspan=\"2\"><bold> &lt; 0.001</bold></td></tr><tr><td align=\"left\">Heart failure (no)</td><td align=\"left\">70 (9.0%)</td><td align=\"left\">708 (91.0%)</td><td align=\"left\">778 (81.2%)</td></tr><tr><td align=\"left\">Total</td><td align=\"left\">144</td><td align=\"left\">814</td><td align=\"left\">958</td><td align=\"left\"/></tr><tr><td align=\"left\">Diabetes (yes)</td><td align=\"left\">89 (15.7%)</td><td align=\"left\">478 (84.3%)</td><td align=\"left\">567 (59.2%)</td><td align=\"left\" rowspan=\"2\">0.488</td></tr><tr><td align=\"left\">Diabetes (no)</td><td align=\"left\">55 (14.1%)</td><td align=\"left\">336 (85.9%)</td><td align=\"left\">391 (40.8%)</td></tr><tr><td align=\"left\">Total</td><td align=\"left\">144</td><td align=\"left\">814</td><td align=\"left\">958</td><td align=\"left\"/></tr><tr><td align=\"left\">Systolic blood pressure</td><td align=\"left\">(131.27 ± 20.55)</td><td align=\"left\">(133.13 ± 19.04)</td><td align=\"left\">(132.85 ± 19.28)</td><td align=\"left\">0.313</td></tr><tr><td align=\"left\">Diastolic blood pressure</td><td align=\"left\">(76.33 ± 10.13)</td><td align=\"left\">(78.09 ± 10.61)</td><td align=\"left\">(77.82 ± 10.55)</td><td align=\"left\">0.066</td></tr><tr><td align=\"left\">Left atrial size</td><td align=\"left\">(4.2 ± 0.461)</td><td align=\"left\">(3.91 ± 0.297)</td><td align=\"left\">(3.96 ± 0.342)</td><td align=\"left\"><bold> &lt; 0.001</bold></td></tr><tr><td align=\"left\">Left ventricular ejection fraction (%)</td><td align=\"left\">(47.85 ± 11.63)</td><td align=\"left\">(52.0 ± 9.0)</td><td align=\"left\">(51.40 ± 9.55)</td><td align=\"left\"><bold> &lt; 0.001</bold></td></tr><tr><td align=\"left\">LV dimension in diastole</td><td align=\"left\">(5.32 ± 0.537)</td><td align=\"left\">(5.17 ± 0.491)</td><td align=\"left\">(5.19 ± 0.501)</td><td align=\"left\"><bold>0.002</bold></td></tr><tr><td align=\"left\">LV dimension in systole</td><td align=\"left\">(4.04 ± 0.732)</td><td align=\"left\">(3.85 ± 0.612)</td><td align=\"left\">(3.88 ± 0.634)</td><td align=\"left\"><bold>0.005</bold></td></tr><tr><td align=\"left\">Serum Creatinine µmol/L</td><td align=\"left\">(141.21 ± 118.0)</td><td align=\"left\">(118.32 ± 126.45)</td><td align=\"left\">(121.41 ± 125.04)</td><td align=\"left\"><bold>0.044</bold></td></tr></tbody></table></table-wrap>", "<table-wrap id=\"Tab2\"><label>Table 2</label><caption><p>The relationship of atrial fibrillation with variables of interest using binary logistic regression.</p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th align=\"left\">Parameter</th><th align=\"left\">Estimate</th><th align=\"left\">Standard error</th><th align=\"left\">Odds ratio</th><th align=\"left\" colspan=\"2\">Wald 95% confidence limits</th><th align=\"left\">P value</th></tr></thead><tbody><tr><td align=\"left\">LVEF% (average)</td><td char=\".\" align=\"char\">– 0.033</td><td char=\".\" align=\"char\">0.024</td><td char=\".\" align=\"char\">0.968</td><td char=\".\" align=\"char\">0.923</td><td char=\".\" align=\"char\">1.015</td><td char=\".\" align=\"char\">0.183</td></tr><tr><td align=\"left\">LV dimension in diastole</td><td align=\"left\">– 0.244</td><td char=\".\" align=\"char\">0.427</td><td char=\".\" align=\"char\">0.784</td><td char=\".\" align=\"char\">0.340</td><td char=\".\" align=\"char\">1.808</td><td char=\".\" align=\"char\">0.568</td></tr><tr><td align=\"left\">LV dimension in systole</td><td align=\"left\">– 0.741</td><td char=\".\" align=\"char\">0.417</td><td char=\".\" align=\"char\">0.476</td><td char=\".\" align=\"char\">0.210</td><td char=\".\" align=\"char\">1.079</td><td char=\".\" align=\"char\">0.076</td></tr><tr><td align=\"left\">Age</td><td align=\"left\">0.046</td><td char=\".\" align=\"char\">0.011</td><td char=\".\" align=\"char\">1.047</td><td char=\".\" align=\"char\">1.026</td><td char=\".\" align=\"char\">1.069</td><td char=\".\" align=\"char\"> &lt; 0.0001</td></tr><tr><td align=\"left\">LA size (largest)</td><td align=\"left\">1.164</td><td char=\".\" align=\"char\">0.309</td><td char=\".\" align=\"char\">3.204</td><td char=\".\" align=\"char\">1.749</td><td char=\".\" align=\"char\">5.870</td><td char=\".\" align=\"char\"> &lt; 0.0001</td></tr><tr><td align=\"left\">Coronary artery disease</td><td align=\"left\">0.722</td><td char=\".\" align=\"char\">0.227</td><td char=\".\" align=\"char\">2.058</td><td char=\".\" align=\"char\">1.319</td><td char=\".\" align=\"char\">3.211</td><td char=\".\" align=\"char\">0.001</td></tr><tr><td align=\"left\">Heart failure</td><td align=\"left\">1.743</td><td char=\".\" align=\"char\">0.267</td><td char=\".\" align=\"char\">5.715</td><td char=\".\" align=\"char\">3.389</td><td char=\".\" align=\"char\">9.637</td><td char=\".\" align=\"char\"> &lt; 0.0001</td></tr></tbody></table></table-wrap>", "<table-wrap id=\"Tab3\"><label>Table 3</label><caption><p>Reasons of patients’ admissions in the cardiac units of KAUH.</p></caption><table frame=\"hsides\" rules=\"groups\"><tbody><tr><td align=\"left\" rowspan=\"10\">Reason of admission – all causes (n = 958)</td><td align=\"left\">Acute coronary syndrome (ACS)</td><td align=\"left\">N = 491</td></tr><tr><td align=\"left\">Heart failure (HF)</td><td align=\"left\">N = 180</td></tr><tr><td align=\"left\">Atrial fibrillation</td><td align=\"left\">N = 144</td></tr><tr><td align=\"left\">Supraventricular tachycardia (VT)</td><td align=\"left\">N = 75</td></tr><tr><td align=\"left\">Hypertensive crises or uncontrolled hypertension</td><td align=\"left\">N = 22</td></tr><tr><td align=\"left\">Ventricular premature contraction</td><td align=\"left\">N = 16</td></tr><tr><td align=\"left\">Complete heart block or mobitz type 2 heart block</td><td align=\"left\">N = 12</td></tr><tr><td align=\"left\">HF + AF</td><td align=\"left\">N = 9</td></tr><tr><td align=\"left\">Uncontrolled BP with HF</td><td align=\"left\">N = 5</td></tr><tr><td align=\"left\">VT with HF</td><td align=\"left\">N = 4</td></tr><tr><td align=\"left\" colspan=\"2\">Total hypertensive patients</td><td align=\"left\">N = 958</td></tr><tr><td align=\"left\" colspan=\"2\">Re-admissions: twice or 3 times</td><td align=\"left\">N = 276</td></tr></tbody></table></table-wrap>" ]

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[ "<table-wrap-foot><p>Categorical variables were analyzed using chi – square test of independence.</p><p>Continuous variables were analyzed using student t-test.</p><p>*The percentage in total column represents total percentage of each row.</p><p>Significant values are in bold.</p></table-wrap-foot>", "<table-wrap-foot><p>The model controlled for the following variables: gender, smoking, and diabetes.</p></table-wrap-foot>", "<fn-group><fn><p><bold>Publisher's note</bold></p><p>Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.</p></fn></fn-group>" ]

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[ "<title>Conclusion and outlook</title>", "<p id=\"Par20\">There is ample evidence that efferocytosis by neutrophils plays an important role in the response to dead cell accumulation. During inflammation, they contribute to the clearance of aggregates of apoptotic neutrophils. In cancer, they participate in the removal of dead tumour cell aggregates. The neutrophil response after engulfing apoptotic cells contributes to the resolution of inflammation and tissue regeneration. However, in the case of cancer, this can be harmful. A meta-analysis of expression signatures from more than 18,000 human tumours found that neutrophils are the tumour-associated cell type linked with the worst prognosis [##REF##26193342##58##]. Neutrophilic efferocytosis might contribute to this situation. As professional phagocytes, neutrophils have the full machinery for engulfment and express many receptors for the detection and binding of dead cells. However, it is also still unclear whether neutrophils distinguish between the different types of cell death from which their target cells have died. Although an increasingly clear picture is emerging on efferocytosis by neutrophils, there are still many unanswered questions awaiting exploration.</p>" ]

[ "<p id=\"Par1\">When a cell dies of apoptosis, it is eliminated either by neighbouring cells or by attracted professional phagocytes. Although it was generally believed that neutrophils also have the ability to perform efferocytosis, their contribution to the clearance of apoptotic cells was considered less important compared with macrophages. Therefore, this ability of neutrophils remained unexplored for a long time. Over the past decade, it has been shown that during inflammation, neutrophils contribute significantly to the clearance of apoptotic neutrophils that accumulate in large numbers at the site of tissue damage. This “neutrophil cannibalism” is accompanied by inhibition of pro-inflammatory activities of these cells, such as respiratory burst and formation of neutrophil extracellular traps (NETs). Furthermore, efferocytosing neutrophils secrete anti-inflammatory mediators and mitogens including hepatocyte growth factor (HGF), fibroblast growth factor 2 (FGF2), vascular endothelial growth factors (VEGF), and transforming growth factor beta (TGFβ). Thus, efferocytosis by neutrophils is involved in resolution of inflammation. Recent research indicates that it plays also a role in cancer. Many different solid tumours contain aggregates of dead tumour cells that have undergone spontaneous apoptosis. Their extent correlates with poor clinical outcome in most cancer types. These clusters of apoptotic tumour cells are strongly infiltrated by tumour-associated neutrophils (TANs) that acquired an anti-inflammatory and pro-resolving polarization state. This review summarizes the potential consequences discussed in the current literature. Although the picture of the role of efferocytosis by neutrophils in inflammation and cancer is becoming clearer, many questions are still unexplored.</p>", "<title>Subject terms</title>" ]

[ "<title>FACTS</title>", "<p id=\"Par2\">\n<list list-type=\"bullet\"><list-item><p id=\"Par3\">Apoptotic cells release “find-me” signals which attract predominantly neutrophils.</p></list-item><list-item><p id=\"Par4\">Neutrophils accumulate in areas of tissue damage and apoptotic tumour cells.</p></list-item><list-item><p id=\"Par5\">After engulfment of apoptotic cells, neutrophils block respiratory burst and NETosis.</p></list-item><list-item><p id=\"Par6\">Efferocytosing neutrophils secrete a variety of soluble mediators such as cytokines, chemokines, and mitogens which create a pro-resolving and tumorigenic microenvironment.</p></list-item></list>\n</p>", "<title>OPEN QUESTIONS</title>", "<p id=\"Par7\">\n<list list-type=\"bullet\"><list-item><p id=\"Par8\">Several of the molecules known to be involved in the process of efferocytosis in macrophages are also expressed in neutrophils, but for many of them there is still a lack of evidence that they also fulfil this function there.</p></list-item><list-item><p id=\"Par9\">The exact mechanism by which neutrophils adopt a regenerative and tumourigenic phenotype after the uptake of apoptotic cells is still largely unexplored.</p></list-item></list>\n</p>", "<title>Neutrophils in inflammation</title>", "<p id=\"Par10\">Neutrophils represent the first line of cellular innate immune response to infection and tissue damage. Recent evidence indicate that this short-lived myeloid cell population exhibits a great phenotypic and functional diversity [##REF##31160207##1##]. It not only plays an important role in triggering inflammation in reaction to pathogens, but also contributes to its subsequent resolution after their clearance. Neutrophils accumulate quickly at the site of tissue damage through a multi-step process called “neutrophil swarming” [##REF##27558329##2##]. Damage-associated molecular patterns (DAMPs) activate resident cells to release short-range chemoattractants for neutrophils. Pioneer neutrophils from around the damage site migrate to the tissue injury within minutes. The contact with pathogen-associated molecular patterns (PAMPs) stimulates them to deploy a plethora of antimicrobial weapons [##REF##30530726##3##]. They form neutrophil extracellular traps (NETs) to entrap invading pathogens. They release a variety of antimicrobial and pro-inflammatory molecules from their granules and produce reactive oxygen species to kill bacteria. Finally, they clear pathogens by phagocytosis. Neutrophil-derived leukotriene B4 (LTB4) enhances the radius of recruitment of further neutrophils from distant tissue sites [##REF##27558329##2##]. Notably, neutrophils also support the resolution of inflammation right from the start. They release anti-inflammatory, resolving and angiogenic mediators such as IL-10, transforming growth factor β (TGFβ), lipoxin 4A, resolvins, protectins, defensins, and vascular endothelial growth factor (VEGF) [##REF##31205028##4##]. Neutrophils that have engulfed pathogens die through phagocytosis-induced apoptosis [##UREF##0##5##]. Cytokine receptors such as IL-1R on their surface, which are no longer functional, scavenge their pro-inflammatory ligands from the microenvironment [##REF##27021499##6##]. Neighbourhood macrophages that phagocytose dying neutrophils adopt an anti-inflammatory, resolving and reparative M2-like phenotype [##REF##23592557##7##–##REF##32351713##9##]. Such resolving mechanisms begin to gain the upper hand as soon as the infection is pushed back. The accumulation of neutrophils at the site of tissue damage thus enables the restructuring of the extracellular matrix, the formation of dense aggregates that seal the wound tightly, and finally the initiation of tissue repair processes. It must be emphasized that signals of tissue damage are sufficient to trigger the attraction of neutrophils, which accordingly can also be observed in response to sterile injury without pathogens [##REF##30530726##3##].</p>", "<title>Efferocytosis by neutrophils</title>", "<p id=\"Par11\">Especially during the early phase of neutrophil swarming, the number of resident macrophages is still very low and probably insufficient to clear all apoptotic neutrophils. Kristina Rydell-Törmänen showed in a mouse model of sterile lung inflammation that almost 50% of neutrophils at the side of injury have phagosomes that contained material from other neutrophils [##REF##27551482##10##]. The authors termed this process “neutrophil cannibalism”. The efferocytotic capacity of neutrophils is similar to that of blood derived DCs, but clearly lower as compared to blood-derived macrophages [##REF##19949068##11##]. It increases in response to pro-inflammatory cytokines such as TNF-α, interferon-gamma (IFN-γ) and granulocyte-macrophage colony-stimulating factor (GM-CSF) and to ligands of TLR2 (Malp2, Pam3CSK4), TLR4 (LPS), TLR7/TLR8 (R848), and TLR9 (ODN 2006) [##REF##19949068##11##, ##REF##22125470##12##]. The efferocytotic ability of neutrophils is not exclusively limited to apoptotic conspecifics but also includes remnants of other cell types.</p>", "<title>Detection of apoptotic cells</title>", "<p id=\"Par12\">Apoptotic cells in general release various “find-me” signals that specifically attract neutrophils, including CCL3, CXCL1, CXCL5, CXCL8/IL8, tyrosyl tRNA synthetase (TyrRS) and endothelial monocyte activating polypeptide II (EMAPII) [##REF##23434371##13##–##REF##35121727##15##] (Fig. ##FIG##0##1##). This suggests that neutrophils may be intentionally recruited to help clear apoptotic cell debris. It has to be noted that apoptotic cells release also lactoferrin, which inhibits neutrophil migration [##REF##19033648##16##]. However, this “keep-out” signal seems not to be sufficient to antagonize other chemoattractants in vivo. Garg and co-workers induced apoptotic cell death in a lung carcinoma cell line before injecting them intradermally into the mice ear pinna [##REF##28234357##17##]. Cells exposed to the immunogenic apoptosis inducer mitoxantrone stimulated rapid recruitment of neutrophils, which in comparison to other leucocyte subsets constituted the predominant immune cell population accumulating at sites of apoptosis. Similarly, neutrophils accumulate at sites of apoptotic hepatocytes of patients with hepatocellular carcinoma [##REF##37728612##18##].</p>", "<p id=\"Par13\">The predominant “eat me” signal on the surface of apoptotic cells is phosphatidylserine (PS). There is an extensive literature on the different mechanisms that macrophages use to detect PS-positive cell debris (reviewed in [##REF##24481336##8##, ##REF##31822793##19##]). They include directly binding receptors such as adhesion G protein-coupled receptor B1 (ADGRB1), stabilin-2 or T-cell membrane protein 4 (TIM-4). Furthermore, PS is also detected indirectly via soluble “bridging factors” like growth-arrest-specific gene-6 (GAS6) or Milk fat globule-EGF factor 8 protein (MFG-E8), which bind to PS and are then themselves detected by specific receptors on the macrophage. The MFG-E8 receptor αVβ3 integrin is also highly expressed in neutrophils [##REF##25632307##20##]. However, neutrophils do not express any direct PS receptor (ADGRB1, stabilin-2, or TIM-4) or any receptor of GAS6. Besides PS, there are also other ‘eat-me’ signals exposed on apoptotic cells, including calreticulin, annexin A1, thrombospondin 1 binding sites, and complement proteins C1q or C3b binding sites [##REF##35650427##21##]. They are recognized by CD91, formyl peptide receptor 2, CD47, CD93, and CD35, respectively. All of them are highly expressed in neutrophils [##UREF##1##22##–##REF##33644062##26##]. However, their functional role in efferocytosis by these cells is still unexplored.</p>", "<title>Clearance of apoptotic cells</title>", "<p id=\"Par14\">The subsequent events in efferocytosis comprise the engulfment of apoptotic cellular corpses, followed by the formation and maturation of the phagosome, culminating with the degradation of the cargo within the phagolysosomal compartment. The molecular mechanisms of this tightly regulated multi-step process have been investigated in detail in macrophages (reviewed in [##REF##32251387##27##]). After activation of “eat-me” receptors, the submembranous actin cortex undergoes specific rearrangements which activates the Rho family of small GTPase RAC1 and promotes the formation of a phagosome [##REF##32251387##27##, ##REF##26586571##28##]. The processing of engulfed cellular cargo requires a non-canonical LC3-asscociated phagocytosis (LAP) [##REF##18097414##29##]. LAP represents a specialized mechanism that utilizes components of the autophagic machinery to enhance the degradation of phagocytosed material in an immunologically silent manner [##REF##21969579##30##]. LAP is triggered by the recruitment of LC3 (microtubule-associated protein 1A/1B-light chain 3) proteins to the single-membrane phagosomes (or LAPosome). For that, PI3KC3 complex needs to be assembled, which consists of Rubicon, vps34, beclin-1, and vps15. This complex converts the LAPosome-bound phosphatidylinositol into the signalling lipid phosphatidylinositol 3-phosphate (PI3P) [##REF##32251387##27##, ##REF##26586571##28##]. The PI3P-coated LAPosome stabilizes the NOX2 complex which is responsible for ROS generation, leading to LC3 ligation machinery activation and LC3-II recruitment to the LAPosome [##REF##26098576##31##]. This last step facilitates the LAPosme-lysosome fusion, resulting in a rapid degradation of the cargo. Recently, Prajsnar et al. identified the LAP machinery in neutrophils, but unfortunately its activation upon efferocytosis of apoptotic cells has not been investigated [##REF##32174246##32##].</p>", "<p id=\"Par15\">Cunha et al. noted that engulfment of apoptotic corpses per se does not result in immunosuppression, but it is rather the subsequent accumulation of digested products which induces immune tolerance [##REF##30245008##33##]. The phagocytes overload with lipids and cholesterol stimulates the activation of nuclear steroid receptors from the liver X receptors (LXRs) and peroxisome proliferator-activated receptors (PPARs) families [##REF##19646905##34##, ##REF##19838202##35##]. Apart from mediating lipid homoeostasis, LXRs and PPARs induce the clearance of apoptotic cells via expression of phagocytic receptors and opsonins, resembling a positive feedback loop. The anti-inflammatory effects attributed to efferocytosis are also mediated by these pathways, by promoting upregulation of the anti-inflammatory cytokines TGFβ and IL-10 whereas the pro-inflammatory cytokines TNFα, IL-1β, and IL-6 are downregulated [##REF##31822793##19##]. A similar response has been observed in neutrophils that had phagocytosed apoptotic neutrophils [##REF##19949068##11##, ##REF##23280587##36##]. They showed an elevated expression of anti-inflammatory TGFβ and of neutrophil chemoattractants CXCL1 and CXCL8/IL8, and a lower secretion of pro-inflammatory cytokines TNFα and CXCL10/IP-10. Furthermore, they downregulate respiratory burst due to a reduced phosphorylation of p38 MAPK and PKCδ, the kinases involved in NADPH oxidase activation [##REF##28187163##37##]. Incubation with anti-TGFβ1 antibodies restores respiratory burst [##REF##23280587##36##]. The inhibitory effect of neutrophil cannibalism on respiratory burst is exploited by invading bacteria to their own advantage. For instance, <italic>Leishmania major</italic>-infected neutrophils acquire enhanced capacity to engulf apoptotic cells. The uptake of apoptotic cells inhibits respiratory burst, protecting thereby the bacteria [##REF##28187163##37##]. Manfredi et al. found that MPO and elastase are translocated into phagolysosomes during the process of efferocytosis to facilitate cargo degradation, making these enzymes unavailable for participating in chromatin decondensation – a prerequisite for NET formation [##REF##29515586##38##]. Thus, neutrophilic efferocytosis impedes NETosis and primes these cells towards a non-inflammatory and resolving phenotype. We could demonstrate recently that neutrophils engulf apoptotic cell-derived extracellular vesicles (aEV) from hepatocytes and several cancer cell lines [##REF##35985549##39##]. This is associated with an increase of cell surface activation markers CD11b, CD16, CD45, CD66b, CD62L, and secretion of various mitogens, including hepatocyte growth factor (HGF), fibroblast growth factor 2 (FGF2), VEGF, and transforming growth factor alpha (TGFα). Neutrophils express HGF mRNA and store the active protein in secretory vesicles and gelatinase granules [##REF##11929792##40##]. The release of HGF and other mitogens in response to aEV results in an elevated metabolic activity and proliferation of co-cultured hepatocytes [##REF##35985549##39##]. This indicates that efferocytosing neutrophils induce tissue regeneration in response to an uptake of apoptotic cells. Strong corroboration for this hypothesis comes from an observation in patients undergoing partial hepatectomy, a surgical procedure that results in massive local apoptosis at the resection margins of the remaining liver lobes [##REF##35985549##39##]. Free HGF as well as neutrophil-bound HGF in the circulation of these patients correlate with the degree of apoptosis. Notably, higher levels of HGF are associated with improved liver regeneration.</p>", "<title>Neutrophils in cancer</title>", "<p id=\"Par16\">Fridlender et al. identified two distinct populations of tumour-associated neutrophils (TANs): anti-tumourigenic N1 and pro-tumourigenic N2 TANs [##REF##19732719##41##]. The latter type prevails in many human cancers [##REF##31160735##42##]. It releases a variety of cytokines, chemokines, and growth factors that promote tumour cell survival and proliferation, such as prostaglandin E2 (PGE2), CCL17, interleukin-6 (IL-6), tumour necrosis factor-alpha (TNF-α), VEGF, and epidermal growth factor (EGF) (Fig. ##FIG##1##2##) [##REF##32694624##43##]. N2 TANs also secrete collagenase (MMP8) and gelatinase B (MMP9), which facilitate the invasion of tumour cells by remodelling the extracellular matrix [##REF##18077379##44##]. In addition, their arginase-1 (ARG1) degrades extracellular arginine, which dampens the proliferation of T cells [##REF##15313928##45##]. Thus, N2 TANs resemble in many respects to neutrophils after uptake of apoptotic cells.</p>", "<p id=\"Par17\">Spontaneous apoptosis of single tumour cells can be observed in many treatment-naive patients. It was shown already more than 25 years ago in prostate cancer that an elevated frequency of tumour cell apoptosis correlates with a higher 5-years progression rate [##REF##7529128##46##]. Similarly, colorectal cancer patients with a higher number of apoptotic cancer cells have a worse overall survival [##REF##12733138##47##]. Table ##TAB##0##1## summarizes numerous studies investigating the relationship between cancer apoptosis rate and clinical outcome in 18 different cancer types. A positive association between apoptosis and poor prognosis was found in most cancers types. Only thyroid carcinoma, neuroblastoma, and glioblastoma showed an inverse relationship. Thus, tumour cell apoptosis promotes the progression of remaining viable tumour cells. Apoptotic cells release the growth factor FGF-β, PGE2, and VEGF, which have a direct promoting effect on the proliferation of adjacent tumour cells (recently reviewed in [##UREF##3##48##]). However, there is strong evidence that the pro-tumourigenic effect of apoptosis is mainly mediated by the phagocyte response during apoptotic cell clearance [##REF##27558817##49##]. Most studies focussed on macrophages, whereas the contribution of efferocytosing neutrophils to tumour growth is much less investigated. Dead tumour cells are not equally distributed throughout the tumour tissue. Many solid cancers show dense cribriform nests or pseudoluminal structures with central aggregates of disintegrated dead tumour cells [##REF##28382188##50##]. We found recently in colorectal cancer patients that such massive dead cell accumulations stain positive for caspase-cleaved cytokeratin 18 and CXCL8/IL-8, indicating that they derive from apoptotic tumour cells, which release a neutrophil chemoattractant (Fig. ##FIG##1##2##) [##REF##35121727##15##]. Accordingly, the great majority of aggregates is highly infiltrated with neutrophils and anti-inflammatory polarized TAMs. Blocking the apoptotic cell-derived CXCL8/IL-8 prevents neutrophil-induced anti-inflammatory macrophage polarization. These data fit to the above-proposed concept that neutrophils play a major role in efferocytosis in cases of massive accumulations of apoptotic dead cell remnants.</p>", "<p id=\"Par18\">Interestingly, also activation of LAP promotes tumour immune tolerance. LAP-sufficient tumour animal models revealed accumulation of M2 macrophages which support the pro-tumorigenic effects of tumour-associated macrophages (TAMs) [##REF##30245008##33##]. Consequently, T cell differentiation is skewed towards regulatory T cells that support inflammation resolution [##REF##31822793##19##, ##REF##30245008##33##]. Indeed, LAP-deficient TAMs trigger STING-mediated type I interferon responses inducing a pro-inflammatory gene expression and increasing CD8⁺ T cell function. Remarkably, the overexpression of Rubicon in cancer patients, which is required for LAP but not autophagy, have been suggested as a potential poor prognostic marker [##REF##32850405##51##]. In line with that, evidence suggest that specifically targeting LAP within the tumour microenvironment through pharmaceutical means promotes an anti-tumour response in a T cell-dependent manner [##REF##30245008##33##]. Hence, development of therapies targeting efferocytosis-related pathways, in macrophages as well as neutrophils, could present a promising approach for cancer treatment.</p>", "<p id=\"Par19\">Neutrophils support tumour growth and spreading not only in the tissue but also in the blood stream. They form heterologous clusters with circulating tumour cells (CTCs) which prolongs their half-life (Fig. ##FIG##1##2##) [##REF##26563367##52##]. CTC-neutrophils clusters support cell cycle progression, proliferation and survival of tumour cells resulting in extended metastatic potential [##REF##30728496##53##]. Patients with CTC–neutrophil clusters have poorer outcomes compared to those with homotypic CTC clusters [##REF##25186613##54##]. CTC-neutrophil clusters may also include NETs, which promote adhesion and extravasation of CTCs at the site of metastasis [##REF##26759232##55##, ##REF##28187288##56##]. However, sequencing analysis of CTC-associated neutrophils revealed a N2-like gene expression profile, indicating that not all neutrophils in the clusters form NETs [##REF##19732719##41##]. N2 neutrophils offer CTCs protection from immune surveillance by inhibition of CD8⁺ T cells NK cells [##REF##27072748##57##]. The blood-stream is a harsh environment for CTCs and single tumour cells may die quickly. They may be efferocytosed by adjacent neutrophils in the cluster, leading to mitogenic support of remaining tumour cell in the cluster. In summary, anti-inflammatory neutrophils support tumour growth and spreading in a variety of ways. Although there is growing evidence that efferocytosis contributes to tumorigenic TAN polarization, further research is needed to confirm this concept.</p>" ]

[ "<title>Acknowledgements</title>", "<p>This study was supported by the Medical University of Vienna.</p>", "<title>Author contributions</title>", "<p>CR and RO performed writing, review and revision of the paper. Both authors read and approved the final paper.</p>", "<title>Data availability</title>", "<p>Data sharing not applicable to this article as no datasets were generated or analyzed during the current study.</p>", "<title>Competing interests</title>", "<p id=\"Par21\">The authors declare no competing interests.</p>" ]

[ "<fig id=\"Fig1\"><label>Fig. 1</label><caption><title>Efferocytosis by neutrophils.</title><p>In response to sterile injury or local infection, neutrophils migrate to the site of tissue damage in a multistep process termed “neutrophils swarming”. Neutrophils undergo apoptosis after phagocytosis of invading pathogens and contribute to local apoptotic cells. Apoptotic cells release “keep-out” signals as well as “find-me” signals, many of which are strong chemoattractants for neutrophils. Neutrophils detect “find-me” signals on the surface of apoptotic cells and engulf the cell remains. This leads to a blockade of signalling pathways responsible for respiratory burst and NETosis. Furthermore, efferocytosing neutrophils expose specific cell surface activation markers and secrete a variety of soluble mediators.</p></caption></fig>", "<fig id=\"Fig2\"><label>Fig. 2</label><caption><title>Efferocytosis by neutrophils in cancer.</title><p>Spontaneous or therapy-induced tumour apoptosis leads to attraction of tumour-associated neutrophils (TANs). They are exposed to tumour-derived factors (dotted line) and engulf apoptotic cell remains. Both polarizes efferocytosing neutrophils towards an anti-inflammatory and pro-resolving N2-like phenotype. These TANs secrete numerous soluble mediators, which modulate tumour cells (Tu), tumour-associated macrophages (TAMs), tumour infiltrating lymphocytes (TILs), endothelial cells (ECs) and the extracellular matrix (ECM) in a pro-tumourigenic way. In addition, cancer may intravasate into adjacent vessels resulting in circulating tumour cells (CTCs). The blood stream is a harsh environment and many of CTCs have a short half-life. CTCs may form clusters with high abundant blood cells such as platelets or neutrophils (polymorphonuclear leucocytes or PMNs). They form NETs protecting CTCs and supporting extravasation and metastasis. Apoptotic CTCs within clusters are expected to bind neutrophils for efferocytosis, which would further support CTC survival and proliferation. However, experimental proof for this role of efferocytosis by neutrophils is still pending.</p></caption></fig>" ]

[ "<table-wrap id=\"Tab1\"><label>Table 1</label><caption><p>Correlative assessment of spontaneous apoptosis in situ with cancer patients’ outcome in several tumour types.</p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th>Tumour type</th><th>Method to detect apoptosis</th><th>OS^a</th><th>Comment</th><th>Refs.</th></tr></thead><tbody><tr><td rowspan=\"2\">Non-Hodgkin’s lymphoma</td><td>TUNEL</td><td>↓</td><td>High grade vs low grade</td><td>[##REF##16696087##59##]</td></tr><tr><td>TUNEL + H&amp;E</td><td>↓</td><td>High versus low tumour cell turnover</td><td>[##REF##8568587##60##]</td></tr><tr><td rowspan=\"5\">Breast carcinoma</td><td>IHC/Caspase-3</td><td>↓</td><td>OE is 75% of invasive BC</td><td>[##REF##12107344##61##]</td></tr><tr><td>H&amp;E</td><td>↓*</td><td>*Recurrence-free survival</td><td>[##REF##7857705##62##]</td></tr><tr><td>TUNEL + H&amp;E</td><td>↓</td><td>&gt;0.50% (shorter OS)</td><td>[##REF##10037180##63##]</td></tr><tr><td>FC + H&amp;E</td><td>↓</td><td/><td>[##REF##7857705##62##]</td></tr><tr><td>IHC/Cleaved caspase-3</td><td>↓</td><td/><td>[##REF##21725296##64##]</td></tr><tr><td rowspan=\"5\">Ovarian carcinoma</td><td>IHC/HtrA20</td><td>↑</td><td>Increased response to chemotherapy</td><td>[##REF##27832666##65##]</td></tr><tr><td>ELISA (Smac/DIABLO)</td><td>↑</td><td>Serum concentrations</td><td>[##REF##25577253##66##]</td></tr><tr><td>FC + IHC (Caspase-3)</td><td>↑</td><td>Metastasis</td><td>[##REF##19157506##67##]</td></tr><tr><td>IHC /Cleaved caspase-3</td><td>↓</td><td/><td>[##REF##25197379##68##]</td></tr><tr><td>TUNEL</td><td>↓</td><td>ovarian serous carcinoma</td><td>[##REF##9497913##69##]</td></tr><tr><td>Cervical cancer</td><td>IHC/Cleaved caspase-3</td><td>↓</td><td/><td>[##REF##25197379##68##]</td></tr><tr><td rowspan=\"4\">Colorectal carcinoma</td><td>IHC/Cleaved caspase-3</td><td>↓</td><td/><td>[##REF##25197379##68##]</td></tr><tr><td>IHC (M30)</td><td>↓</td><td>Higher turnover tumours</td><td>[##REF##12733138##47##]</td></tr><tr><td>TUNEL + H&amp;E</td><td>↓</td><td/><td>[##REF##8625232##70##]</td></tr><tr><td>TUNEL + H&amp;E</td><td>↑</td><td/><td>[##REF##9306959##71##]</td></tr><tr><td rowspan=\"2\">Lung carcinoma</td><td>TUNEL + H&amp;E</td><td>↓</td><td>Non-small cell lung carcinoma</td><td>[##REF##7585640##72##]</td></tr><tr><td>TUNEL</td><td>↓</td><td>Non-small cell lung carcinoma</td><td>[##REF##10561347##73##]</td></tr><tr><td rowspan=\"2\">Gastric carcinoma</td><td>IHC /Cleaved caspase-3</td><td>↓</td><td/><td>[##REF##25197379##68##]</td></tr><tr><td>TUNEL</td><td>↑</td><td>advanced gastric carcinoma</td><td>[##REF##10357401##74##]</td></tr><tr><td rowspan=\"3\">Prostate carcinoma</td><td>H&amp;E</td><td>↓</td><td/><td>[##REF##17214354##75##]</td></tr><tr><td>H&amp;E</td><td>↓*</td><td>*Actuarial progression rate at 5 years</td><td>[##REF##7529128##46##]</td></tr><tr><td>TUNEL</td><td>↓*</td><td>*Disease recurrence</td><td>[##REF##9428494##76##]</td></tr><tr><td>Thyroid carcinoma</td><td>TUNEL</td><td>↑</td><td>papillary thyroid carcinoma (PTC)</td><td>[##REF##17325701##77##]</td></tr><tr><td rowspan=\"2\">Bladder carcinoma</td><td>H&amp;E</td><td>↓</td><td/><td>[##REF##7965393##78##]</td></tr><tr><td>IHC</td><td>↓</td><td>Invasive transitional cell carcinoma</td><td>[##REF##10962336##79##]</td></tr><tr><td rowspan=\"2\">Pancreatic carcinoma</td><td>TUNEL</td><td>↓</td><td>Higher AI in undifferentiated vs. differentiated cancers</td><td>[##REF##10707925##80##]</td></tr><tr><td>TUNEL</td><td>↓</td><td/><td>[##REF##16173075##81##]</td></tr><tr><td>Salivary glands</td><td>TUNEL</td><td>↓</td><td/><td>[##REF##17325701##77##]</td></tr><tr><td rowspan=\"2\">Hepatocellular carcinoma</td><td>H&amp;E</td><td>↑</td><td>low growth index</td><td>[##REF##8624258##82##]</td></tr><tr><td>H&amp;E</td><td>↓*</td><td>*Disease-free survival</td><td>[##REF##10574266##83##]</td></tr><tr><td>Neuroblastoma</td><td>TUNEL</td><td>↑</td><td/><td>[##REF##7576945##84##]</td></tr><tr><td rowspan=\"2\">Mesothelioma</td><td>TUNEL + H&amp;E</td><td>↓</td><td/><td>[##REF##10919383##85##]</td></tr><tr><td>TUNEL</td><td>↓</td><td>Pleural mesothelioma</td><td>[##REF##10962437##86##]</td></tr><tr><td>Tongue carcinoma</td><td>TUNEL</td><td>↓</td><td>Early stage squamous carcinoma</td><td>[##REF##11169941##87##]</td></tr><tr><td>Laryngeal carcinoma</td><td>TUNEL</td><td>↓</td><td>Squamous cell carcinoma</td><td>[##REF##10448265##88##]</td></tr><tr><td>Glioblastoma</td><td>TUNEL + H&amp;E</td><td>↑</td><td/><td>[##REF##12084348##89##]</td></tr></tbody></table></table-wrap>" ]

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[ "<table-wrap-foot><p>Patient-derived histological samples were analyzed by using either DNA end-labelling techniques (TUNEL), plain morphology combined with hematoxylin and eosin (H&amp;E), immunohistochemistry (IHC), flow cytometry (FC), or enzyme-linked immunosorbent assay (ELISA). Clinical studies in which patients were treated before assessment were excluded from the analysis.</p><p>^aOverall survival (if not otherwise indicated in the comments field) at increased apoptosis.</p></table-wrap-foot>", "<fn-group><fn><p><bold>Publisher’s note</bold> Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.</p></fn></fn-group>" ]

[ "<graphic xlink:href=\"41420_2024_1809_Fig1_HTML\" id=\"d32e269\"/>", "<graphic xlink:href=\"41420_2024_1809_Fig2_HTML\" id=\"d32e402\"/>" ]

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{ "acronym": [], "definition": [] }

[ "<title>Introduction</title>", "<p id=\"Par3\">In a non-centrosymmetric material, light-matter interactions can generate a finite DC photocurrent under homogeneous illumination in absence of external bias and spatial inhomogeneity. This photovoltaic effect, governed by the intrinsic symmetry properties of materials, is referred to the intrinsic photovoltaic effect (IPVE) or bulk photovoltaic effect (BPVE)^{##REF##34127824##1##–##UREF##2##4##}. Hence, the unique physics of IPVE offers an effective approach to surpass the Shockley-Queisser limit in traditional photovoltaic devices^{##REF##34127824##1##–##REF##29674433##9##}, which attracts growing attention recently. Initial studies on IPVE mainly focused on ferroelectric insulators, such as LiNbO₃^{##UREF##0##2##}, BiFeO₃^{##REF##19228998##10##} and BaTiO₃^{##UREF##3##8##,##REF##20062051##11##}. Later, researchers found that reducing bandgap size and lowering dimensionality could further enhance the efficiency of IPVE^{##REF##31217597##5##–##REF##36109522##7##,##REF##32632282##12##–##REF##34625541##17##}. For example, the IPVE photocurrents observed in narrow bandgap semiconductors (including one-dimensional/1D WS₂ nanotubes^{##REF##31217597##5##}) and Weyl semimetals with broken inversion symmetry are orders of larger than those in wide bandgap ferroelectric insulators^{##REF##28120823##6##,##REF##36109522##7##,##REF##32632282##12##–##REF##31844067##15##}. On the other hand, van der Waals (vdW) layered materials meet all merits for IPVE investigations due to its low dimensionality, tunable bandgap, flexibility, easy manipulation, and rich species^{##REF##31844067##15##–##REF##33795452##26##}. For example, strain-gradient-engineered MoS₂ shows a strong IPVE with photocurrent density over 10²A cm⁻², which is comparable to that in 1D WS₂ nanotube^{##REF##31217597##5##,##REF##34140672##20##}; The external quantum efficiency of 3R-MoS₂ with spontaneous out-of-plane polarization shows the highest reported value of 16%^{##UREF##6##22##}; IPVE observed in the in-plane strained 3R-MoS₂ is over two orders of magnitude higher than the unstrained one^{##REF##34140672##20##}; The non-centrosymmetric nano-antennas in centrosymmetric graphene can result in artificial IPVE^{##REF##33335090##23##–##UREF##8##25##}; Moiré-pattern in twisted bilayer graphene and WSe₂/BP interface can lead to the emergence of spontaneous IPVE^{##REF##35418636##21##,##REF##33795452##26##}; Low-dimensional vdW structures such as quasi-1D edges of Weyl semimetal WTe₂ can generate strong IPVE-induced photocurrents, attributing to the strong symmetry breaking and low dimensionality of edges^{##REF##31844067##15##}; Robust IPVE-induced photocurrents are observed in topological insulator monolayer WTe₂^{##UREF##4##16##}. Here, we introduce an alternative low-dimensional system, one-dimensional grain boundary (GB) with non-centrosymmetric crystalline structure, for IPVE investigations. Distinct from previous IPVE systems, GBs widely exist in all kinds of materials. For example, GBs have been uncovered in various vdW layered materials regardless of their crystalline symmetry, including graphene^{##REF##20348912##27##}, MoS₂^{##REF##23644523##28##}, ReS₂^{##REF##35293650##29##–##REF##30226744##33##}, and MoSe₂^{##UREF##11##34##}.</p>", "<p id=\"Par4\">1 T′-ReS₂ GBs are ideal for IPVE investigations due to following reasons. (1) Anisotropic optical properties of ReS₂ allow to identify positions of GBs and subdomains simply using polarization-resolved optical microscopy; (2) GBs in ReS₂ have well-defined structures free of dangling bonds. In this work, we uncover strong and robust IPVE in 1D vdW GBs in ReS₂. Symmetry analysis and experimental results demonstrate that inversion symmetry is broken near GBs, which results in a DC photocurrent that propagates along GBs without any voltage bias. We demonstrate that this IPVE-induced photocurrent is gate tunable and possesses a pronounced polarization-independent component. Furthermore, the IPVE-induced photocurrent densities in 1D ReS₂ GBs are among the highest values compared with reported material systems.</p>" ]

[ "<title>Methods</title>", "<title>Sample preparation</title>", "<p id=\"Par16\">ReS₂ samples were prepared on silicon substrate covered with 300 nm SiO₂ through standard mechanical exfoliation method. Angle-resolved polarized optical microscope is used to identify GBs and domains in ReS₂. Electrodes (5/35 nm Cr/Au) were patterned via standard photolithography process (MicroWriter ML3, Durham Magneto Optics Ltd).</p>", "<title>STEM characterizations</title>", "<p id=\"Par17\">The STEM/HAADF images were obtained using a JEOL ARM200F equipped with a CEOS aberration corrector. The microscope featured a cold field emission gun and was operated at an accelerating voltage of 200 kV. The convergence angle was ~28 mrad.</p>", "<title>Optical characterizations</title>", "<p id=\"Par18\">The devices were characterized using a semiconductor parameter analyzer (FS-Pro) under vacuum (~10⁻⁶ mbar) at room temperature. For short-circuit photocurrent measurements, 532 nm lasers were used as excitation sources with laser power of 200 μW, respectively. The lasers were focused by a 50× microscope objective lens (0.5 N.A.). The size of laser spot with a Gaussian profile was ~3 μm for 532 nm laser. Angle-resolved polarized Raman spectroscopy was performed using a 532 nm laser with a spectrometer (Andor SR-500i-D2). A linear polarizer and half-wave plate (Thorlabs) were used to adjust the orientation of the laser polarization.</p>", "<title>Theoretical calculation</title>", "<p id=\"Par19\">Density functional theory (DFT) calculations for structure optimization and electronic properties were performed using the Vinna ab initio simulation package (VASP)^{##REF##10004490##49##}. Exchange-correction functional was treated within the generalized gradient approximation of Perdew, Burke, and Ernzerhof^{##REF##10062328##50##}. The electronic wave functions were expanded using a planewave basis set with an energy cutoff of 300 eV, and the tolerance for the total energy was &lt;10⁻⁴eV. A 1 × 10 × 1 k-mesh was utilized for self-consistent calculations of the supercell structure.</p>" ]

[ "<title>Results</title>", "<title>Characterization of 1D vdW GBs in ReS₂</title>", "<p id=\"Par5\">Bulk 1 T′-ReS₂ is a vdW semiconductor with a centrosymmetric crystalline structure under the inversion symmetric space group of <italic>P</italic>^{##REF##26280493##35##–##REF##26390381##38##}, as demonstrated by the scanning transmission electron microscope (STEM) image in Fig. ##FIG##0##1a##. Thus, IPVE is not allowed in thin-film ReS₂. In addition, we find an abundance of 1D GBs in ReS₂. The in-plane orientations, signified by the direction of Re chains, of the two neighboring subdomains form a 120° angle (see Fig. ##FIG##0##1b## and Supplementary Fig. ##SUPPL##0##1##), which aligns with prior STEM research findings^{##UREF##9##30##,##REF##30226744##33##}. Figure ##FIG##0##1c## shows a top view schematic of ReS₂ crystalline structure with GBs. Here, we use A and B to represent two adjacent subdomains. BA and AB GBs are denoted by “↑” and “↓” arrows, respectively. Using polarization-resolved optical microscopy (see Fig. ##FIG##0##1d, e## and Supplementary Fig. ##SUPPL##0##2##), we can clearly identify the positions of GBs in ReS₂ flakes due to the anisotropic optical reflection and different Re-chain directions of subdomains. As shown in Fig. ##FIG##0##1e##, the ReS₂ flake is separated by multiple GBs (along the <italic>y</italic>-direction) and forms multi-domain structures. Angle-resolved polarized Raman spectroscopy was further performed to identify the crystalline orientations of ReS₂ subdomains (see Fig. ##FIG##0##1f##). The intensity of <italic>A</italic>_g2 mode (212 cm⁻¹) is maximum when the light polarization direction is parallel to the Re-chain direction of ReS₂^{##UREF##9##30##,##REF##35605130##32##,##REF##26799768##36##}. Raman result indicates that there is ~117° difference between Re-chain directions of two adjacent subdomains, which is very close to the angle ~120° observed in STEM. The ~3° deviation of Raman characterization is within the permissible range of our instruments.</p>", "<title>IPVE theory in 1D vdW GBs</title>", "<p id=\"Par6\">The only symmetry present near GB regions is the two-fold rotation along the <italic>y</italic>-directional axis. Expanded analysis in Supplementary Note ##SUPPL##0##1## and Supplementary Fig. ##SUPPL##0##3## suggests that the GB region is characterized by a point group of <italic>C</italic>₂ and associated with a broken inversion symmetry, resulting in nonzero second-order nonlinear light-matter-interaction tensors , where <italic>w</italic> is the angular frequency of incident light, is the wave vector and <italic>l</italic>/<italic>j</italic>/<italic>k</italic> represents <italic>x</italic>-, <italic>y-</italic>, or <italic>z</italic>-directions. Under linearly polarized light, a finite DC photocurrent density along <italic>l</italic>-direction can be generated (only consider the -independent term)^{##REF##34140672##20##,##REF##30131488##39##}where . Since only depends on the intrinsic physical properties of materials, this effect is called IPVE or BPVE. Here, we mainly focus on IPVE-induced photocurrent along GB (<italic>y</italic>-direction) . For incident light normal to the two-dimensional plane of ReS₂ flake (<italic>E</italic>_z = 0), can be written as</p>", "<p id=\"Par7\">Equation ##FORMU##9##2## can be further simplified utilizing the rotation symmetry in GB region as shown in Fig. ##FIG##0##1b## (see Supplementary Note ##SUPPL##0##2## for details). Under rotation symmetry (<italic>x</italic>, <italic>y</italic>, <italic>z</italic> → -<italic>x</italic>, <italic>y</italic>, -<italic>z</italic>), which makes. If we write <italic>E</italic> = [<italic>E</italic>₀sin<italic>θ</italic>, <italic>E</italic>₀cos<italic>θ</italic>, 0], where <italic>θ</italic> is the angle between <italic>y</italic>-direction and light polarization direction, then we have</p>", "<p id=\"Par8\">Here, and are the polarization-dependent and polarization-independent terms, respectively. Furthermore, IPVE-induced photocurrents along two adjacent ↑ and ↓ GBs should have opposite directions restricted by their reversed orientations.</p>", "<title>Experimental observation of IPVE in 1D vdW GBs</title>", "<p id=\"Par9\">To study the IPVE in ReS₂ GBs, photodetectors with channel parallel to GBs are fabricated. Figure ##FIG##1##2a## shows the optical image of a device under polarized white light (the angle between light polarization and <italic>y</italic>-axis is ~30°). The optical images at other polarized angles are shown in Supplementary Fig. ##SUPPL##0##2##. The thickness of ReS₂ flake is ~180 nm determined by atomic force microscope (AFM) (see Supplementary Fig. ##SUPPL##0##4##). The ↑ and ↓ GBs are denoted by blue and red dash lines, respectively. The detailed fabrication process can be found in the Method section. The device shows linear current-voltage (<italic>I</italic>_ds-<italic>V</italic>_ds) characteristic, indicating a good Ohmic contact between ReS₂ and metal electrodes (see Supplementary Fig. ##SUPPL##0##5##). A linearly polarized 532 nm laser with a diameter around 3 μm was focused on the channel and short-circuit photocurrents (<italic>I</italic>_ph) were collected (the angle between laser polarization and <italic>y</italic>-axis is ~30°). As shown in Fig. ##FIG##1##2b##, <italic>I</italic>_ph(<italic>y</italic>) at pristine region (<italic>x</italic> = 0 μm) shows ordinary shape with vanishing value in the middle of channel. The finite photocurrents near electrodes can be attributed to extrinsic photovoltaic effect, such as built-in Schottky junction between ReS₂ and electrodes and photo-thermoelectric effect. This indicates that the pristine region of ReS₂ does not support IPVE due to the preservation of inversion symmetry. This observation is further confirmed in a device based on ReS₂ without GBs (see Supplementary Fig. ##SUPPL##0##6##). On the other hand, we observed very robust photocurrents in the middle regions of GBs with negative values at ↑ GB (along blue dashed line in Fig. ##FIG##1##2a##) and positive values at ↓ GB (along red dashed line in Fig. ##FIG##1##2a##) in sharp contrast to vanishing photocurrents in pristine regions. This phenomenon is reproducible in other samples (see Supplementary Fig. ##SUPPL##0##7##). Scanning photocurrent spectroscopy of total <italic>I</italic>_ph further confirms the observation as shown in Fig. ##FIG##1##2c##. Moreover, we measured photocurrent along <italic>x</italic>-direction <italic>I</italic>_ph(<italic>x</italic>) at fixed <italic>y</italic> position (indicated by the black dashed line in Fig. ##FIG##1##2a##). As shown in Fig. ##FIG##1##2d##, consistent and robust valley and peak features are observed at ↑ and ↓ GBs, respectively. The above results show excellent agreement with IPVE theory.</p>", "<p id=\"Par10\">To further demonstrate the effectiveness of IPVE theory, we check if a large polarization-independent photocurrent term exists in ReS₂ GBs. Figure ##FIG##1##2e## shows the polarization-resolved photocurrents in middle of ↑ and ↓ GBs. The polarization-dependent term is complicated since it is influenced by both anisotropic properties of ReS₂ domains and IPVE. Hence, we mainly focused on the polarization-independent term . Besides, polarization-independent term is more appealing for energy harvesting applications due to the unpolarized nature of sunlight. We further extract and show the mapping results in Fig. ##FIG##1##2f##. The opposite directions of are observed in ↑ and ↓ GBs, consistent with IPVE theory.</p>", "<p id=\"Par11\">We then investigated the electrical tunability of IPVE in ReS₂ GBs using gate bias. Figure ##FIG##2##3a## shows a device based on few-layer ReS₂ with GBs. The thickness of ReS₂ flake is 8 nm. The few-layer ReS₂ phototransistor exhibits n-type characteristics (see Supplementary Fig. ##SUPPL##0##8##), consistent with previous reports^{##REF##26390381##38##,##REF##25947630##40##}. Opposite directional photocurrents are observed at ↑ and ↓ GBs (see Fig. ##FIG##2##3b##), showing good agreement with other samples. Moreover, IPVE-induced photocurrents can be effectively tuned by gate voltage with ~ 28% and 60% enhancement from −40 to 40 V for ↑ and ↓ GBs, respectively. To understand this gate tunability of IPVE-induced photocurrent, we first examine whether it simply originates from the tuned Schottky barrier at metal-ReS₂ interface which may affect the collection efficiency of carriers. As shown in Supplementary Fig. ##SUPPL##0##9##, the 8 nm-thick ReS₂ device shows a good linear current-voltage characteristic at various gate voltages, indicating a good Ohmic contact between ReS₂ and metal electrodes. Hence, if there exists Schottky barrier, it would be very low which is unlikely to significantly affect the photocurrent intensities. On the other hand, IPVE-induced photocurrents have two contributions which are shift and ballistic currents^{##REF##28120823##6##,##REF##36411374##19##,##REF##34140672##20##,##REF##30746451##41##,##REF##27386523##42##}. Shift and ballistic currents strongly depend on the properties of nonequilibrium carriers excited by polarized lights. Thermalization of nonequilibrium carriers can be caused by electron-defect, electron-phonon, and electron-electron interactions. At different gate voltages, electron concentration changes which probably affects the thermalization processes of excited nonequilibrium carriers, such as their mean free bath length and mobility, and hence affects induced IPVE photocurrent densities. This is one plausible explanation. Further studies can be conducted to fully understand this phenomenon.</p>", "<p id=\"Par12\">We compared the strength of IPVE in ReS₂ GBs with other materials. Although structures of ReS₂ GBs are well defined, the effective width of GBs, which denotes the active region with strong inversion symmetry breaking for generating IPVE photocurrent, is unknown. Here, we give a photocurrent density range when effective width varies from 3 to 300 nm. As shown in Fig. ##FIG##3##4##, the photocurrent densities in ReS₂ GBs are comparable to those in 1D WS₂ nanotube^{##REF##31217597##5##}, strained 3R-MoS₂^{##REF##36411374##19##} and MoS₂^{##REF##34140672##20##} and orders of magnitude higher than those in ferroelectric materials^{##UREF##1##3##,##UREF##2##4##,##REF##34625541##17##,##REF##28785049##43##,##UREF##12##44##}.</p>" ]

[ "<title>Discussion</title>", "<p id=\"Par13\">To better understand the giant IPVE photocurrent densities in ReS₂ GBs and its underlying physical mechanism, the first-principles calculations of band properties are performed. Detailed information about calculations can be found in the Method Section and Supplementary Information (see Supplementary Figs. ##SUPPL##0##10##, ##SUPPL##0##11##). As shown in Fig. ##FIG##4##5a##, ReS₂ near GBs has a lower conduction band minimum and higher valence band maximum compared with those of pristine ReS₂. In addition, GBs have significant influence to the band structures of ReS₂ near GBs through introducing significant number of new states (see Supplementary Figs. ##SUPPL##0##10##, ##SUPPL##0##11##). These new states might improve the light absorption and enhance the IPVE photocurrent. Importantly, a quantum-well structure is formed along <italic>x</italic>-direction (normal to GB direction) due to lower conduction band minimum and higher valence band maximum near GBs as shown in Fig. ##FIG##4##5b##. This indicates that carriers generated near GBs tends to be caught into the quantum well and transport along GBs (carrier collection direction of electrodes) is more preferred than other directions. This further enhances the IPVE photocurrent. Besides, the well-defined structures of GBs without any dangling bonds and the indirect bandgap of ReS₂ near GBs could further suppress scatterings and recombination of photo-excited carriers (see Supplementary Fig. ##SUPPL##0##10##). These are possible reasons that lead to the giant IPVE photocurrent density in ReS₂ GBs. As shown in Supplementary Fig. ##SUPPL##0##12##, we still can observe pronounced IPVE photocurrent at ReS₂ GBs with channel length over 100 μm. In addition, we also studied IPVE at ReS₂ edges for comparison, since edges are non-centrosymmetric with broken periodic structures. We fabricated and measured three ReS₂ samples in which we did not found detectable IPVE-induced photocurrent at edges (see Supplementary Fig. ##SUPPL##0##13##). High density of defect states at edges, such as dangling bonds, might induce strong electron-defect scatterings and suppress the IPVE photocurrent^{##REF##37454221##45##}.</p>", "<p id=\"Par14\">Power- and wavelength- dependent photocurrents are measured to further clarify the physical mechanism of IPVE observed in ReS₂ GBs. As shown in Fig. ##FIG##4##5c##, the power-dependent photocurrent at GBs shows a transition from linear to square-root dependence when power increases which is consistent with the prediction of theoretical shift current model and previous experimental reports^{##REF##31217597##5##,##REF##36411374##19##,##REF##34140672##20##,##REF##27386523##42##,##UREF##13##46##}. We theoretically calculated the shift current in ReS₂ GBs. Detailed calculation process and discussion are shown in Supplementary Information (see Supplementary Note ##SUPPL##0##3## and Supplementary Fig. ##SUPPL##0##14##). As shown in Fig. ##FIG##4##5d##, our shift current model shows good agreement with experimental results at different excitation wavelengths. All above results suggest that shift current dominates the photocurrent generation process of IPVE in ReS₂ GBs.</p>", "<p id=\"Par15\">Finally, we conclude through discussing the distinctive aspects of GB-induced symmetry breaking and its potential implications relative to prior research. Firstly, GBs widely exist in all kinds of materials and have a variety of configuration, which provides a capacious platform for IPVE and physics investigations. Secondly, GBs are embedded in bulk materials and there is no symmetry requirement for the crystalline structure of bulk material to induce symmetry breaking in GBs. Thirdly, formation of the quantum-well structure makes GBs a good 1D/quasi-1D system for IPVE investigation which can effectively suppress carrier dissipation to other directions. Fourthly, compared with edges^{##REF##37454221##45##}, GBs with well-defined crystalline structures are free of dangling bonds. The reduced electron-defect scatterings in GBs with well-defined structures might suppress scatterings of photo-excited carriers and enhance IPVE photocurrent. Lastly, structures and densities of GBs can be generated and controlled through adjusting material growth conditions^{##REF##35777352##47##,##REF##31896753##48##}. Other approaches, such as external strain, can also generate and control GBs in materials^{##REF##35293650##29##}. The ability to control formation and structures of GBs is important for making efficient optoelectronic devices. Hence, we believe the rich species and configurations, well-defined 1D/quasi-1D structures, and potential controllability make GBs a promising optoelectronic platform for novel physics and device applications.</p>" ]

[]

[ "<p id=\"Par1\">The photovoltaic effect lies at the heart of eco-friendly energy harvesting. However, the conversion efficiency of traditional photovoltaic effect utilizing the built-in electric effect in p-n junctions is restricted by the Shockley-Queisser limit. Alternatively, intrinsic/bulk photovoltaic effect (IPVE/BPVE), a second-order nonlinear optoelectronic effect arising from the broken inversion symmetry of crystalline structure, can overcome this theoretical limit. Here, we uncover giant and robust IPVE in one-dimensional (1D) van der Waals (vdW) grain boundaries (GBs) in a layered semiconductor, ReS₂. The IPVE-induced photocurrent densities in vdW GBs are among the highest reported values compared with all kinds of material platforms. Furthermore, the IPVE-induced photocurrent is gate-tunable with a polarization-independent component along the GBs, which is preferred for energy harvesting. The observed IPVE in vdW GBs demonstrates a promising mechanism for emerging optoelectronics applications.</p>", "<p id=\"Par2\">The intrinsic photovoltaic effect (IPVE) in noncentrosymmetric materials has the potential to overcome the limitations of traditional photovoltaic devices. Here, the authors report the observation of a strong and gate-tunable IPVE in 1D grain boundaries of a van der Waals semiconductor, ReS₂.</p>", "<title>Subject terms</title>" ]

[ "<title>Supplementary information</title>", "<p>\n\n\n</p>" ]

[ "<title>Supplementary information</title>", "<p>The online version contains supplementary material available at 10.1038/s41467-024-44792-4.</p>", "<title>Acknowledgements</title>", "<p>The work was financially supported by the National Natural Science Foundation of China (62275117, X.C.; 62261136552, J.M.; 52273279, X.Zhao), Shenzhen Excellent Youth Program (RCYX20221008092900001, X.C.), Open research fund of State Key Laboratory of Infrared Physics, Shanghai Institute of Technical Physics, Chinese Academy of Sciences (SITP-NLIST-YB-2022-05, X.C.), Shenzhen Basic Research Program (20220815162316001, X.C.), Natural Science Foundation of Guangdong Province (2023A1515011852, X.L.Y.), Guangdong Major Talent Project (2019QN01C177, X.C.; 2019CX01X014, T.W.), Fundamental Research Funds for the Central Universities (X.Zhao), and Beijing Natural Science Foundation (Z220020 X.Zhao).</p>", "<title>Author contributions</title>", "<p>X.C. conceived and supervised the projects. Y.Z. fabricated ReS₂ samples and devices with the assistance of Z.L. Y.Z. characterized photocurrent of devices with the assistance of Z.L. and T.W. X.L.Y. did the theoretical calculations. X.Zhou conducted the STEM characterizations with the assistance of X.Zhao, X.C., J.M. and Y.Z. proposed the IPVE/BPVE mechanisms. Y.Z. and X.C. drafted the manuscript with assistance of X.L.Y., X.Zhou and J.M. All authors discussed and commented the manuscript.</p>", "<title>Peer review</title>", "<title>Peer review information</title>", "<p id=\"Par20\"><italic>Nature Communications</italic> thanks the anonymous reviewers for their contribution to the peer review of this work. A peer review file is available.</p>", "<title>Data availability</title>", "<p>Relevant data supporting the key findings of this study are available within the article and the Supplementary Information file. All raw data generated during the current study are available from the corresponding authors upon request.</p>", "<title>Competing interests</title>", "<p id=\"Par21\">The authors declare no competing interests.</p>" ]

[ "<fig id=\"Fig1\"><label>Fig. 1</label><caption><title>Characterization of 1D van der Waals (vdW) grain boundaries (GBs) in ReS2.</title><p><bold>a</bold> Atomic-resolution scanning transmission electron microscope (STEM)/high angle annular dark field (HAADF) image of bulk ReS₂, accompanied with a lattice structure schematic. The blue and pink spheres in schematic represent the Re and S atoms, respectively. The unit cell of ReS₂ is delineated by a white rhombus. Scale bar is 1 nm. <bold>b</bold> The STEM/HAADF image of a ReS₂ GB, indicated by a white dashed line. There is 120° between Re-chain directions of two adjacent subdomains. Scale bar is 1 nm. Insets show Fourier transform patterns of two adjacent subdomains with scale bars of 10 nm⁻¹. <bold>c</bold> Top view schematic of lattice structures near GBs. The inversion symmetry is broken near GBs. <bold>d</bold>, <bold>e</bold> Unpolarized (<bold>d</bold>) and polarized (<bold>e</bold>) optical images of a ReS₂ flake with subdomain structures. Two adjacent subdomains are marked by A (red) and B (blue), respectively. BA and AB GBs are denoted by ↑ and ↓ arrows, respectively. Scale bar is 10 μm. <bold>f</bold> Angle-resolved polarized Raman spectroscopy results of <italic>A</italic>_g2 mode (212 cm⁻¹) in A (red dots) and B (blue dots) subdomains. The Re-chain directions of A and B subdomains are denoted by red and blue arrows, respectively. Solid lines are fitting curves.</p></caption></fig>", "<fig id=\"Fig2\"><label>Fig. 2</label><caption><title>Observation of intrinsic photovoltaic effect (IPVE) in 1D ReS2 GBs.</title><p><bold>a</bold> Optical image of a photodetector with multiple domains under polarized white light. Two adjacent GBs with reversed orientations are denoted by “↑” and “↓” and marked by blue and red dash lines, respectively. Scale bar is 10 μm. <bold>b</bold> Short-circuit photocurrents (<italic>I</italic>_ph) measured along <italic>y</italic>-direction at pristine region (gray dots), ↑ (blue dots) and ↓ (red dots) GBs. Robust <italic>I</italic>_ph are observed at ↑ and ↓ GBs. Electrodes are marked by yellow regions. <bold>c</bold> Scanning photocurrent spectroscopy of total photocurrent <italic>I</italic>_ph. Incident laser is polarized along <italic>x</italic>-axis. The ↑ and ↓ GBs are marked by blue and red dash lines, respectively. <bold>d</bold>\n<italic>I</italic>_ph measured along <italic>x-</italic>direction as indicated by the black dashed line in <bold>a</bold>. Valley and peak features are observed at ↑ and ↓ GBs, respectively. <bold>e</bold> Polarization-resolved photocurrents in middle positions of ↑ and ↓ GBs. The polarization degree is the angle between laser polarization and <italic>y</italic>-axis. <bold>f</bold> Spatial distribution of the polarization-independent term . The ↑ and ↓ GBs are marked by blue and red dash lines, respectively.</p></caption></fig>", "<fig id=\"Fig3\"><label>Fig. 3</label><caption><title>Electrical tunability of IPVE in few-layer ReS2 with GBs.</title><p><bold>a</bold> Optical image of a phototransistor based on few-layer ReS₂ with GBs under polarized white light. Orange dashed lines enclose the electrode areas. Scale bar is 5 μm. <bold>b</bold> Measured short-circuit photocurrent as a function of gate voltage. The ↑ and ↓ GBs are marked by blue and red dashed lines, respectively. The light spots illuminated at ↑ and ↓ GBs are marked by blue and red circles (presented in <bold>a</bold>), respectively.</p></caption></fig>", "<fig id=\"Fig4\"><label>Fig. 4</label><caption><title>IPVE-induced photocurrent density in various materials.</title><p>The orange and blue solid dots indicate IPVE-induced photocurrent density in 8 nm-thick ReS₂ GBs samples. The orange and blue dashed lines are linear fitting curves to the experimental data. The shaded area indicates the photocurrent density range when effective width varies from 3 to 300 nm. Data for other materials are taken from the literature (BaTiO₃ (BTO), ref. ^{##UREF##1##3##}; Pb(ZrTi)O₃ (PZT), ref. ^{##UREF##1##3##,##UREF##2##4##}; Mn-doped BiFeO₃ (BFO:Mn), ref. ^{##REF##28785049##43##}; Bi-Mn-O composite, ref. ^{##UREF##12##44##}; thin-film CuInP₂S₆ with thickness of 60 and 90 nm (CIPS-60 and −90 nm), ref. ^{##REF##34625541##17##}; strained MoS₂, ref. ^{##REF##34140672##20##}; strained 3R-MoS₂, ref. ^{##REF##36411374##19##}; WS₂ NTs, ref. ^{##REF##31217597##5##}). NTs is short for nanotubes; W is short for effective width.</p></caption></fig>", "<fig id=\"Fig5\"><label>Fig. 5</label><caption><title>Mechanism of giant IPVE in ReS2 GBs.</title><p><bold>a</bold> Calculated density of states of ReS₂ near GBs and pristine ReS₂. <bold>b</bold> Schematic of band diagram of ReS₂ near GBs. Quantum well structures are formed along the <italic>x</italic>-direction. The photo-excited electrons and holes are represented by filled and empty circles, respectively. The transport directions of these carriers are marked by black arrows. <bold>c</bold> The power-dependence of IPVE-induced photocurrent of GBs in ReS₂ samples. Dashed lines serve as guidelines for linear and square-root dependence. <bold>d</bold> Wavelength-dependent photocurrents at ↑ and ↓ GBs. The calculated and experimental results at ↑ (↓) GBs are represented by blue (red) solid lines and dots, respectively. Error bars denote the standard deviation of the photocurrent measured at GBs.</p></caption></fig>" ]

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mathvariant=\"normal\">LBPVE</mml:mi></mml:mrow></mml:msubsup><mml:mo>=</mml:mo><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:mfrac><mml:munder><mml:mrow><mml:mo>∑</mml:mo></mml:mrow><mml:mrow><mml:mi>j</mml:mi><mml:mo>,</mml:mo><mml:mi>k</mml:mi></mml:mrow></mml:munder><mml:msub><mml:mrow><mml:mi>χ</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">ljk</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mrow><mml:mi>E</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">j</mml:mi></mml:mrow></mml:msub><mml:mi>E</mml:mi><mml:msub><mml:mrow><mml:mspace width=\"0.25em\"/></mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">k</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mrow><mml:mspace width=\"0.25em\"/></mml:mrow><mml:mrow><mml:mo>*</mml:mo></mml:mrow></mml:msup><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mi>E</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">k</mml:mi></mml:mrow></mml:msub><mml:mi>E</mml:mi><mml:msub><mml:mrow><mml:mspace width=\"0.25em\"/></mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">j</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mrow><mml:mspace width=\"0.25em\"/></mml:mrow><mml:mrow><mml:mo>*</mml:mo></mml:mrow></mml:msup></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq5\"><alternatives><tex-math id=\"M11\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\chi }_{{{{{{\\rm{ljk}}}}}}}={\\sigma }_{{{{{{\\rm{ljk}}}}}}}^{(2)}(w,0)$$\\end{document}</tex-math><mml:math id=\"M12\"><mml:msub><mml:mrow><mml:mi>χ</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">ljk</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:msubsup><mml:mrow><mml:mi>σ</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">ljk</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mn>2</mml:mn></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:msubsup><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>w</mml:mi><mml:mo>,</mml:mo><mml:mn>0</mml:mn></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq6\"><alternatives><tex-math id=\"M13\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${J}_{{{{{{\\rm{l}}}}}}}^{{{{{{\\rm{LBPVE}}}}}}}$$\\end{document}</tex-math><mml:math id=\"M14\"><mml:msubsup><mml:mrow><mml:mi>J</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">l</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">LBPVE</mml:mi></mml:mrow></mml:msubsup></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq7\"><alternatives><tex-math id=\"M15\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${J}_{{{{{{\\rm{y}}}}}}}^{{{{{{\\rm{LBPVE}}}}}}}$$\\end{document}</tex-math><mml:math id=\"M16\"><mml:msubsup><mml:mrow><mml:mi>J</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">y</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">LBPVE</mml:mi></mml:mrow></mml:msubsup></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq8\"><alternatives><tex-math id=\"M17\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${J}_{{{{{{\\rm{y}}}}}}}^{{{{{{\\rm{LBPVE}}}}}}}$$\\end{document}</tex-math><mml:math id=\"M18\"><mml:msubsup><mml:mrow><mml:mi>J</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">y</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">LBPVE</mml:mi></mml:mrow></mml:msubsup></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ2\"><label>2</label><alternatives><tex-math id=\"M19\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${J}_{{{{{{\\rm{y}}}}}}}^{{{{{{\\rm{LBPVE}}}}}}}={\\chi }_{{{{{{\\rm{yxx}}}}}}}|{E}_{x}{|}^{2}+{\\chi }_{{{{{{\\rm{yyy}}}}}}}|{E}_{y}{|}^{2}+{\\chi }_{{{{{{\\rm{yxy}}}}}}}{E}_{{{{{{\\rm{x}}}}}}}{E}_{{{{{{\\rm{y}}}}}}}^{\\ast }+{\\chi }_{{{{{{\\rm{yxy}}}}}}}{E}_{y}{E}_{x}^{\\ast }$$\\end{document}</tex-math><mml:math id=\"M20\"><mml:msubsup><mml:mrow><mml:mi>J</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">y</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">LBPVE</mml:mi></mml:mrow></mml:msubsup><mml:mo>=</mml:mo><mml:msub><mml:mrow><mml:mi>χ</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">yxx</mml:mi></mml:mrow></mml:msub><mml:mo>∣</mml:mo><mml:msub><mml:mrow><mml:mi>E</mml:mi></mml:mrow><mml:mrow><mml:mi>x</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mrow><mml:mo>∣</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mi>χ</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">yyy</mml:mi></mml:mrow></mml:msub><mml:mo>∣</mml:mo><mml:msub><mml:mrow><mml:mi>E</mml:mi></mml:mrow><mml:mrow><mml:mi>y</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mrow><mml:mo>∣</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mi>χ</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">yxy</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>E</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">x</mml:mi></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mi>E</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">y</mml:mi></mml:mrow><mml:mrow><mml:mo>*</mml:mo></mml:mrow></mml:msubsup><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mi>χ</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">yxy</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mi>E</mml:mi></mml:mrow><mml:mrow><mml:mi>y</mml:mi></mml:mrow></mml:msub><mml:msubsup><mml:mrow><mml:mi>E</mml:mi></mml:mrow><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mo>*</mml:mo></mml:mrow></mml:msubsup></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq9\"><alternatives><tex-math id=\"M21\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${J}_{{{{{{\\rm{y}}}}}}}^{{{{{{\\rm{LBPVE}}}}}}}(x,\\, y,\\, z)={J}_{{{{{{\\rm{y}}}}}}}^{{{{{{\\rm{LBPVE}}}}}}}(-x,\\, y,\\, -z)$$\\end{document}</tex-math><mml:math id=\"M22\"><mml:msubsup><mml:mrow><mml:mi>J</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">y</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">LBPVE</mml:mi></mml:mrow></mml:msubsup><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mspace width=\"0.25em\"/><mml:mi>y</mml:mi><mml:mo>,</mml:mo><mml:mspace width=\"0.25em\"/><mml:mi>z</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:msubsup><mml:mrow><mml:mi>J</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">y</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">LBPVE</mml:mi></mml:mrow></mml:msubsup><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mo>−</mml:mo><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mspace width=\"0.25em\"/><mml:mi>y</mml:mi><mml:mo>,</mml:mo><mml:mspace width=\"0.25em\"/><mml:mo>−</mml:mo><mml:mi>z</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq10\"><alternatives><tex-math id=\"M23\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\chi }_{{{{{{\\rm{yxy}}}}}}}=0$$\\end{document}</tex-math><mml:math id=\"M24\"><mml:msub><mml:mrow><mml:mi>χ</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">yxy</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ3\"><label>3</label><alternatives><tex-math id=\"M25\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${J}_{{{{{{\\rm{y}}}}}}}^{{{{{{\\rm{LBPVE}}}}}}}=\\frac{{{E}_{0}}^{2}}{2}({\\chi }_{{{{{{\\rm{yyy}}}}}}}-{\\chi }_{{{{{{\\rm{yxx}}}}}}})\\cos 2{\\theta }+\\frac{{{E}_{0}}^{2}}{2}({\\chi }_{yyy}+{\\chi }_{{{{{{\\rm{yxx}}}}}}})={J}_{{{{{{\\rm{y}}}}}}}^{{{{{{\\rm{Pol}}}}}}-{{{{{\\rm{dp}}}}}}}+{J}_{{{{{{\\rm{y}}}}}}}^{{{{{{\\rm{Pol}}}}}}-{{{{{\\rm{indp}}}}}}}$$\\end{document}</tex-math><mml:math id=\"M26\"><mml:msubsup><mml:mrow><mml:mi>J</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">y</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">LBPVE</mml:mi></mml:mrow></mml:msubsup><mml:mo>=</mml:mo><mml:mfrac><mml:mrow><mml:msup><mml:mrow><mml:msub><mml:mrow><mml:mi>E</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:mfrac><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mrow><mml:mi>χ</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">yyy</mml:mi></mml:mrow></mml:msub><mml:mo>−</mml:mo><mml:msub><mml:mrow><mml:mi>χ</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">yxx</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mi>cos</mml:mi><mml:mn>2</mml:mn><mml:mi>θ</mml:mi><mml:mo>+</mml:mo><mml:mfrac><mml:mrow><mml:msup><mml:mrow><mml:msub><mml:mrow><mml:mi>E</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:mfrac><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mrow><mml:mi>χ</mml:mi></mml:mrow><mml:mrow><mml:mi>y</mml:mi><mml:mi>y</mml:mi><mml:mi>y</mml:mi></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mi>χ</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">yxx</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:msubsup><mml:mrow><mml:mi>J</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">y</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">Pol</mml:mi><mml:mo>−</mml:mo><mml:mi mathvariant=\"normal\">dp</mml:mi></mml:mrow></mml:msubsup><mml:mo>+</mml:mo><mml:msubsup><mml:mrow><mml:mi>J</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">y</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">Pol</mml:mi><mml:mo>−</mml:mo><mml:mi mathvariant=\"normal\">indp</mml:mi></mml:mrow></mml:msubsup></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq11\"><alternatives><tex-math id=\"M27\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${J}_{{{{{{\\rm{y}}}}}}}^{{{{{{\\rm{Pol}}}}}}-{{{{{\\rm{dp}}}}}}}$$\\end{document}</tex-math><mml:math id=\"M28\"><mml:msubsup><mml:mrow><mml:mi>J</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">y</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">Pol</mml:mi><mml:mo>−</mml:mo><mml:mi mathvariant=\"normal\">dp</mml:mi></mml:mrow></mml:msubsup></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq12\"><alternatives><tex-math id=\"M29\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${J}_{{{{{{\\rm{y}}}}}}}^{{{{{{\\rm{Pol}}}}}}-{{{{{\\rm{indp}}}}}}}$$\\end{document}</tex-math><mml:math id=\"M30\"><mml:msubsup><mml:mrow><mml:mi>J</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">y</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">Pol</mml:mi><mml:mo>−</mml:mo><mml:mi mathvariant=\"normal\">indp</mml:mi></mml:mrow></mml:msubsup></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq13\"><alternatives><tex-math id=\"M31\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${J}_{{{{{{\\rm{y}}}}}}}^{{{{{{\\rm{Pol}}}}}}-{{{{{\\rm{indp}}}}}}}$$\\end{document}</tex-math><mml:math id=\"M32\"><mml:msubsup><mml:mrow><mml:mi>J</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">y</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">Pol</mml:mi><mml:mo>−</mml:mo><mml:mi mathvariant=\"normal\">indp</mml:mi></mml:mrow></mml:msubsup></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq14\"><alternatives><tex-math id=\"M33\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${J}_{{{{{{\\rm{y}}}}}}}^{{{{{{\\rm{Pol}}}}}}-{{{{{\\rm{indp}}}}}}}$$\\end{document}</tex-math><mml:math id=\"M34\"><mml:msubsup><mml:mrow><mml:mi>J</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">y</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">Pol</mml:mi><mml:mo>−</mml:mo><mml:mi mathvariant=\"normal\">indp</mml:mi></mml:mrow></mml:msubsup></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq15\"><alternatives><tex-math id=\"M35\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${J}_{{{{{{\\rm{y}}}}}}}^{{{{{{\\rm{Pol}}}}}}-{{{{{\\rm{dp}}}}}}}$$\\end{document}</tex-math><mml:math id=\"M36\"><mml:msubsup><mml:mrow><mml:mi>J</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">y</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">Pol</mml:mi><mml:mo>−</mml:mo><mml:mi mathvariant=\"normal\">dp</mml:mi></mml:mrow></mml:msubsup></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq16\"><alternatives><tex-math id=\"M37\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} 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[ "<supplementary-material content-type=\"local-data\" id=\"MOESM1\"></supplementary-material>", "<supplementary-material content-type=\"local-data\" id=\"MOESM2\"></supplementary-material>" ]

[ "<fn-group><fn><p><bold>Publisher’s note</bold> Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.</p></fn><fn><p>These authors contributed equally: Yongheng Zhou, Xin Zhou.</p></fn></fn-group>" ]

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[ "<media xlink:href=\"41467_2024_44792_MOESM1_ESM.pdf\"><caption><p>Supplementary Information</p></caption></media>", "<media xlink:href=\"41467_2024_44792_MOESM2_ESM.pdf\"><caption><p>Peer Review File</p></caption></media>" ]

{ "acronym": [], "definition": [] }

[ "<title>Introduction</title>", "<p id=\"Par2\">Diabetes mellitus (DM) is a major public health problem worldwide and one of the major diseases threatening human health and survival. Diabetic retinopathy (DR) is the most common microvascular complication of diabetes and one of the leading causes of blindness in people with diabetes in clinical practice, affecting people’s quality of life^{##REF##25104599##1##–##REF##20596693##3##}. The biochemical mechanisms underpinning the development of DR are poorly understood and may be related to metabolic abnormalities, cellular signaling pathway transduction, glucotoxicity, and oxidative stress^{##REF##33316144##4##}. Although accumulating epidemiological evidence has linked serum metal exposure to diabetes, the association between heavy metals and DR remains elusive.</p>", "<p id=\"Par3\">Due to the rapid development of industrialization and urbanization, the pollution concentration of lead (Pb), cadmium (Cd), mercury (Hg), selenium (Se), and manganese (Mn) in the environment is higher than that of other heavy metals^{##REF##14757716##5##,##REF##27619212##6##}. Epidemiologic studies that have evaluated the relationship between these heavy metals and the risk of type 2 diabetes mellitus (T2DM) and its complications. An observational study showed that higher Mn intake was directly associated with a lower risk of type 2 diabetes in a population of postmenopausal women^{##REF##32295807##7##}. Previous study found a negative association between typical levels of serum Cd and diabetes risk in a United States of America population aged 20 years or older^{##REF##36570173##8##}. Kornhauser et al. reported that plasma Se was negatively associated with the severity of diabetic nephropathy in patients with type 2 diabetes^{##REF##25695036##9##}. There exists scientific evidence indicating that low levels of Cd exposure exacerbate the progression of kidney disease in diabetics^{##REF##37240395##10##}. However, the relationship between these heavy metals and diabetic retinopathy is poorly studied. Serum albumin (ALB) is a crucial physiological transporter of vital metal ions Pb²⁺, Cd²⁺, Hg²⁺, Se²⁺, and Mn²⁺ in the bloodstream^{##REF##23811338##11##,##REF##5490241##12##}. Nonetheless, the evidence as to whether heavy metals influence the incidence of DR by binding to ALB and thereby affecting the incidence of DR evidence on the relationship between serum ALB and heavy metals and DR is unclear.</p>", "<p id=\"Par4\">Therefore, we hypothesized that the relationship between blood levels of these five heavy metals and DR is nonlinear. The present sought to explore the association between blood levels of Pb, Cd, Hg, Se, Mn, and DR using data from a representative sample of people with T2DM who participated in the National Health and Nutrition Examination Survey (NHANES) between 2011 and 2020.</p>" ]

[ "<title>Methods</title>", "<title>Sample and overall</title>", "<p id=\"Par5\">The National Center for Health Statistics (NCHS) conducted a population-based cross-sectional NHANES to collect data on the nutritional and health conditions of children and adults in the United States of America via interviews, physical examinations, and laboratory tests. NHANES employs a categorized, multistage probability sampling design to select a sample of an unstructured population to evaluate health and nutritional status. NHANES procedures and protocols were approved by the NCHS Research Ethics Review Committee and written consent was obtained.</p>", "<title>Inclusion and exclusion criteria</title>", "<p id=\"Par6\">The study population was mainly over 30 years of age. This cross-sectional analysis used data from five NHANES survey cycles (each lasting two years) between 2011 and 2020. Of the 45,462 participants in this study, participants aged &lt; 30 years (n = 23,484), those with unknown diabetes-related conditions, and those lacking information on the glycated hemoglobin (HbA1c) state or the HbA1c level was less than 6.5% (n = 18,955) were eliminated. Diabetic patients aged &lt; 30 years (n = 910) were eliminated to lower the likelihood of having type 1 diabetes. Besides, participants lacking information on serum levels of heavy metals (n = 517) and DR (n = 13) were disqualified. We filled in the remaining missing values in using the average of the entire sequence in the spss software. Finally, 1583 participants were included in the analysis. The study design flow is depicted in Fig. ##FIG##0##1##.</p>", "<title>Diagnosis of DR</title>", "<p id=\"Par7\">The following criteria were used to identify diabetes: (1) a prior diagnosis from a medical expert, (2) fasting blood glucose (FBG) levels of &gt; 7.0 mmol/L, (3) HbA1c levels of &gt; 6.5%, or (4) receiving diabetes medication^{##REF##29222373##13##}. DR was self-reported using a dichotomous classification, whereby the respondent had been informed by their physician that diabetes was affecting their eyes.</p>", "<title>Determination of heavy metals</title>", "<p id=\"Par8\">Whole blood specimens were collected by physicians at the NHANES Mobile Examination Center (MEC) via venipuncture in EDTA-coated tubes, centrifuged on-site, and stored frozen at − 30 °C for transportation to the Centers for Disease Control and Prevention (CDC) laboratory in California, where they were stored in a frozen state until further analysis. First, in the sample dilution step, a small amount of whole blood is taken from a larger sample and mixed to distribute the cellular components evenly. This mixing is critical because certain metals (e.g., lead) are primarily associated with red blood cells. Samples with clots or microclots are identified and excluded from analysis due to concerns about sample inhomogeneity. Diluted blood samples are prepared by mixing 1 part of the sample, 1 part of water, and 48 parts of diluent. The diluent contains chemicals that release metals from red blood cells, reduce ionization inhibition, prevent clogging, and enable the use of internal standards. The diluted sample is then passed through an inductively coupled plasma (ICP) ion source into the mass spectrometer. The liquid blood sample is converted into aerosol droplets that are vaporized, atomized, and ionized in the plasma region. The resulting ions enter the mass spectrometer with argon gas for analysis. The Dynamic Reaction Cell (DRC) plays a vital role in the selective reaction by removing interferences or enhancing the ion signal of specific elements. The ions passing through the DRC are electrically selected and directed to the analytical quadrupole. The electrical signal generated by the ion impact detector is processed into digital information to determine the elemental concentration. For the values under LODs, an imputed fill value (LOD/√2) was adopted to fill up the vacancy in data preprocessing. Quality assessment and quality control (QA/QC) processes for data collection from blood samples complied with the Therapeutic Laboratory Development Act of 1988^{##REF##30261124##14##}. Detail information of the NHANES laboratory procedure is available at <ext-link ext-link-type=\"uri\" xlink:href=\"https://www.cdc.gov/nchs/nhanes/index.htm\">https://www.cdc.gov/nchs/nhanes/index.htm</ext-link>.</p>", "<title>Covariates</title>", "<p id=\"Par9\">Data on demographics (age, gender, and race/ethnicity), academic achievement, poverty-to-income proportion, smoking, body mass index (BMI), waist circumference, serum ALB, FBG, HbA1c, triglycerides, and total cholesterol were collected using standardized questionnaires. Participants were divided into Caucasian, Black, Hispanic, Mexican American, and other race categories. Height (H, m) and weight (W, kg) were measured according to norms, and BMI was calculated as W/H² (kg/m²). Smokers were defined as individuals who smoked ≥ 100 cigarettes^{##REF##28597109##15##}. Smoking status was divided into three categories: never smoked (&lt; 100 cigarettes in their lifetime), gave up smoking (previously smoked &gt; 100 cigarettes but no longer smoke), and still smoking (previously smoked &gt; 100 cigarettes and still smoking). Hypertension was defined as systolic blood pressure &gt; 140 mmHg or diastolic blood pressure &gt; 90 mmHg, use of blood pressure drugs, or self-reported high blood pressure^{##REF##32371787##16##}.</p>", "<title>Statistical analysis</title>", "<p id=\"Par10\">Categorical variables were expressed as proportions and frequencies and were statistically analyzed using Chi-square analyses. Continuous variables were expressed as mean and standard errors and were analyzed using linear regression models. Multifactorial logistic regression analysis was used to examine the relationship between heavy metal exposure and DR. Additionally, subgroup analysis of heavy metal concentration and DR was performed to test whether the effect of serum levels of heavy metal on DR could be altered by age, sex, and BMI. All data analyses were performed using R (version 4.0.2) and SPSS software (version 24.0). p &lt; 0.05 was considered statistically significant.</p>", "<title>Institutional review board statement</title>", "<p id=\"Par11\">Our study used five cycles of open NHANES database (2011–2020), National Center for Health Statistics granted the study procedures of Ethics Review Board. These data could be accessible at the following URL: <ext-link ext-link-type=\"uri\" xlink:href=\"https://www.cdc.gov/nchs/nhanes/irba98.htm#print\">https://www.cdc.gov/nchs/nhanes/irba98.htm#print</ext-link>.</p>", "<title>Ethics statement</title>", "<p id=\"Par12\">All survey participants signed a declaration of consent form after being made aware of the nature of the poll. The informed consent was approved after evaluation by the Committee of the National Center for Statistics on Health Ethics Assessment Board. To make the best use of these resources, all of the data is made publicly accessible after formal anonymization. These data could be accessible at the following URL: <ext-link ext-link-type=\"uri\" xlink:href=\"https://www.cdc.gov/nchs/nhanes/irba98.htm#print\">https://www.cdc.gov/nchs/nhanes/irba98.htm#print</ext-link>.</p>" ]

[ "<title>Results</title>", "<title>Participant characteristics</title>", "<p id=\"Par13\">This study included 1583 T2DM patients, of which 331 had concomitant DR and 1252 had no concomitant DR. The demographic details of participants with or without DR are shown in Table ##TAB##0##1##. DR participants were older and had higher HbA1c levels and lower waist circumference, BMI, serum ALB, and serum Mn levels than non-DR participants. No statistically significant difference was observed between DR and non-DR groups in terms of race, gender, education, household income-to-poverty ratio, triglycerides, total cholesterol, smoking status, blood Pb, Cd, Hg, and Se, and hypertension status (p &gt; 0.05).</p>", "<title>Relationship between serum level of heavy metal and DR</title>", "<p id=\"Par14\">Multifactorial logistic regression analysis was performed to evaluate the relationship between serum heavy metal exposure and DR, as shown in Table ##TAB##1##2##. The data from Model 1 was left unadjusted, Model 2 was adjusted for gender, age, and race, and Model 3 was adjusted for education, household income-to-poverty ratio, smoking, hypertension, and BMI based on Model 2. The blood Mn exposure in Model 1 decreased the incidence of DR (p = 0.002). The blood Mn exposure was significantly negatively associated with DR in Models 2 and 3. However, no association was observed between blood Cd, Se, Hg, or Pb exposure and DR in all models (p &gt; 0.05).</p>", "<title>Subgroup analysis</title>", "<p id=\"Par15\">A significant negative correlation between blood Mn content and DR was observed after adjusting for age, sex, ethnicity, and other factors. The subgroups were further stratified by age, sex, and BMI, as shown in Table ##TAB##2##3##. In the age ≥ 60 years group, blood Mn was significantly negatively associated with DR prevalence without adjusting for confounders (p &lt; 0.001) and even after adjusting for all potential variables. However, no significant association between blood Mn and DR in the 30–44 and 45–60 age categories (p &gt; 0.05). After adjusting for all variables, the data were stratified by gender, and the results showed there was a significant negative association between blood Mn concentration and DR prevalence in the male population compared with the female population (p = 0.013). When stratified by BMI, no significant correlations were observed between blood Mn and DR in 0–25 and 25–30 kg/m² groups after adjusting for confounders. Additionally, a significant inverse correlation was found between blood Mn and DR in the BMI ≥ 30 kg/m² group (p = 0.014).</p>", "<title>Effect of mediating factors on the association between serum Mn and DR</title>", "<p id=\"Par16\">The Baron and Kenny causal step method combined with multiple linear regression models was utilized to analyze the mediating role of serum ALB in the association between serum Mn and DR. As shown in Fig. ##FIG##1##2##, after adjusting for Model 3, serum Mn had a direct impact on the prevalence of DR (p = 0.010), which was partially mediated by ALB (p = 0.008). Serum ALB mediated 12.80% of the association between blood Mn and DR.</p>" ]

[ "<title>Discussion</title>", "<p id=\"Par17\">The present study aimed to elucidate whether there was an association between serum levels of heavy metals and DR among individuals with T2DM in the United States of America. We found a substantial negative relationship between blood Mn and DR, which was partially mediated by serum ALB. Only individuals aged 60 years exhibited a significant inverse association between blood Mn levels and DR after age stratification. Under sex stratification, a significant negative association between blood Mn levels and DR prevalence was observed in the male population compared with the female population. Moreover, a statistically significant association between blood Mn and DR was observed in the BMI ≥ 30 kg/m² group under BMI stratification.</p>", "<p id=\"Par18\">DR, a common microvascular complication of diabetes with a global prevalence of 34.6%, is a major cause of blindness^{##REF##32117292##17##}. The onset and evolution of DR are influenced by various mechanisms, such as inflammatory processes, high blood pressure, abnormal lipid metabolism, and insulin resistance^{##REF##29600185##18##–##REF##36582230##20##}. Only a few studies have examined the association between exposure to heavy metals and DR, and the findings are controversial. Pb, Hg, Se, and Cd are widespread environmental heavy metal toxicants. Handan et al. found that Se exerts a protective effect on retinal pigment epithelium (ARPE-19) and primary human retinal microvascular endothelial (ACBRI 181) cells against high glucose (HG)-induced oxidative stress and apoptosis^{##REF##37630213##21##}. A previous retrospective study found that blood Cd levels were higher in the DR group than in the DM group^{##REF##33217819##22##}. These findings are inconsistent with our results, perhaps due to different populations, races, disease states, and health conditions that affect the absorption, transport, and metabolism of Se and Cd in the body, affecting the association between Se and Cd and the disease.</p>", "<p id=\"Par19\">Mn is one of the essential components of the Mn super antioxidant dismutase (MnSOD), which scavenges reactive oxygen species (ROS) under mitochondrial oxidative stress. MnSOD genes and Mn levels may impact MnSOD activity^{##REF##12618592##23##}. It was previously demonstrated that Mn administration improved MnSOD activation and shielded people from T2DM and related consequences^{##REF##35270089##24##}. Kowluru et al. demonstrated that Mn-SOD overexpression prevented glucose-induced increased oxidative stress and apoptosis of retinal endothelial cells, suggesting a protective role for Mn-SOD in the pathogenesis of diabetic retinopathy^{##REF##16565397##25##}. On the other hand, previous studies have found that insulin synthesis and secretion in the pancreas are impaired, and insulin degradation and glucagon release are accelerated in Mn-deficient mice, whereas other studies have found that Mn reduces oxidative stress through antioxidant enzyme systems and non-enzymatic pathways to protect pancreatic islet cells from reactive oxygen species (ROS)^{##REF##23372018##26##,##UREF##0##27##}. The current study suggested that blood Mn may act as a barrier against the onset of DR. This is consistent with our findings. However, whether Mn can influence the development of DR through other metabolic pathways has yet to be investigated.</p>", "<p id=\"Par20\">In plasma, approximately 80% of the oxidized state of Mn²⁺ is bound to ALB and globulin and transported to the liver, kidney, small intestine, endocrine glands, pancreas, brain, bone, muscle, and hair^{##REF##5490241##12##,##REF##4759569##28##}. ALB is a glycosylated protein synthesized and secreted by hepatocytes, which has anti-oxidative stress and anti-inflammatory^{##REF##30439428##29##}. Previous studies have shown reduced serum ALB levels are associated with DR risk^{##REF##29615813##30##,##REF##18323672##31##}. The present study found that serum ALB significantly mediated the association between blood Mn and DR. This implies that blood manganese combined with serum albumin may reduce the development of DR through its antioxidant and anti-inflammatory effects.</p>", "<p id=\"Par21\">In this study, we found a negative correlation between blood Mn and DR only in people aged ≥ 60 years. This may be because aging is linked to a decline in Mn levels. This generally agrees with a large cross-sectional study^{##REF##27529280##32##}, which found that serum Mn levels in people aged ≥ 60 years may contribute to preventing and controlling prediabetes and diabetes. Compared with females, males in the current study exhibited an adverse correlation between serum Mn levels and DR. According to a survey of dietary intake of Mn, females received considerably more Mn from their diets than males^{##REF##7985639##33##}. The survey also showed that the biological half-life of Mn was substantially lower in women than in men. These results imply that Mn absorption and metabolism differ between males and females, which may help to explain some of the observed gender disparities in our study.</p>", "<p id=\"Par22\">Our study is the first study to explore the relationship between blood levels of heavy metals and DR in T2DM patients older than 30 in the United States of America. Nonetheless, the current study has some limitations. First, we could not demonstrate the causal link between specific molecular mechanisms in blood Mn and DR due to the cross-sectional design of the study, which should be further validated in a prospective cohort study and fundamental research. Second, the outcome variable was based on self-reported history of DR, which may not be entirely accurate. Third, although some of the confounders were adjusted in this study, there were still many confounders affecting DR, such as drug use and family history of dyslipidemia, that we were unable to obtain. Finally, because the NHANES-based study population was used, it was difficult to assess the validity of the findings from various ethnic studies.</p>" ]

[ "<title>Conclusions</title>", "<p id=\"Par23\">In summary, the present study found a significant negative association between blood Mn levels and DR, suggesting that an appropriate increase in Mn intake may delay the onset and development of DR.</p>" ]

[ "<p id=\"Par1\">The present study utilized the National Health and Nutrition Examination Survey (NHANES) database to examine the relationship between serum levels of heavy metals and Diabetic retinopathy (DR) in individuals aged over 30 years with type 2 diabetes mellitus (T2DM) in the United States. A cross-sectional analysis was conducted on 1583 individuals with T2DM from the NHANES 2011–2020, including 331 individuals in the DR group and 1252 individuals in the non-DR group. We collected data on serum levels of heavy metals, DR, and serum albumin for descriptive statistics, linear regression, and logistical regression analysis. After adjusting for age, gender, race and other factors, there was no statistically significant association between blood cadmium, selenium, mercury, or lead and DR. However, serum manganese (Mn) and DR had a significant negative association (β = − 0.2045, 95% CI = − 0.3484, − 0.0606). Serum albumin partially modulated the indirect influence of serum Mn on the incidence of DR, accounting for 12.80% of the association between serum Mn and DR. There was a negative association between serum Mn levels and the prevalence of DR in people with T2DM. Mn intake at least in this study has a little influence on the onset and development of DR.</p>", "<title>Subject terms</title>" ]

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[ "<title>Author contributions</title>", "<p>This study’s data gathering, curation, analysis of statistics, and writing of the publication were all jointly completed by Y.Z. Input from X.K.L. was used in the statistical analysis. W.X. oversaw the research and helped with the editing and revision of the manuscript. All authors contributed to the article and approved the submitted version.</p>", "<title>Funding</title>", "<p>This work was supported by generous grants from the Xuzhou Medical University Affiliated Hospital Development Fund (XYFZ2020008), National Nature Science Foundation of China (NSFC-81870534), the China International Medical Exchange Foundation Endocrinology and Metabolism Elite Research Fund (2021-N-03).</p>", "<title>Data availability</title>", "<p>The data used in the present research were obtained from publicly accessible sources. These data could be accessible at the following URL: <ext-link ext-link-type=\"uri\" xlink:href=\"https://www.cdc.gov/nchs/nhanes/\">https://www.cdc.gov/nchs/nhanes/</ext-link>.</p>", "<title>Competing interests</title>", "<p id=\"Par24\">The authors declare no competing interests.</p>" ]

[ "<fig id=\"Fig1\"><label>Figure 1</label><caption><p>Flowchart of study target population NHANES (2011–2020).</p></caption></fig>", "<fig id=\"Fig2\"><label>Figure 2</label><caption><p>Effect of the serum albumin (mediator) on the relationship between blood manganese (exposure) and diabetic retinopathy (DR outcome); <italic>DE</italic> direct effect, <italic>IE</italic> indirect effect, <italic>TE</italic> total effect.</p></caption></fig>" ]

[ "<table-wrap id=\"Tab1\"><label>Table 1</label><caption><p>Characteristics of study sample with and without diabetic retinopathy.</p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th align=\"left\"/><th align=\"left\">Non-diabetic retinopathy (n = 1252)</th><th align=\"left\">Diabetic retinopathy (n = 331)</th><th align=\"left\">p-value</th></tr></thead><tbody><tr><td align=\"left\">Age, years</td><td align=\"left\">62.85 ± 11.05</td><td align=\"left\">64.5 ± 10.68</td><td char=\".\" align=\"char\">0.015</td></tr><tr><td align=\"left\" colspan=\"3\">Sex, n (%)</td><td char=\".\" align=\"char\">0.700</td></tr><tr><td align=\"left\"> Male</td><td align=\"left\">707 (56.5%)</td><td align=\"left\">183 (55.3%)</td><td char=\".\" align=\"char\"/></tr><tr><td align=\"left\"> Female</td><td align=\"left\">545 (43.5%)</td><td align=\"left\">148 (44.7%)</td><td char=\".\" align=\"char\"/></tr><tr><td align=\"left\" colspan=\"3\">Race, n (%)</td><td char=\".\" align=\"char\">0.281</td></tr><tr><td align=\"left\"> Mexican American</td><td align=\"left\">195 (15.6%)</td><td align=\"left\">48 (14.5%)</td><td char=\".\" align=\"char\"/></tr><tr><td align=\"left\"> Other Hispanic</td><td align=\"left\">124 (9.9%)</td><td align=\"left\">44 (13.3%)</td><td char=\".\" align=\"char\"/></tr><tr><td align=\"left\"> White</td><td align=\"left\">393 (31.4%)</td><td align=\"left\">89 (26.9%)</td><td char=\".\" align=\"char\"/></tr><tr><td align=\"left\"> Black</td><td align=\"left\">351 (28.0%)</td><td align=\"left\">97 (29.3%)</td><td char=\".\" align=\"char\"/></tr><tr><td align=\"left\"> Other race</td><td align=\"left\">189 (15.1%)</td><td align=\"left\">53 (16.0%)</td><td char=\".\" align=\"char\"/></tr><tr><td align=\"left\" colspan=\"3\">Educational level, n (%)</td><td char=\".\" align=\"char\">0.463</td></tr><tr><td align=\"left\"> Less than high school</td><td align=\"left\">372(29.7%)</td><td align=\"left\">109(32.9%)</td><td char=\".\" align=\"char\"/></tr><tr><td align=\"left\"> High school</td><td align=\"left\">296(23.6%)</td><td align=\"left\">82(24.8%)</td><td char=\".\" align=\"char\"/></tr><tr><td align=\"left\"> More than high school</td><td align=\"left\">582(46.5%)</td><td align=\"left\">140(42.3%)</td><td char=\".\" align=\"char\"/></tr><tr><td align=\"left\"> Don’t know</td><td align=\"left\">2(0.2%)</td><td align=\"left\">0(0.0%)</td><td char=\".\" align=\"char\"/></tr><tr><td align=\"left\">Ratio of family income to poverty</td><td align=\"left\">2.36 ± 1.56</td><td align=\"left\">2.16 ± 1.51</td><td char=\".\" align=\"char\">0.054</td></tr><tr><td align=\"left\" colspan=\"3\">Smoking status</td><td char=\".\" align=\"char\">0.263</td></tr><tr><td align=\"left\"> Current smoker</td><td align=\"left\">194 (15.5%)</td><td align=\"left\">40 (12.1%)</td><td char=\".\" align=\"char\"/></tr><tr><td align=\"left\"> Never smoker</td><td align=\"left\">630 (50.3%)</td><td align=\"left\">178 (53.8%)</td><td char=\".\" align=\"char\"/></tr><tr><td align=\"left\"> Former smoker</td><td align=\"left\">428 (34.2%)</td><td align=\"left\">113 (34.1%)</td><td char=\".\" align=\"char\"/></tr><tr><td align=\"left\" colspan=\"3\">Hypertension</td><td char=\".\" align=\"char\">0.384</td></tr><tr><td align=\"left\"> Yes</td><td align=\"left\">860(68.7%)</td><td align=\"left\">235 (71.0%)</td><td char=\".\" align=\"char\"/></tr><tr><td align=\"left\"> No</td><td align=\"left\">389(31.1%)</td><td align=\"left\">94 (28.4%)</td><td char=\".\" align=\"char\"/></tr><tr><td align=\"left\"> Don’t know</td><td align=\"left\">3 (0.2%)</td><td align=\"left\">2 (0.6%)</td><td char=\".\" align=\"char\"/></tr><tr><td align=\"left\">Waist circumference (cm)</td><td align=\"left\">110.17 ± 16.00</td><td align=\"left\">107.56 ± 14.75</td><td char=\".\" align=\"char\">0.011</td></tr><tr><td align=\"left\">BMI (kg/m²)</td><td align=\"left\">32.38 ± 7.28</td><td align=\"left\">31.23 ± 6.47</td><td char=\".\" align=\"char\">0.007</td></tr><tr><td align=\"left\">Serum glucose (mg/dL)</td><td align=\"left\">167.84 ± 75.86</td><td align=\"left\">177.61 ± 84.42</td><td char=\".\" align=\"char\">0.036</td></tr><tr><td align=\"left\">Glycohemoglobin (%)</td><td align=\"left\">8.07 ± 1.68</td><td align=\"left\">8.33 ± 1.65</td><td char=\".\" align=\"char\">0.013</td></tr><tr><td align=\"left\">Triglyceride (mg/dL)</td><td align=\"left\">1.81 ± 1.61</td><td align=\"left\">1.60 ± 1.27</td><td char=\".\" align=\"char\">0.119</td></tr><tr><td align=\"left\">Total-cholesterol (mg/dL)</td><td align=\"left\">4.63 ± 1.20</td><td align=\"left\">4.59 ± 1.32</td><td char=\".\" align=\"char\">0.546</td></tr><tr><td align=\"left\">Serum albumin (g/dL)</td><td align=\"left\">4.08 ± 0.35</td><td align=\"left\">4.00 ± 0.41</td><td char=\".\" align=\"char\">&lt; 0.001</td></tr><tr><td align=\"left\">Blood lead (μmol/L)</td><td align=\"left\">0.06 ± 0.06</td><td align=\"left\">0.06 ± 0.05</td><td char=\".\" align=\"char\">0.416</td></tr><tr><td align=\"left\">Blood cadmium (nmol/L)</td><td align=\"left\">4.16 ± 4.90</td><td align=\"left\">4.00 ± 4.34</td><td char=\".\" align=\"char\">0.586</td></tr><tr><td align=\"left\">Blood mercury, total (μmol/L)</td><td align=\"left\">7.44 ± 14.79</td><td align=\"left\">7.16 ± 14.21</td><td char=\".\" align=\"char\">0.758</td></tr><tr><td align=\"left\">Blood selenium (μmol/L)</td><td align=\"left\">2.46 ± 0.40</td><td align=\"left\">2.42 ± 0.35</td><td char=\".\" align=\"char\">0.125</td></tr><tr><td align=\"left\">Blood manganese (μmol/L)</td><td align=\"left\">173.20 ± 69.02</td><td align=\"left\">162.99 ± 62.80</td><td char=\".\" align=\"char\">0.015</td></tr></tbody></table></table-wrap>", "<table-wrap id=\"Tab2\"><label>Table 2</label><caption><p>Association between blood metal levels and DR.</p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th align=\"left\"/><th align=\"left\">Model 1, β (95% CI)</th><th align=\"left\">p</th><th align=\"left\">Model 2, β (95% CI)</th><th align=\"left\">p</th><th align=\"left\">Model 3, β (95% CI)</th><th align=\"left\">p</th></tr></thead><tbody><tr><td align=\"left\" colspan=\"7\">DR</td></tr><tr><td align=\"left\"> Blood lead (μmol/L)</td><td align=\"left\">0.0062 (− 0.0665, 0.0788)</td><td char=\".\" align=\"char\">0.867</td><td align=\"left\">− 0.0160 (− 0.0923, 0.0602)</td><td align=\"left\">0.680</td><td align=\"left\">− 0.0307 (− 0.1179, 0.0565)</td><td align=\"left\">0.490</td></tr><tr><td align=\"left\"> Blood cadmium (nmol/L)</td><td align=\"left\">0.0024 (− 0.0554, 0.0602)</td><td char=\".\" align=\"char\">0.934</td><td align=\"left\">− 0.0067 (− 0.0657, 0.0523)</td><td align=\"left\">0.825</td><td align=\"left\">− 0.0201 (− 0.0892, 0.0490)</td><td align=\"left\">0.569</td></tr><tr><td align=\"left\"> Blood mercury, total (μmol/L)</td><td align=\"left\">− 0.0203 (− 0.0652, 0.0246)</td><td char=\".\" align=\"char\">0.375</td><td align=\"left\">− 0.0216 (− 0.0682, 0.0249)</td><td align=\"left\">0.362</td><td align=\"left\">− 0.0168 (− 0.0697, 0.0362)</td><td align=\"left\">0.535</td></tr><tr><td align=\"left\"> Blood selenium (μmol/L)</td><td align=\"left\">− 0.2291 (− 0.5376, 0.0794)</td><td char=\".\" align=\"char\">0.145</td><td align=\"left\">− 0.189962 (− 0.5005, 0.1206)</td><td align=\"left\">0.230</td><td align=\"left\">− 0.1766 (− 0.5240, 0.1709)</td><td align=\"left\">0.319</td></tr><tr><td align=\"left\"> Blood manganese (μmol/L)</td><td align=\"left\">− 0.2016 (− 0.3309, − 0.0724)</td><td char=\".\" align=\"char\">0.002</td><td align=\"left\">− 0.19230 (− 0.3230, − 0.0629)</td><td align=\"left\">0.004</td><td align=\"left\">− 0.2045 (− 0.3484, − 0.0606)</td><td align=\"left\">0.005</td></tr></tbody></table></table-wrap>", "<table-wrap id=\"Tab3\"><label>Table 3</label><caption><p>Association between serum manganese and stratified by age, sex, and BMI.</p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th align=\"left\"/><th align=\"left\">Model 1, β (95% CI)</th><th align=\"left\">p-value</th><th align=\"left\">Model 2, β (95% CI)</th><th align=\"left\">p-value</th><th align=\"left\">Model 3, β (95% CI)</th><th align=\"left\">p-value</th></tr></thead><tbody><tr><td align=\"left\" colspan=\"7\">DR</td></tr><tr><td align=\"left\" colspan=\"7\"> Stratified by age</td></tr><tr><td align=\"left\">  30–44 years</td><td align=\"left\">0.1198 (− 0.3538, 0.5935)</td><td char=\".\" align=\"char\">0.617</td><td align=\"left\">0.2014 (− 0.3115, 0.714)</td><td char=\".\" align=\"char\">0.438</td><td align=\"left\">0.3312 (− 0.1980, 0.8603)</td><td char=\".\" align=\"char\">0.216</td></tr><tr><td align=\"left\">  45–59 years</td><td align=\"left\">0.0010 (− 0.2336, 0.2536)</td><td char=\".\" align=\"char\">0.936</td><td align=\"left\">0.0206 (− 0.2256, 0.2668)</td><td char=\".\" align=\"char\">0.869</td><td align=\"left\">− 0.0180 (− 0.3003, 0.2644)</td><td char=\".\" align=\"char\">0.901</td></tr><tr><td align=\"left\">  ≥ 60 years</td><td align=\"left\">− 0.3092 (− 0.4704, − 0.1480)</td><td char=\".\" align=\"char\">&lt; 0.001</td><td align=\"left\">− 0.3119 (− 0.4735, − 0.1503)</td><td char=\".\" align=\"char\">&lt; 0.001</td><td align=\"left\">− 0.3122 (− 0.4903, − 0.1342)</td><td char=\".\" align=\"char\">0.001</td></tr><tr><td align=\"left\" colspan=\"7\"> Stratified by sex</td></tr><tr><td align=\"left\">  Male</td><td align=\"left\">− 0.2215 (− 0.3960, − 0.0469)</td><td char=\".\" align=\"char\">0.013</td><td align=\"left\">− 0.2203 (− 0.3949, − 0.0456)</td><td char=\".\" align=\"char\">0.013</td><td align=\"left\">− 0.2533 (− 0.4526, − 0.0540)</td><td char=\".\" align=\"char\">0.013</td></tr><tr><td align=\"left\">  Female</td><td align=\"left\">− 0.1857 (− 0.3801, 0.0087)</td><td char=\".\" align=\"char\">0.061</td><td align=\"left\">− 0.1615 (− 0.3574, 0.0345)</td><td char=\".\" align=\"char\">0.106</td><td align=\"left\">− 0.1418 (− 0.3501, 0.0665)</td><td char=\".\" align=\"char\">0.182</td></tr><tr><td align=\"left\" colspan=\"7\"> Stratified by BMI (kg/m²) (%)</td></tr><tr><td align=\"left\">  0–25</td><td align=\"left\">− 0.0544 (− 0.3948, 0.2860)</td><td char=\".\" align=\"char\">0.753</td><td align=\"left\">− 0.1397 (− 0.5133, 0.2339)</td><td char=\".\" align=\"char\">0.462</td><td align=\"left\">− 0.1570 (− 0.5634, 0.2494)</td><td char=\".\" align=\"char\">0.447</td></tr><tr><td align=\"left\">  25–30</td><td align=\"left\">− 0.1441 (− 0.3954, 0.1072)</td><td char=\".\" align=\"char\">0.261</td><td align=\"left\">− 0.1303 (− 0.3827, 0.1221)</td><td char=\".\" align=\"char\">0.311</td><td align=\"left\">− 0.1667 (− 0.4411, 0.1078)</td><td char=\".\" align=\"char\">0.233</td></tr><tr><td align=\"left\">  ≥ 30</td><td align=\"left\">− 0.2633 (− 0.4333, − 0.0934)</td><td char=\".\" align=\"char\">0.002</td><td align=\"left\">− 0.2553 (− 0.4289, − 0.0817)</td><td char=\".\" align=\"char\">0.004</td><td align=\"left\">− 0.2422 (− 0.4351, − 0.0493)</td><td char=\".\" align=\"char\">0.014</td></tr></tbody></table></table-wrap>" ]

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[ "<table-wrap-foot><p>Continuous variables are presented as mean ± SD, p-value was calculated by a linear regression model. Categorical variables are presented as %, p-value was calculated by Chi-square test.</p></table-wrap-foot>", "<table-wrap-foot><p>Model 1: no modification variables; Model 2: only adjusts for age, gender, and race; Model 3: adjusts age, gender, race, educational level, ratio of family income to poverty, skiing status, hypertension, BMI, and waist circumference.</p><p>Logarithmic conversion of all blood metal levels data.</p></table-wrap-foot>", "<table-wrap-foot><p>Model 1: no modification variables; Model 2: only adjusts for age, gender, and race; Model 3: adjusts age, gender, race, educational level, ratio of family income to poverty, smoking status, hypertension, BMI, and waist circumference.</p><p>The model is not adjusted for the stratification variable itself in the subgroup analysis.</p><p>Logarithmic conversion of all serum manganese data.</p></table-wrap-foot>", "<fn-group><fn><p><bold>Publisher's note</bold></p><p>Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.</p></fn></fn-group>" ]

[ "<graphic xlink:href=\"41598_2024_51749_Fig1_HTML\" id=\"MO1\"/>", "<graphic xlink:href=\"41598_2024_51749_Fig2_HTML\" id=\"MO2\"/>" ]

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{ "acronym": [], "definition": [] }

[ "<title>Introduction</title>", "<p id=\"Par3\">Acute lung injury (ALI) and its more severe form, acute respiratory distress syndrome (ARDS) are common, life-threatening critical illnesses that lead to significant morbidity and mortality [##REF##34217425##1##]. Despite prominent breakthroughs in the pathophysiology of ALI/ARDS, the hospital mortality rate of these disorders remains high (46.1%), and effective pharmacological treatments are still lacking [##REF##36070788##2##]. ALI/ARDS is characterized by dysregulated lung parenchymal inflammation, leading to diffuse alveolar damage and edema, ultimately contributing to acute hypoxemic respiratory failure [##REF##36070787##3##]. Uncontrolled local or systemic inflammation is believed to be the predominant cause of ALI/ARDS [##REF##33188794##4##]. Activated macrophages, especially recruited circulating monocyte-derived macrophages further release pro-inflammatory cytokines, which give rise to an inflammation cascade [##REF##28306336##5##].</p>", "<p id=\"Par4\">The NOD-like receptor pyrin domain-containing protein 3 (NLRP3) inflammasome is excessively activated in macrophages during ALI/ARDS progression [##REF##34115964##6##]. The NLRP3 inflammasome, which consists of a sensor (NLRP3), an adaptor apoptosis-associated speck-like protein containing a caspase-recruitment domain (ASC), and an effector caspase (caspase-1), is involved in the production of pro-inflammatory cytokines, interleukin (IL)-1β and IL-18 [##REF##33850310##7##]. NLRP3 inflammasome activation reportedly involves two steps: priming and activation [##UREF##0##8##]. The priming step of NLRP3 inflammasome activation is regulated via transcriptional and post-translational mechanisms [##REF##31036962##9##]. NF-κB signaling induces the transcriptional activation of NLRP3 priming by upregulating the gene expression of NLRP3 inflammasome components [##REF##19570822##10##, ##UREF##1##11##]. Post-translational modifications (PTMs) of NLRP3, such as ubiquitination, phosphorylation, and SUMOylation, may stabilize NLRP3 in an auto-suppressed inactive state [##REF##28615462##12##]. The activation step is induced by various pathogen-associated molecular patterns (PAMPs) and damage-associated molecular patterns (DAMPs), including extracellular ATP, pore-forming toxins, RNA viruses, and particulate matter [##REF##32531835##13##]. However, post-transcriptional regulation of NLRP3 inflammasome activation in ALI/ARDS remains unclear.</p>", "<p id=\"Par5\">N⁶-methyladenosine (m⁶A) modification, the most abundant modification of messenger RNA (mRNA), may reversibly regulate target genes at the post-transcriptional level, thereby affecting almost all crucial biological processes [##REF##37068987##14##]. This dynamic and reversible process is primarily regulated by the m⁶A methyltransferase complex, which contains methyltransferase-like 3 (METTL3), methyltransferase-like 14 (METTL14), Wilms tumor suppressor-1-associated protein (WTAP), and demethylases, including fat mass and obesity-related protein (FTO) and ALKB homolog 5 protein (ALKBH5) [##REF##31520073##15##]. Meanwhile, RNA-binding proteins that identify and bind to m⁶A sites, such as the YT521-B homology domain (YTHD) family, and the insulin-like growth factor 2 mRNA-binding protein (IGF2BP) family, serve as m⁶A readers and direct the fate of target RNAs by influencing alternative pre-mRNA splicing, RNA stability, and translation efficiency [##REF##32276589##16##].</p>", "<p id=\"Par6\">m⁶A facilitates the progression of several inflammatory diseases, such as non-alcoholic fatty liver disease, autoimmune diseases, and infections [##REF##36578520##17##–##REF##30559377##20##]. Evidently, global m⁶A levels are significantly increased in alveolar epithelial cells, mediated by the upregulation of METTL3, which is closely associated with ALI [##REF##35637949##21##]. Nonetheless, the effects of METTL14-regulated m⁶A methylation in ALI/ARDS remain unclear and the precise molecular targets of METTL14 in ALI/ARDS remain to be elucidated. Therefore, we sought to determine the functional role of METTL14 and its target in ALI/ARDS.</p>", "<p id=\"Par7\">Herein, we first elucidated that RNA m⁶A modification in macrophages is involved in the progression of ALI/ARDS, and then verified the role of METTL14 in this process. Further mechanistic studies revealed that NLRP3 is the methylated target of METTL14 and that IGF2BP2 stabilizes NLRP3 mRNA during NLRP3 inflammasome activation in ALI/ARDS. These results indicated that METTL4 together with IGF2BP2 may be promising therapeutic targets in ALI/ARDS.</p>" ]

[ "<title>Materials and methods</title>", "<title>Animals</title>", "<p id=\"Par25\">Male specific-pathogen-free C57/BL6 mice (8–10 weeks old) were obtained from the Guangdong Medical Laboratory Animal Center (Guangzhou, China). All mice were housed under controlled, pathogen-free conditions at the Laboratory Animal Center of Sun Yat-sen University Cancer Center. All animals were housed in separate cages in a temperature- (24 ± 1 °C) and humidity-controlled (50–60%) room under a 12/12-h light/dark cycle. All experiments were conducted in accordance with the guidelines defined by the Sun Yat-sen University Cancer Center. The study was approved by the Animal Care and Ethics Committee of Sun Yat-sen University Cancer Center (Permit Number: 2021-000043).</p>", "<title>Animal models and treatments</title>", "<p id=\"Par26\">A mouse LPS-induced ALI model was established as previously described [##REF##32265704##47##]. Briefly, mice from the ALI groups were treated with a single intraperitoneal dose of 15 mg/kg LPS obtained from Escherichia coli 055: B5 (Sigma-Aldrich, St. Louis, MO, USA) in saline, whereas mice injected with the same volume of saline served as controls. After 24 h, the mice were killed and the lung lobes were harvested for further analysis. This in vivo study was performed via six series of experiments. Mice in the first series were randomly divided into control and ALI groups. Mice in the second series were randomly assigned to receive the following treatments: (1) control + si-NC, (2) control + si-METTL14, (3) control + MCC950, (4) ALI + si-NC, (5) ALI + si-METTL14 and (6) ALI + MCC950 (an NLRP3 inflammasome inhibitor, 50 mg/kg, i.p., Selleck, Shanghai, China). Mice in the third series were grouped as follows: (1) control + si-NC, (2) control + si-IGF2BP2, (3) ALI + si-NC, or (4) ALI + si-IGF2BP2. Each group received a dose of 20 nmol siRNA (either si-NC, si-METTL14 or si-IGF2BP2) in 200 μl of saline via the tail vein 2 d before being challenged with LPS or saline. Mice in the fourth series were grouped as follows: (1) control + AAV-GFP, (2) control + AAV-METTL14, (3) ALI + AAV-GFP, or (4) ALI + AAV-METTL14. Each group received a dose of 50 μl viral solution (either AAV-GFP or AAV-METTL14 with 10¹² vg/ml titer) via intranasal instillation 4 weeks before being challenged with LPS or saline. Mice in the fifth series were grouped as follows: (1) ALI + AAV-GFP + DMSO, (2) ALI + AAV-METTL14 + DMSO, (3) ALI + AAV-GFP + MCC950, or (4) ALI + AAV-METTL14 + MCC950. Mice in the sixth series were grouped as follows: (1) ALI + AAV-GFP, (2) ALI + AAV-METTL14, (3) ALI + AAV-METTL14 + si-NC, or (4) ALI + AAV-METTL14 + si-IGF2BP2. The treatments of each group were performed according to the mentioned above.</p>", "<title>Histopathological analysis</title>", "<p id=\"Par27\">Left lung lobes were fixed in 4% paraformaldehyde for 48 h, dehydrated, embedded in paraffin, and sliced into 5-µm-thick sections. The sections were then stained with hematoxylin and eosin (H&amp;E) according to the manufacturer’s instructions, to evaluate lung histopathology. The damage to the lung tissues was scored using a previously described semiquantitative scoring system [##REF##32265704##47##]. Images were captured using a microscope (NIKON Eclipse Ni-U; NIKON, Tokyo, Japan).</p>", "<title>Lung wet/dry ratio</title>", "<p id=\"Par28\">Any blood present on the isolated right lungs was blotted with filter paper before the weights of these lungs were recorded as wet weight. Then, the lungs were then stored in an incubator at 60 °C for 48 h, following which the weight of the lungs was recorded as dry weight. The lung wet/dry (W/D) ratio was used to evaluate the degree of pulmonary edema.</p>", "<title>Bronchoalveolar lavage</title>", "<p id=\"Par29\">At the time of lavage, the mice were anesthetized with an i.p. injection of 1% pentobarbital sodium (50 mg/kg). The chest cavity was exposed, then the trachea was intubated, and a whole lung lavage was performed by employed sterile PBS (1 mL). The collected lavage fluid was centrifuged at 1000×<italic>g</italic> for 10 min at 4 °C, and the cell-free supernatants were harvested and stored at −80^◦C for further analysis. The total protein concentration of bronchoalveolar lavage fluid (BALF) was measured using the BCA Protein Assay Kit (Thermo Fisher Scientific).</p>", "<title>Cell culture and treatments</title>", "<p id=\"Par30\">RAW264.7 cells, a mouse macrophage cell line, was obtained from ATCC and cultured in Dulbecco’s modified Eagle’s medium (Gibco from Thermo Fisher Scientific, Waltham, MA, USA) with 10% fetal bovine serum (Gibco) in an incubator at 37 °C and 5% CO₂. To establish an NLRP3 inflammasome activation model in vitro, RAW264.7 cells were stimulated with LPS (1 μg/mL) for 6 h, then treated with nigericin (10 μM, InvivoGen, San Diego, CA, USA) for 30 min.</p>", "<p id=\"Par31\">For transient transfection purpose, cells were seeded at 30–40% confluence and cultured overnight, following which si-METTL14, si-IGF2BP2, and negative control (si-NC) were diluted in Opti-MEM^® medium (Thermo Fisher Scientific, Waltham, MA, USA) and transfected using Lipofectamine 3000 transfection reagent (Invitrogen, Carlsbad, CA, USA) according to the manufacturer’s instructions. After 48 h of transfection, the cells were treated with or without LPS (1 μg/mL) for 6 h. Three siRNA sequences targeting METTL14 and IGF2BP2 were designed and synthesized by RiboBio (Guangzhou, China) and these were listed in Supplemental Table ##SUPPL##0##1##. Specific siRNA with the best knockdown effect was used for further research and in vivo assays.</p>", "<title>ELISA analysis and myeloperoxidase activity</title>", "<p id=\"Par32\">Mice were anesthetized with an intraperitoneal injection of 1% pentobarbital sodium (50 mg/kg). Blood samples were collected from the retro-orbital sinus after the mice lost consciousness. Subsequently, blood samples were allowed to clot by leaving them undisturbed at 25 °C for 30 min. The clots were then removed to obtain serum via centrifugation at 1000 × g for 10 min at 4 °C. Part of the right lung from each mouse was homogenized with ELISA buffer and centrifuged to obtain lung tissue supernatants. Samples of murine serum, lung tissue supernatants, and cell culture supernatants were used to quantify the concentrations of IL-1β (R&amp;D System, Minneapolis, MN, USA) and IL-18 (R&amp;D System) by using murine ELISA kits, according to the manufacturer’s instructions. The MPO activity in the lung tissue was assessed by an MPO assay kit (R&amp;D System).</p>", "<title>RNA m⁶A dot blot assay</title>", "<p id=\"Par33\">Total RNA and poly-A RNA were isolated from the lung tissue or RAW264.7 macrophages using a RNeasy mini kit (Qiagen, Düsseldorf, Germany) and a Dynabeads^® mRNA purification kit (Ambion, Austin, TX, USA), according to the manufacturer’s instructions. RNA was quantified using a Nanodrop, and equal amounts of RNA were crosslinked onto Hybond-N discs (Cytiva, USA) using a UV crosslinker (Spectroliner, Long Island, NY, USA). The membrane was quickly washed and blocked using 5% nonfat dry milk in 0.1% phosphate-buffered saline with Tween-20 (PBST) supplemented with RNaseOUT (Thermo Fisher Scientific). The membrane was incubated overnight at 4 °C with rabbit anti-m⁶A antibody (1:500, cat#A-1802-100, EpiGentek, Farmingdale, NY, USA), followed by incubation with horseradish peroxidase (HRP)-conjugated secondary anti-rabbit antibody. Membranes were washed and visualized using an enhanced chemiluminescence detection system. Images were acquired using a ChemiDoc™ Touch Imaging System (Bio-Rad, Berkeley, CA, USA). Finally, the membranes were stained with methylene blue as a loading control. The signal intensity of the dot blot was analyzed using ImageJ software (NIH, Bethesda, MD, USA).</p>", "<title>RNA m⁶A modification quantification</title>", "<p id=\"Par34\">The levels of m⁶A in lung tissues and RAW264.7 macrophages were quantified using an EpiQuik m⁶A RNA Methylation Quantification Kit (EpiGentek), according to the manufacturer’s recommendations.</p>", "<title>Western blotting</title>", "<p id=\"Par35\">Mouse lung tissues or cells were lysed using RIPA lysis buffer (Beyotime, Shanghai, China) containing a protein inhibitor cocktail (Roche, Mannheim, Baden Württemberg, Germany). The total protein concentration was quantified using a BCA kit (Thermo Fisher Scientific). Samples were denatured at 100 °C for 10 min and separated on 10–12% SDS-PAGE gels with a molecular weight standard. Proteins from SDS-PAGE gel were transferred onto PVDF membranes (Merck Millipore, Darmstadt, Germany), blocked using 5% non-fat milk for 2 h, and incubated at 4 °C with the following primary antibodies overnight: METTL3 (1:1000, 15073-1-AP, Proteintech, Wuhan, China), METTL14 (1:1000, A8530, Abclonal, Boston, MA, USA), METTL16 (1:1000, 17676 S, Cell Signaling Technology, Danvers, MA, USA), WTAP (1:1000, 60188-1-Ig, Proteintech), FTO (1:1000, 45980 S, Cell Signaling Technology), ALKBH5 (1:1000, 16837-1-AP, Proteintech), NLRP3 (1:1000, AG-20B-0006-C100, AdipoGen, San Diego, CA, USA), Caspase-1 (1:1000, AG-20B-0042-C100, AdipoGen), IL-1β (1:500, AF-401-NA, R&amp;D System), IGF2BP1 (1:1000, A1517, Abclonal), IGF2BP2 (1:1000, 11601-1-AP, Proteintech), IGF2BP3 (1:1000, A23295, Abclonal). After three washes, the membranes were incubated with the corresponding HRP-conjugated secondary antibody (1:1000, Abcam, Cambridge, UK) at room temperature for 1 h. Protein bands were detected using ECL and visualized using a ChemiDoc™ Touch Imaging System (Bio-Rad). The band intensities were analyzed by using ImageJ software.</p>", "<title>Immunofluorescence</title>", "<p id=\"Par36\">Left lung lobes were fixed with 4% paraformaldehyde for 48 h, dehydrated, embedded in paraffin, and sectioned into 5-µm slices. After deparaffinated, dehydration, and antigen recovery, sections were incubated in blocking solution (Beyotime) for 1 h at room temperature and then incubated with primary antibodies overnight at 4 °C, followed by incubation at 25 °C for 1 h with fluorescently labeled secondary antibodies. Nuclei were stained for 10 min with DAPI. Images from six representative non-overlapping high-power fields (HPFs) of individual mice were taken on a fluorescent microscope (Leica, Wetzlar, Germany) in a blinded manner. The following antibodies were used: anti-METTL14 (1:200, A8530, Abclonal), anti-CD68 (18985-1-AP, 1:100, Proteintech), anti-F4/80 (18985-1-AP, 1:100, Proteintech), anti-Siglec F (18985-1-AP, 1:100, Proteintech), and anti-CK7 (16001- 1-AP, 1:100; Proteintech).</p>", "<title>Quantitative real-time RT-PCR</title>", "<p id=\"Par37\">Total RNA was extracted using TRIzol reagent (Invitrogen), and subsequently reverse-transcribed to cDNA according to the instructions of manufacturer of the qPCR transcription kit (EZ Bioscience, Roseville, NM, USA). Quantitative PCR was performed using SYBR Green Mix (EZ Bioscience) and a CFX96 Real-Time PCR Detection System (Bio-Rad). Target mRNA expression was calculated via the 2^-ΔΔCt method using GAPDH as an endogenous control. Primer sequences are listed in Supplemental Table ##SUPPL##0##2##.</p>", "<title>RNA stability assay</title>", "<p id=\"Par38\">RAW264.7 macrophages were cultured in six-well culture plates until they reached 80% confluence. Actinomycin D (Abmole, Houston, TX, USA) was added at a final concentration of 5 μg/ml. Cells were collected at 0, 0.5, 1, 2, 4, and 6 h. Total RNA was extracted, RT–qPCR was performed as described above, and GAPDH was used as the loading control for normalization. Then, the RNA half-life was calculated.</p>", "<title>RNA immunoprecipitation assay</title>", "<p id=\"Par39\">RIP was performed according to the manufacturer’s instructions (Merck Millipore). Briefly, lung tissues of equal weight were mechanically sheared into a single-cell suspension using a homogenizer and resuspended in RIP lysis buffer containing protease inhibitor and RNase inhibitors. The mixture was centrifuged at 14 000 rpm at 4 °C for 10 min to obtain the supernatant, which was then divided into three fractions: Input, IP, and IgG. Each fraction was incubated overnight with the corresponding primary antibody at 4 °C, followed by protein A/G bead incubation at room temperature for 30 min. After six washes, the beads were incubated with 150 μl proteinase K buffer at 55 °C for 30 min. Total RNA was extracted, and the expression of related genes was detected via RT-qPCR.</p>", "<title>m⁶A RNA Immunoprecipitation assay</title>", "<p id=\"Par40\">The MeRIP assay was performed using a Magna MeRIP m⁶A Kit (Merck Millipore). Briefly, total RNA was extracted from lung tissues and RAW264.7 macrophages using TRIzol reagent. Approximately 20 μg of purified RNA was incubated with RNA fragmentation buffer. Then, 1 μg of the fragmented mRNA was used as input, while the remaining RNA was incubated overnight with anti-m⁶A antibody (Synaptic Systems, Gottingen, Germany) or anti-IgG antibody in 500 μl of IP buffer at 4 °C. The following procedure was performed in the same manner as that described for the RIP assay.</p>", "<title>RNA pull-down assay</title>", "<p id=\"Par41\">An RNA pull-down assay was performed using a Magnetic RNA-Protein Pull-down Kit (Thermo Fisher Scientific) according to the manufacturer’s instructions. The 3’-end Biotin-TEG modified-DNA probes against NLRP3 were synthesized by RiboBio. The biotinylated NLRP3 probe (50 pmol) was incubated with streptavidin beads to generate probe-coated beads. Lung tissue homogenates with probe-coated beads was incubated at 4 °C overnight. After three washes, proteins bound to the beads was boiled and used for the immunoblotting.</p>", "<title>Statistical analysis</title>", "<p id=\"Par42\">All sample size information is shown (figure legends). Statistical analyses were conducted using GraphPad Prism 8.0 (GraphPad software Inc, San Diego, CA, USA). Quantitative data were assessed for normality test and presented as the means ± SD. Differences between two normally distributed groups were analyzed using a two-tailed unpaired Student’s <italic>t</italic> test. Multiple comparisons of parametric data were performed using one-way ANOVA. Exact <italic>P</italic> values are indicated in all figures, and statistical significance was set at <italic>P</italic> &lt; 0.05.</p>" ]

[ "<title>Results</title>", "<title>Global m⁶A levels and METTL14 expression are increased in ALI mice</title>", "<p id=\"Par8\">To confirm the role of m⁶A modification in ALI/ARDS, we evaluated the global m⁶A levels in the lung tissues of control and ALI mice. Both dot blot assay and colorimetric quantification showed that global m⁶A levels in the total RNA were significantly increased in ALI lungs compared to the control group (Fig. ##FIG##0##1A–C##). We then detected the expression of m⁶A methyltransferases (METTL3, METTL14, METTL16, and WTAP) and demethylases (FTO and ALKBH5) in lung tissues. The expression of METTL14 mRNA and METTL14 protein was markedly upregulated in ALI mice, whereas no significant differences were found in the expression of other regulators (Fig. ##FIG##0##1D–F##). These results indicated that METTL14-mediated m⁶A methylation may play a functional role in ALI/ARDS. Subsequently, we employed immunofluorescence staining to identify the specific cell types involved in ALI/ARDS that express METTL14. Our findings revealed co-localization of METTL14 not only with CD68-labeled macrophages but also with CK7-labeled pulmonary epithelial cells. Interestingly, compared with sham group, ALI lungs exhibited an increased number of METTL14-expressed CD68⁺ macrophages, while the number of METTL14-expressed CK7⁺ epithelial cells did not show a significant change (Figs. ##FIG##0##1G, H## and ##SUPPL##3##S1A, B##). To clarify the origins of elevated METTL14⁺ macrophages, quantitative analysis unveiled that METTL14⁺/ F4/80⁺, rather than METTL14⁺/Siglec-F⁺ (a marker for resident alveolar macrophage) cells increased in ALI mice compared with the corresponding sham mice (Fig. ##FIG##0##1I–L##). Collectively, these findings indicated that the expression level of METTL14 is elevated in LPS-induced ALI model, particularly in recruited circulating monocyte-derived macrophages within the lung.</p>", "<title>Global m⁶A levels and METTL14 expression are increased in LPS-activated macrophages</title>", "<p id=\"Par9\">The NLRP3 inflammasome is a crucial factor in triggering the activation of macrophages during this pathogenesis [##REF##34641966##22##]. We subsequently assessed the expression level of METTL14 within a NLRP3 inflammasome activation model in RAW264.7 macrophages. As expected, treatment of RAW264.7 macrophages with LPS and nigericin significantly increased the release of NLRP3-inflammasome-dependent cytokines, including IL-1β p17, Caspase-1 p20, IL-1β and IL-18 (Fig. ##FIG##1##2A, E, F##). The mRNA and protein levels of METTL14 and NLRP3 in RAW264.7 macrophages were observably enhanced following stimulation with LPS whether combined with nigericin or not, indicating METTL14 may participate in the priming step (Fig. ##FIG##1##2A–D##). Both dot blot assay and colorimetric quantification showed that global m⁶A levels of total RNA in activated macrophages were notably elevated (Fig. ##FIG##1##2G–I##). Our findings pointed towards a possible role for METTL14-mediated m⁶A methylation in the process of NLRP3 inflammasome activation.</p>", "<title>Knocking down METTL14 inhibits the activation of NLRP3 inflammasome and alleviates lung injury in vitro and in vivo</title>", "<p id=\"Par10\">To determine the function of METTL14, we knocked down METTL14 expression in RAW264.7 macrophages using small interfering RNA (siRNA). Western blotting, real-time PCR, and colorimetric m⁶A quantification were used to verify the knockdown effect (Fig. ##FIG##2##3A–D##). Considering si-METTL14 #3 exhibited the superior knockdown efficiency, it was selected for further experiments. Compared with the negative control (si-NC) cells, knockdown of METTL14 exhibited a significant decrease in NLRP3 protein expression, as well as a reduced release of IL-1β and IL-18 cytokines in LPS-stimulated macrophages (Fig. ##FIG##2##3E–H##). Interestingly, METTL14 knockdown only downregulated the mRNA expression of NLRP3, but not that of IL-1b or IL-18 in LPS-treated cells (Fig. ##FIG##2##3I–K##). These results suggested that METTL14 may mediate NLRP3 inflammasome activation via regulating NLRP3.</p>", "<p id=\"Par11\">We next employed METTL14 siRNA to determine the in vivo function of METTL14 in ALI. Total m⁶A levels in ALI mice were reduced after METTL14 was knocked down (Fig. ##FIG##2##3L##), suggesting that m⁶A modification occurring in ALI/ARDS was mainly mediated by METTL14. Both METTL14 siRNA and MCC950 (NLRP3 inhibitor) administration decreased the lung wet/dry ratio in ALI mice, indicating an alleviation of pulmonary edema associated with ALI/ARDS. (Fig. ##FIG##2##3M##). Compared with ALI group, the total protein concentrations in BALF and myeloperoxidase (MPO) activity in lung tissues of si-METTL14 + ALI and MCC950 + ALI group were notably lower (Fig. ##FIG##2##3N, O##). Similarly, H&amp;E staining showed relatively intact alveolar structure and less inflammatory cell infiltration in si-METTL14 + ALI and MCC950 + ALI group than those in the ALI group (Fig. ##FIG##2##3P, Q##). Consistent with the in vitro results, we found that METTL14 knockdown inhibited the activation of NLRP3 inflammasome in ALI mice via regulating the mRNA levels of NLRP3, rather than IL-1b and IL-18 (Figs. ##FIG##2##3R–T## and S2A–D). Collectively, these results suggested that METTL14 knockdown may inhibit NLRP3 inflammasome activation and alleviate lung injury in vitro and in vivo, confirming that METTL14 plays a vital role in NLRP3 inflammasome activation in ALI/ARDS.</p>", "<title>Elevated METTL14 promotes the activation of NLRP3 inflammasome and aggravates lung injury in vitro and in vivo</title>", "<p id=\"Par12\">To further elucidate the function of METTL14 in ALI, we performed gain-of-function assay by overexpressing METTL14 in RAW264.7 macrophages (Fig. ##FIG##3##4A–C##). METTL14 overexpression in RAW264.7 macrophages increased m⁶A levels (Fig. ##FIG##3##4D##) and activated NLRP3 inflammasome in macrophages by upregulating the mRNA levels of NLRP3, rather than IL-1b and IL-18 (Fig. ##FIG##3##4E–K##). We subsequently explored the in vivo function of METTL14 in ALI using AAV9 that expressed full-length METTL14. AAV-GFP was used as a control. As shown in Fig. ##FIG##3##4L##, a marked increase in the level of m⁶A modification was detected in lung tissue of ALI mice. AVV-METTL14 aggravated pulmonary edema of ALI mice, as revealed by lung wet/dry ratio (Fig. ##FIG##3##4M##). METTL14 overexpression also increased the total protein concentrations in BALF and MPO activity (Fig. ##FIG##3##4N, O##). Likewise, H&amp;E staining showed thicker alveolar walls and more inflammatory infiltration in AVV-METTL14 + ALI group than those in the ALI group (Fig. ##FIG##3##4P, Q##). Indeed, METTL14 activated NLRP3 inflammasome via upregulating the mRNA levels of NLRP3 (Fig. ##FIG##3##4R–V##). Taken together, these data supported that the elevation of METTL14 contributes to the activation of the NLRP3 inflammasome and the exacerbation of lung injury in vitro and in vivo.</p>", "<title>NLRP3 is the direct target of METTL14-mediated m⁶A modification</title>", "<p id=\"Par13\">NLRP3 is present in low concentrations under resting conditions, which is insufficient to activate the inflammasome [##UREF##1##11##]. Based on previous results showing that METTL14 regulated the mRNA expression of NLRP3 both in vivo and in vitro, we surmised that NLRP3 may be a direct target of METTL14. To validate the role of m⁶A methylation modulated by METTL14 in NLRP3 transcript, we analyzed potential m⁶A targeting motifs using SRAMP (<ext-link ext-link-type=\"uri\" xlink:href=\"http://www.cuilab.cn/sramp\">http://www.cuilab.cn/sramp</ext-link>). A total of 24 RRACH m⁶A-binding motifs were identified in NLRP3 mRNA sequence (Fig. ##FIG##4##5A## and Supplemental Table ##SUPPL##2##3##). The m⁶A RNA immunoprecipitation (MeRIP) assays confirmed that NLRP3 mRNA m⁶A modification was enhanced in ALI mice and LPS-treated macrophages (Fig. ##FIG##4##5B, D##), but significantly decreased after METTL14 knockdown (Fig. ##FIG##4##5C, E##). Next, we used RNA pull-down and RNA immunoprecipitation (RIP) assays to determine whether there is a direct interaction between NLRP3 mRNA and METTL14. RNA pull-down assays showed that METTL14 interacts with the NLRP3 transcript, and that this interaction was enhanced in ALI mice (Fig. ##FIG##4##5F##). RIP analysis with the METTL14 antibody further confirmed the interaction between METTL14 and NLRP3 mRNA both in vivo and in vitro (Fig. ##FIG##4##5G, H##). Moreover, rescue assays were performed by using MCC950, an NLRP3 inhibitor, in AAV-METTL14 mice. Our data revealed that the extent of lung injury in AAV-METTL14 ALI mice was restored by MCC950 treatment, as revealed by lung wet/dry ratio, BALF protein content, MPO activity and histological injury score (Fig. ##FIG##4##5I–M##). The over-release of IL-1β and IL-18 in lung tissues and serum in AAV-METTL14 ALI mice was reversed by MCC950 treatment (Fig. ##FIG##4##5N–Q##). These findings indicated that NLRP3 is a direct and functional target of METTL14-mediated m⁶A modification during NLRP3 inflammasome activation in ALI/ARDS.</p>", "<title>IGF2BP2 is upregulated and enhances the stability of NLRP3 mRNA in ALI</title>", "<p id=\"Par14\">Considering that METTL14 induces NLRP3 mRNA m⁶A methylation and that the loss of m⁶A in NLRP3 mRNA mediated by METTL14 knockdown leads to a decrease in NLRP3 mRNA and protein expression in ALI mice, we sought to determine whether METTL14-mediated m⁶A modification affects the NLRP3 mRNA stability. We treated RAW264.7 macrophages with the transcription inhibitor actinomycin D (ActD) and found that NLRP3 decay in si-METTL14-treated macrophages was faster than that in corresponding controls when stimulated with LPS (Fig. ##FIG##5##6A##), suggesting that METTL14 regulates NLRP3 expression via an m⁶A-dependent mRNA decay mechanism. Therefore, we identified m⁶A readers that may participate in the regulation of NLRP3 mRNA stability. The IGF2BP family regulates the stability of methylated mRNA by acting as m⁶A readers [##REF##29954406##23##]. First, we detected the protein expression of IGF2BP1, IGF2BP2, and IGF2BP3 using western blotting. We found that protein expression of IGF2BP2 was distinctly upregulated in ALI mice (Fig. ##FIG##5##6B, C##), which was consistent with its mRNA expression (Fig. ##FIG##5##6D##). Similarly, the protein and mRNA expression levels of IGF2BP2 were notably augmented in LPS-treated RAW264.7 macrophages (Fig. ##FIG##5##6E–G##). To further validate the direct interaction between NLRP3 mRNA and IGF2BP2, we performed an in vivo RNA precipitation assay using a biotinylated NLRP3 probe. RNA pull-down assay detected that specific binding of IGF2BP2 was enhanced in ALI mice (Fig. ##FIG##5##6H##). RIP analysis with the IGF2BP2 antibody further confirmed that their interaction was facilitated in vivo and in vitro during ALI (Fig. ##FIG##5##6I, J##). These results implied that the stability of NLRP3 mRNA might be regulated by IGF2BP2 via METTL14-mediated m⁶A modification.</p>", "<title>IGF2BP2 knockdown decreases NLRP3 mRNA stability and inhibits NLRP3 inflammasome activation in LPS-activated macrophages</title>", "<p id=\"Par15\">To examine whether IGF2BP2 regulates NLRP3 expression, we used siRNA to knockdown IGF2BP2 in RAW264.7 macrophages (Fig. ##FIG##6##7A–C##), and si-IGF2BP2 #2 with the best knockdown effect was selected for further experiments. As shown, IGF2BP2 knockdown significantly downregulated the mRNA expression of NLRP3 (Fig. ##FIG##6##7D##), but not IL-1b (Fig. ##FIG##6##7E##) or IL-18 (Fig. ##FIG##6##7F##), in LPS-treated RAW264.7 macrophages. To further determine the mechanism underlying IGF2BP2-induced regulation of NLRP3, we examined the effect of IGF2BP2 knockdown on the lifetime of NLRP3 mRNA. We found that the stability of NLRP3 mRNA in LPS-treated RAW264.7 macrophages was reduced by IGF2BP2 knockdown (Fig. ##FIG##6##7G##). As expected, IGF2BP2 knockdown inhibited NLRP3 expression (Fig. ##FIG##6##7H, I##) and the release of IL-1β and IL-18 (Fig. ##FIG##6##7J, K##) in LPS-treated RAW264.7 macrophages. We proceeded to silence IGF2BP2 in METTL14-overexpressing cells. Our data showed that IGF2BP2 knockdown reduced NLRP3 mRNA lifespan (Fig. ##FIG##6##7L##) and restored the over-release of IL-1β and IL-18 (Fig. ##FIG##6##7M, N##) in METTL14-overexpressing macrophages. These results confirmed that IGF2BP2 participates in METTL14-mediated NLRP3 inflammasome activation by enhancing the stability of NLRP3 mRNA in macrophages.</p>", "<title>Knocking down IGF2BP2 inhibits the activation of NLRP3 inflammasome and alleviates lung injury in ALI mice</title>", "<p id=\"Par16\">We further determined the therapeutic potential of IGF2BP2 against mouse ALI models by applying siRNA to knock down IGF2BP2 in vivo. Compared to mice treated with control siRNA, the si-IGF2BP2 group showed significantly alleviated lung wet/dry ratio in ALI mice (Fig. ##FIG##7##8A##). IGF2BP2 inhibition also decreased the total protein levels in BALF and MPO activity in ALI lung (Fig. ##FIG##7##8B, C##). Similar effects of IGF2BP2 knockdown on alleviating lung injury in ALI mice were revealed H&amp;E staining (Fig. ##FIG##7##8D, E##). Disruption of IGF2BP2 downregulated the mRNA expression of NLRP3 in the lung tissues of ALI mice (Fig. ##FIG##7##8F##). The dramatic increase in the IL-1β and IL-18 levels were efficiently diminished in ALI mice after treated with si-IGF2BP2 (Fig. ##FIG##7##8G–J##). We further performed IGF2BP2 inhibition in AAV-METTL14 mice. The deterioration of lung wet/dry ratio, BALF protein content, MPO activity in AAV-METTL14 ALI mice were restored by IGF2BP2 knockdown (Fig. ##FIG##7##8K-M##). IGF2BP2 inhibition also reduced the upregulation of IL-1β and IL-18 levels in lung tissues and serum (Fig. ##FIG##7##8N–Q##) and NLRP3 mRNA (Fig. ##FIG##7##8R##) in AAV-METTL14 ALI mice. These results manifested that IGF2BP2 knockdown may relieve ALI via inhibiting the NLRP3 inflammasome activation in vivo.</p>" ]

[ "<title>Discussion</title>", "<p id=\"Par17\">In this study, we discovered that the contents of m⁶A and METTL14 in lung tissues of ALI mice subjected to LPS were enhanced. METTL14-mediated NLRP3 mRNA m⁶A modification increases NLRP3 mRNA stability in injured lungs in an IGF2BP2-dependent manner. Thus, knocking down METTL14 or IGF2BP2 may play a protective role in ALI by inhibiting NLRP3 inflammasome activation. This finding may provide new pathophysiological insights into potential therapeutic strategies for ALI/ARDS.</p>", "<p id=\"Par18\">N⁶-Methyladenosine (m⁶A) is the most abundant epigenetic mRNA modification and exerts different biological effects in various diseases via post-transcriptional regulation [##REF##32183948##24##–##REF##33956076##26##]. Emerging evidence indicates that m⁶A may play an indispensable role in some inflammatory diseases [##REF##34150765##27##]. RNA m⁶A modification is mediated by m⁶A writers (methyltransferases), erasers (demethylases), and readers. METTL14, a key component of the m⁶A methyltransferase complex, stabilizes the structure of METTL3 and enhances its enzymatic activity by binding to RNA, which ultimately increases m⁶A level [##REF##29290617##28##]. Our study showed that m⁶A modification and the m⁶A methyltransferase METTL14 were increased in ALI mice. Further analysis confirmed METTL14 is mainly elevated in recruited circulating monocyte-derived macrophages of ALI mice. Interestingly, some studies have shown that neutrophil extracellular traps (NETs) induced ferroptosis in alveolar epithelial cells of cecal ligation and puncture (CLP)-mouse model by activating METTL3, while a few studies showed METTL3-mediated m⁶A modification alleviated ALI via inhibiting endothelial injury, indicating that m⁶A may exert different effects on ALI/ARDS owing to cell types and challenges [##REF##37715457##29##–##REF##35693774##31##].</p>", "<p id=\"Par19\">Alveolar macrophages (AMs) consist of two subpopulations, including resident AMs and recruited AMs [##REF##32647933##32##]. The resident AMs serve as an immunosuppressive subpopulation and mainly present the M2 phenotype, whereas the recruited AMs, which are derived from circulating monocytes, prefer to differentiate into pro-inflammatory M1 phenotype [##REF##25225663##33##, ##REF##37042701##34##]. Consistent with the enrichment of METTL14 in recruited macrophages, we found the m⁶A levels and METTL14 expression were also increased in a RAW264.7 macrophage NLRP3 inflammasome activation model. This finding was in line with recent studies showing that METTL14 activated M1 polarization of macrophages in ischemic stroke and coronary heart disease, indicating that METTL14 may play a vital role in the functional regulation of macrophages [##REF##37541353##35##, ##REF##35598196##36##].</p>", "<p id=\"Par20\">Uncontrolled inflammatory responses mediated by pulmonary macrophages are indeed crucial in the pathogenesis of ALI/ARDS [##REF##36264361##37##]. In the present study, we found that METTL14 knockdown alleviated lung injury via inhibiting NLRP3 inflammasome activation in macrophages, consistent with the result in sepsis-associated myocardial dysfunction [##REF##36864027##38##]. The NLRP3 inflammasome, which acts as the core of the inflammatory response, mediates caspase-1 activation and the secretion of proinflammatory cytokines, IL-1β/IL-18 [##REF##30026524##39##]. Enhanced activation of the NLRP3 inflammasome in alveolar macrophage is involved in the pathogenesis of ALI/ARDS caused by various pathogenic factors [##REF##34612516##40##, ##REF##35432726##41##]. Inhibition NLRP3 inflammasome using the specific inhibitor MCC950 has achieved satisfactory therapeutic results not only in ALI model but also other inflammatory conditions including autoimmune diseases [##REF##35124417##42##]. However, the liver toxicity of MCC950 was found in a phase II clinical trial for rheumatoid arthritis, which casts a shadow over the future clinical application of MCC950 [##REF##35714692##43##]. In our study, we found that the protective effects of METTL14 knockdown were similar to MCC950 in ALI. Therefore, it is promising to develop a specific inhibitor of METTL14 for treatment on ALI/ARDS and other inflammatory diseases.</p>", "<p id=\"Par21\">Although some studies have revealed an association between METTL14 and NLRP3 inflammasome activation [##REF##35013106##44##, ##REF##34412584##45##], whether METTL14 plays a direct role in regulating NLRP3 expression remains unclear. NLRP3 in low concentrations is inadequate for initiating inflammasome activation under resting conditions [##REF##31036962##9##]. Our results revealed that METTL14 knockdown markedly downregulated the mRNA expression of NLRP3, but not that of IL-1b or IL-18, both in vivo and in vitro. Therefore, we suspected that NLRP3 mRNA may be the m⁶A methylation target of METTL14 in ALI/ARDS. Hence, we analyzed potential m⁶A targeting motifs in SRAMP and identified 24 RRACH m⁶A-binding motifs in NLRP3 mRNA. We further confirmed that the loss of METTL14 abolished the increase in m⁶A methylation levels of NLRP3 mRNA in ALI mice and RAW264.7 macrophages treated with LPS. RNA pull-down and RIP assays confirmed that METTL14 directly interacted with NLRP3 mRNA, and that such binding was enhanced in ALI mice. These results indicated that NLRP3 mRNA is a direct m⁶A methylation target of METTL14 during NLRP3 inflammasome activation in ALI/ARDS.</p>", "<p id=\"Par22\">The biological functions of m⁶A modifications rely on m⁶A readers, which regulate RNA metabolism, including translation, splicing, export, and degradation [##REF##32276589##16##]. Elevated m⁶A modification mediated by METTL14 increases NLRP3 mRNA and protein expression in ALI mice. Furthermore, an ActD RNA stability assay showed that the half-life of NLRP3 transcripts had decreased following METTL14 knockdown, indicating that NLRP3 expression was modulated via an m⁶A-dependent mRNA decay mechanism. The m⁶A readers, IGF2BP1/2/3, are involved in regulating the stability of methylated mRNA [##REF##29476152##46##]. Based on our data indicating that only IGF2BP2 was upregulated in ALI mice and LPS-treated RAW264.7 macrophages, we hypothesized that IGF2BP2 may act as the potential binding protein of NLRP3 mRNA via an m⁶A-dependent mRNA decay mechanism. Indeed, RNA pull-down and RIP assays confirmed that IGF2BP2 directly binds to NLRP3 transcripts. Moreover, our findings suggested that IGF2BP2 knockdown may decrease the NLRP3 mRNA stability and inhibit NLRP3 inflammasome activation in ALI mice and LPS-treated RAW264.7 macrophages, thereby alleviating lung injury. Collectively, these results suggest that IGF2BP2 specifically binds to the NLRP3 transcripts and enhances NLRP3 mRNA stability in an m⁶A-dependent manner during ALI/ARDS.</p>", "<p id=\"Par23\">This study has some limitations. We investigated the role of METTL14/IGF2BP2 in NLRP3 inflammasome activation in ALI mice and RAW264.7 macrophages. However, the role of METTL14/IGF2BP2 in clinical patients of ARDS remains to be elucidated. In addition, although we verified that m⁶A modification of NLRP3 mRNA was mediated by METTL14, the specific motif of NLRP3 transcripts methylated by METTL14 has not yet been elucidated and may have to be confirmed in future research. Third, although we found that METTL14 and IGF2BP2 were upregulated in ALI mice and RAW264.7 macrophages, upstream mechanisms underlying this process have not yet been explored and require further investigation via a follow-up study.</p>", "<p id=\"Par24\">Overall, our study provides robust in vitro and in vivo evidence supporting the critical roles of METTL14/IGF2BP2 in NLRP3 inflammasome activation during ALI/ARDS. Mechanistically, METTL14-catalyzed NLRP3 mRNA m⁶A methylation enhances the stability of NLRP3 mRNA in an IGF2BP2-dependent manner in ALI/ARDS. Our findings indicate that METTL14/IGF2BP2 shows potential as therapeutic targets in the treatment of ALI/ARDS.</p>" ]

[]

[ "<p id=\"Par1\">Acute lung injury (ALI) as well as its more severe form, acute respiratory distress syndrome (ARDS), frequently leads to an uncontrolled inflammatory response. N⁶-methyladenosine (m⁶A) modification was associated with the progression of several inflammatory diseases. However, the role of methyltransferase-like 14 (METTL14)-mediated m⁶A methylation in ALI/ARDS remains unclear. Here, we reported an increase in overall expression levels of m⁶A and METTL14 in circulating monocyte-derived macrophages recruited to the lung following ALI, which is correlated with the severity of lung injury. We further demonstrated the critical function of METTL14 in activating NOD-like receptor pyrin domain-containing protein 3 (NLRP3) inflammasome in vitro and in mouse models of ALI/ARDS, and validated NLRP3 as the downstream target of METTL14 by the m⁶A RNA immunoprecipitation (MeRIP) and RIP assays. Mechanistically, METTL14-methylated NLRP3 transcripts were subsequently recognized by insulin-like growth factor 2 mRNA-binding protein 2 (IGF2BP2), an m⁶A reader, which stabilized NLRP3 mRNA. Furthermore, we observed that IGF2BP2 knockdown diminished LPS-induced ALI in mice by downregulating NLRP3 expression. In summation, our study revealed that the molecular mechanism underlying the pathogenesis of ALI/ARDS involves METTL14-mediated activation of NLRP3 inflammasome in an IGF2BP2 dependent manner, thereby demonstrating the potential of METTL14 and IGF2BP2 as promising biomarkers and therapeutic targets for ALI/ARDS treatment.</p>", "<p id=\"Par2\">\n\n</p>", "<title>Subject terms</title>" ]

[ "<title>Supplementary information</title>", "<p>\n\n\n\n\n\n\n\n</p>" ]

[ "<title>Supplementary information</title>", "<p>The online version contains supplementary material available at 10.1038/s41419-023-06407-6.</p>", "<title>Acknowledgements</title>", "<p>We thank our colleagues for technical help and stimulating discussion. This work was supported by grants from the National Natural Science Foundation of China (81870878, 8217102207, 82101348), Guangdong Basic and Applied Basic Research Foundation (2019B151502010, 2022B1515120026, and 2021A1515220117).</p>", "<title>Author contributions</title>", "<p>JX, WZ, and FC conceived and designed research; FC, GC, YX, and XW performed research; FC, YX, and JX performed writing, review, and revision of the paper; XT analyzed data; WZ, XS, and XY provided technical and material support. All authors read and approved the final paper.</p>", "<title>Data availability</title>", "<p>Data are available from the corresponding author on reasonable request.</p>", "<title>Competing interests</title>", "<p id=\"Par43\">The authors declare no competing interests.</p>", "<title>Ethical approval</title>", "<p id=\"Par44\">The animal study was approved by the Institutional Animal Care and Use Committee of Sun Yat-Sen University Cancer Center and carried out under the guidelines of the Guide for the Care and Use of Laboratory Animals of the China National Institutes of Health.</p>" ]

[ "<fig id=\"Fig1\"><label>Fig. 1</label><caption><title>Global m⁶A levels and METTL14 expression are increased in ALI mice.</title><p>Mice were injected intraperitoneally with LPS (15 mg/kg) or an equal volume of saline. <bold>A</bold>, <bold>B</bold> m⁶A dot-blot assays of lung tissues from Ctr and ALI mice. <bold>C</bold> A colorimetric assay measured m⁶A mRNA methylation in lung tissues of each group. <bold>D</bold> RT-qPCR analysis and (<bold>E</bold>, <bold>F</bold>) Western blot analysis of m⁶A regulators (METTL3, METTL14, METTL16, WTAP, FTO, and ALKBH5) in lung tissues of Ctr and ALI mice. <bold>G</bold>–<bold>L</bold> The immunofluorescence co-staining analysis of METTL14 (green) and CD68 (a macrophage marker, red) (<bold>G</bold>, <bold>H</bold>), F4/80 (a macrophage marker, red) (<bold>I</bold>, <bold>J</bold>), or Siglec-F (a resident alveolar macrophage marker, red) (<bold>K</bold>, <bold>L</bold>) in mouse lungs. The graphs depict mean ± SD, <italic>n</italic> = 6 mice per group.</p></caption></fig>", "<fig id=\"Fig2\"><label>Fig. 2</label><caption><title>Global m⁶A levels and METTL14 expression are increased in LPS-activated macrophages.</title><p><bold>A</bold>, <bold>B</bold> Western blot analysis of IL-1β p17, Caspase-1 p20 in supernatants (SN) and pro-IL-1β, pro-Caspase-1, NLRP3, METTL14 in cell extracts (Lysate) of RAW264.7 cells treated with or without nigericin for 30 min after LPS pretreatment for 6 h, when compared with control. <bold>C</bold>, <bold>D</bold> RT-qPCR analysis of NLRP3 and METTL14 mRNA levels in RAW264.7 cells. <bold>E</bold>, <bold>F</bold> The IL-1β and IL-18 concentrations in supernatants of the indicated RAW264.7 cells. <bold>G</bold>, <bold>H</bold> m⁶A dot-blot assays and (<bold>I</bold>) A colorimetric assay measured m⁶A mRNA methylation in RAW264.7 cells treated with or without nigericin for 30 min after LPS pretreatment for 6 h, when compared with control. All data are represented as mean ± SD, <italic>n</italic> = 6 replicates of each condition.</p></caption></fig>", "<fig id=\"Fig3\"><label>Fig. 3</label><caption><title>Knocking down METTL14 inhibits the activation of NLRP3 inflammasome and alleviates lung injury in vitro and in vivo.</title><p><bold>A</bold> RT-qPCR and (<bold>B</bold>, <bold>C</bold>) Western blot analysis of METTL14 in RAW264.7 cells following si-NC or si-METTL14 transfection to validated knockdown effect (<italic>n</italic> = 3). <bold>D</bold> A colorimetric assay measured the levels of m⁶A modification in RAW264.7 cells (<italic>n</italic> = 6). <bold>E</bold>, <bold>F</bold> The IL-1β and IL-18 concentrations in supernatants of si-NC and si-METTL14 RAW264.7 cells treated with or without nigericin for 30 min after LPS pretreatment for 6 h (<italic>n</italic> = 6). <bold>G</bold>, <bold>H</bold> Western blot analysis of NLRP3 in the indicated RAW264.7 cells (<italic>n</italic> = 6). <bold>I</bold>–<bold>K</bold> The mRNA expression of NLRP3, IL-1b, IL-18 were detected by RT-qPCR (<italic>n</italic> = 6). <bold>L</bold> The levels of m⁶A mRNA methylation in mouse lungs were measured by colorimetric assays (<italic>n</italic> = 6). <bold>M</bold> Pulmonary edema was assessed by lung wet/dry ratios (<italic>n</italic> = 6). <bold>N</bold> The total levels of protein in BALF. <bold>O</bold> Neutrophil accumulation measured by MPO activity assay in lung tissues. <bold>P</bold>, <bold>Q</bold> Representative H&amp;E-stained lung sections (Scale bars, 100 μm) and histological injury score of each group. <bold>R</bold>, <bold>S</bold> The IL-1β, IL-18 concentrations in mouse lungs (<italic>n</italic> = 6) (<bold>T</bold>) NLRP3 mRNA expression in lung tissues (<italic>n</italic> = 6). All data are represented as mean ± SD. The unit for <italic>n</italic> is ‘replicates’(<bold>A</bold>–<bold>K</bold>) or ‘samples’ (<bold>L</bold>–<bold>T</bold>).</p></caption></fig>", "<fig id=\"Fig4\"><label>Fig. 4</label><caption><title>Elevated METTL14 promotes the activation of NLRP3 inflammasome and aggravates lung injury in vitro and in vivo.</title><p><bold>A</bold> RT-qPCR analysis (<italic>n</italic> = 4) and (<bold>B</bold>, <bold>C</bold>) Western blot analysis (<italic>n</italic> = 5) of METTL14 in RAW264.7 cells with or without METTL14 overexpression. <bold>D</bold> Quantification of m⁶A methylation in RAW264.7 cells (<italic>n</italic> = 6). <bold>E</bold>, <bold>F</bold> The IL-1β and IL-18 concentrations in supernatants of AAV-GFP and AAV-METTL14 RAW264.7 cells treated with or without nigericin for 30 min after LPS pretreatment for 6 h (<italic>n</italic> = 6). <bold>G</bold>, <bold>H</bold> Western blot analysis of NLRP3, and (<bold>I</bold>–<bold>K</bold>) RT-qPCR analysis of NLRP3, IL-1b, IL-18 in the indicated RAW264.7 cells (<italic>n</italic> = 6). <bold>L</bold> Quantification of m⁶A methylation in mouse lungs (<italic>n</italic> = 6). <bold>M</bold> Wet/dry ratios of mouse lungs were assessed (<italic>n</italic> = 6). <bold>N</bold>, <bold>O</bold> The levels of total protein in BALF and MPO activity in lung were assessed by biochemical kits. <bold>P</bold>, <bold>Q</bold> Representative H&amp;E-stained sections (Scale bars, 100μm) and histological injury score of mouse lungs. <bold>R</bold> NLRP3 mRNA expression in lung tissues (<italic>n</italic> = 6). <bold>S</bold>–<bold>V</bold> The IL-1β, IL-18 concentrations in mouse lungs and serum (<italic>n</italic> = 6). All data are represented as mean ± SD. The unit for n is ‘replicates’(<bold>A</bold>–<bold>K</bold>) or ‘samples’ (<bold>L</bold>–<bold>V</bold>).</p></caption></fig>", "<fig id=\"Fig5\"><label>Fig. 5</label><caption><title>NLRP3 is the direct target of METTL14-mediated m6A modification.</title><p><bold>A</bold> Potential targeted m⁶A sites in NLRP3 mRNA according to SRAMP online website. <bold>B</bold>, <bold>C</bold> MeRIP analysis of m⁶A modified NLRP3 mRNA in (<bold>B</bold>) Ctr and ALI mice (<italic>n</italic> = 3), and (<bold>C</bold>) ALI mice with or without METTL14 knockdown (<italic>n</italic> = 3). <bold>D</bold>, <bold>E</bold> m⁶A enrichment of NLRP3 mRNA in (<bold>D</bold>) Ctr and LPS-treated RAW264.7 cells (<italic>n</italic> = 3), and (<bold>E</bold>) LPS-treated RAW264.7 cells with or without METTL14 knockdown (<italic>n</italic> = 3). <bold>F</bold> RNA pull-down assay was performed in ALI and Ctr mice, and the METTL14 pulled down by NLRP3 RNA probe in mouse lungs were detected by western blot, LacZ RNA probe served as the RNA control (<italic>n</italic> = 3). <bold>G</bold>, <bold>H</bold> RT-qPCR analysis of RIP assays showed the direct binding between the METTL14 protein and NLRP3 mRNA in (<bold>G</bold>) lung tissues of Ctr and ALI mice (<italic>n</italic> = 3), and (<bold>H</bold>) Ctr and LPS-treated RAW264.7 cells (<italic>n</italic> = 3). <bold>I</bold> Lung wet/dry ratios was used to assess lung fluid content (<italic>n</italic> = 6). <bold>J</bold> The concentrations of total protein in BALF and (<bold>K</bold>) MPO activity were detected in mouse lung. <bold>L</bold>, <bold>M</bold> Presentative H&amp;E-stained sections (Scale bars, 100μm) and histological injury score for each group of mice were shown. <bold>N</bold>–<bold>Q</bold> The IL-1β, IL-18 concentrations in mouse lungs and serum (<italic>n</italic> = 6). All data are represented as mean ± SD. The unit for n is ‘replicates’ (<bold>D</bold>, <bold>E</bold>, <bold>H</bold>) or ‘samples’ (<bold>B</bold>, <bold>C</bold>, <bold>F</bold>, <bold>G</bold>, <bold>I</bold>–<bold>Q</bold>).</p></caption></fig>", "<fig id=\"Fig6\"><label>Fig. 6</label><caption><title>IGF2BP2 is upregulated and interacts with NLRP3 mRNA both in vivo and in vitro during ALI.</title><p><bold>A</bold> METTL14 siRNA reduced NLRP3 stability in LPS-treated RAW264.7 macrophages. Decay rate of NLRP3 mRNA in si-NC or si-METTL14 transfected RAW264.7 cells treated with or without LPS following actinomycin D (5 μg/ml) treatment (LPS si-NC compared with Ctr si-NC: ***<italic>P</italic> &lt; 0.001, ****<italic>P</italic> &lt; 0.0001; LPS si-METTL14 compared with LPS si-NC: ^#<italic>P</italic> &lt; 0.05, ^###<italic>P</italic> &lt; 0.001, ^####<italic>P</italic> &lt; 0.0001, <italic>n</italic> = 3). <bold>B</bold>, <bold>C</bold> Western blot analysis of m⁶A readers (IGF2BP1-3) in lung tissues of ALI and Ctr mice (<italic>n</italic> = 6). <bold>D</bold> RT-qPCR analysis of IGF2BP2 in lung tissues of ALI and Ctr mice (<italic>n</italic> = 6). <bold>E</bold>, <bold>F</bold> Western blot and (<bold>G</bold>) RT-qPCR analysis of IGF2BP2 in RAW264.7 cells (<italic>n</italic> = 6). <bold>H</bold> RNA pull-down assay was performed in lung tissues of ALI and Ctr mice, and the IGF2BP2 pulled down by NLRP3 RNA probe was significantly elevated in ALI mice, LacZ RNA probe was set as the RNA control (<italic>n</italic> = 3). RT-qPCR analysis of RIP assays in (<bold>I</bold>) lung tissues of ALI and Ctr mice and (J) LPS-treated RAW264.7 cells, showing the direct binding between the IGF2BP2 protein and NLRP3 mRNA was significantly increased during ALI (<italic>n</italic> = 3). All data are represented as mean ± SD. The unit for n is ‘replicates’ (<bold>E</bold>–<bold>G</bold>, <bold>J</bold>) or ‘samples’ (<bold>A</bold>-<bold>D</bold>, <bold>H, I</bold>).</p></caption></fig>", "<fig id=\"Fig7\"><label>Fig. 7</label><caption><title>IGF2BP2 knockdown decreases NLRP3 mRNA stability and inhibits NLRP3 inflammasome activation in LPS-activated macrophage.</title><p><bold>A</bold> RT-qPCR analysis and (<bold>B</bold>, <bold>C</bold>) Western blot analysis of IGF2BP2 in RAW264.7 following si-NC or si-IGF2BP2 transfection to verify the knockdown effect (<italic>n</italic> = 3). <bold>D</bold>–<bold>F</bold> The mRNA expression of NLRP3, IL-1b, IL-18 in RAW264.7 cells by RT-qPCR (<italic>n</italic> = 6). <bold>G</bold> Decay rate of NLRP3 mRNA in LPS-treated RAW264.7 with or without IGF2BP2 knockdown following actinomycin D (5 μg/ml) treatment (*<italic>P</italic> &lt; 0.05, ***<italic>P</italic> &lt; 0.001, ****<italic>P</italic> &lt; 0.0001). <bold>H</bold>, <bold>I</bold> Western blot analysis of NLRP3 in RAW264.7 treated with or without LPS after siRNA transfection (<italic>n</italic> = 6). <bold>J</bold>, <bold>K</bold> The IL-1β and IL-18 concentrations in supernatants of si-NC or si-IGF2BP2 transfected RAW264.7 cells treated with or without nigericin for 30 min after LPS pretreatment for 6 h (<italic>n</italic> = 6). <bold>L</bold> Decay rate of NLRP3 mRNA, and (<bold>M</bold>, <bold>N</bold>) the IL-1β and IL-18 concentrations in AAV-GFP and AAV-METTL14 LPS-treated RAW264.7 cells with or without IGF2BP2 knockdown (<italic>n</italic> = 6). The graphs depict mean ± SD, <italic>n</italic> = 6 replicates of each condition.</p></caption></fig>", "<fig id=\"Fig8\"><label>Fig. 8</label><caption><title>Knocking down IGF2BP2 inhibits the activation of NLRP3 inflammasome and alleviates lung injury in ALI mice.</title><p><bold>A</bold> Lung wet/dry ratios, (<bold>B</bold>) total protein content in BALF, and (<bold>C</bold>) MPO activity were detected in Ctr and ALI mice treated with si-NC or si-IGF2BP2 (<italic>n</italic> = 6). <bold>D</bold> Lung injury score and (<bold>E</bold>) presentative images from H&amp;E staining-sections of lung tissues were shown (Scale bars, 100μm) (<italic>n</italic> = 6). <bold>F</bold> RT-qPCR analysis of NLRP3 mRNA expression (<italic>n</italic> = 6). <bold>G</bold>, <bold>I</bold> The IL-1β and (<bold>H</bold>, <bold>J</bold>) IL-18 concentrations in lung tissue and serum of Ctr and ALI mice treated with si-NC or si-IGF2BP2 is shown (<italic>n</italic> = 6). <bold>K</bold> Lung wet/dry ratios, (<bold>L</bold>) total protein content in BALF, and (<bold>M</bold>) MPO activity were detected in AAV-GFP and AAV-METTL14 ALI mice with or without IGF2BP2 knockdown (<italic>n</italic> = 6). The IL-1β and IL-18 concentrations in (<bold>N</bold>, <bold>O</bold>) lung tissue and (<bold>P</bold>, <bold>Q</bold>) serum were measured by ELISA (<italic>n</italic> = 6). <bold>R</bold> RT-qPCR analyzed NLRP3 mRNA expression level. All data are represented as mean ± SD, <italic>n</italic> = 6 mice per group.</p></caption></fig>" ]

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[ "<supplementary-material content-type=\"local-data\" id=\"MOESM1\"></supplementary-material>", "<supplementary-material content-type=\"local-data\" id=\"MOESM2\"></supplementary-material>", "<supplementary-material content-type=\"local-data\" id=\"MOESM3\"></supplementary-material>", "<supplementary-material content-type=\"local-data\" id=\"MOESM4\"></supplementary-material>", "<supplementary-material content-type=\"local-data\" id=\"MOESM5\"></supplementary-material>", "<supplementary-material content-type=\"local-data\" id=\"MOESM6\"></supplementary-material>", "<supplementary-material content-type=\"local-data\" id=\"MOESM7\"></supplementary-material>" ]

[ "<fn-group><fn><p>Edited by Sudan He</p></fn><fn><p><bold>Publisher’s note</bold> Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.</p></fn><fn><p>These authors contributed equally: Fei Cao, Guojun Chen, Yixin Xu.</p></fn></fn-group>" ]

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[ "<media xlink:href=\"41419_2023_6407_MOESM1_ESM.docx\"><caption><p>Supplementary Table 1</p></caption></media>", "<media xlink:href=\"41419_2023_6407_MOESM2_ESM.docx\"><caption><p>Supplementary Table 2</p></caption></media>", "<media xlink:href=\"41419_2023_6407_MOESM3_ESM.docx\"><caption><p>Supplementary Table 3</p></caption></media>", "<media xlink:href=\"41419_2023_6407_MOESM4_ESM.tif\"><caption><p>Supplementary Fgure 1</p></caption></media>", "<media xlink:href=\"41419_2023_6407_MOESM5_ESM.tif\"><caption><p>Supplementary Fgure 2</p></caption></media>", "<media xlink:href=\"41419_2023_6407_MOESM6_ESM.docx\"><caption><p>Supplementary Figure Legends</p></caption></media>", "<media xlink:href=\"41419_2023_6407_MOESM7_ESM.pptx\"><caption><p>Original western blots</p></caption></media>" ]

{ "acronym": [], "definition": [] }

[ "<title>Introduction</title>", "<p id=\"Par2\">One of the most important evolutionary characteristics of human beings is their social nature. The ability to develop social bonds in the environment in which we live largely depends on the ability of individuals of the same species to recognize and accurately interpret the emotional state of other group members^{##UREF##0##1##,##UREF##1##2##}. The ability to correctly identify emotions, as well as to demonstrate emotions through facial expressions, is crucial for communication^{##REF##12899416##3##,##REF##12270585##4##}. Facial expressions are, therefore, a major contributor to social connections, as they not only express emotional states but also convey intentions^{##UREF##2##5##,##UREF##3##6##}. This was demonstrated by Bradley et al.^{##REF##12934687##7##}, in which pictures of families and babies were rated as highly pleasant, and were associated with the activation of neural systems that lead to closeness and evoked greater activity of the zygomatic major muscle (related to a smile) compared to the activities evoked by neutral or unpleasant pictures. Likewise, social bonding stimuli, such as pictures of people interacting, are perceived as more pleasant (high valence) and more arousing (greater activation) than pictures depicting the same people not making direct social contact^{##REF##28740473##8##}. Pictures with social interaction cues were recently shown to promote larger EMG activation of the zygomatic major muscle in healthy individuals compared to that for pictures without social interaction. Pictures with social interaction cues also increased the expectation of closeness and reduced fear of rejection in these individuals^{##REF##33414495##9##}.</p>", "<p id=\"Par3\">People with depression have difficulties in social interaction processes^{##REF##10830147##10##–##REF##10779900##13##} and have differentiated muscle activity in response to emotional stimuli. Evidence indicates that individuals with depressive symptoms express fewer facial expressions than those without these symptoms, except when the expression is related to sadness^{##REF##10779900##13##,##REF##28024224##14##}. Low facial responsiveness occurs mainly in the activity of the zygomatic major muscle associated with happy facial expressions^{##REF##12270585##4##,##REF##18006196##15##–##REF##15950175##18##}. Low facial expressiveness, together with deficits in the recognition of facial emotions, can compromise interaction initiation and subsequent development^{##UREF##2##5##,##UREF##3##6##}. Consequently, individuals with depression have fewer interactions and greater social rejection by non-depressed individuals. Thus, these individuals may assess themselves and be judged by others to be less socially competent^{##REF##10779900##13##}.</p>", "<p id=\"Par4\">Although individuals with depression show reduced zygomatic activity when viewing pleasant pictures in general^{##REF##12270585##4##,##REF##18006196##15##–##REF##15950175##18##}, studies have not yet investigated the differences in the EMG reactivity of the zygomatic major in these individuals with pleasant pictures in a context of social interaction. Therefore, we investigated whether the smiling facial expression of individuals with high levels of depressive symptoms differed from those of individuals without depressive symptoms when viewing pictures with and without social interaction clues, with the same valence and activation, when the only difference between pictures is the presence or absence of social interaction cues. We also evaluated whether, as observed in individuals without depression, social interaction cues were relevant enough to increase the emotional states of sociability and altruism in individuals with depression.</p>", "<p id=\"Par5\">We believe that this is the first study to compare the reactivity of the zygomatic major muscle, affiliative state and altruism behavior evoked by visual stimuli that portray the same people, in the same background scene and environment, with valence and activation pairing, where the only difference between the stimuli is the context of social bond (pictures with and without social interaction) in individuals with and without depressive symptoms. Therefore, we aimed to investigate whether social interaction pictures differently modulated the facial EMG reactivity of the major zygomatic muscle, affiliative state, and altruistic behavior between individuals with and without depression. We hypothesized that the group with depression would exhibit less zygomatic muscle reactivity and lower levels of closeness expectation and altruism when viewing affiliative pictures compared to those in the group without depression. We also hypothesized that the group with depression would not show altered fear of rejection, while the group without depression would show a lower level when viewing interaction pictures. Our general hypothesis was that individuals with depression are less sensitive to the social interaction cues displayed in pictures of social interaction stimuli (affiliative pictures) compared to non-social interaction (control pictures) and neutral image stimuli and, therefore, do not exhibit changes in the expression of smile, emotional state and altruism behavior compared to individuals without depression.</p>" ]

[ "<title>Methods</title>", "<title>Participants</title>", "<p id=\"Par26\">The cohort comprised 85 individuals (66 women; 77%) aged between 18 and 35 years (average = 23.9, SD = 3.7). The exclusion criteria were: a medical diagnosis of psychiatric diseases; history of peripheral or central facial paralysis; and use of continuous medications (except for contraceptives).</p>", "<p id=\"Par27\">The participants answered the Beck Depression Inventory II (BDI-II)^{##UREF##17##56##}, and based on the final score, the participants were separated into groups without symptoms of depression (non-depressive group: n = 69) and with symptoms of depression (depressive group: n = 16). The study followed the recommendations of the Declaration of Helsinki and the experimental protocol was approved by the Research Ethics Committee of the Federal University of Ouro Preto – Brazil (CAAE 90012318.10000.5150). All participants provided written informed consent. Data were collected before the COVID-19 outbreak.</p>", "<title>Visual stimuli</title>", "<p id=\"Par28\">The participants were exposed to three blocks containing 28 pictures each (neutral, affiliative and control). The neutral block consisted of neutral pictures taken from the <italic>International Affective Picture System</italic> (IAPS)^{##REF##15925028##25##} (valence: mean = 4.96; SD = 0.49; activation: mean = 2.76; SD = 0.8). The pictures of the affiliative and control blocks were extracted from an image^{##REF##28740473##8##} and their valence and activation values were classified as described by Ref.^{##REF##15925028##25##}. These blocks depicted pairs composed of an adult and a child, or two children. For each image in the affiliative block depicting two individuals interacting socially, there was an image in the control block that portrayed the same pair of individuals, in the same scenario, nevertheless without direct social interaction between them. The affiliative and control blocks had approximately the same number of smiles (affiliative: 87.5% and control: 71.4%) and were paired by valence (affiliative: average = 7.17; standard deviation = 0.39; control: average = 7.02; standard deviation = 0.38; t = 1.39; p = 0.17) and activation (affiliative: average = 3.69; standard deviation = 0.50; control: average = 3.92; standard deviation = 0.59; t = − 1.58; p = 0.41). The block order was fixed across participants to reduce the number of sequences and because we previously tested the block difference in a similar experimental design^{##REF##33414495##9##}. The pictures taken from the IAPS catalog were: 2880; 5510; 5520; 6150; 7000; 7002; 7004; 7006; 7009; 7010; 7025; 7050; 7080; 7130; 7090; 7150; 7170; 7175; 7207; 7211; 7217; 7233; 7235; 7490; 7550; 7595; 7705; and 7950. For examples of pictures affiliative and control block, see Refs.^{##REF##28740473##8##} and ^{##REF##33414495##9##}.</p>", "<title>Psychometric assessments</title>", "<p id=\"Par29\">The participants completed the following questionnaires:</p>", "<p id=\"Par30\">BDI-II^{##UREF##17##56##,##REF##8736107##57##}: This 21-item self-report measure assesses depressive symptoms in the cognitive, affective, somatic, and motivational dimensions. The BDI-II scores range from 0 to 64, with the choice of cut-off point depending on the sample and study objectives^{##UREF##18##58##,##UREF##19##59##}. In clinically undiagnosed samples the term “depression” should be used only for scores above 20, preferably with a concomitant clinical diagnosis^{##UREF##19##59##–##UREF##20##61##}. Thus, in the present study, individuals with scores above 21 were included in the depressive group^{##UREF##19##59##,##REF##8559894##60##}.</p>", "<p id=\"Par31\"><italic>Affiliative status scale</italic>^{##REF##17010974##62##}. This scale comprises 27 adjectives that are divided into two subscales: 13 adjectives form the expectation of approximation subscale, which assesses the individual’s motivational state in performing social interactions, and 14 adjectives form the fear of rejection subscale, which predicts the motivation to be kept away from contact with others. The scores on the two subscales range from 13 to 52 points for the expectation of closeness and 13 to 52 points for the fear of rejection.</p>", "<p id=\"Par32\"><italic>Altruistic behavior scale</italic>^{##REF##17237779##63##}. This scale includes 16 items, 8 of which assess the individual’s altruistic behavior towards a friend and eight which assess behavior towards a stranger. The score for each subscale ranges from 0 to 32 points. The final score is the sum of the two subscales, defined as the total altruism score, where the higher the index, the higher the individual’s level of altruism.</p>", "<title>Display of visual stimuli and signal processing</title>", "<p id=\"Par33\">Visual stimuli were displayed on a 23″ S23C550H Samsung TV monitor placed 94 cm in front of the participants. All pictures were viewed full-screen. E-Prime version 2.0 (Professional Psychology Software Tools Inc., Pittsburgh, PA), was used to generate the visual stimuli shown on the screen and to generate the pulses related to the beginning of these stimuli. The triggers generated by E-Prime were sent by parallel cable to the EMG signal acquisition system. A BIOPAC MP100 amplifier (Biopac Systems, Inc., Goleta, CA; <ext-link ext-link-type=\"uri\" xlink:href=\"https://www.biopac.com/\">https://www.biopac.com/</ext-link>) was used, with a sampling rate of 1000 Hz and a gain of 1000 for the EMG100C electromyographic module.</p>", "<p id=\"Par34\">The EMG signal was filtered online using a 10-Hz high-pass filter and a 500-Hz low-pass filter. The amplifier was connected directly to Acknowledge software (<ext-link ext-link-type=\"uri\" xlink:href=\"https://www.biopac.com/\">https://www.biopac.com/</ext-link>) to acquire the raw electromyographic signals in real time. The raw signals were pre-processed off-line in MATLAB version 7.0 (MATricesLABoratory). We applied a 2th-order Butterworth filter with a cutoff frequency of 20 Hz followed by a constant detrending to remove artifacts in the EMG signal. After we used the full-wave rectified signal. Spreadsheets with the signs of each volunteer were prepared in Microsoft Excel 2016.</p>", "<p id=\"Par35\">The total window analysis of zygomaticus major muscle was defined from − 2 to 6 s. Mean amplitude of activation of EMG muscle activity in the interval between − 2 s and time zero (beginning of the stimulus) was used as the baseline and was applied for baseline correction. Mean amplitude of activation of EMG muscle activity in the interval between 0 and 6 s (picture visualization) in bins of 500 ms was used as the picture windows analysis, totaling 12 intervals. Processing was performed as described by Refs.^{##REF##33414495##9##,##REF##21855602##20##,##REF##20678982##21##}.</p>", "<title>Experimental procedure</title>", "<p id=\"Par36\">All procedures were performed in a room with controlled lighting, acoustic isolation, and controlled temperature (21° C). The participants remained seated in a comfortable armchair positioned in front of a table upon which was placed a computer monitor. To preserve privacy and avoid discomfort when completing the scales, the participants remained alone in the room while completing the questionnaires and viewing the pictures.</p>", "<p id=\"Par37\">The participants started the experiment by completing the BDI-II^{##UREF##17##56##}. Afterward, each participant was asked to clean their face with running water and neutral soap and were advised to use the bathroom and drink water, if desired. Then, the area on which the electrodes were placed on the face of the participants was cleaned with 70% alcohol and a small exfoliation with a paper towel was performed. To collect the EMG signal, two 4-mm silver chloride (Ag–AgCl) electrodes were placed on the zygomatic major muscle on the left side of the face, as described by Ref.^{##REF##3809364##64##}.</p>", "<p id=\"Par38\">The participants received brief instructions on the dynamics of the experiment and were informed that all guidance regarding the procedures would appear on the computer screen. To ensure that the participants’ attention was maintained on the stimuli, the participants were instructed to keep their gaze fixed on the central point and carefully observe the pictures without looking away. Each block contained 28 pictures and each picture was displayed for 6 s, separated by a black screen, with a fixation point, which was displayed for a random duration (4–5 s). The experiment began with the presentation of the 28 pictures of the neutral block, followed by the blocks of interest; namely, the affiliate and control blocks. Between each block was an interval during which the participants responded to the mood state scales (affiliative state and altruistic behavior scales). After completing these scales, the display of pictures were continued, thus totaling 3 blocks and three applications of the mood scales. All three blocks had the same time settings. The duration of each of the three blocks varied between 280 s (4 min 40 s) and 308 s (5 min 8 s). At the end of the experimental session, the electrodes were removed and each participant was thanked. See Fig. ##FIG##4##5## for the experimental sequence.</p>", "<title>Statistical analyses</title>", "<p id=\"Par39\">Statistical analyses were performed using the web-based development environment of SAS software (SAS Institute Inc., 2015), version 9.04, and the graphs were plotted using GraphPad Prism 6.0.1 (GraphPad Software, Inc.).</p>", "<p id=\"Par40\">For sample characterization, the two groups were compared using t-tests (age and symptoms of depression) and chi-squared test (gender).</p>", "<p id=\"Par41\">The EMG data and the Mood State Scale data were analyzed using repeated measures models (ANOVA) and residual dependence was modeled considering a compound symmetry structure, which is typical of repeated measures data analysis. In the case of the Mood State Scale data, a model was used for each scale (expectation of approximation, fear of rejection, altruistic behavior to friends, altruistic behavior to strangers) considering group (depressive and non-depressive) as the between-subjects factor, and block (neutral, affiliative and control) as the within-subjects factor.</p>", "<p id=\"Par42\">For the EMG data, a model was used considering group (depressive and non-depressive) as the between-subjects factor, and block (neutral, affiliative and control) and time (12 bins of 0.5 s) as the within-subjects factor. In this repeated measures model, interactions among these three factors were considered, and, since the number of times within a block was relatively high (12), it was found more appropriate to model residual dependence by means of an autoregressive structure of order 1. Under this structure, the residual covariance between the two time points tends to be lower as the time points are farther apart. This and other structures of residual dependence are easily handled in the 'Mixed' procedure of the SAS software. Moreover, for EMG data, the Box-Cox transformation^{##UREF##21##65##} was applied, using the boxcox function of the MASS package^{##UREF##22##66##} of the R software. For a given statistical model, the Box-Cox method estimates the best power transformation such that the distribution of residuals approaches a normal distribution.In both repeated measures models (for Mood State Scale and EMG data), whenever a pertinent null hypothesis was rejected (of an interaction, or of a single factor that does not interact with other factors), a post hoc multiple comparison test was carried out using the Tukey–Kramer method of adjustment. This method controls the familywise error rate, considering all pairs of means to be compared as a family of comparisons^{##UREF##23##67##}.</p>", "<p id=\"Par43\">The level of significance adopted in this study was α = 0.05.</p>" ]

[ "<title>Results</title>", "<title>Cohort characterization</title>", "<p id=\"Par6\">The participants were divided into the non-depressive (n = 69; 55 women; mean age = 24.29; SD = 3.4) and depressive (n = 16; 11 women; mean age = 25.6; SD = 4.08) groups. The groups were similar in age (t = 0.69; p = 0.48) and sex (p = 0.11). The depressive group showed higher levels of depression compared to the non-depressive group (p &lt; 0.001). The values for the self-depreciation, affection and cognition, and somatic dimensions of the depression scale in the depressive group were significantly higher than those in the non-depressive group (Table ##TAB##0##1##).</p>", "<title>Social interaction pictures increase zygomatic major muscle activation in non-depressive but not in depressive individuals</title>", "<p id=\"Par7\">The ANOVA showed a main effect for pictures block (F(2, 345) = 20.83, p &lt; 0.0001), and time (F(11, 1683) = 11.74, p &lt; 0.0001), but not for group (F(1, 125) = 2.69, p = 0.10). It also showed an interaction between group and block (F(2, 337) = 6.74, p = 0.001), group and time (F(11, 1,694,345) = 1.81, p = 0.047), and block and time (F(22, 1618) = 2.98, p &lt; 0.0001), but it did not show a triple interaction among group, block and time (F(22, 1614) = 1.10, p = 0.34).</p>", "<p id=\"Par8\">The Tukey–Kramer post hoc tests of the interaction between group and block showed that the non-depressive group had greater activation of the zygomatic major muscle during visualization of the affiliative block than the neutral (p &lt; 0.0001) and control (p &lt; 0.0001) blocks. In addition, zygomatic major muscle activation was greater in the non-depressive group than in the depressive group during visualization of the affiliative block (p = 0.02) (Fig. ##FIG##0##1##).</p>", "<p id=\"Par9\">With regard to the interaction between group and time, the post hoc tests showed greater activation of the zygomatic major muscle in the non-depressive group during the second time compared to the first (p = 0.02) and during the third time compared to the second (p &lt; 0.001) (Fig. ##FIG##1##2##).</p>", "<p id=\"Par10\">Finally, in the interaction between block and time, the Tukey–Kramer tests showed that the affiliative block of pictures was associated with a greater activation of the zygomatic major muscle during the second time compared to the first (p = 0.049) and similarly when comparing the third time to the second (p = 0.002) (Fig. ##FIG##2##3##).</p>", "<title>Social interaction pictures reduce the fear of rejection in non-depressive individuals and increase it in depressive individuals</title>", "<p id=\"Par11\">The ANOVA with the scale of fear of rejection showed a main effect of group (F(1, 71) = 15.41, p = 0.0002) and block (F(2, 142) = 3.53, p = 0.0 3) and an interaction between group and block (F(2, 142) = 4.92, p = 0.009). The Tukey–Kramer post hoc tests showed that fear of rejection marginally reduced after the affiliative block compared to the neutral block in the non-depressive group (p = 0.06). The fear of rejection was significantly greater in the depressive group than in the non-depressive group after the affiliative (p = 0.002) and control (p = 0.0004) blocks (Fig. ##FIG##3##4##).</p>", "<p id=\"Par12\">The ANOVA with the scale of expectation of approximation showed a main effect of group (F(1, 71) = 9.87, p = 0.03), but no main effect of block (F(2, 142) = 2.68, p = 0.07) or interaction between group and block (F(2, 142) = 2.01, p = 0.14). The non-depressive group had more expectation of approximation than the depressive group (p = 0.03) independently of the block of pictures.</p>", "<p id=\"Par13\">The ANOVA with the scale of altruistic behavior to friends showed a main effect of group (F(1, 69) = 4.00, p = 0.049), but no main effect of block (F(2, 138) = 0.40, p = 0.67) or interaction between group and block (F(2, 138) = 0.06, p = 0.94). The non-depressive group had more altruistic behavior to friends than the depressive group (p = 0.49) independently of the block of pictures.</p>", "<p id=\"Par14\">The ANOVA with the scale of altruistic behavior to strangers showed no main effect of group (F(1, 69) = 1.90, p = 0.17) or block (F(2, 138) = 1.95, p = 0.15) nor for the interaction between group and block (F(2, 138) = 0.15, p = 0.86).</p>" ]

[ "<title>Discussion</title>", "<p id=\"Par15\">The present study evaluated the differences in EMG reactivity of the zygomatic major muscle and emotional states in depressive and non-depressive individuals upon exposure to visual stimuli that were neutral, with social interaction (affiliative), and without social interaction (control). An important control performed in the present study was that the blocks of pictures with and without social interaction were pairs of pictures showing the same people and background scene, with the same mean valence and activation. The only difference between the blocks was the context of social interaction. When viewing the affiliative pictures, the non-depressive individuals showed greater zygomatic EMG reactivity compared to the control and neutral pictures, thus demonstrating increased smile expression. In contrast, the depressive group showed no difference in EMG reactivity while viewing the three image blocks. In addition, greater sociability (less fear of rejection) was observed in the non-depressive group compared to those in the depressive group. These results suggested that individuals with depression have decreased smile and sociability to social interaction scenes compared to that in individuals without depression, which demonstrated their difficulty in reacting to pleasant stimuli with social content.</p>", "<p id=\"Par16\">Higher valence stimuli increase the EMG activity of the zygomatic major muscle, while lower valence stimuli show lower activation of this musculature^{##REF##12934687##7##,##REF##26151956##19##}. The results observed in non-depressive individuals corroborate those reported previously and are consistent with those of a study showing increased zygomatic activity for social interaction pictures (preceded or not by a priming text) compared to control pictures also pared by valence and arousal^{##REF##33414495##9##}. Additionally, the temporal analysis during image visualization showed that the affiliative pictures promoted increased smile expression from 0.5 s onwards compared to the control pictures, a difference that persisted until the end of the image visualization. Thus, the social interaction clues not only promoted greater smile expression but also induced a sustained positive emotional mood^{##REF##33414495##9##,##REF##21855602##20##,##REF##20678982##21##}.</p>", "<p id=\"Par17\">Among the high-valence visual stimuli, those that evoked sociability, such as pictures of people interacting socially and pictures of babies, increased zygomatic EMG activity by activating brain circuits that “prepare” individuals for social interaction^{##REF##17683225##22##–##REF##15925028##25##}. However, most studies on psychophysiological reactions using social interaction scenes are carried out in healthy individuals and do not present stimuli matched for valence and activation^{##REF##12270585##4##,##REF##11228851##26##}. Therefore, when observing the increase in zygomatic EMG activity during exposure to pictures with scenes of social interaction (affiliative block) compared to pictures with the same pairs of people, but without social interaction (control block), the results of the present study suggested that the context of social interaction promoted a smile expression, which acted as a facilitator for social interactions in healthy people^{##REF##33414495##9##,##REF##18662717##27##}. However, this effect did not occur in people with depression.</p>", "<p id=\"Par18\">Depressed individuals show less smile expression than control individuals^{##UREF##5##28##,##REF##25378765##29##}. A reduction in zygomatic EMG activity was also observed during the visualization of high-valence pictures depicting happy faces, indicating that social pictures of high pleasantness provoke negative responses in individuals with depression compared to those in control individuals^{##REF##12270585##4##,##UREF##6##30##}. A review of 39 studies showed alterations in emotional facial expressions across different mental disorders. The majority of studies point towards decreased facial emotional expressivity in individuals with depression, specifically, the decrease in facial expression is mainly evident for positive stimuli^{##REF##26915928##31##}. Our results corroborate these findings and additionally show that individuals with depression are not responsive to highly pleasant social stimulus. In the present study, the affiliative and control image sets had similar valence and arousal, number of people and background scene; the only difference between the blocks was the social content. Thus, affiliative pictures promote changes in relevant facial expressions that drive social interactions^{##REF##33414495##9##} but not in individuals with depressive symptoms.</p>", "<p id=\"Par19\">Facial expressions have an important influence in social contexts, especially the smile, which can evoke feelings of affiliation. Zygomatic activity has been proposed to act as a social facilitator^{##REF##30626917##32##} indicating a willingness to make connections^{##REF##28837957##33##–##UREF##7##35##}. We suggest that individuals with depression have low responsiveness to the social context, as reflected by apathy in their facial expressions. Therefore, this factor may be a determinant for poorer social interactions and avoidance of contact with other people among individuals with depression^{##REF##10779900##13##}.</p>", "<p id=\"Par20\">Studies on the social interactions have reported sadness or anguish in the facial expressions of depressed individuals, including low zygomatic activity to avoid arouse sympathy, help, and proximity to those around them^{##REF##12003451##36##}. However, other studies emphasize that the interpersonal behaviors of individuals with depression lead to their rejection by the people with whom they live. The behaviors of people with depression are categorized as aversive, which, in turn, reinforces the cycle of loneliness and further accentuates the depression. While some studies claim that people with depression have low accuracy in correctly identifying happy faces, others^{##UREF##0##1##,##REF##12270585##4##,##UREF##5##28##,##REF##12003451##36##} observed that depressed individuals could differentiate pictures with happy expressions from pictures from other categories (for example, pictures with sad expressions), with similar precision to non-depressed individuals. However, the picture blocks in the present study were matched for valence and activation and had similar proportions of smiles (affiliative block 87.5%, control block: 71.4%); the only differences between them were the subtle cues of social interaction. Thus, we supposed that for individuals with depression, social interaction is not a pleasant enough stimulus, resulting in the lack of changes in smile expression.</p>", "<p id=\"Par21\">Another component that could influence the facial expressiveness in depressed individuals is facial mimicry. Mimicry facilitates the creation and maintenance of sociability and plays an important role in social interactions seeing that it creates empathy, linking and affiliation between people^{##REF##12807406##37##,##REF##23020640##38##}. Hess and Fisher suggested that mimicry functions as a social regulator^{##REF##35920780##39##,##UREF##8##40##}; thus, when mimicry is hindered or disturbed, it can impair emotion recognition^{##REF##18633815##41##} and lead to elevated stress reactions in the interaction partner^{##UREF##9##42##}. In a review, Kampf^{##REF##36762295##43##} showed that mimicry is decreased in depressive states^{##UREF##10##44##–##UREF##12##46##}, whereas mimicry is increased in positive mood states^{##UREF##12##46##}. Research has shown that patients with depression show less mimicry of pictures of happy and sad faces compared to the non-clinical control group^{##REF##8022947##47##}, and symptom severity of depression is associated with fewer affiliative and higher non-affiliative facial expressions^{##REF##25378765##29##}. Moreover, acutely depressed patients compared to remitted patients and non-clinical participants showed less mimicry of happy faces and were less accurate in recognizing happy faces, yet reduced mimicry did not mediate these deficits^{##REF##28024224##14##}. This suggests that people with affective disorders might show less mimicry behavior during depressive episodes, which may in turn influence their social relationships^{##REF##35920780##39##}. After exposure to the affiliative block of pictures, the non-depressive group showed less fear of rejection compared to neutral block. No modulation was observed in the depressive group. Therefore, in the non-depressive group, the affiliative pictures motivated social interactions, increasing states of sociability (reducing the fear of rejection scores), an effect not observed in the depressive group. These results corroborate those described in the literature, which showed a reduced fear of rejection after viewing affiliative pictures in non-depressive individuals^{##REF##33414495##9##,##REF##25674068##23##}. It is known that people with depression often have the expectation that they might be rejected or that it is too exhausting to engage with others^{##REF##24636342##48##,##UREF##13##49##}, which may also lead to less mimicry behavior and less states of sociability. However, to the best of our knowledge, this result has not been demonstrated in people with high levels of depressive symptoms during the visualization of positive social stimuli.</p>", "<p id=\"Par22\">Our results can also be explained by the “social risk hypothesis” of depression^{##REF##25378765##29##,##REF##14599287##50##}. This hypothesis proposes that people with high levels of depressive symptoms tend to distance themselves from other people to protect themselves from rejection, contempt, and social exclusion. This happens through the signaling of submission in socially competitive environments, in addition to distancing in exchange-oriented social contexts, where requests for help could be ignored or ridiculed^{##UREF##5##28##}. As depressive symptoms disappear, individuals send more signals that indicate their willingness to interact socially^{##REF##14599287##50##}. Our results are consistent with this hypothesis, in which the affiliative pictures did not reduce the fear of social exclusion, in contrast to the observations in the non-depressive group. Furthermore, according to the social risk hypothesis, help-seeking behavior is reserved for contexts oriented towards reciprocity with friends and family, who are more likely to provide the requested help^{##REF##25378765##29##,##REF##14599287##50##}. Since our stimuli comprised of pairs of pictures depicting individuals unknown to the volunteers, we can assume that this may also have contributed to the non-modulation of affiliative behaviors in individuals with depression, as they tend to direct their affiliative behaviors to known people in whom the risk of exclusion is minimized^{##REF##25378765##29##,##REF##14599287##50##}. Adaptations in future experiments should be considered to assess the responsiveness of individuals with depression exposed to visual stimuli from people in their social circle, as described by Ref.^{##UREF##14##51##} in a non-depressed sample.</p>", "<p id=\"Par23\">Altruistic behaviors are defined as voluntary actions performed without interest in receiving internal or external rewards and intended to improve the well-being of others^{##REF##19156513##52##,##UREF##15##53##}. We expected that non-depressive group would increase the altruistic behavior after visualizing the affiliative block. But in the non-depressive and depressive group, no changes were observed in altruistic behavior towards friends or strangers.</p>", "<p id=\"Par24\">The present study has some limitations. First, our sample, despite high scores on the BDI-II, is not a clinical sample and the participants were not diagnosed with depression based on the diagnostic and statistical manual of mental disorders, Fifth Edition (DSM-5) by a psychiatry^{##UREF##16##54##}. Therefore, future studies in a clinical sample should be considered. Secondly, although the groups had the same internal proportions of men and women, the sample mostly comprised women. Additional studies with balanced sex distributions in the sample overall are needed. Thirdly, our groups are unbalanced. We have much more non-depressive participants. Fourthly, our visual stimuli had a low ethnic diversity. As the feeling of belonging to a group favors facial mimicry^{##REF##18164534##55##}, visual stimuli that cover all ethnic diversity are needed to increase participant sense of belonging in the experimental context.</p>", "<p id=\"Par25\">We concluded that individuals with high levels of depressive symptoms are not susceptible to emotional modulation promoted by social interaction cues; that is, they are not able to promote the sustained expression of a smile and to increase their feelings of sociability. Through self-report and facial EMG measurements, we showed that affiliative pictures increased the somatic responses of smile expression and reduced the fear of being rejected in individuals without depressions, in contrast to individuals with high levels of depressive symptoms, who did not present these responses.</p>" ]

[]

[ "<p id=\"Par1\">Individuals with severe depressive symptoms present diminished facial expressions compared to healthy individuals. This reduced facial expression, which occurs in most depressive patients could impair social relationships. The current study sought to investigate whether pictures with social interaction cues could elicit different modulations of facial expressions and mood states in individuals with depressive symptoms compared to healthy individuals. A total of 85 individuals were divided into depressive and non-depressive groups based on their beck depression inventory scores. Participants viewed pictures containing neutral (objects), affiliative (people interacting socially), and control (people not interacting) scenes. Electromyographic signals were collected during the entire period of visualization of the blocks, and emotional questionnaires were evaluated after each block to assess sociability and altruism (prosocial states). In non-depressed individuals, affiliative pictures increased the activity of the zygomatic muscle compared to both neutral and control pictures and reduced fear of rejection compared to neutral pictures. During the visualization of the affiliative block, zygomatic major muscle activation was higher and fear of rejection was lower in the non-depressive individuals than in the depressive. These effects reflected the low expressions of smiling and sociability to affiliative pictures in depressive individuals. These findings highlight the importance of smiling and prosocial states in social interactions, especially in these individuals.</p>", "<title>Subject terms</title>" ]

[]

[ "<title>Acknowledgements</title>", "<p>This work was supported by the National Council for Scientific and Technological Development, Brazil (<italic>Conselho Nacional de Desenvolvimento Científico e Tecnológico</italic>—CNPq); Coordination for the Improvement of Higher Education Personnel, Brazil (<italic>Coordenação de Aperfeiçoamento de Pessoal de Nível Superior</italic>—CAPES); Research Foundation of the State of Minas Gerais (<italic>Fundação de Amparo à Pesquisa do Estado de Minas Gerais</italic>—FAPEMIG); and the Federal University of Ouro Preto—Brazil (<italic>Universidade Federal de Ouro Preto</italic>—UFOP).</p>", "<title>Author contributions</title>", "<p>K.C.D.L. Developed the study, conducted data acquisition, wrote the manuscript draft, analyzed and interpreted the data and prepared the figures. F.C.O.S. Conducted data acquisition and critically reviewed the final manuscript. C.R.A. and B.E.F.M. Contributed with data analysis and interpretation and critically reviewed the final manuscript. P.M.G, W.B. and L. V. critically reviewed the final manuscript. E.B analyzed the data and critically reviewed the final manuscript. G.G.L.S. Developed the study concept and study design, and supervised and administered the project. Contributed with data analysis and critically reviewed the final manuscript. All authors contributed to and have approved the final manuscript. All authors have approved the final version of the manuscript.</p>", "<title>Data availability</title>", "<p>The datasets used and/or analyzed during the current study are available from the corresponding author on reasonable request.\n</p>", "<title>Competing interests</title>", "<p id=\"Par44\">The authors declare no competing interests.</p>" ]

[ "<fig id=\"Fig1\"><label>Figure 1</label><caption><p>Least square means of electromyographic activation of the zygomatic major muscle in the non-depressive (white circles) and depressive (black squares) groups during the visualization of neutral, affiliative and control blocks. The points represent the mean value of each volunteer in microvolts transformed by Box-cox. *p &lt; 0.05.</p></caption></fig>", "<fig id=\"Fig2\"><label>Figure 2</label><caption><p>Temporal course of the least square means of electromyographic activity of the zygomaticus major over 12 half-seconds times in microvolts transformed by Box-cox considering the neutral, affiliative and control blocks collapsed for depressive (black squares) and non-depressive (white circles). *p &lt; 0.05 represents the comparison between time 1 and 2, and time 2 and 3 in the non-depressive group.</p></caption></fig>", "<fig id=\"Fig3\"><label>Figure 3</label><caption><p>Temporal course of the least square means of electromyographic activity of the zygomaticus major over 12 half-seconds times in microvolts transformed by Box-cox. of neutral (black circles), affiliative (gray triangles) and control (white squares) pictures considering that both groups collapsed. *p &lt; 0.05 represents the comparison between times 1 and 2, and times 2 and 3 for affiliative pictures.</p></caption></fig>", "<fig id=\"Fig4\"><label>Figure 4</label><caption><p>Fear of rejection scores after viewing the picture blocks. Self-reported values in the non-depressive (white circles) and depressive groups (black squares). The points represent the mean value of the fear of rejection score for each participant. *p &lt; 0.05 and ^#p = 0.06.</p></caption></fig>", "<fig id=\"Fig5\"><label>Figure 5</label><caption><p>Sequence describing the order of the events throughout the experiment. The two experimental groups were subjected to the same sequence of experiments. Initially, they completed the scale of depression. In sequence, the participants started viewing the 28 pictures of the neutral block, 28 pictures of the affiliative block, and 28 pictures of the control block. At the end of each block of pictures, the volunteers completed the affiliative state scales and the altruistic behavior scale. The state scales were applied a total of three times (once after each block). All pictures were displayed for 6 s. Between each image, a black screen with a cross was displayed for 4–5 s. For more details about the complete picture catalogue and its standardization, see Silva et al.^{##REF##28740473##8##}. All pictures from the catalogue are property of the present research group, and their reproduction is authorized for scientific purposes only.</p></caption></fig>" ]

[ "<table-wrap id=\"Tab1\"><label>Table 1</label><caption><p>Sample characterization.</p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th align=\"left\"/><th align=\"left\">Non-depressive, n = 69</th><th align=\"left\">Depressive, n = 16</th><th align=\"left\">p value*</th></tr></thead><tbody><tr><td align=\"left\">Gender (F/M)</td><td align=\"left\">55/14</td><td align=\"left\">11/5</td><td align=\"left\">p = 0.11</td></tr><tr><td align=\"left\">Age (years old)</td><td align=\"left\">24.29 (3.4)</td><td align=\"left\">25.6 (4.08)</td><td align=\"left\">p = 0.48</td></tr><tr><td align=\"left\">Depression</td><td align=\"left\">10.82 (5.53)</td><td align=\"left\">26.7 (10.7)</td><td align=\"left\">p &lt; 0.001*</td></tr><tr><td align=\"left\"> F1. Self-denigration</td><td align=\"left\">3.5 (2.25)</td><td align=\"left\">8.23 (4.64)</td><td align=\"left\">p &lt; 0.001*</td></tr><tr><td align=\"left\"> F2. Affection and cognition</td><td align=\"left\">4.34 (2.84)</td><td align=\"left\">10.06 (4.33)</td><td align=\"left\">p &lt; 0.001*</td></tr><tr><td align=\"left\"> F3. Somatic</td><td align=\"left\">2.03 (1.44)</td><td align=\"left\">4.70 (1.68)</td><td align=\"left\">p &lt; 0.001*</td></tr></tbody></table></table-wrap>" ]

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[ "<table-wrap-foot><p>Data expressed as mean (SD).</p><p>*Statistically significant difference (p &lt; 0.05).</p></table-wrap-foot>", "<fn-group><fn><p><bold>Publisher's note</bold></p><p>Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.</p></fn></fn-group>" ]

[ "<graphic xlink:href=\"41598_2024_51813_Fig1_HTML\" id=\"MO1\"/>", "<graphic xlink:href=\"41598_2024_51813_Fig2_HTML\" id=\"MO2\"/>", "<graphic xlink:href=\"41598_2024_51813_Fig3_HTML\" id=\"MO3\"/>", "<graphic xlink:href=\"41598_2024_51813_Fig4_HTML\" id=\"MO4\"/>", "<graphic xlink:href=\"41598_2024_51813_Fig5_HTML\" id=\"MO5\"/>" ]

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[ "<title>Introduction</title>", "<p id=\"Par2\">Bacterial biofilms pose a formidable global health challenge, exhibiting resilience against antibiotics, phagocytosis, and environmental stressors, perpetuating chronic infections, particularly by Staphylococcal and Pseudomonas strains^{##REF##23880136##1##}. Biofilm formation entails an intricate, multistep process where bacteria adhere to surfaces, leading to heightened antibiotic resistance. The ensuing microbial consortium releases extracellular polysaccharides (EPS), evading immune surveillance and instigating pathogenesis^{##REF##12867469##2##–##REF##20929376##4##}. Recent research reveals a profound connection between biofilms and various cancers, going beyond conventional links to colorectal and gastric cancers. In vivo, malignant cells rely on a complex tumor microenvironment (TME) for crucial support, where communication within the TME leads to significant cellular and genetic diversity. This dynamic environment triggers changes in the extracellular matrix (ECM), epithelial-to-mesenchymal transition (EMT), and immune modulation, creating a microenvironment conducive to cancer progression^{##REF##32010611##5##}.</p>", "<p id=\"Par3\">Concurrently, cancer cells and biofilms undergo metabolic adaptations, notably the Warburg effect, optimizing nutrient distribution for ATP generation and heightened glutamine levels essential for growth. Biofilms manifest analogous metabolic strategies, efficiently metabolizing glucose for sustained energy in oxygen-limited conditions^{##REF##28393116##6##,##REF##36324588##7##}. The coming together of these common metabolic processes presents a crucial challenge with significant implications for treatment approaches. Biofilms impact cancer through various ways, including triggering inflammation, altering immune responses, producing carcinogenic toxins, and changing host metabolism. The developing concept of the tumor microbiome within the tumor microenvironment closely links with cancer progression, highlighting the need to understand bacterial interactions in cancer and uncover the mechanisms of biofilms for improved diagnostic accuracy and innovative therapeutic strategies^{##REF##30515135##8##}. Current scientific investigations seek innovative approaches for precise therapeutic agent delivery, enhancing bacterial mortality and impeding cancer cell viability. Employing nanoparticles is a key method to target specific tissues, such as tumor tissues and biofilms, due to their small size and unique properties. This enables direct drug delivery to biofilm-infected tumor sites, overcoming a major challenge in biofilm treatment—limited drug penetration.</p>", "<p id=\"Par4\">Nanotechnology involves engineering atoms and molecules to synthesize nanoparticles (NPs), solid particles typically 1 to 100 nm in diameter and 1 to 1000 nm in length. These NPs, with amorphous or crystalline structures, serve as versatile carriers for liquids or gases, bridging bulk materials and molecular structures^{##REF##29719757##9##,##UREF##0##10##}. The nanoparticle size varies based on synthesis methods: chemical, physical, and biological. It is crucial to note that chemical and physical methods may pose environmental and organismal risks due to potential toxicity. Hence, the adoption of a green synthesis approach, utilizing biological components like plant extracts, bacteria, yeast, algae, and fungi, is increasingly favored. Biological elements, particularly alkaloids and flavonoids, play pivotal roles in reducing NPs during green synthesis, rendering it more environmentally friendly than chemical and physical methods^{##UREF##1##11##,##REF##35163607##12##}.</p>", "<p id=\"Par5\">NPs fall into two main categories: monometallic nanoparticles (MNPs) with a single metal and bimetallic nanoparticles (BNPs) with two different metals. Recent research highlights the superior attributes of BNPs, driven by their increased surface area, making them particularly significant. BNPs leverage the distinct properties of both metals, resulting in unique combined attributes and diverse applications^{##UREF##2##13##–##REF##25000181##16##}. Silver (Ag) and copper (Cu) NPs, synthesized from biological sources like plants, fungi, bacteria, and algae, exhibit antibacterial, anticancer, anti-biofilm, and antioxidant activities. They emerge as promising candidates in various fields, especially biomedicine, demonstrating potential as carriers for diagnostics, hydrophobic medicines, hyperthermia, and cancer therapeutics. The combination of Ag and Cu into bimetallic NPs has piqued interest for its antimicrobial efficacy against various pathogens and preferential toxicity to cancer cells, positioning it as a valuable tool in combatting multidrug resistance and advancing cancer treatment^{##REF##30727758##17##–##UREF##5##20##}.</p>", "<p id=\"Par6\">This study marks the initial step in elucidating a sustainable methodology for synthesizing Ag–Cu bimetallic nanoparticles (NPs) using <italic>A. lanata</italic> plant extract. The NPs undergo comprehensive characterization, encompassing optical, morphological, and elemental analyses. Of specific interest is the assessment of their biological functionalities, namely antibacterial and antioxidant activities, with a targeted focus on elucidating their pronounced antibiofilm and cytotoxic activities.</p>" ]

[ "<title>Materials and methods</title>", "<title>Chemicals and materials</title>", "<p id=\"Par7\">Silver nitrate and copper sulfate (analytical grade) were obtained from Sigma Aldrich Bangalore, India. All bacteriological media components such as crystal violet, tryptone, yeast extract, glucose, and sodium chloride, and media such as nutrient both were procured from Hi-Media Laboratories Pvt. Ltd., Mumbai, India. The HeLa &amp; HEK293 cell lines utilized in the experiment was acquired from American Type Culture Collection (ATTC, LG Promochem, Barce-lona, Spain).</p>", "<title>Preparation of <italic>Aerva lanata</italic> plant extract</title>", "<p id=\"Par8\">The <italic>Aerva lanata</italic> plant was sourced from the I-AIM herbal garden and nursery in Bengaluru, India, following the relevant guidelines and regulations set by CHRIST (Deemed to be University). To ensure the reproducibility of the study, the voucher representing the specimen has been deposited within the herbarium at the Department of Life Sciences, Christ (Deemed to be University), and is assigned the identification number CULS_AS_001<italic>.</italic> The collected plant material, representing the entire plant, was washed with double distilled deionized water and subsequently shade-dried for a period of 5 days at room temperature. The dried plants were then finely chopped and coarsely powdered. To prepare the extract, 5 g of the whole plant powder was boiled with 100 mL of double distilled deionized water at 70 °C for 20 min. This solution was further filtered using Whatman No.1 filter paper and centrifuged at 6000 <italic>rpm</italic> for 5 min and the supernatant was stored at 4 °C for experimental use.</p>", "<title>Synthesis of bimetallic Ag–Cu NP’s</title>", "<p id=\"Par9\">Silver nitrate and copper sulphate were used as precursor salts for the biosynthesis of bimetallic Ag–Cu NPs. The synthesis process was performed by mixing a stock solution of both the precursor salts (AgNO₃ and CuSO₄) in an equal volume of 60 mL with the defined molar concentration of 0.01 M and boiled 10 min at 80 °C. About 30 mL of plant extract was added. With constant mixing on magnetic stirrer at 40 °C for 1 h. Variation in color was observed from green to dark brown and then finally to black precipitate of bimetallic Ag–Cu NPs. These NPs were collected by centrifuging at 10,000 <italic>rpm</italic> and the pellet was washed thoroughly with double-distilled deionized water, followed by the final rinse with ethanol. Nanoparticles were dried in a hot air oven at 80 °C and stored at 4 °C for further use^{##REF##37520120##21##}.</p>", "<title>Characterization</title>", "<p id=\"Par10\">Various analytical tools were employed to investigate the key properties of bimetallic NPs synthesized using <italic>A. lanata</italic>. Figure ##FIG##0##1## depicts that the schematic study plan and objective. To determine the crystal structure, the Ag–Cu NPs were analyzed using Bragg–Brentano geometry with a fine focus of cu-anode working at 40 kV and 30 mA, utilizing a Shimadzu model XRD 6000 X-ray diffractometer. The spectrum of the bimetallic Ag–Cu NPs was recorded in the 2θ range, with the detector positioned from 10 to 80 degrees in a step scan mode of 0.02°. Fourier-transform infrared spectroscopy (FT-IR) was employed to study the functional groups present in the green-synthesized Ag–Cu NPs. The KBr pellet method was adopted for sample preparation to avoid spectrum adsorption in the IR region and enable the distinction of various functional groups. The FTIR spectrum was measured in the wave number range of 4000–400 cm^–1. The surface morphology and elemental composition of the synthesized NPs were examined using Field Emission-Scanning Electron Microscope (FE-SEM, model JEOL JSM-6390) and EDX (EDX Oxford Instrument, INCA PentaFETX3). To prevent agglomeration, the Ag–Cu NPs were suspended in a suitable solvent and subjected to sonication. A silicon wafer substrate was then immersed in the colloidal suspension and covered with a spin-coating process, ensuring secure attachment of the particles for SEM and EDX analysis. High Resolution—Transmission Electron Microscope (HR-TEM) analysis was conducted using a Philips TEM (CM200; Eindhoven, The Netherlands) operating at a potential of 120 kV to determine the average particle size and diffraction pattern. The NPs were sonicated and diluted, and the diluted sample was placed on a copper grid and dried at room temperature, HR-TEM images were captured at different magnification powers.</p>", "<title>Determination of minimum inhibitory and minimum bactericidal concentration (MIC and MBC)</title>", "<p id=\"Par11\">The MIC and MBC of the Ag–Cu NPs were tested against <italic>Staphylococcus aureus</italic> and <italic>Pseudomonas aeruginosa</italic>, and results were recorded as per Clinical and Laboratory Standards Institute guidelines. The assay was performed by diluting the concentration of NPs with sterile LB broth to finally obtain series of concentration (15, 30, 60, 120, 240 μg mL⁻¹) inoculated with <italic>S. aureus</italic> and <italic>P. aeruginosa</italic> adjusted to the turbidity of 5 × 10⁵ CFU/mL. These incubated broths were incubated on shaker incubator maintained at 37 °C for 24 h. The concentration at which there was no cell density or turbidity was recorded as MIC^{##REF##32939454##22##}. Following the Ag–Cu NPs MIC measurement, 30 μL aliquots from the tubes that were plated on MH agar plates and incubated overnight at 37 °C overnight. The lowest concentration of NPs that reported bacterial killing was reported to be the MBC value.</p>", "<title>Well diffusion assay</title>", "<p id=\"Par12\">Agar well diffusion technique was adopted to study the anti-microbial activity of the Ag–Cu NPs. This assay was performed with two bacterial species <italic>S. aureus</italic> and <italic>P. aeruginosa</italic>. A revived mid log phase bacterial cultures were spread plated on Muller Hinton agar. Followed by punching of the wells in the solidified media using gel puncher that resulted in wells with a diameter of 5 mm. Under aseptic conditions the wells were loaded with different concentrations of Ag–Cu NPs from 15, 30, 60, 120 μg mL⁻¹ and positive control ampicillin. The sample loaded culture plate was then incubated at 37 °C for overnight to observe and record zone of clearance in the plates and the diameter of the zones was measured^{##REF##25313307##23##}.</p>", "<title>Effect of Ag–Cu NPs on biofilm formation</title>", "<title>Biofilm formation assay</title>", "<p id=\"Par13\">The test organism in LB broth was inoculated and allowed to grow until reaching a constant turbidity of 0.5 McFarland standards (5 × 10⁵ CFU/mL). Different concentrations (15, 30, 60, 120, 240 μg mL⁻¹) of Ag–Cu NPs were added to separate tubes containing the bacterial suspension and incubated at 37 °C for 24 h. A negative control without the addition of NPs was also maintained. After incubation, the media was removed, and the tubes were washed three times with phosphate buffer saline (PBS, pH 7.2). Subsequently, the tubes were stained with 0.1% crystal violet dye for 30 min. Excess stain was washed off with deionized water, and the tubes were dried at room temperature. The biofilm formation was assessed by observing the presence of a thin blue film on the walls of the tubes^{##REF##21860999##24##}.</p>", "<p id=\"Par14\">To quantitatively estimate the effectiveness of Ag–Cu NPs in inhibiting biofilm formation, a method proposed by Kalishwaralal et al. was employed using a 96-well microtiter plate^{##REF##20493674##25##}. Briefly, 10 μL of overnight grown and diluted (1:100) culture was added to sterile wells containing 180 μL of LB broth. Ag–Cu NPs were then added at concentrations of 15, 30, 60, 120, 240 μg mL⁻¹. After incubation at 37 °C for 24 h, the LB broth was discarded, and the wells were washed three times with phosphate buffer saline (PBS, pH 7.2) to remove any unbound cells. The biofilm adhered to the well walls was fixed with 2% w/v sodium acetate and stained with 0.1% crystal violet dye for 10 min. The stained cells were washed with sterile double distilled deionized water, and the crystal violet stain on the surface of the biofilm was solubilized by adding 30% acetic acid and incubating at room temperature for 10–15 min. The solubilized crystal violet was transferred to a new 96-well plate, and the absorbance at 570 nm (OD 570) was measured using an ELISA plate reader. The percentage of biofilm inhibition was calculated using the provided formula:</p>", "<title>Effect of Ag–Cu NPs on the production of extracellular polymeric substance (EPS)</title>", "<p id=\"Par15\">The inhibitory influence of Ag–Cu NPs on the production of EPS was quantitatively analyzed slightly modified assay^{##REF##29090281##26##}. An overnight culture of <italic>P. aeruginosa</italic> and <italic>S. aureus</italic> aseptically inoculated in 50 mL sterile LB broth and each of these cultures were subjected to different sub-inhibitory MIC concentrations of NPs (15, 30, 60, 120, 240 μg mL⁻¹) and one without NPs was maintained as negative control. All the tubes were incubated at 37 °C for 24 h in a shaker incubator at 140 <italic>rpm</italic>. After incubation period, the control and the treated samples were centrifuged at 6000 <italic>rpm</italic> for 15 min at 4 °C and the supernatant was collected. Supernatant obtained was treated with double the volumes of ice-cold ethanol followed by refrigeration at 4 °C to facilitate the precipitation of EPS. This precipitation was separated and concentrated by centrifugation at 6000 <italic>rpm</italic> for 20 min at 4 °C. The obtained pellet was used for the quantification by measuring carbohydrates utilizing anthrone method^{##REF##13286391##27##}. The percentage of EPS inhibition was calculated using Eq. (##FORMU##0##1##).</p>", "<title>Analysis of antioxidant activity of the synthesized bimetallic Ag–Cu NPs</title>", "<title>Reducing power assay</title>", "<p id=\"Par16\">The reducing power of the NPs was determined according to the method of Alavi and Karimi with some modifications^{##REF##29233039##28##}. The concentrations of the bimetallic NPs from 15, 30, 60, 120, 240 μg mL⁻¹ were prepared using 0.2 M of phosphate buffer of pH 6.4. The aliquots were mixed with 2.5 mL of 10 mL^–1 concentration of potassium ferricyanide and incubated at 50 °C for 30 min. After the incubation, 2.5 mL of 100 mL^-1 concentration of trichloro acetic acid was added and centrifuged at 10,000 <italic>rpm</italic> for 12 min. The supernatant was mixed with equal volume of distilled water (3 mL), 0.6 mL of 1.0 mL concentration of ferric chloride was added. With BHT as control setup, the solution was read for absorbance at 700 nm. The results indicate that more the absorbance, the higher is its reducing power.</p>", "<title>Hydrogen peroxide scavenging assay</title>", "<p id=\"Par17\">The hydrogen peroxide scavenging assay was performed by modifying the methods of Vilas et al.^{##UREF##6##29##}, with ascorbic acid as control, various concentration of the bimetallic NPs (15, 30, 60, 120, 240 μg mL⁻¹) was aliquoted and added with 100 μL of 3 mM hydrogen peroxide solution. The solution was incubated at room temperature for 30 min and the absorbance was measured at 610 nm. The percentage of scavenging was calculated using Eq. (##FORMU##0##1##).</p>", "<title>Cytotoxicity assay</title>", "<title>MTT assay for testing cell viability</title>", "<p id=\"Par18\">The effect of biogenic Ag–Cu NPs on HeLa and HEK293 cell lines was investigated using the MTT assay (3-(4,5-dimethylthiazol-2-yl)-2,5-diphenyltetrazolium bromide) at various concentrations. Briefly, the metabolic rate and viability of the treated cells were assessed by measuring the reduction of the yellow tetrazolium salt MTT to a blue formazan product by mitochondrial dehydrogenase. The cells were seeded in a 96-well plate at a uniform density of 1 × 10⁴ cells/well and allowed to adhere and proliferate overnight in a 5% CO₂ incubator. After a 24 h incubation period, the monolayer was washed with fresh medium, and the adhered cells were exposed to Ag–Cu NPs at concentrations ranging from 0 to 240 µg mL^–1 for an additional 24 h. To monitor cell viability, 10 μL of MTT prepared with PBS was added to each well. The plates were then incubated in darkness at 37 °C in a humidified atmosphere containing 5% CO₂ for 4 h. The supernatant was mixed with 100 μL of DMSO to solubilize the formazan product. The absorbance of the plates was measured at 550 nm^{##UREF##7##30##}. A blank well containing medium without bimetallic nanoparticles at the corresponding concentrations was included. The percentage of inhibition was calculated using a specific formula, and the concentration of nanoparticles required to inhibit cell growth by 50% (IC₅₀) values was determined from the dose–response curves for each cell line using GraphPad Prism 7.0.</p>", "<title>Statistical analysis</title>", "<p id=\"Par19\">The experiments were conducted in triplicate and analyzed using Student’s <italic>t</italic> test and two-way ANOVA, with statistical significance defined as p-values &lt; 0.01, utilizing GraphPad Prism software (GraphPad Software Tools, Inc., La Jolla, CA, USA).</p>" ]

[ "<title>Results and discussions</title>", "<title>Synthesis of Ag–Cu NPs</title>", "<p id=\"Par20\">The plant chosen for this research belongs to the <italic>Amaranthaceae</italic> family and is found extensively in tropical and subtropical regions around the world. Due to its rich composition of phytochemicals, including alkaloids, terpenoids, sterols, flavonoid glycosides, and polyphenols, it holds significant importance in ayurveda and is utilized for various medicinal purposes such as antimicrobial, antiasthmatic, anti-urolithiasis and anti-hyperglycemic^{##REF##22279378##31##,##UREF##8##32##}. In recent studies, numerous investigations have confirmed the viability of utilizing <italic>A. lanata</italic> plant extract for the synthesis of different metal nanoparticles. The presence of phytochemicals in the plant extract acts as an essential reducing or capping agent, facilitating the synthesis of metallic nanoparticles^{##REF##27900878##33##–##REF##25459695##36##}. Upon combining the plant extract with the precursor salt, the color of the solution undergoes a discernible change from green to black after an hour of incubation. This color transformation serves as confirmation of the reduction process facilitated by the plant's metabolites. Eventually, the resulting precipitate settles down undisturbed, indicating the successful reduction and formation of bimetallic Ag–Cu NPs.</p>", "<title>Characterization of bimetallic Ag–Cu NPs</title>", "<title>X-ray diffraction</title>", "<p id=\"Par21\">The phase—purity and the crystallinity of the prepared bimetallic Ag–Cu NPs were studied using XRD technique. The XRD pattern of the prepared sample is depicted in Fig. ##FIG##1##2##. It is observed that, XRD pattern exhibits sharp, well defined high peaks. In detail, the peaks observed at 38.3°, 44.1° and 64.1° in 2θ are assigned to the (111), (200) and (220) planes which corresponds to the JCPDS card no. 04 - 0783 belongs to the face centered cubic (fcc) crystal structures of Ag^{##UREF##10##37##}. The crystalline peaks observed at 43.0° and 50.1° are attributed to the (111) and (200) planes of fcc cubic structured Cu NPs corresponds to JCPDS card no. 04 - 0836^{##UREF##11##38##}. In addition to the Ag and Cu crystalline peaks, certain unassigned peaks (marks with *) are also observed. The existence of these peaks could be due to the crystalline organic matter induced during the synthesis via plant extract. These findings show that Cu–Ag NPs produced using <italic>A. lanata</italic> extract results in the nano crystalline particles. The average crystallite size was estimated using Debye – Scherrer equation; D = kλ/βcosθ; where β is the full width at half maximum and λ is the x-ray wavelength (1.5418 Å) and k is the constant. All the peaks obtained in the study were used to estimate the size crystallites and the average size of the crystallite found be 14.6 nm that is in correspondence to previously reported Cu–Ag NPs^{##UREF##12##39##}.</p>", "<title>FT-IR</title>", "<p id=\"Par22\">The FT-IR study was conducted to analyze the functional group of the tethered capping biomolecule that aided in the synthesis and stabilization of the bimetallic Ag–Cu NPs. Both the extract and the NPs were subjected to IR radiation within the range from 400 to 4000 cm^–1. FT-IR spectrum of <italic>A. lanata</italic> extract exhibited absorption bands at 3345, 1639, 1224, 1033, 925, 844, 702 cm^–1 which corresponds to stretching vibration of medium N–H stretching (aliphatic primary amine), bending of –OH, C=O stretching of strong alkyl aryl ether, C–N variable stretching, O–H stretching and N–H wag^{##UREF##12##39##}. For the spectrum of Ag–Cu/Al, the characteristic peaks of the <italic>A. lanata</italic> extract were found together with a new medium peak of 615 cm^–1 correlating the C–H bond indicating nanoparticles of Ag–Cu and validating that Ag–Cu NPs were capped with phytoconstituents of <italic>A. lanata</italic>. Furthermore, when the <italic>A. lanata</italic> extract and Ag–Cu NPs were compared (Fig. ##FIG##2##3##), the peaks of hydroxyl, carbonyl, and amino groups were found to slightly shifted in the NPs, confirming the interaction and participation of bioactive molecules in the production of Ag–Cu NPs. These findings corroborated the utilization of bioactive <italic>A. lanata</italic> components on the surface of Ag–Cu NPs as capping, reducing, and stabilizing agents^{##UREF##1##11##,##UREF##13##40##}.</p>", "<title>Field emission—scanning electron microscope (FE-SEM) and EDX</title>", "<p id=\"Par23\">The SEM images, presented in Fig. ##FIG##3##4##, depict the bimetallic Ag–Cu NPs that were synthesized using a green method. The images reveal that these NPs have a varied size distribution and form semi-spherical agglomerated clusters with a combination of different shapes. Additionally, a significant portion of the NPs exhibit rough surfaces. The EDAX spectra were used to perform a semi-quantitative elemental mapping of the bimetallic Ag–Cu NPs. The results from EDAX confirm the presence of both Ag and Cu particles, indicating their bimetallic nature^{##UREF##14##41##}.</p>", "<title>High resolution—transmission electron microscope (HR-TEM)</title>", "<p id=\"Par24\">The HR-TEM technique was employed to analyze the size distribution and morphology of the bimetallic NPs. The HR-TEM images, along with the corresponding size distribution histogram, are presented in Fig. ##FIG##4##5##. Upon examining the HR-TEM images (Fig. ##FIG##4##5##a,b), it can be observed that the particles are uniformly dispersed and exhibit a wide size distribution. The particle sizes were measured using Image J software, and the corresponding size distribution histogram is shown in Fig. ##FIG##4##5##c. The histogram reveals that the particle sizes range from 5 to 30 nm, with the majority falling within the 7 nm to 12 nm range, this was further confirmed by dynamic light scattering analysis Fig. ##FIG##4##5##d. The average size of the particles is approximately 9.5 nm, which aligns well with the crystallite size estimated from XRD analysis. It is worth noting that the bimetallic Ag–Cu NPs synthesized through the green method exhibit a spherical shape. The presence of larger particles may be attributed to the coagulation or overlapping of smaller particles, a phenomenon that has been reported in similar studies conducted by Al Tamimi, S. et al.^{##UREF##15##42##}.</p>", "<title>Determination of minimum inhibitory concentration</title>", "<p id=\"Par25\">The MIC and MBC of the biosynthesized NPs against the desired organisms was evaluated. It was shown that the MIC and MBC of Ag–Cu NPs against <italic>P. aeruginosa</italic> and <italic>S. aureus</italic> was 240 and 120 μg mL⁻¹ respectively. The findings demonstrate that the effectiveness of Ag–Cu NPs was more potent for <italic>S. aureus</italic> than <italic>P. aeruginosa</italic>. This is due to the presence of positive charged silver in the bimetallic NPs that trap and block the lipopolysaccharides core component of bacterial cell wall of Gram-negative bacteria <italic>P. aeruginosa</italic>, finally them susceptible for bimetallic NPs. Along with this morphology and surface-to-mass ratio of NPs influence their reactivity against microbes^{##REF##20952651##43##}. It was also implied that, due to its biological and chemical reactive edges, uneven and irregular particles have various binding features with the microbial surface. The plant-based synthesis resulted in a wide range of particle sizes. The findings of this study are in moderate correlation with NPs XRD results. A prominent peak of 111 facets can be seen in the bimetallic NPs. The XRD pattern is closely related to TEM analysis, emphasizing that that majority of synthesized NPs are spherical shaped nanoparticles in the sample with high 111 facets, which enhanced their antibacterial activity^{##REF##17261510##44##}.</p>", "<title>Agar well diffusion method</title>", "<p id=\"Par26\">The anti-bacterial activity of the green synthesized Ag–Cu NPs was qualitatively evaluated using well diffusion method against Gram positive (<italic>S. aureus</italic>) and Gram negative (<italic>P. aeruginosa</italic>) organisms. The results of the antibacterial activity are presented in Fig. ##FIG##5##6##A, Table ##TAB##0##1## and Figs. ##SUPPL##0##S1##, ##SUPPL##0##S2##. Figure ##FIG##5##6##A illustrates that the highest inhibitory activity of the Ag–Cu NPs was observed at a concentration of 120 μg mL^–1 against both the organisms. When compared to the control antibiotic, Ampicillin, the activity of the NPs against <italic>P. aeruginosa</italic> was found to be more effective, while against <italic>S. aureus</italic>, it was recorded as moderate. In comparison to the antibacterial activity of the plant extract as reported by Al-Ansari et al., as well as the monometallic Ag and Cu NPs reported by Naidu et al. and Thanganadar Appapalam &amp; Panchamoorthy et al. our bimetallic nanoparticles exhibited superior efficacy^{##UREF##16##45##–##UREF##17##47##}. They demonstrated significantly larger zones of inhibition at lower concentrations. This enhanced activity attributed to its unique property of large surface area ratio (less particle size) of the NPs which enables accumulation and penetration of the NPs through the external barrier cell wall followed by the generation of reactive oxygen species (ROS) a stress marker which could damage the cell integrity by the disruption of the cell membrane leading to the leakage of membrane proteins and cell lysis^{##UREF##18##48##} (Fig. ##FIG##5##6##B). According to Slavin et al. the copper ions have more affinity towards the gram-positive organisms mainly because of projected functional groups such as carboxyl and amine functional groups on the cell surface^{##UREF##19##49##}.</p>", "<title>Crystal violet blue assay</title>", "<p id=\"Par27\">The objective of this assay was to investigate the impact of Ag–Cu nanoparticles (NPs) on biofilm formation, both quantitatively and qualitatively, using <italic>S. aureus</italic> and <italic>P. aeruginosa</italic> as test microbes. Sub-inhibitory concentrations of Ag–Cu NPs were used, and the results demon-started their inhibitory effect on biofilm formation. A comparative study was conducted, showing that the presence of bimetallic NPs inhibited biofilm formation without affecting the control group, and this effect was concentration-dependent. Qualitative evaluation involved observing the presence of a thin layer of biofilms on the walls of the culture tubes after staining with crystal violet. The findings indicated that <italic>S. aureus</italic> did not exhibit visible biofilm formation in the culture tubes containing Ag–Cu NPs concentrations of 120 μg/mL and higher. Similarly, <italic>P. aeruginosa</italic> showed no visible biofilm formation at concentrations of 240 μg/mL (Table ##TAB##1##2##). The quantitative assessment of biofilm inhibition was conducted using a microtiter plate assay at concentrations ranging from 15 to 240 μg/mL, with the inhibition percentage re-ported in Fig. ##FIG##6##7##. These results align with a study conducted by Ghosh, S. et al. which examined the effect of AucoreAgshell NPs synthesized using <italic>Dioscorea bulbifera</italic> plant extract on biofilm production by <italic>P. aeruginosa</italic>^{##UREF##20##50##}. The Au_coreAg_shell NPs showed an efficiency of 18.93% inhibition of biofilms at a concentration of 100 μg/mL.</p>", "<title>Effect of Ag–Cu NPs on the production of extracellular polymeric substance (EPS)</title>", "<p id=\"Par28\">Carbohydrates are considered as the major constituents for EPS in a pure culture that can mediate the process of biofilm formation. Current study data obtained clearly suggests that the bimetallic Ag–Cu NPs have potent antibiofilm activity by disrupting the EPS matrix. These Ag–Cu NPs exhibited potent EPS degradation against both the bacterial strains in a dose de-pendent manner (15, 30, 60, 120, and 240 μg mL^–1). The qualitative observation was done by observing the formation of precipitate after the incubation time of 48 h. At a higher concentration of 120 and 240 μg mL⁻¹ Ag–CuNPs had manifested a prominent inhibitory effect on the EPS formation with <italic>S. aureus</italic> moderate effect observed with <italic>P. aeruginosa</italic> however the highest concentration of 240 μg mL⁻¹ manifested the disruption of EPS against <italic>P. aeruginosa</italic> as well. Anthrone method was found to be a reliable and highly sensitive method to estimate EPS (carbohydrate) and to interpret the inhibitory effect of Ag–Cu NPs (Figs. ##FIG##7##8##a,b). The ratio of exopolysaccharide was comparatively less after the exposure of the bacterial cultures with the sub-inhibitory concentrations of NPs. From Table ##TAB##2##3## it was shown that the highest percentage reduction of EPS was observed against <italic>S. aureus</italic> and <italic>P. aeruginosa</italic> with respect to the concentration of 120 and 240 μg mL⁻¹ which is considered to be crucial for biofilm formation. In contrast to our findings, Borcherding et al. reported that when <italic>P. aeruginosa</italic> culture was treated with super paramagnetic iron oxide nanoparticles it demonstrated a significant increase in biofilm biomass^{##REF##25221673##51##}. This was attributed to the nutritional role of the iron NPs which supports microbial growth leading to the development of increased microbial biofilm. This demands extensive research to understand the differential effects of various NPs towards cell density and biofilm.</p>", "<title>Assessment of Ag–Cu Nps cytotoxic activity</title>", "<p id=\"Par29\">The cytotoxicity of <italic>A. Lanata</italic> aqueous extract and Ag–Cu NPs was investigated on HeLa and HEK293 cell lines, with doxorubicin serving as the positive control, and untreated cells as the negative control, using the MTT assay. The resulting MTT assay results are graphically depicted in Fig. ##FIG##8##9##. Markedly, the IC50 values of HeLa cells were 15.86 μg mL⁻¹for the extract and 17.63 μg mL⁻¹ for Ag–Cu NPs, respectively. In stark contrast, significantly higher IC50 values of 35.89 μg mL⁻¹ and 21.78 μg mL⁻¹ were obtained from the treatment of normal human embryonic kidney (HEK293) cells with the respective materials. Meanwhile, similar in range IC50 values were observed among the different cell lines treated with Doxorubicin, with values of 12.52 and 16.490 μM mL⁻¹ for HeLa and HEK293 cell lines, respectively. This is consistent with the microscopic examination data, through which the cytotoxic activity could be distinguished by features such as membrane blebbing, cell swelling, or shrinkage, as shown in Figs. ##SUPPL##0##S3##, ##SUPPL##0##S4##. This study strongly suggests that the synergistic effect arising from the combined presence of silver and copper in the nanoparticles significantly contributes to their efficacy. When comparing the response of HeLa and normal HEK 293 cells, it was observed that the bimetallic nature of the nanoparticles exhibited lower toxicity towards non-cancerous cells, particularly at a concentration of 120 μg mL⁻¹. This phenomenon can be attributed to the antioxidant properties of the NPs, resulting in higher toxicity towards cancerous cells and lower toxicity towards healthy cells, possibly through the expression of apoptotic molecules^{##UREF##21##52##,##UREF##22##53##}. Further comprehensive investigations are warranted to gain a deeper understanding of the underlying mechanisms responsible for the anticancer activity of the synthesized bimetal NPs.</p>", "<title>Antioxidant activity</title>", "<p id=\"Par30\">Antioxidant activity is a complex process influenced by various mechanisms and factors, making it challenging to fully comprehend using a single methodology. To capture the diverse mechanisms of antioxidant action, it is necessary to employ multiple evaluation methods for assessing antioxidant capacity. In this study, two complementary tests, namely the reducing power assay and the hydroxyl radical scavenging assay, were utilized to evaluate the anti-oxidant activity of bimetallic Ag–Cu NPs. The reducing power assay measures antioxidant activity by reducing ferric cyanide complex (Fe³⁺) to the ferrocyanide form (Fe²⁺). The presence of antioxidative properties in the NPs enables them to donate an electron, resulting in a color change from green to blue, which indicates their scavenging ability^{##UREF##6##29##}. The results of this study revealed that the NPs exhibited a reduction ability, as evidenced by the color change to Perl's Prussian blue observed at 700 nm. The scavenging activity was assessed in comparison to ascorbic acid, a well-known standard antioxidant. Notably, the nanoparticles exhibited the most substantial reduction in Fig. ##FIG##9##10##A and B when employed at a concentration of 240 µg mL⁻¹. Importantly, it is worth highlighting that, in accordance with Al-Ansari et al. research, the plant extract exhibited 58.5% antioxidant activity at its highest concentration (100 µg), whereas the synthesized Ag–Cu nanoparticles achieved ~ 60 ± 5% of activity at a lower concentration (60 µg mL⁻¹)^{##REF##31516340##46##}. These findings indicates that the nanoparticles are capable of converting the ferric cyanide complex (Fe³⁺) into the ferrocyanide form (Fe²⁺) and effectively neutralizing hydroxyl (OH–) free radicals^{##UREF##23##54##}. Moreover, the effectiveness of this scavenging activity is influenced by the dosage of the nanoparticles administered.</p>" ]

[ "<title>Results and discussions</title>", "<title>Synthesis of Ag–Cu NPs</title>", "<p id=\"Par20\">The plant chosen for this research belongs to the <italic>Amaranthaceae</italic> family and is found extensively in tropical and subtropical regions around the world. Due to its rich composition of phytochemicals, including alkaloids, terpenoids, sterols, flavonoid glycosides, and polyphenols, it holds significant importance in ayurveda and is utilized for various medicinal purposes such as antimicrobial, antiasthmatic, anti-urolithiasis and anti-hyperglycemic^{##REF##22279378##31##,##UREF##8##32##}. In recent studies, numerous investigations have confirmed the viability of utilizing <italic>A. lanata</italic> plant extract for the synthesis of different metal nanoparticles. The presence of phytochemicals in the plant extract acts as an essential reducing or capping agent, facilitating the synthesis of metallic nanoparticles^{##REF##27900878##33##–##REF##25459695##36##}. Upon combining the plant extract with the precursor salt, the color of the solution undergoes a discernible change from green to black after an hour of incubation. This color transformation serves as confirmation of the reduction process facilitated by the plant's metabolites. Eventually, the resulting precipitate settles down undisturbed, indicating the successful reduction and formation of bimetallic Ag–Cu NPs.</p>", "<title>Characterization of bimetallic Ag–Cu NPs</title>", "<title>X-ray diffraction</title>", "<p id=\"Par21\">The phase—purity and the crystallinity of the prepared bimetallic Ag–Cu NPs were studied using XRD technique. The XRD pattern of the prepared sample is depicted in Fig. ##FIG##1##2##. It is observed that, XRD pattern exhibits sharp, well defined high peaks. In detail, the peaks observed at 38.3°, 44.1° and 64.1° in 2θ are assigned to the (111), (200) and (220) planes which corresponds to the JCPDS card no. 04 - 0783 belongs to the face centered cubic (fcc) crystal structures of Ag^{##UREF##10##37##}. The crystalline peaks observed at 43.0° and 50.1° are attributed to the (111) and (200) planes of fcc cubic structured Cu NPs corresponds to JCPDS card no. 04 - 0836^{##UREF##11##38##}. In addition to the Ag and Cu crystalline peaks, certain unassigned peaks (marks with *) are also observed. The existence of these peaks could be due to the crystalline organic matter induced during the synthesis via plant extract. These findings show that Cu–Ag NPs produced using <italic>A. lanata</italic> extract results in the nano crystalline particles. The average crystallite size was estimated using Debye – Scherrer equation; D = kλ/βcosθ; where β is the full width at half maximum and λ is the x-ray wavelength (1.5418 Å) and k is the constant. All the peaks obtained in the study were used to estimate the size crystallites and the average size of the crystallite found be 14.6 nm that is in correspondence to previously reported Cu–Ag NPs^{##UREF##12##39##}.</p>", "<title>FT-IR</title>", "<p id=\"Par22\">The FT-IR study was conducted to analyze the functional group of the tethered capping biomolecule that aided in the synthesis and stabilization of the bimetallic Ag–Cu NPs. Both the extract and the NPs were subjected to IR radiation within the range from 400 to 4000 cm^–1. FT-IR spectrum of <italic>A. lanata</italic> extract exhibited absorption bands at 3345, 1639, 1224, 1033, 925, 844, 702 cm^–1 which corresponds to stretching vibration of medium N–H stretching (aliphatic primary amine), bending of –OH, C=O stretching of strong alkyl aryl ether, C–N variable stretching, O–H stretching and N–H wag^{##UREF##12##39##}. For the spectrum of Ag–Cu/Al, the characteristic peaks of the <italic>A. lanata</italic> extract were found together with a new medium peak of 615 cm^–1 correlating the C–H bond indicating nanoparticles of Ag–Cu and validating that Ag–Cu NPs were capped with phytoconstituents of <italic>A. lanata</italic>. Furthermore, when the <italic>A. lanata</italic> extract and Ag–Cu NPs were compared (Fig. ##FIG##2##3##), the peaks of hydroxyl, carbonyl, and amino groups were found to slightly shifted in the NPs, confirming the interaction and participation of bioactive molecules in the production of Ag–Cu NPs. These findings corroborated the utilization of bioactive <italic>A. lanata</italic> components on the surface of Ag–Cu NPs as capping, reducing, and stabilizing agents^{##UREF##1##11##,##UREF##13##40##}.</p>", "<title>Field emission—scanning electron microscope (FE-SEM) and EDX</title>", "<p id=\"Par23\">The SEM images, presented in Fig. ##FIG##3##4##, depict the bimetallic Ag–Cu NPs that were synthesized using a green method. The images reveal that these NPs have a varied size distribution and form semi-spherical agglomerated clusters with a combination of different shapes. Additionally, a significant portion of the NPs exhibit rough surfaces. The EDAX spectra were used to perform a semi-quantitative elemental mapping of the bimetallic Ag–Cu NPs. The results from EDAX confirm the presence of both Ag and Cu particles, indicating their bimetallic nature^{##UREF##14##41##}.</p>", "<title>High resolution—transmission electron microscope (HR-TEM)</title>", "<p id=\"Par24\">The HR-TEM technique was employed to analyze the size distribution and morphology of the bimetallic NPs. The HR-TEM images, along with the corresponding size distribution histogram, are presented in Fig. ##FIG##4##5##. Upon examining the HR-TEM images (Fig. ##FIG##4##5##a,b), it can be observed that the particles are uniformly dispersed and exhibit a wide size distribution. The particle sizes were measured using Image J software, and the corresponding size distribution histogram is shown in Fig. ##FIG##4##5##c. The histogram reveals that the particle sizes range from 5 to 30 nm, with the majority falling within the 7 nm to 12 nm range, this was further confirmed by dynamic light scattering analysis Fig. ##FIG##4##5##d. The average size of the particles is approximately 9.5 nm, which aligns well with the crystallite size estimated from XRD analysis. It is worth noting that the bimetallic Ag–Cu NPs synthesized through the green method exhibit a spherical shape. The presence of larger particles may be attributed to the coagulation or overlapping of smaller particles, a phenomenon that has been reported in similar studies conducted by Al Tamimi, S. et al.^{##UREF##15##42##}.</p>", "<title>Determination of minimum inhibitory concentration</title>", "<p id=\"Par25\">The MIC and MBC of the biosynthesized NPs against the desired organisms was evaluated. It was shown that the MIC and MBC of Ag–Cu NPs against <italic>P. aeruginosa</italic> and <italic>S. aureus</italic> was 240 and 120 μg mL⁻¹ respectively. The findings demonstrate that the effectiveness of Ag–Cu NPs was more potent for <italic>S. aureus</italic> than <italic>P. aeruginosa</italic>. This is due to the presence of positive charged silver in the bimetallic NPs that trap and block the lipopolysaccharides core component of bacterial cell wall of Gram-negative bacteria <italic>P. aeruginosa</italic>, finally them susceptible for bimetallic NPs. Along with this morphology and surface-to-mass ratio of NPs influence their reactivity against microbes^{##REF##20952651##43##}. It was also implied that, due to its biological and chemical reactive edges, uneven and irregular particles have various binding features with the microbial surface. The plant-based synthesis resulted in a wide range of particle sizes. The findings of this study are in moderate correlation with NPs XRD results. A prominent peak of 111 facets can be seen in the bimetallic NPs. The XRD pattern is closely related to TEM analysis, emphasizing that that majority of synthesized NPs are spherical shaped nanoparticles in the sample with high 111 facets, which enhanced their antibacterial activity^{##REF##17261510##44##}.</p>", "<title>Agar well diffusion method</title>", "<p id=\"Par26\">The anti-bacterial activity of the green synthesized Ag–Cu NPs was qualitatively evaluated using well diffusion method against Gram positive (<italic>S. aureus</italic>) and Gram negative (<italic>P. aeruginosa</italic>) organisms. The results of the antibacterial activity are presented in Fig. ##FIG##5##6##A, Table ##TAB##0##1## and Figs. ##SUPPL##0##S1##, ##SUPPL##0##S2##. Figure ##FIG##5##6##A illustrates that the highest inhibitory activity of the Ag–Cu NPs was observed at a concentration of 120 μg mL^–1 against both the organisms. When compared to the control antibiotic, Ampicillin, the activity of the NPs against <italic>P. aeruginosa</italic> was found to be more effective, while against <italic>S. aureus</italic>, it was recorded as moderate. In comparison to the antibacterial activity of the plant extract as reported by Al-Ansari et al., as well as the monometallic Ag and Cu NPs reported by Naidu et al. and Thanganadar Appapalam &amp; Panchamoorthy et al. our bimetallic nanoparticles exhibited superior efficacy^{##UREF##16##45##–##UREF##17##47##}. They demonstrated significantly larger zones of inhibition at lower concentrations. This enhanced activity attributed to its unique property of large surface area ratio (less particle size) of the NPs which enables accumulation and penetration of the NPs through the external barrier cell wall followed by the generation of reactive oxygen species (ROS) a stress marker which could damage the cell integrity by the disruption of the cell membrane leading to the leakage of membrane proteins and cell lysis^{##UREF##18##48##} (Fig. ##FIG##5##6##B). According to Slavin et al. the copper ions have more affinity towards the gram-positive organisms mainly because of projected functional groups such as carboxyl and amine functional groups on the cell surface^{##UREF##19##49##}.</p>", "<title>Crystal violet blue assay</title>", "<p id=\"Par27\">The objective of this assay was to investigate the impact of Ag–Cu nanoparticles (NPs) on biofilm formation, both quantitatively and qualitatively, using <italic>S. aureus</italic> and <italic>P. aeruginosa</italic> as test microbes. Sub-inhibitory concentrations of Ag–Cu NPs were used, and the results demon-started their inhibitory effect on biofilm formation. A comparative study was conducted, showing that the presence of bimetallic NPs inhibited biofilm formation without affecting the control group, and this effect was concentration-dependent. Qualitative evaluation involved observing the presence of a thin layer of biofilms on the walls of the culture tubes after staining with crystal violet. The findings indicated that <italic>S. aureus</italic> did not exhibit visible biofilm formation in the culture tubes containing Ag–Cu NPs concentrations of 120 μg/mL and higher. Similarly, <italic>P. aeruginosa</italic> showed no visible biofilm formation at concentrations of 240 μg/mL (Table ##TAB##1##2##). The quantitative assessment of biofilm inhibition was conducted using a microtiter plate assay at concentrations ranging from 15 to 240 μg/mL, with the inhibition percentage re-ported in Fig. ##FIG##6##7##. These results align with a study conducted by Ghosh, S. et al. which examined the effect of AucoreAgshell NPs synthesized using <italic>Dioscorea bulbifera</italic> plant extract on biofilm production by <italic>P. aeruginosa</italic>^{##UREF##20##50##}. The Au_coreAg_shell NPs showed an efficiency of 18.93% inhibition of biofilms at a concentration of 100 μg/mL.</p>", "<title>Effect of Ag–Cu NPs on the production of extracellular polymeric substance (EPS)</title>", "<p id=\"Par28\">Carbohydrates are considered as the major constituents for EPS in a pure culture that can mediate the process of biofilm formation. Current study data obtained clearly suggests that the bimetallic Ag–Cu NPs have potent antibiofilm activity by disrupting the EPS matrix. These Ag–Cu NPs exhibited potent EPS degradation against both the bacterial strains in a dose de-pendent manner (15, 30, 60, 120, and 240 μg mL^–1). The qualitative observation was done by observing the formation of precipitate after the incubation time of 48 h. At a higher concentration of 120 and 240 μg mL⁻¹ Ag–CuNPs had manifested a prominent inhibitory effect on the EPS formation with <italic>S. aureus</italic> moderate effect observed with <italic>P. aeruginosa</italic> however the highest concentration of 240 μg mL⁻¹ manifested the disruption of EPS against <italic>P. aeruginosa</italic> as well. Anthrone method was found to be a reliable and highly sensitive method to estimate EPS (carbohydrate) and to interpret the inhibitory effect of Ag–Cu NPs (Figs. ##FIG##7##8##a,b). The ratio of exopolysaccharide was comparatively less after the exposure of the bacterial cultures with the sub-inhibitory concentrations of NPs. From Table ##TAB##2##3## it was shown that the highest percentage reduction of EPS was observed against <italic>S. aureus</italic> and <italic>P. aeruginosa</italic> with respect to the concentration of 120 and 240 μg mL⁻¹ which is considered to be crucial for biofilm formation. In contrast to our findings, Borcherding et al. reported that when <italic>P. aeruginosa</italic> culture was treated with super paramagnetic iron oxide nanoparticles it demonstrated a significant increase in biofilm biomass^{##REF##25221673##51##}. This was attributed to the nutritional role of the iron NPs which supports microbial growth leading to the development of increased microbial biofilm. This demands extensive research to understand the differential effects of various NPs towards cell density and biofilm.</p>", "<title>Assessment of Ag–Cu Nps cytotoxic activity</title>", "<p id=\"Par29\">The cytotoxicity of <italic>A. Lanata</italic> aqueous extract and Ag–Cu NPs was investigated on HeLa and HEK293 cell lines, with doxorubicin serving as the positive control, and untreated cells as the negative control, using the MTT assay. The resulting MTT assay results are graphically depicted in Fig. ##FIG##8##9##. Markedly, the IC50 values of HeLa cells were 15.86 μg mL⁻¹for the extract and 17.63 μg mL⁻¹ for Ag–Cu NPs, respectively. In stark contrast, significantly higher IC50 values of 35.89 μg mL⁻¹ and 21.78 μg mL⁻¹ were obtained from the treatment of normal human embryonic kidney (HEK293) cells with the respective materials. Meanwhile, similar in range IC50 values were observed among the different cell lines treated with Doxorubicin, with values of 12.52 and 16.490 μM mL⁻¹ for HeLa and HEK293 cell lines, respectively. This is consistent with the microscopic examination data, through which the cytotoxic activity could be distinguished by features such as membrane blebbing, cell swelling, or shrinkage, as shown in Figs. ##SUPPL##0##S3##, ##SUPPL##0##S4##. This study strongly suggests that the synergistic effect arising from the combined presence of silver and copper in the nanoparticles significantly contributes to their efficacy. When comparing the response of HeLa and normal HEK 293 cells, it was observed that the bimetallic nature of the nanoparticles exhibited lower toxicity towards non-cancerous cells, particularly at a concentration of 120 μg mL⁻¹. This phenomenon can be attributed to the antioxidant properties of the NPs, resulting in higher toxicity towards cancerous cells and lower toxicity towards healthy cells, possibly through the expression of apoptotic molecules^{##UREF##21##52##,##UREF##22##53##}. Further comprehensive investigations are warranted to gain a deeper understanding of the underlying mechanisms responsible for the anticancer activity of the synthesized bimetal NPs.</p>", "<title>Antioxidant activity</title>", "<p id=\"Par30\">Antioxidant activity is a complex process influenced by various mechanisms and factors, making it challenging to fully comprehend using a single methodology. To capture the diverse mechanisms of antioxidant action, it is necessary to employ multiple evaluation methods for assessing antioxidant capacity. In this study, two complementary tests, namely the reducing power assay and the hydroxyl radical scavenging assay, were utilized to evaluate the anti-oxidant activity of bimetallic Ag–Cu NPs. The reducing power assay measures antioxidant activity by reducing ferric cyanide complex (Fe³⁺) to the ferrocyanide form (Fe²⁺). The presence of antioxidative properties in the NPs enables them to donate an electron, resulting in a color change from green to blue, which indicates their scavenging ability^{##UREF##6##29##}. The results of this study revealed that the NPs exhibited a reduction ability, as evidenced by the color change to Perl's Prussian blue observed at 700 nm. The scavenging activity was assessed in comparison to ascorbic acid, a well-known standard antioxidant. Notably, the nanoparticles exhibited the most substantial reduction in Fig. ##FIG##9##10##A and B when employed at a concentration of 240 µg mL⁻¹. Importantly, it is worth highlighting that, in accordance with Al-Ansari et al. research, the plant extract exhibited 58.5% antioxidant activity at its highest concentration (100 µg), whereas the synthesized Ag–Cu nanoparticles achieved ~ 60 ± 5% of activity at a lower concentration (60 µg mL⁻¹)^{##REF##31516340##46##}. These findings indicates that the nanoparticles are capable of converting the ferric cyanide complex (Fe³⁺) into the ferrocyanide form (Fe²⁺) and effectively neutralizing hydroxyl (OH–) free radicals^{##UREF##23##54##}. Moreover, the effectiveness of this scavenging activity is influenced by the dosage of the nanoparticles administered.</p>" ]

[ "<title>Conclusion</title>", "<p id=\"Par31\">The present study successfully demonstrates the development of an eco-friendly, rapid, and cost-efficient method for synthesizing Ag–Cu NPs using <italic>Aerva lanata</italic> extract, fabricated through a phytofabrication process, exhibit great potential for applications in therapeutics as antibiofilm treatments. The FT-IR analysis indicates the presence of hydroxyl, carbonyl, and amino groups in the <italic>A. lanata</italic> extract, which play a significant role as capping agents for the Ag–Cu NPs. The green synthesis of Ag–Cu NPs resulted in a face-centered cubic crystal structure with an average particle size of 9.5 nm. Moreover, these nanoparticles exhibit promising activities in terms of cytotoxicity, antibacterial, and antioxidant properties, attributed to the intrinsic hydroxy and phenol capping, thus suggesting their potential for broader applications. These bimetallic nanoparticles hold promise for applications in combating bio-films. which are crucial factors in various fields such as healthcare and industry. Additionally, the Ag–Cu NPs demonstrate remarkable anti-cancer, antibacterial, and antioxidant activities, making them potential candidates for further research and development in the pursuit of novel therapeutic interventions.</p>" ]

[ "<p id=\"Par1\">In this study, we demonstrate the green synthesis of bimetallic silver-copper nanoparticles (Ag–Cu NPs) using <italic>Aerva lanata</italic> plant extract. These NPs possess diverse biological properties, including in vitro antioxidant, antibiofilm, and cytotoxic activities. The synthesis involves the reduction of silver nitrate and copper oxide salts mediated by the plant extract, resulting in the formation of crystalline Ag–Cu NPs with a face-centered cubic structure. Characterization techniques confirm the presence of functional groups from the plant extract, acting as stabilizing and reducing agents. The synthesized NPs exhibit uniform-sized spherical morphology ranging from 7 to 12 nm. They demonstrate significant antibacterial activity against <italic>Staphylococcus aureus</italic> and <italic>Pseudomonas aeruginosa</italic>, inhibiting extracellular polysaccharide secretion in a dose-dependent manner. The Ag–Cu NPs also exhibit potent cytotoxic activity against cancerous HeLa cell lines, with an inhibitory concentration (IC₅₀) of 17.63 µg mL⁻¹. Additionally, they demonstrate strong antioxidant potential, including reducing capability and H₂O₂ radical scavenging activity, particularly at high concentrations (240 µg mL⁻¹). Overall, these results emphasize the potential of <italic>A. lanata</italic> plant metabolite-driven NPs as effective agents against infectious diseases and cancer.</p>", "<title>Subject terms</title>", "<p>Open access funding provided by North-West University.</p>" ]

[ "<title>Supplementary Information</title>", "<p>\n</p>" ]

[ "<title>Supplementary Information</title>", "<p>The online version contains supplementary material available at 10.1038/s41598-024-51647-x.</p>", "<title>Acknowledgements</title>", "<p>All the authors are thankful to their respective universities and institutes for their support. The work was supported by the researchers supporting project number (RSP2024R261) King Saud University, Riyadh, Saudi Arabia.</p>", "<title>Author contributions</title>", "<p>This research article was produced through collaboration between the authors. Conceptualization, V.V.L., K.R.R.R., B.B.; writing-original manuscript, G.T., A.N; Methodology, data curation, and formal analysis, G.T., J.A.G., A.N., P.N., S.N., M.P., A.M.; Project ad-ministration and Supervision, V.V.L., A.N; Review and editing, B.B., A.M.A., and K.R.R.R.; Interpretation, and review/revision, V.V.L., B.B., K.R.R.R. and A.M.A. All authors have read and agreed to the published version of the manuscript.</p>", "<title>Funding</title>", "<p>Open access funding provided by North-West University.</p>", "<title>Data availability</title>", "<p>The data presented in this study are available on request from the corresponding authors.</p>", "<title>Competing interests</title>", "<p id=\"Par32\">The authors declare no competing interests.</p>" ]

[ "<fig id=\"Fig1\"><label>Figure 1</label><caption><p>Schematic representation of the study.</p></caption></fig>", "<fig id=\"Fig2\"><label>Figure 2</label><caption><p>The XRD pattern of green synthesized bimetallic Ag–Cu nanoparticles.</p></caption></fig>", "<fig id=\"Fig3\"><label>Figure 3</label><caption><p>The FTIR analysis spectra of <italic>Aerva lanata</italic> extract and green synthesized Ag–Cu NPs.</p></caption></fig>", "<fig id=\"Fig4\"><label>Figure 4</label><caption><p>(<bold>A</bold>–<bold>C</bold>) SEM images at different magnifications and (<bold>D</bold>) EDAX spectra of green synthesized bimetallic Ag–Cu NPs.</p></caption></fig>", "<fig id=\"Fig5\"><label>Figure 5</label><caption><p>(<bold>a</bold>) and (<bold>b</bold>) HR-TEM images of Ag–Cu NPs; (<bold>c</bold>) showing the histogram with the Gaussian fitted particle size distribution; (<bold>d</bold>) DLS graph exhibiting the hydrodynamic diameter of the nanoparticles.</p></caption></fig>", "<fig id=\"Fig6\"><label>Figure 6</label><caption><p>(<bold>A</bold>) Inhibition zones observed for Ag–Cu NPs and Ampicillin (at concentrations ranging from 15 to 120 µg mL^–1) against <italic>P. aeruginosa</italic> and <italic>S. aureus</italic>. (<bold>B</bold>) Mechanism of action underlying the antibacterial activity of Ag–Cu nanoparticles (NPs).</p></caption></fig>", "<fig id=\"Fig7\"><label>Figure 7</label><caption><p>Effect of Ag–Cu NPs on biofilm production by <italic>S. aureus</italic> and <italic>P. aeruginosa</italic>.</p></caption></fig>", "<fig id=\"Fig8\"><label>Figure 8</label><caption><p>In comparison with the control the inhibition of carbohydrates in EPS produced from <italic>S. aureus</italic> (<bold>A</bold>) was found to be at the concentration of 120 and 150 μg mL⁻¹and for <italic>P. aeruginosa</italic> (<bold>B</bold>) it was found to be at the concentration of 150 μg mL⁻¹, followed by a gradual decrease in both the bacterial strains.</p></caption></fig>", "<fig id=\"Fig9\"><label>Figure 9</label><caption><p>IC₅₀ of <italic>A. Lanata</italic> aqueous extract, Ag–Cu NPs and Doxorubicin on the HeLa cancer cells compared to HEK293 normal cells.</p></caption></fig>", "<fig id=\"Fig10\"><label>Figure 10</label><caption><p>The highest OH– free radical scavenging activity was observed in the concentration of 240 μg mL⁻¹ in both Hydrogen peroxide scavenging assay (<bold>A</bold>) and reducing power (<bold>B</bold>); however, the activity was less when compared with the standard AA.</p></caption></fig>" ]

[ "<table-wrap id=\"Tab1\"><label>Table 1</label><caption><p>Summary of the anti-bacterial activity of Ag–Cu NPs.</p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th align=\"left\" rowspan=\"3\">Organisms</th><th align=\"left\" colspan=\"4\">Zone of inhibition (mm)</th></tr><tr><th align=\"left\" colspan=\"4\">Different concentrations</th></tr><tr><th align=\"left\">15 μg mL⁻¹</th><th align=\"left\">30 μg mL⁻¹</th><th align=\"left\">60 μg mL⁻¹</th><th align=\"left\">120 μg mL⁻¹</th></tr></thead><tbody><tr><td align=\"left\"><italic>Staphylococcus aureus</italic></td><td char=\".\" align=\"char\">13.0 ± 0.3</td><td char=\".\" align=\"char\">15.3 ± 0.4</td><td char=\".\" align=\"char\">17.1 ± 0.6</td><td char=\".\" align=\"char\">18.0 ± 0.2</td></tr><tr><td align=\"left\"><italic>Pseudomonas aeruginosa</italic></td><td char=\".\" align=\"char\">12.2 ± 0.4</td><td char=\".\" align=\"char\">14.0 ± 0.7</td><td char=\".\" align=\"char\">15.6 ± 0.5</td><td char=\".\" align=\"char\">17.7 ± 0.2</td></tr></tbody></table></table-wrap>", "<table-wrap id=\"Tab2\"><label>Table 2</label><caption><p>Percentage (%) inhibition of biofilm by Ag–Cu NPs.</p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th align=\"left\">Organisms</th><th align=\"left\">15 μg mL⁻¹</th><th align=\"left\">30 μg mL⁻¹</th><th align=\"left\">60 μg mL⁻¹</th><th align=\"left\">120 μg mL⁻¹</th><th align=\"left\">240 μg mL⁻¹</th></tr></thead><tbody><tr><td align=\"left\"><italic>Staphylococcus aureus</italic></td><td char=\".\" align=\"char\">25.09</td><td char=\".\" align=\"char\">40.08</td><td char=\".\" align=\"char\">71.72</td><td char=\".\" align=\"char\">97.43</td><td align=\"left\">100</td></tr><tr><td align=\"left\"><italic>Pseudomonas aeruginosa</italic></td><td char=\".\" align=\"char\">20.52</td><td char=\".\" align=\"char\">37.65</td><td char=\".\" align=\"char\">68.71</td><td char=\".\" align=\"char\">85.55</td><td align=\"left\">100</td></tr></tbody></table></table-wrap>", "<table-wrap id=\"Tab3\"><label>Table 3</label><caption><p>Comparison of carbohydrate concentration and carbohydrate inhibition &amp; of the samples with control.</p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th align=\"left\" rowspan=\"2\">Organisms</th><th align=\"left\" colspan=\"11\">Carbohydrate concentration (µg) &amp; Carbohydrate Inhibition (%) in control and treated samples</th></tr><tr><th align=\"left\">C</th><th align=\"left\">%</th><th align=\"left\">15 μg mL⁻¹</th><th align=\"left\">%</th><th align=\"left\">30 μg mL⁻¹</th><th align=\"left\">%</th><th align=\"left\">60 μg mL⁻¹</th><th align=\"left\">%</th><th align=\"left\">120 μg mL⁻¹</th><th align=\"left\">%</th><th align=\"left\">240 μg mL⁻¹</th></tr></thead><tbody><tr><td align=\"left\"><italic>P. aeruginosa</italic></td><td char=\".\" align=\"char\">608.4</td><td align=\"left\">–</td><td char=\".\" align=\"char\">448.6</td><td char=\".\" align=\"char\">25.43</td><td char=\".\" align=\"char\">282.1</td><td char=\".\" align=\"char\">53.12</td><td char=\".\" align=\"char\">204.1</td><td char=\".\" align=\"char\">64.62</td><td char=\".\" align=\"char\">146.4</td><td char=\".\" align=\"char\">75.00</td><td align=\"left\">–</td></tr><tr><td align=\"left\"><italic>S. aureus</italic></td><td char=\".\" align=\"char\">531.4</td><td align=\"left\">–</td><td char=\".\" align=\"char\">379.3</td><td char=\".\" align=\"char\">28.21</td><td char=\".\" align=\"char\">217.6</td><td char=\".\" align=\"char\">58.21</td><td char=\".\" align=\"char\">184.9</td><td char=\".\" align=\"char\">68.28</td><td char=\".\" align=\"char\">–</td><td char=\".\" align=\"char\">–</td><td align=\"left\">–</td></tr></tbody></table></table-wrap>" ]

[ "<disp-formula id=\"Equ1\"><label>1</label><alternatives><tex-math id=\"M1\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\text{Inhibition (\\%) = }\\frac{\\left(\\left({\\text{OD}}_{\\text{control}}\\text{ - }{\\text{OD}}_{\\text{treated}}\\right)\\right)}{{\\text{OD}}_{\\text{control}}} \\, \\times \\text{100 .}$$\\end{document}</tex-math><mml:math id=\"M2\" display=\"block\"><mml:mrow><mml:mtext>Inhibition (\\%) =</mml:mtext><mml:mspace width=\"0.333333em\"/><mml:mfrac><mml:mfenced close=\")\" open=\"(\"><mml:mfenced close=\")\" open=\"(\"><mml:msub><mml:mtext>OD</mml:mtext><mml:mtext>control</mml:mtext></mml:msub><mml:mspace width=\"0.333333em\"/><mml:mtext>-</mml:mtext><mml:mspace width=\"0.333333em\"/><mml:msub><mml:mtext>OD</mml:mtext><mml:mtext>treated</mml:mtext></mml:msub></mml:mfenced></mml:mfenced><mml:msub><mml:mtext>OD</mml:mtext><mml:mtext>control</mml:mtext></mml:msub></mml:mfrac><mml:mspace width=\"0.166667em\"/><mml:mo>×</mml:mo><mml:mtext>100.</mml:mtext></mml:mrow></mml:math></alternatives></disp-formula>" ]

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[ "<supplementary-material content-type=\"local-data\" id=\"MOESM1\"></supplementary-material>" ]

[ "<table-wrap-foot><p><italic>C</italic> control.</p></table-wrap-foot>", "<fn-group><fn><p><bold>Publisher's note</bold></p><p>Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.</p></fn><fn><p>These authors contributed equally: Gopishankar Thirumoorthy and Balamuralikrishnan Balasubramanian.</p></fn></fn-group>" ]

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[ "<media xlink:href=\"41598_2024_51647_MOESM1_ESM.docx\"><caption><p>Supplementary Figures.</p></caption></media>" ]

{ "acronym": [], "definition": [] }

[ "<title>Introduction</title>", "<p id=\"Par2\">Glioblastoma (GBM) is the most aggressive brain tumor with extensive angiogenesis and poor clinical outcome [##REF##34185076##1##]. Abnormally increased vascularization results in tumor progression and recurrence. Abundant pro-angiogenic factors in the tumor environment contributed greatly to endothelial cell (EC) proliferation. Among these factors, vascular endothelial growth factor (VEGF) is the most well-known and effective cytokine. Targeting blood vessels has now been an alternative treatment approach in GBM, especially recurrent GBM. Bevacizumab, the first anti-angiogenesis agent approved for glioma, neutralizing VEGF, prolongs progression-free survival (PFS) in GBM patient, however, showed minimal efficacy for overall survival (OS) [##UREF##0##2##]. These evidences indicate that further exploration in GBM angiogenesis mechanism is urgently needed.</p>", "<p id=\"Par3\">A subpopulation of tumor cells, termed GBM stem cells (GSCs), were verified to be responsible for GBM progression, vascularization and recurrence [##REF##26109046##3##]. GSCs serve as progenitors for self-renew and proliferation in glioma tissue [##REF##32938908##4##]. There is emerging evidence indicating that cancer stem cells (CSCs) might support tumor progression by enhancing tumor angiogenesis. The proliferation of adjacent blood vessels or the migration of BM-derived stem cells to the tumor result in tumor endothelial cells [##REF##25527451##5##]. There are considerable evidences that GSCs would promote pro-angiogenic factors in tumor microenvironment. For instance, glioma stem cells produce high level of stromal-derived factor 1, which eventually promotes local endothelial activity and systemic angiogenic processes involving bone marrow-derived endothelial progenitor cells (EPC) [##REF##19738068##6##]. A study reported by Ping and et al. revealed that glioma cells co-expressing CD133 and CXCR4 promotes angiogenesis by producing VEGF [##REF##21618540##7##]. By activating histamine H1 receptor (H1R) and Ca2⁺-NF-κB axis, GSCs-secreted histamine promotes angiogenesis and the progression of GBM [##REF##36265493##8##]. Notably, several markers such as Sox2 and Nestin indicated that stem cells are increased in bevacizumab-resistant patients [##REF##22965162##9##]. These results indicated glioma stem cell as a potential target in tumor angiogenesis. Understanding molecular mechanism involved in cancer stem cell-mediated angiogenesis is instrumental for anti-angiogenic therapy in glioma.</p>", "<p id=\"Par4\">Interferon-inducible transmembrane proteins (IFITM) comprise a family of interferon-induced molecules, consisting primarily of IFITM1, IFITM2, IFITM3 and IFITM5 [##REF##31792954##10##]. During endothelial cells sprouting process in vitro, IFITM proteins were induced expeditiously and were essential for lumenized vessel formation [##REF##19887652##11##]. IFITM 1–3 have been implicated in viral pathogen restriction [##REF##34321474##12##]. IFITM3 was firstly identified to restrict influenza A virus infection and influence the adaptive immune response [##REF##20064371##13##]. Recent findings have indicated IFITM3 as a key role in a number of pathological process, particularly in neoplasms. IFITM3 in prostate cancer cells promotes tumor progression and bone metastasis by activating TGFβ pathways [##REF##31273201##14##]. Latest research indicated that IFITM3 phosphorylation accounts for amplification of PI3K signaling, facilitating malignant transformation of B cells [##REF##33149299##15##]. In fact, human ESCs (hESCs) express high levels of IFITM1 and their expression decreases with differentiation into hepatocyte-like cells [##REF##29249360##16##]. And Fragilis 2 (encoding IFITM1 in mouse) was reported to be expressed in murine pluripotent embryonic stem cells and a vital downstream mediator of Wnt/β-catenin [##REF##15857914##17##]. These evidences suggest that IFITM family might play a crucial role in stem cells, including cancer stem cells. Considering these evidences that IFITM proteins are critical for cell stemness and angiogenesis, we attempted to explore the role of IFITM3 in GSCs and glioma angiogenesis and understand the potentially underlying mechanism.</p>", "<p id=\"Par5\">In the present study, we report a GSCs-mediated angiogenesis manner, in which IFITM3 expressed in GSCs regulates JAK/STAT3 signaling pathway that preferentially stimulates bFGF production, leading to tumor angiogenesis. Blocking IFITM3 expression in GSCs substantially suppresses angiogenesis, especially when in combination with anti-VEGF agent.</p>" ]

[ "<title>Materials and methods</title>", "<title>Glioblastoma specimen and cell culture</title>", "<p id=\"Par6\">Twenty-eight paraffin-embedded samples from human glioma patients (WHO I-IV) with corresponding clinic-pathological data were collected in the Department of Neurosurgery at Zhujiang Hospital from 2017 to 2019. Informed consent was obtained from all of patients. Both study protocol and informed consent were approved by the Ethical Committee of Zhujiang Hospital.</p>", "<p id=\"Par7\">For GBM stem cell culture, tumor samples were collected from consenting patients diagnosed as GBM. GBM tissues were treated as previously reported [##REF##28783169##18##]. Digested cells were cultured in DMEM/F12 (Gibco, USA) medium supplemented with EGF (20 ng/ml, Peprotech, USA), bFGF (20 ng/ml, Peprotech) and B27 (1:50, Gibco). GSCs were cultured and expanded in two methods, suspension culture of neuro-spheres in low-attached wells, and adherent culture on Laminin (Corning Biosciences, USA) -coated plates [##REF##19497285##19##]. Low passage (2–4) GSCs were used in this study.</p>", "<p id=\"Par8\">U87 and U251 cells were purchased from Chinese Academy of Sciences Cell Bank (Shanghai, China). Glioma cell lines were cultured in high glucose DMEM medium (Gibco) containing 10% fetal bovine serum (FBS, Gibco). Human brain micro-vessel endothelial cells (hBMECs) were purchased from Procell Life Science &amp; Technology (Wuhan, China) and cultured in supplemented endothelial growth medium (EGM-2, Lonza, Walkersville, MD, USA).</p>", "<title>Reagents, shRNA and transfection</title>", "<p id=\"Par9\">shRNA and lentiviral vector for IFITM3 were designed and constructed by Genechem (Shanghai, PR China). The shRNA sequences include IFITM3 shRNA-1 (5′-GCTTCATAGCATTCGCCTACT-3′), IFITM3 shRNA-2 (5′-CCTGTTCAACAC CCTCTTCAT-3′) and shRNA Scrambled control (5′-GATATGTGCGTACCTAGCA T-3′). Glioma cells were transfected with 50 nmol/L shRNAs by using Lipofectamine (Invitrogen, Carlsbad, CA, USA) according to the manufacturer’s instructions.</p>", "<title>Bio-informatical analysis</title>", "<p id=\"Par10\">Gene expression profiles and corresponding clinical data in Glioma/GBM data were obtained from TCGA and CGGA databases. Primary glioma patients with complete survival data, mRNA sequencing data, and WHO Grade classification data were collected as including patients. Gene expression based on tumor grade and survival analysis were performed using R studio, GEPIA2, and Gliovis [##REF##31114875##20##, ##REF##28031383##21##]. GBM samples from TCGA of CGGA were divided into high and low expression groups according to the median IFITM3 expression level. Differentially expressed genes (DEGs) and Gene set enrichment analysis (GSEA) were examined using R packages DESeq2 and clusterProfiler to pick out the significantly enriched pathways based on Gene ontology terms. The summary information for the patients is presented in supplementary table ##SUPPL##5##S2##.</p>", "<title>Cell Counting Kit-8 (CCK-8) and EdU cell proliferation assay</title>", "<p id=\"Par11\">CCK-8 and EdU assays were performed to examine endothelial cell proliferative capacity. For CCK-8 assay, endothelial cells were seeded onto 96-well plates (Costar, Cambridge, MA, USA) and cultivated for 7 days. Viable cells were analyzed using Cell Counting Kit-8 (Dojindo, Kumamoto, Japan) according to the manufacturer’s instructions. For Edu assay, endothelial cells to be stained were added with EdU solution, followed by incubation, fixation, permeabilization and Edu staining (Abcam, USA).</p>", "<title>Cell invasion assay</title>", "<p id=\"Par12\">Cell invasion assay was conducted using cell culture insert with 8-um pores in 24-well plates (Costar, USA). Insert was pre-coated with Matrigel (Corning) and the bottom chamber was filled with 0.5 mL medium containing 10% FBS. Cells (1 × 10⁵) suspended in 100 μl DMEM medium were placed at the upper chamber and incubated for 24 h. Migrated cells on the underside of the insert were stained with crystal violet.</p>", "<title>Tube formation assay</title>", "<p id=\"Par13\">Tube formation assay was carried out as previously stated [##REF##35676251##22##]. 96-well plate was coated with Growth factor reduced Matrigel and incubated at 37 °C for 30 min. Endothelial cells were seeded at 2 × 10⁴ cells/well and incubated for 24 h. Tube quantification was examined with ImageJ software.</p>", "<title>Endothelium spheroid-based sprouting angiogenesis assay</title>", "<p id=\"Par14\">Sprouting assay was performed according to protocol reported by Korff with minor modification [##REF##10504330##23##]. Dissociated endothelial cells were suspended in complete EGM-2 medium containing 0.25% methylcellulose and seeded on low attachment 96-well plates (Corning). About 800–1000 cells per well bond together within 48 h to form a single spheroid, which were then transferred to 24-well plate and cultivated in Collagen I solution (Enzo Life Sciences, USA) supplemented with 20% FBS. Sprouting capacity was measured via counting the total sprout number using NeuronJ plugin of ImageJ software.</p>", "<title>Immunoblotting and immunological analysis</title>", "<p id=\"Par15\">The immunoblotting assay was performed as previously described [##REF##30368528##24##]. Human ELISA kit (eBioscience) was used to measure the concentration of bFGF from culture medium according to manufacturer’s protocol. Cells were cultured in 6-well plates for 72 h and supernatants were collected for further analysis. Experiments were performed in triplicate. Antibodies used in the present study were listed in the supplementary table ##SUPPL##6##S3##.</p>", "<title>Tissue Immunohistochemical (IHC) and Immunofluorescence staining (IF)</title>", "<p id=\"Par16\">Specimens sections of surgical GBM tissues and intracranial mice xenografts were prepared in a routine procedure [##UREF##1##25##] and immuno-stained with targeted proteins using a protocol as previously described [##UREF##2##26##].</p>", "<title>In vivo xenograft assay</title>", "<p id=\"Par17\">BALB/C nude mice in the present research were procured from Experimental Animal Center of Southern Medical University. Orthotopic xenograft model was established using Balb/c nude mice (6-week-old) with GBM cells (1 × 10⁵ cells in 0.1 ml PBS) stably transfected with mCherry-LUC vector based on Ozawa’s protocol [##UREF##3##27##]. Intracranial tumor growth was examined by in-vivo imaging system (IVIS Lumina II, Caliper, USA). The protocol has been registered and approved by the Animal Care and Use Committee of Southern Medical University.</p>", "<title>Statistical analysis</title>", "<p id=\"Par18\">All experiments in the study were carried out at least 3 time and analyzed using Prism 8 (GraphPad Software Inc., USA) or R software. Results were displayed as Mean ± SD. Statistical significance was determined via using Student’s <italic>t</italic>-test or one-way ANOVA with Bonferroni correction for multiple comparisons. <italic>P</italic> &lt; 0.05 was considered statistically significant.</p>" ]

[ "<title>Results</title>", "<title>IFITM3 expression is elevated in human GBM tissue</title>", "<p id=\"Par19\">To examine the expression of IFITM3 in GBM, we investigated IFITM3 gene expression level in glioma datasets from The Cancer Genome Atlas (TCGA) and Chinese Glioma Genome Atlas (CGGA). Results showed a markedly elevated expression of IFITM3 in GBM, compared to lower grade gliomas (grade II, III) (Fig. ##FIG##0##1A##). Data from GEPIA2 web tool also revealed an obviously higher IFITM3 expression in GBM sample than that in normal brain tissue (Fig. ##FIG##0##1B##). Mesenchymal subtype of GBM has been associated with therapy resistant and more aggressive features than other subtypes [##REF##33762019##28##]. Our data suggested that GBM sample with mesenchymal subtype displayed a markedly upregulated IFITM3 expression (Supplementary Fig. ##SUPPL##2##1a##). IFITM3 also correlated with IDH1 (Supplementary Fig. ##SUPPL##2##1b##) and MGMT (Supplementary Fig. ##SUPPL##2##1C##) status in GBM samples. And upregulated IFITM3 significantly correlated with poor outcome in both GBM and LGG samples (Fig. ##FIG##0##1C##). We examined IHC staining with IFITM3 from human protein atlas (HPA) and observed a strong intensity in high grade glioma, while a weak intensity was documented in low-grade glioma (Fig. ##FIG##0##1D##). Furthermore, we collected 28 glioma specimens with clinicopathologic data. IHC staining of glioma samples indicated a positive relation between IFITM3 expression and tumor grade (Fig. ##FIG##0##1E, F##). Moreover, patients were divided into two groups based on IFITM3 expression. Surprisingly, it revealed that IFITM3 expression was only significantly related to tumor grade, rather than gender, age and tumor size (Fig. ##FIG##0##1G##). Taken together, these data indicated that IFITM3 expression was elevated in glioma samples, especially in GBM sample.</p>", "<title>IFITM3 is enriched in GBM stem cells</title>", "<p id=\"Par20\">Cancer stem cells (CSCs), a subpopulation inside the heterogeneous tumor tissues, are characterized by multi-lineage differentiation potential [##REF##31430455##29##]. Evidences from various studies indicated that CSCs were essential for tumor progression, recurrence and metastasis [##UREF##4##30##]. In the present study, GBM stem cells (GSCs) were derived from fresh GBM samples and cultured in vitro. GSCs possessed the sphere-forming capacity (Fig. ##FIG##1##2A##) and neural stem cell markers (Fig. ##FIG##1##2B##) as reported previously. Immunoblotting confirmed that GSCs within the culture universally express IFITM3, despite some variations in levels between cells (Fig. ##FIG##1##2C##). While GSCs were enriched for IFITM3, GSC serum-differentiated cells (GSDCs) exhibited reduced expression (Fig. ##FIG##1##2D##), indicating that IFITM3 was solely expressed in neural stem cells or cancer stem cells. Moreover, we observed a positive association between IFITM3 and Nestin expression in samples from GBM patients (Fig. ##FIG##1##2E##). Likewise, in CGGA GBM dataset, IFITM3 expression closely correlated with stem cell markers Nestin and Sox2 (Fig. ##FIG##1##2F##). Sufficient evidence has established a role for hypoxia environment in stem cell maintenance [##REF##20375133##31##]. Interestingly, low oxygen level induced increased IFITM3 and HIF1α expression in GSCs, rather than glioma cell lines (Fig. ##FIG##1##2G##). When GSCs were exposed to 2% oxygen, protein level of IFITM3 gradually increased over time (Fig. ##FIG##1##2H##). When exposed to varying levels of O₂ for 24 h, GSCs exhibited rising IFITM3 expression (Fig. ##FIG##1##2H##). Similarly, GSC spheres were cultured under hypoxic environment, IFITM3 and HIF1α expression were elevated (Fig. ##FIG##1##2I##). And in CGGA and Gravendeel GBM datasets, IFITM3 was significantly correlated with HIF1α mRNA levels (Fig. ##FIG##1##2J##).</p>", "<title>IFITM3 in GBM stem cells regulates endothelium proliferation and sprouting</title>", "<p id=\"Par21\">To further determine the role of IFITM3 on biological function of GSCs. We examined sphere-forming competency of GSCs and found that downregulation of IFITM3 had no impact on GSC sphere-forming capacity (Fig. ##FIG##2##3A##). Since sufficient angiogenesis is essential for solid tumor expansion [##REF##2480145##32##], we co-cultured GSCs with brain microvascular endothelial cells (hBMECs), and examined cell proliferation and sprouting capacity (Fig. ##FIG##2##3B##). It suggested that IFITM3 knockdown in GSCs reduced tube formation and sprouting capacity of hBMECs (Fig. ##FIG##2##3C##). Moreover, proliferative ability of endothelial cells was impaired when IFITM3 was downregulated (Fig. ##FIG##2##3D, E##). We established intracranial xenografts in nude mice through implanting GBM stem cells to evaluate the effect of IFITM3 on angiogenesis in vivo. Results indicated that IFITM3 downregulation led to reduced vessel density as assessed by CD34, while exhibited no effect on stem cell population (Fig. ##FIG##2##3F##). Together, these findings suggested that IFITM3 in GBM stem cells would be responsible for increased angiogenesis in tumor microenvironment.</p>", "<title>IFITM3 induces JAK/STAT3 activation in GSCs</title>", "<p id=\"Par22\">GBM patients from TCGA dataset were categorized into high and low expression groups by using median expression value of IFITM3 as cut-off point. Differentially gene expression with RNA-seq data was analyzed (Fig. ##FIG##3##4A##). Gene set enrichment analysis (GSEA) was utilized for assessment of key pathways between two groups (Fig. ##FIG##3##4B##). Among these enriched pathways, angiogenesis-related pathways including PI3K/Akt, JAK/STAT and Notch were selected (Supplementary Table ##SUPPL##0##1##). GSCs were treated with scrambled shRNA targeting IFITM3, and key protein expressions of these pathways were determined. Immunoblotting assay indicated that IFITM3 in GSCs regulate key protein expression in JAK/STAT3 signaling pathway (Fig. ##FIG##3##4C##). Next, GSCs were transfected with lentiviral vector encoding IFITM3, and co-incubated with endothelial cells. In vitro angiogenic assays displayed enhanced tube formation and sprouting capacities in hBMECs co-cultured with IFITM3-GSCs (Fig. ##FIG##3##4D##). Moreover, when WP1066 (JAK inhibitor) was added into GSCs culture medium, IFITM3-induced angiogenesis was significantly attenuated (Fig. ##FIG##3##4D##). In human GBM specimen, we observed elevated p-STAT expression in robust IFITM3-staining region (Fig. ##FIG##3##4E##). When fractionated IFITM3⁺ or IFITM3^- GSCs were injected orthopically into nude mice, IFITM3 expression was also consistent with p-STAT3 expression (Fig. ##FIG##3##4F##).</p>", "<title>GSC-derived IFITM3 contributes to angiogenesis via producing bFGF</title>", "<p id=\"Par23\">The GO term molecular function (MF) analysis indicated that cytokine activity was significantly enriched in IFITM3-regulated genes (Supplementary Fig. ##SUPPL##3##2##). Secreted cytokines have powerful influence on angiogenesis either in normal tissue or malignant tissue. To identify the angiogenic factors involved in IFITM3-mediated angiogenesis, an angiogenesis array was performed. The expression of bFGF and TGFβ1 markedly decreased in shIFITM3 GSCs as compared with shCon-GSCs (Fig. ##FIG##4##5A, C##). When GSC#2 cells were fractionated into two groups with high or low IFITM3 expression. Culture media from high-IFITM3 cells showed increased bFGF level (Fig. ##FIG##4##5B, C##). In various GSCs, IFITM3-regulated bFGF production was further validated (Fig. ##FIG##4##5D, E##). Interestingly, a co-cultivation of GSCs with hBMECs revealed that blocking bFGF would substantially mitigate pro-angiogenic effect by IFITM3 (Fig. ##FIG##4##5F##). To explore IFITM3-mediated angiogenesis in vivo, we established intracranial mice model by implanting Luc labeled GSC#1 into nude mice. Tumor growth was monitored using in vivo Imaging System. Results indicated that IFITM3 knockdown exhibited attenuated tumor growth (Fig. ##FIG##4##5G##) and declined bFGF secretion (Fig. ##FIG##4##5H##). In human GBM samples, our data showed that IFITM3 expression was associated with bFGF production and enhanced angiogenesis (Fig. ##FIG##4##5I##). These findings indicated that GSC-derived IFITM3 contributed significantly to angiogenesis in vivo via regulating bFGF.</p>" ]

[ "<title>Discussion</title>", "<p id=\"Par24\">In the present study, we describe a novel mechanism of GSCs-induced GBM angiogenesis through expression of IFITM3 and bFGF (Fig. ##FIG##4##5J##). IFITM3 is a interferon-induced protein mediating antiviral activity via suppressing the entry of viruses [##REF##30951861##33##]. Interestingly, it has been gradually unraveled that IFITM3 correlates with tumor progression, including proliferation [##UREF##5##34##], invasion [##REF##28544512##35##], chemoresistance [##REF##35941699##36##], metastasis [##REF##30272306##37##], as well as angiogenesis [##REF##18281150##38##]. A Recent finding indicated that TGFβ-regulated IFITM3 expression facilitates glioma cell invasion [##REF##31637620##39##]. However, further investigation regarding molecular mechanism, by which IFITM3 influences glioma progression remains unclear. Conclusive evidence is now provided that IFITM3 functions as an angiogenesis inducer. Angiogenesis is the formation of new blood vessels including multiple components, such as the endothelial cell proliferation, cell migration, cell adherence and in vitro tube formation [##REF##11166264##40##]. In the present research, we focused our investigations on GBM stem cell-induced alterations on endothelial cells. we utilize a co-cultivation system to demonstrate that IFITM3 is capable to stimulate endothelial cell proliferation, migration and sprouting through interaction between GSCs and endothelial cells. The observation that conditioned media from GSCs, as well as physically separated GSCs modulates human brain endothelial cell phenotypes indicates that GSCs release pro-angiogenic cytokines in a IFITM3-dependent method.</p>", "<p id=\"Par25\">It has gradually become clear that cancer stem cells (CSCs) exist in various types of tumors tissues, even though the controversy about exact markers to identify and isolate CSCs [##REF##19760641##41##]. Recent researches have unveiled that GSCs share similar characteristics. Both cell types express stem cell markers and the ability of self-renewal [##REF##22469978##42##]. CD133 and Nestin have been widely considered as markers in GSCs [##REF##20225923##43##]. However, there is also evidence that CD133-negative cells cultured from GBM patients harbored a self-renewal ability [##REF##20385361##44##]. The present study demonstrates that IFITM3 is preferentially expressed in GSCs and decreased markedly in differentiated glioma cells, yet has no significant impact on GSC self-renewal ability (Fig. ##FIG##2##3A##). In fact, IFITM3 has been reported to exert no effect on cell proliferation and invasion in glioma cell lines [##REF##27835870##45##], indicating that IFITM3 had no direct effect on glioma cells. These evidences suggest that IFITM3 might not be an indispensable factor in stemness maintenance of cancer cells, despite the close correlation between protein expression and stemness status. Furthermore, hypoxic situation led to a dramatic elevation of IFITM3 expression (Fig. ##FIG##1##2G, H##). Hypoxia is known to affect the CSCs maintenance and functions. Stem cells tend to residence in hypoxic regions within tumors, which probably benefit the preservation of tumor stemness [##REF##33472628##46##]. And hypoxia could regulate a variety of targets related to tumor stemness and invasiveness. Our data is consistent with a prior study by Cai that high-IFITM3 expression correlated with high hypoxia score in the bladder cancer using bio-informatic method [##REF##34456915##47##]. Rajan et al. demonstrated that gene expression of IFITM3 was elevated in microglia when rat brain suffered a transient cerebral ischemia [##REF##30485549##48##]. Data from a study by Harmon and colleagues showed that ischemic injury in aged brains following stroke resulted in the induction of IFITM3 proteins [##UREF##6##49##]. Therefore, IFITM3 would act as a hypoxia-induced intermediate factor in GSCs, and subsequently the affects the tumor microenvironment surrounding GSCs.</p>", "<p id=\"Par26\">Besides from tumor initiation, CSCs are shown to interact with endothelial cells and be closely linked to vasculature formation in perivascular niche [##REF##18286393##50##]. Recent findings indicate that GSCs contributed to tumor vasculature through direct differentiation or releasing pro-angiogenic cytokines. GSCs cultured from GBM patients differentiated into vascular endothelial cells and pericytes when induced by serum in vitro [##REF##22973019##51##, ##UREF##7##52##]. These GSCs-derived vascular cells harbored the tube formation capacity and could be identified in human GBM tissues [##REF##21102434##53##]. In our study, IFITM3^high GSCs confers endothelial cells proliferation, migration and sprouting through paracrine.</p>", "<p id=\"Par27\">Through human angiogenesis antibody array, we identified bFGF as a key motivator in GSC-mediated angiogenesis. High bFGF level has been reported in individuals with various categories of neoplasms and predicted a poor prognosis [##UREF##8##54##]. Elevated bFGF levels have been identified in human glioma, suggesting its importance in tumor growth progression [##REF##24333730##55##]. Besides, one of the most prominent reason for the resistance to anti-VEGF treatment is the compensatory mechanism of other growth factors, with the bFGF being at the top of the list [##UREF##8##54##]. These results showed that IFITM3-bFGF axis between GSCs and endothelial cells might be the compensatory angiogenic approach when VEGF-dependent angiogenesis is interrupted.</p>", "<p id=\"Par28\">In summary, the present study discloses a functional role IFITM3 in regulating GBM angiogenesis and identifying the downstream effectors, which are important for GBM progression. And bFGF is shown to be a downstream target of IFITM3 that modulates brain endothelial cell proliferation, migration and sprouting. Our findings establish the rationale for developing anti-vascular therapy based on the targeted disruption of IFITM3.</p>" ]

[]

[ "<p id=\"Par1\">Interferon-induced transmembrane protein 3 (IFITM3) has been previously verified to be an endosomal protein that prevents viral infection. Recent findings suggested IFITM3 as a key factor in tumor invasion and progression. To clarify the role and molecular mechanism of IFITM3 in Glioblastoma multiforme (GBM) progression, we investigated the expression of IFITM3 in glioma datasets culled from The Cancer Genome Atlas (TCGA) and Chinese Glioma Genome Atlas (CGGA). Primary GBM stem cells (GSCs) were cultured and identified in vitro. Loss-of-function and gain-of-function experiments were established by using shRNAs and lentiviral vectors targeting IFITM3. Co-culture system of GSCs and vascular endothelial cells was constructed in a Transwell chamber. Tube formation and spheroid-based angiogenesis assays were performed to determine the angiogenic capacity of endothelial cells. Results revealed that IFITM3 is elevated in GBM samples and predictive of adverse outcome. Mechanistically, GSCs-derived IFITM3 causes activation of Jak2/STAT3 signaling and leads to robust secretion of bFGF into tumor environment, which eventually results in enhanced angiogenesis. Taken together, these evidence indicated IFITM3 as an essential factor in GBM angiogenesis. Our findings provide a new insight into mechanism by which IFITM3 modulates GBM angiogenesis.</p>", "<title>Subject terms</title>" ]

[ "<title>Supplementary information</title>", "<p>\n\n\n\n\n\n\n\n\n</p>" ]

[ "<title>Supplementary information</title>", "<p>The online version contains supplementary material available at 10.1038/s41419-023-06416-5.</p>", "<title>Acknowledgements</title>", "<p>This work was supported by the National Natural Science Foundation of China (No. 82002631, No. 82072762, and No. 82203135) and President Foundation of Zhujiang Hospital, Southern Medical University (No. yzjj2022ms07).</p>", "<title>Author contributions</title>", "<p>Conceptualization: YL and YK; Methodology: XX, ZX, and HC; Data curation: ZX, XX, YZ, and TC; Original draft preparation: ZX and XX; Supervision: YL; All authors have read and agreed to the published version of the manuscript.</p>", "<title>Data availability</title>", "<p>All data generated in the present study are included either in the main article or in the ##SUPPL##0##supplementary information## files.</p>", "<title>Competing interests</title>", "<p id=\"Par29\">The authors declare no competing interests.</p>", "<title>Ethics approval and consent to participate</title>", "<p id=\"Par30\">Animal study was performed with the permission of the Animal Care and Use Committee of Southern Medical University. And study protocol and informed consent were approved by the Ethical Committee of Zhujiang Hospital.</p>" ]

[ "<fig id=\"Fig1\"><label>Fig. 1</label><caption><title>IFITM3 correlates with glioma grade and prognosis of GBM patients.</title><p><bold>A</bold> Gene expression profile of IGFBP4 in TCGA/CGGA glioma datasets stratified by tumor grade. <bold>B</bold> Gene expression analysis of IFITM3 in GBM (TCGA) and normal brain tissues (GTEx) was conducted using GEPIA2 web tool. <bold>C</bold> Kaplan–Meier survival analysis of GBM/LGG patients with high or low IFITM3 expression. <bold>D</bold> Representative images of IHC staining for IFITM3 in patients with high or low-grade glioma from Human Protein Atlas datasets. <bold>E</bold> Representative IHC images for IFITM3 in patients diagnosed as grade IV or Grade II glioma from Zhujiang Hospital. <bold>F</bold> IHC score of IFITM3 in human glioma samples stratified by tumor grade. <bold>G</bold> Correlation of IFITM3 expression with gender, age, tumor size, and tumor grade. Scale bar = 25/50 μm. Data are expressed as Mean ± SD. **<italic>p</italic> &lt; 0.05, **<italic>p</italic> &lt; 0.01, ***<italic>p</italic> &lt; 0.001. TCGA The Cancer Genome Atlas, CGGA Chinese Glioma Genome Atlas, GBM Glioblastoma, LGG Low-grade gliomas, IHC Immunohistochemistry.</p></caption></fig>", "<fig id=\"Fig2\"><label>Fig. 2</label><caption><title>IFITM3 is enriched in glioma stem cells.</title><p><bold>A</bold> Sphere formation ability was identified in human GBM-derived stem cells. <bold>B</bold> Nestin and CD133 were identified in GSCs using immunofluorescence staining. <bold>C</bold> IFITM3, Nestin and CD133 expression were examined via using immunoblotting. <bold>D</bold> IFITM3 was detected in GSC sphere and differentiated cells (GSDCs) via immunofluorescence staining. <bold>E</bold> IFITM3 and molecular markers for stem cells were examined in human GBM specimens. <bold>F</bold> Correlation between IFITM3 expression and stem cell markers in CGGA glioma dataset was evaluated. <bold>G</bold> GSCs and glioma cells were cultured under normoxic or hypoxic conditions, IFITM3 and HIF1α expression were determined by immunoblotting. <bold>H</bold> GSCs were exposed to hypoxia at different time points or varying oxygen levels for 48 h. IFITM3 expression were analyzed using immunoblotting. <bold>I</bold> IFITM3 and HIF1α were examined in cells cultivated under normoxic or hypoxic environment. <bold>J</bold> Correlation between IFITM3 and HIF1A gene expression were assessed in CGGA and Gravendeel GBM datasets. Results are represented as Mean ± SD of biologically triplicate assays. *<italic>p</italic> &lt; 0.05, **<italic>p</italic> &lt; 0.01, ***<italic>p</italic> &lt; 0.001. Scale bar = 100 μm.</p></caption></fig>", "<fig id=\"Fig3\"><label>Fig. 3</label><caption><title>IFITM3 is linked to enhanced angiogenesis.</title><p><bold>A</bold> ShScrambled or shIFITM3 GSCs were cultivated in serum-free media to form spheres. Sphere diameters were measured. <bold>B</bold> Illustration of co-culture systems in which GSCs and hBMECs were cultivated in transwell chamber. <bold>C</bold> Tube formation and sprouting capacity of endothelial cells cultured with shScrambled/shIFITM3 GSCs. Bar charts show tube number and sprout length of corresponding assays. <bold>D</bold> Proliferation of endothelial cells cultured with shScrambled or shIFITM3 GSCs was calculated using CCK-8 assay. <bold>E</bold> endothelial cells cultured with shScrambled or shIFITM3 GSCs were subjected to Edu staining. <bold>F</bold> Intracranial tumor model was established with shScrambled or shIFITM3 GSCs. Xenograft specimen were stained with Nestin and CD34. Bar charts revealed fluorescence intensity of indicated proteins. Scale bar = 100 μm. Results are represented as Mean ± SD of biologically triplicate assays. *<italic>p</italic> &lt; 0.05, **<italic>p</italic> &lt; 0.01, ***<italic>p</italic> &lt; 0.001.</p></caption></fig>", "<fig id=\"Fig4\"><label>Fig. 4</label><caption><title>IFITM3 induces invasiveness of GBM cells via upregulating JAK/STAT signaling.</title><p><bold>A</bold> Volcano plot of gene expression profiles of human glioma cases from TCGA database. <bold>B</bold> Gene set enrichment analysis (GSEA) between high and low IFITM3 expression show enriched pathways associated with angiogenesis. <bold>C</bold> GSCs were treated with scrambled or shRNA targeting IFITM3, indicated proteins of JAK/STAT, PI3K/Akt, and Notch pathways were determined by immunoblotting. <bold>D</bold> IFITM3-expressing GSCs with/without JAK/STAT inhibitor WP1066 were co-cultured with endothelial cells, followed by Matrigel-based tube formation assay (upper panel) or sphere-based sprouting assay (lower panel) to evaluate in vitro angiogenesis capacity of the latter. Bar charts in the right panel show statistical results <bold>E</bold> Representative IHC images of IFITM3 and p-STAT3 from human GBM samples. <bold>F</bold> Representative IF images of IFITM3 and p-STAT3 from intracranial xenograft specimens. Results are represented as Mean ± SD of biologically triplicate assays. *<italic>p</italic> &lt; 0.05, **<italic>p</italic> &lt; 0.01, ***<italic>p</italic> &lt; 0.001. ns not significant.</p></caption></fig>", "<fig id=\"Fig5\"><label>Fig. 5</label><caption><title>IFITM3 regulates GSC-mediated angiogenesis process in a bFGF-dependent manner.</title><p><bold>A</bold> GSC#1-shScr or GSC#1-shIFITM3 were cultivated for 24 h, of which culture media were harvested for cytokine detection using angiogenesis antibody array. <bold>B</bold> GSC#2 cells were fractionated into IFITM3+ or IFITM3-groups, and angiogenic cytokines were measured. <bold>C</bold> Bar charts show markedly altered cytokines of (<bold>A</bold>) and (<bold>B</bold>). <bold>D</bold> Secreted bFGF levels of GSC#1–5 were measured using Elisa assay. <bold>E</bold> Representative immunofluorescent images with bFGF (red) staining of GSC#1-shScr and GSC#1-shIFITM3. F-actin (green) is stained with phalloidin. Nuclei are counterstained with Dapi (blue). Scale bar = 50 μm. <bold>F</bold> Angiogenic abilities of fractionated IFITM3⁻ and IFITM3⁺ GSCs are measured, with the latter cells treated with manipulated bFGF downregulation. <bold>G</bold> Intracranial xenograft models are constructed with GSC#1-shScr and GSC#1-shIFITM3. Tumor sizes are monitored by in vivo imaging. <bold>H</bold> Representative immunofluorescence images staining for IFITM3 (green) and bFGF (red) and in GSC#1-shScr or GSC#1-shIFITM3 xenografts. <bold>I</bold> Representative immunofluorescence images staining for IFITM3 (green), bFGF (red), and CD34 (purple) in human GBM samples. <bold>J</bold> Schematic illustration of GSCs-mediated glioma angiogenesis through IFITM3-JAK/STAT-bFGF signaling pathway. Results are represented as Mean ± SD of biologically triplicate assays. *<italic>p</italic> &lt; 0.05, **<italic>p</italic> &lt; 0.01, ***<italic>p</italic> &lt; 0.001. ns not significant.</p></caption></fig>" ]

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[ "<supplementary-material content-type=\"local-data\" id=\"MOESM1\"></supplementary-material>", "<supplementary-material content-type=\"local-data\" id=\"MOESM2\"></supplementary-material>", "<supplementary-material content-type=\"local-data\" id=\"MOESM3\"></supplementary-material>", "<supplementary-material content-type=\"local-data\" id=\"MOESM4\"></supplementary-material>", "<supplementary-material content-type=\"local-data\" id=\"MOESM5\"></supplementary-material>", "<supplementary-material content-type=\"local-data\" id=\"MOESM6\"></supplementary-material>", "<supplementary-material content-type=\"local-data\" id=\"MOESM7\"></supplementary-material>", "<supplementary-material content-type=\"local-data\" id=\"MOESM8\"></supplementary-material>" ]

[ "<fn-group><fn><p>Edited by Dr. Maria Victoria Niklison Chirou</p></fn><fn><p><bold>Publisher’s note</bold> Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.</p></fn><fn><p>These authors contributed equally: Zhangsheng Xiong, Xiangdong Xu.</p></fn></fn-group>" ]

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[ "<media xlink:href=\"41419_2023_6416_MOESM1_ESM.docx\"><caption><p>Supplementary figures and legends</p></caption></media>", "<media xlink:href=\"41419_2023_6416_MOESM2_ESM.pdf\"><caption><p>Checklist</p></caption></media>", "<media xlink:href=\"41419_2023_6416_MOESM3_ESM.tif\"><caption><p>Supplementary Fig.1</p></caption></media>", "<media xlink:href=\"41419_2023_6416_MOESM4_ESM.tif\"><caption><p>Supplementary Fig.2</p></caption></media>", "<media xlink:href=\"41419_2023_6416_MOESM5_ESM.xlsx\"><caption><p>Supplementary Table S1</p></caption></media>", "<media xlink:href=\"41419_2023_6416_MOESM6_ESM.xlsx\"><caption><p>Supplementary Table S2</p></caption></media>", "<media xlink:href=\"41419_2023_6416_MOESM7_ESM.xlsx\"><caption><p>Supplementary Table S3</p></caption></media>", "<media xlink:href=\"41419_2023_6416_MOESM8_ESM.tif\"><caption><p>Original blots data</p></caption></media>" ]

{ "acronym": [], "definition": [] }

[ "<title>Introduction</title>", "<p id=\"Par2\">Since the beginning of the twenty-first century, industrial big data has experienced rapid development with the improvement of data collection and processing capabilities^{##UREF##0##1##}. The centralized big data processing model centered around cloud computing can no longer support industrial data analysis. Its various drawbacks have become evident, such as difficulty integrating heterogeneous data from multiple sources, handling high broadband loads, and dealing with limited resources^{##UREF##1##2##–##UREF##3##4##}. This is particularly critical for equipment management systems demanding highly real-time data processing. Suppose faults within the equipment are not detected at the earliest opportunity. In that case, it diminishes product processing quality and leads to even more significant losses across the entire industrial production line. In recent years, edge computing technology, based on industrial-grade intelligent hardware, has become a hot research field. Establishing a data bridge between production equipment and cloud-based systems achieves rapid sensing of equipment operating statuses in the industrial Internet of Things (IIoT) and enables intelligent adjustments. This advancement has propelled significant developments in intelligent systems and smart manufacturing^{##UREF##4##5##}. The operating scope of edge computing technology includes downstream data from cloud services and upstream data from the Internet of Things services^{##UREF##5##6##}. It is a novel computing model that performs computations at the network edge^{##UREF##6##7##}.</p>", "<p id=\"Par3\">Edge computing can be traced back to the content distribution network proposed by Akamai in 1998^{##UREF##7##8##}. In 2013, the American scholar Ryan La Mothe first proposed “edge computing” in an internal report^{##UREF##6##7##}. In May 2016, Professor Shi Weisong and his team from Wayne State University in the United States formally defined edge computing^{##UREF##8##9##}. In the same year, China established the Edge Computing Consortium (ECC), which Huawei Technologies Co. Ltd. and the Shenyang Institute of Automation of the Chinese Academy of Sciences founded^{##UREF##6##7##}. The consortium covers various fields, such as scientific research institutions and industrial manufacturing. In the case of the Industrial Internet of Things (IIoT), edge computing meets its requirements for real-time control and edge device security and privacy in practical applications, making it a direction for developing the IIoT industry. In their study, Shi Weisong et al.^{##UREF##5##6##} summarized the current situation and prospects of edge computing and provided a summary from the industrial Internet of Things perspective. Edge computing can address the real-time control of networked production and processing, edge device security and privacy, and localized processing of production data faced by the development of industrial IoT, and has advantages in improving performance, ensuring data security and privacy, and reducing operating costs in practical applications^{##UREF##5##6##,##UREF##6##7##}.</p>", "<p id=\"Par4\">Currently, research on equipment management mainly focuses on three directions: predictive maintenance^{##UREF##9##10##–##UREF##11##12##}, fault diagnosis^{##UREF##12##13##,##UREF##13##14##}, and quality prediction, as shown in Fig. ##FIG##0##1##<bold>.</bold> Equipment predictive maintenance refers to collecting operational data and environmental data during the operation of equipment, using big data and machine learning methods to predict the service life and damage of important components of the equipment, avoiding excessive maintenance of the equipment, reducing the failure rate of the equipment, and lowering the manufacturing cost of products. The main approach to equipment fault diagnosis is to establish the mechanism of equipment faults, study the relationships between various causes of faults, fault characterizations, and fault signals, in order to diagnose equipment faults quickly when they occur. Quality prediction is an essential means to reduce the probability of product quality problems and improve the qualification rate by analyzing the operating parameters of the equipment, obtaining quality characteristics based on equipment parameters, and monitoring and controlling the parameters of the equipment processing process^{##UREF##14##15##}.</p>", "<p id=\"Par5\">Product quality management is an essential issue in intelligent factory information services, and many scholars have elaborated on different aspects such as reliability^{##UREF##15##16##}, helpful life^{##UREF##16##17##}, and retrievability^{##UREF##17##18##}. Among them, the prediction of product quality is also a hot topic. Product quality often requires specialized, expensive, and complex testing equipment, and the testing process can take a long time. Therefore, rapid, effective product quality prediction is significant for providing decision-making services to factory managers.</p>" ]

[ "<title>Active control method for quality prediction</title>", "<p id=\"Par26\">This section first analyzed the product’s quality characteristics and selected criticalquality-related parameters with correlation coefficients greater than the set threshold based on the correlation coefficients of industrial product quality inspection results and quality-related parameters. Established the SMOTE-XGBoost quality prediction model and optimized the hyperparameters. Finally, the active control method for prediction.</p>", "<title>Analysis of quality characteristics</title>", "<p id=\"Par27\">In product quality issues,this paper abstracts the product processing process as a manufacturing processing unit and the process of changing the product quality state as process characteristic data of processing quality. Additionally, it analyzesthe process parameter data during equipment operation.</p>", "<p id=\"Par28\">As shown in Fig. ##FIG##4##5##. In the manufacturing processing unit, represents the product state before the execution of the manufacturing processing unit; represents the product state after the execution of the manufacturing processing unit; From the perspective of quality data, refers to the resource processing data received by the manufacturing processing unit; represents the product quality state data before the manufacturing processing unit processes it; refers to the output product quality state data processed by the manufacturing processing unit; represents the difference between the actual qualified rate of the calculated output product and the qualified rate of the industrial product containing the predicted results, and is the threshold. When the value exceeds a certain threshold, the edge computing layer will generate corresponding process adjustment control instructions ,and send them to the relevant processing equipment, such as adjusting the spindle speed and feed rate^{##UREF##14##15##}.</p>", "<p id=\"Par29\">From the perspective of task execution, process refers to the process of transforming the quality characteristics of a product from state to state through a series of processing methods.</p>", "<p id=\"Par30\">From the perspective of quality characteristics, the current quality characteristic is the result of the current process equipment processing the quality characteristic in the current environment^{##UREF##14##15##}. The process of changing quality characteristics is the process of transforming input data into output data through its processing mechanism.</p>", "<p id=\"Par31\">As the manufacturing processing continues, manufacturing quality-related parameter data is collected one by one at a fixed frequency. The type of equipment process data parameter set for collection is , and each set of equipment quality-related parameter data collected is represented by an array, as shown in Eq. (##FORMU##15##1##).</p>", "<p id=\"Par32\">In Eq. (##FORMU##15##1##), represents an array of equipment quality-related parameters collected at a certain moment. represents the -th parameter of array . As time passes and the processing progresses, more and more data is collected, forming a matrix of quality-related parameter data as shown in Eq. (##FORMU##20##2##).</p>", "<title>Selection of quality-related parameters</title>", "<p id=\"Par33\">Throughout the production process of industrial goods, a large amount of data related to their quality is collected through the equipment perception layer, including quality inspection results and corresponding quality-related parameters. Including quality inspection results and corresponding quality-related parameters. Based on the quality inspection results and corresponding quality-related parameters, important rules for selecting quality-related parameters can be established, as described in section “<xref rid=\"Sec8\" ref-type=\"sec\">Equipment process parameters</xref>”. This article selects the quality-related parameters that affect the indirectly dynamic equipment process data.</p>", "<p id=\"Par34\">The selection rule of quality-related parameters mainly refers to selecting the key quality-related parameters with a correlation coefficient greater than a set threshold through the correlation analysis between the quality inspection results and the quality-related parameters in industrial product manufacturing. Formula for calculating the correlation coefficient between the quality inspection results and the quality-related parameters in industrial product manufacturing is:where represents the -th quality-related parameter, represents the number of nonconforming industrial product manufacturing quality inspection results, and represents the number of conforming industrial product manufacturing quality inspection results. represents the joint distribution of and \n represents the joint distribution of and . , and are the probability distributions of variables , , and ,respectively. and are adjustment coefficients for data imbalance, with a sum of 1, generally determined based on the quality of the data samples obtained.</p>", "<p id=\"Par35\">According to the correlation coefficient between the quality inspection results and the quality-related parameters in industrial product manufacturing, the importance of the features is sorted. Obtain a feature set , where represents the -th feature value.</p>", "<title>SMOTE-XGBoost algorithm for quality prediction</title>", "<title>Data preprocessing based on SMOTE</title>", "<p id=\"Par36\">According to the factory survey results, in the stable production line of brake discs, majority of the final quality is qualified, and only a small number of products have quality problems (unqualified products). The brake disc production line produces more than 1000 products per day, of which over 95% are qualified products. From a data mining perspective, this means that the input labels of the prediction model are imbalanced. Imbalanced label data is a common type of data that is widely present in various industrial fields.</p>", "<p id=\"Par37\">This article adopts the Synthetic Minority Oversampling Technique (SMOTE) algorithm to address the issue of imbalanced data. The core idea of the algorithm is to perform interpolation on the minority class samples in the dataset based on the k-nearest neighbor rule (as shown in Fig. ##FIG##5##6## below). Generating more minority class samples as a result^{##UREF##36##39##}. As the production dataset of brake discs is imbalanced, SMOTE is used in this chapter to balance the dataset. The main steps of the algorithm are as follows:</p>", "<p id=\"Par38\">The dataset of brake disc processing collected by the production line is an imbalanced dataset. Based on the number of minority class samples and majority class samples , the required number of synthesized samples is calculated:</p>", "<p id=\"Par39\">For each unqualified product data sample (minority class) , where . Select nearest neighbors ( is usually set to 5) of the minority class sample randomly with the Euclidean distance as the measurement standard.</p>", "<p id=\"Par40\">Assuming the selected neighboring point is , the new synthetic sample point is generated according to the following formula.where represents a random number between 0 and 1. Generate new minority class samples, merge them with the original data set to get a balanced data set. Then input them into XGboost for identification.</p>", "<title>Predictive model based on XGboost</title>", "<p id=\"Par41\">XGBoost is ensemble learning model framework based on gradient boosting algorithm, which was proposed by Dr. Tianqi Chen and his colleagues^{##UREF##37##40##}. Compared with the traditional Gradient Based Decision Tree (GBDT), both are based on decision trees. However, XGboost effectively controls the complexity of the model and greatly reduces the variance of the model by using second-order Taylor expansion and adding regularization terms. The trained model is also simpler and more stable^{##UREF##38##41##}.</p>", "<p id=\"Par42\">Assuming that the input samples are , The output of the XGboost model can be represented as the sum of weak learner outputs:where represents the output of the -th weak learner.</p>", "<p id=\"Par43\">The model’s bias and variance determine the prediction accuracy of a model. The loss function represents the bias of the model, and to reduce the variance, a regularization term needs to be added to the objective function to prevent overfitting. The objective function comprises the model’s loss function and a regularization term to suppress model complexity. The objective function to minimize in function space is:</p>", "<p id=\"Par44\">Here, represents the loss function, represents the regularization function, is the number of leaf nodes, and is the weight value of leaf nodes. In the XGBoost model, most weak learns are based on Classification and Regression Trees (CART). Therefore, each round of optimization only focuses on the objective function of the -th classification and regression tree based on the previous models.</p>", "<p id=\"Par45\">Next, perform second-order Taylor expansion on the loss function of XGboos:</p>", "<p id=\"Par46\">And in the above equation:</p>", "<p id=\"Par47\">In which, and are the first-order and second-order derivatives of each sample on the loss function, respectively. Therefore, the optimization of the objective function can be transformed into the process of finding the minimum value of a quadratic function.</p>", "<title>SMOTE-XGBoost with jointly optimized hyperparameters</title>", "<p id=\"Par48\">(1) SMOTE-XGBoost model</p>", "<p id=\"Par49\">The hyperparameter optimization methods mainly include grid search, random search, heuristic algorithms, and so on^{##UREF##39##42##}. This article used the gridsearch method to optimize the above three hyperparameters, in order to obtain the optimal predictive model.</p>", "<p id=\"Par50\">The Smote algorithm and the XGboost algorithm both have hyperparameters that need to be set before training the algorithm. The setting of hyperparameters affects the performance of predictive models. Previous research has mainly focused on the hyperparameters in classification or regression models. Therefore, consider Smote and XGboost as a whole and propose a joint optimization method for hyperparameters, called SMOTE-XGboost, to improve the performance of quality prediction models. Specifically, this paper focuses on the optimization of the hyperparameters in SMOTE (Number of nearest neighbors for selecting samples), in XGboost (Number of decision trees), and in XGboost (Number of leaf nodes). Selecting the maximum score as the optimization objective to obtain the best hyperparameters. The principle of joint hyperparameter optimization is as follows: Train the original SMOTE-XGboost model on historical data, which can be represented as:where represents the number of nearest neighbors selected in SMOTE. represents the number of decision trees in XGBoost, and represents the number of leaf nodes in XGBoost. Training process of the SMOTE-XGboost prediction model described in this article includes: To optimize the hyperparameters of the SMOTE-XGboost model with the goal of obtaining the maximum AUC score, the following formula is used:</p>", "<p id=\"Par51\">In the expression: represents the true quality result; represents the predicted quality result; represents a quality prediction function; is a non-analytic function of the decision variable . is the scoring formula; represents the first data points in the test set.</p>", "<p id=\"Par52\">(2) Model evaluation indicators</p>", "<p id=\"Par53\">To effectively evaluate the reliability of predictive models, comparative experiments of different algorithms are conducted using the coefficient of determination () and the AUC as evaluation metrics to assess the relationship between predicted values and true values of the models. AUC is defined as the area enclosed by the coordinate axis under the ROC curve. It is a comprehensive performance classification indicator, which is commonly used to measure classification performance^{##UREF##29##31##,##UREF##40##43##}. The higher the AUC, the better the algorithm performance.</p>", "<p id=\"Par54\">Scoring formula for :</p>", "<p id=\"Par55\">In this expression, represents the true value, represents the predicted value, represents the sample mean, and represents the sample size. A higher value indicates better performance. When the predictive model makes no errors, achieves the maximum value of 1.</p>", "<p id=\"Par56\">Scoring formula for :</p>", "<p id=\"Par57\">In this expression, represents the ranking number of positive samples in the data set, represents the number of positive samples in the data set, and represents the total number of samples in the data set.</p>", "<p id=\"Par58\">(3) Active control methods</p>", "<p id=\"Par59\">In the actual production process, manufacturing process data of industrial products is first transmitted to edge computing nodes through Ethernet. The edge computing nodes use important quality-related parameter selection rules to filter and reduce data, and make real-time quality predictions for products as qualified or non-qualified based on the quality active prediction model deployed on the edge computing nodes.</p>", "<p id=\"Par60\">Active control methods refer to calculating the difference between the actual qualified rate of the produced product and the predicted qualified rate of products. If this difference is greater than a certain threshold, the edge computing layer will generate corresponding process adjustment control instructions and send them to the relevant processing equipment, such as adjusting spindle speed, feed rate, etc.</p>", "<p id=\"Par61\">Edge computing-based proactive control method for industrial product manufacturing quality prediction, It characteristics lie in the calculation formula for the difference between the actual qualified rate of the produced product and the qualified rate of products with prediction results, which is as follows:</p>", "<p id=\"Par62\">In the formula, and respectively represent the number of qualified products in the actual output, and the number of qualified products with prediction results; and respectively represent the total number of products in the actual output, and the total number of products with prediction results.</p>" ]

[ "<title>Experiments and results</title>", "<title>Qualitys correlation parameter selection</title>", "<p id=\"Par67\">Some studies^{##UREF##41##44##,##UREF##42##45##} have pointed out that the equipment process data obtained from the processing equipment, including spindle power (P), spindle current (I), spindle speed (S), feed speed (F), and clamping force (N), were related to the changes of product quality characteristics in the processing process.</p>", "<p id=\"Par68\">In order to validate the effectiveness of the proposed quality prediction method, historical data sets from the edge server were collected as the data source for overall quality prediction analysis. The data set includes 5 quality characteristics and 1 final quality label (qualified or fault product). The quality characteristics are all continuous random variables. Table ##TAB##2##3## shows the specific quality characteristics of the partial samples. There are 1844 samples in the data set, including 1778 samples of qualified products and 66 samples of fault products. The imbalance ratio of the data set is about 26.9:1.</p>", "<p id=\"Par69\">As per the calculation method described in section “<xref rid=\"Sec12\" ref-type=\"sec\">Selection of quality-related parameters</xref>”, calculated the importance of each quality feature and sorted them in descending order, as shown in Fig. ##FIG##6##7##, ultimately selected four quality features, including spindle speed (S) and feed speed (F), spindle power (P), and spindle current (I), and clamping force (N), to construct the prediction model.</p>", "<title>Classification results with other machine learning methods</title>", "<p id=\"Par70\">This paper conducted comparative experiments among different algorithms to validate the effectiveness of the proposed quality prediction model. The relationship between predicted and actual values was evaluated using coefficients () and AUC as assessment metrics. All experiments in this study were deployed in a python3.6 environment and run on a desktop computer with an Intel Core i7 processor, 3.6 GHz, and 16 GB RAM.</p>", "<p id=\"Par71\">First, the data set was extracted based on the sorted quality features. The data description of the training set and test set is shown in Table ##TAB##3##4##. Apply the SMOTE oversampling strategy only in the training set to avoid over-optimism^{##UREF##35##38##,##UREF##43##46##}. The data after SMOTE processing is shown in Table ##TAB##4##5##. Then, this text used the training set to build the SMOTE-XGBoost prediction model and used grid search to jointly optimize the hyperparameters of the brake disc quality prediction model (the hyperparameter optimization range is shown in Table ##TAB##6##7##). The final optimal values for each hyperparameter of the SMOTE-XGBoost were determined to be k = 6, e = 100, and T = 3. The optimized quality prediction model is named SMOTE-XGboost_t, and its prediction results on part of the test data set are shown in Fig. ##FIG##7##8##. This paper designed comparative experiments from the perspectives of classification algorithms and hyperparameter optimization to highlight the superiority of the proposed method.</p>", "<p id=\"Par72\">(1) Comparison experiment of classification algorithms.</p>", "<p id=\"Par73\">To verify the classification performance of the proposed method compared to other classification methods under the same criteria, this study used the same SMOTE method and compared the proposed method with other mainstream machine learning classification methods (Support Vector Machine, SVM; Logistic Regression, LR; Decision Tree, DT; Random Forest, RF). The experimental results are shown in Table ##TAB##5##6##, based on the table, as can be seen that the proposed SMOTE-XGboost_t method has slightly higher and AUC values compared to other classifiers in the experiment using the same SMOTE method. Moreover, the ROC curves of the model’s indicators are shown in Fig. ##FIG##8##9##. AUC is defined as the area enclosed by the coordinate axis under the ROC curve. From the figure, as can be seen that the AUC value of the proposed SMOTE-XGboost_t method is as high as 0.916, which indicates that the proposed method can effectively identify unqualified products and thus better predict the quality of brake discs.</p>", "<p id=\"Par74\">(2) Hyperparameter optimization comparative experiment</p>", "<p id=\"Par75\">In addition, to investigate the impact of hyperparameter optimization on the model, this study conducted four different experiments:In the model named SMOTE-XGboost, the default values were used for the hyperparameters without any hyperparameter optimization; The hyperparameter ( Number of nearest neighbors for selecting samples) in SMOTE was optimized in the model SMOTE-XGboost_s; In the model SMOTE-XGboost_x, only the hyperparameters (Number of decision trees) and (Number of leaf nodes) in XGBoost were optimized; The last experiment involved joint optimization of the hyperparameters (Number of nearest neighbors for selecting samples), (Number of decision trees), and (number of leaf nodes) in both SMOTE and XGboost using grid search in the SMOTE-XGboost_t model. The optimal hyperparameters and optimization ranges for the predictive models in the four experiments are shown in Table ##TAB##5##6##, and the experimental comparison results are shown in Table ##TAB##7##8##.</p>", "<p id=\"Par76\">Based on Table ##TAB##6##7##, see that in the SMOTE-XGboost_t model, the optimal value is 6 instead of the default value of = 5. This indicates that when integrating oversampling algorithms with traditional machine learning classification algorithms, there may be uncertainties in the prediction results due to the hyperparameters of the sampling model and the classification model. Therefore, optimizing the hyperparameters in both the SMOTE sampling algorithm and the XGboost classification model is beneficial to improve the quality prediction performance.</p>", "<p id=\"Par77\">Analysis of the ROC curves for the four experiments based on AUC values is shown in Fig. ##FIG##9##10##. It can be observed that the SMOTE-XGboost_t and SMOTE-XGboost methods are slightly better than the other methods. SMOTE-XGboost_t had the best performance with an AUC value of 0.916.</p>", "<p id=\"Par78\">The analysis from Table ##TAB##7##8## shows that the proposed method performs better than other methods in terms of AUC and scores, indicating that the quality prediction model has a strong ability to identify the quality of brake discs after joint optimization of hyperparameters. Based on the actual operation of the factory, factory managers are more concerned about defective products than the large quantity of qualified products. Therefore, the method proposed in this paper has strong comprehensive prediction ability.</p>" ]

[ "<title>Discussion</title>", "<p id=\"Par79\">In terms of imbalanced data, Table ##TAB##5##6## and Fig. ##FIG##8##9## demonstrate that the SMOTE and XGBoost combination outperforms the combination of SMOTE with other classification algorithms Fig. ##FIG##6##7## displays the importance of quality features; selecting these features is crucial for predicting and analyzing quality issues. Additionally, simultaneous investigation of hyperparameters in joint optimization included k (the number of nearest neighbors in SMOTE), e (the number of decision trees in XGBoost), and T (the number of leaf nodes in XGBoost). Table ##TAB##7##8## and Fig. ##FIG##9##10## indicate that the SMOTE-XGBoost method with jointly optimized hyperparameters can enhance classification performance.</p>", "<p id=\"Par80\">This result also indicates that our proposed method contributes to addressing imbalanced data classification issues. and AUC are two widely used metrics in various classification problems. Additionally, AUC is a comprehensive metric that considers both qualified and defective products. Therefore, AUC is a more critical metric in unbalanced quality prediction scenarios and is widely used in various imbalanced classification problems.</p>", "<p id=\"Par81\">Existing traditional industrial product manufacturing quality has long relied on passive analysis methods such as statistical monitoring. This method primarily involves testing the product quality using quality inspection equipment after the production and processing of the product. The limitations of this method lie in two aspects. Firstly, specific products require particular quality inspection equipment, which takes considerable time and involves expensive equipment. Secondly, it is impossible to forecast whether the product quality will be up to standard. When faults occur in equipment affecting product quality, there is no timely feedback for adjusting the equipment. So, rapid and efficient quality prediction methods can potentially replace specialized equipment, saving on equipment costs and testing time.</p>" ]

[ "<title>Conclusion</title>", "<p id=\"Par82\">This article proposes an Edge computing-based proactive control method for industrial product manufacturing quality prediction, addressing the issue of imbalanced data in the manufacturing process. Firstly, an edge computing-based framework for quality prediction in industrial product manufacturing was proposed. Secondly, a method for selecting quality-related parameters was designed, this provides insights into quality analysis problems. Finally, a SMOTE-XGboost quality forecasting active control method based on joint optimization hyperparameters is proposed to solve the problem of manufacturing quality forecasting of industrial products under category imbalance (Table ##TAB##7##8##).</p>", "<p id=\"Par83\">This paper compared prediction algorithms based on five different classification methods under specific experimental conditions. The experimental results indicate that the proposed SMOTE-XGboost_t method slightly outperforms the other four classifiers in terms of and AUC metrics. This indicates that the proposed method has good performance in predicting the manufacturing quality of industrial products and detecting faulty products. Finally, the optimal values for each hyperparameter of SMOTE-XGboost were determined to be = 6, = 100, and = 3, and the prediction results were better than those obtained through single hyperparameter optimization.</p>", "<p id=\"Par84\">The research in this article enhances the capability for product quality control and provides intelligent information services for enterprises. However, there are still some issues that need further study. This paper only considered the product quality prediction results after processing in a single processing unit. Therefore, future research will focus on predicting product quality for multi-stage processing. Additionally, since the process-related data during manufacturing is incremental, another research direction involves addressing the issue of the source database of the quality prediction model in the edge computing scenario updating over time in the production line. This involves devising an incremental data training strategy for obtaining performance updates by training incremental data on the existing model.</p>" ]

[ "<p id=\"Par1\">With the emergence of intelligent manufacturing, new-generation information technologies such as big data and artificial intelligence are rapidly integrating with the manufacturing industry. One of the primary applications is to assist manufacturing plants in predicting product quality. Traditional predictive models primarily focus on establishing high-precision classification or regression models, with less emphasis on imbalanced data. This is a specific but common scenario in practical industrial environments concerning quality prediction. A SMOTE-XGboost quality prediction active control method based on joint optimization hyperparameters is proposed to address the problem of imbalanced data classification in product quality prediction. In addition, edge computing technology is introduced to address issues in industrial manufacturing, such as the large bandwidth load and resource limitations associated with traditional cloud computing models. Finally, the practicality and effectiveness of the proposed method are validated through a case study of the brake disc production line. Experimental results indicate that the proposed method outperforms other classification methods in brake disc quality prediction.</p>", "<title>Subject terms</title>" ]

[ "<title>Related work</title>", "<p id=\"Par6\">The current research on quality prediction methods is mainly divided into two categories: model-based prediction methods and data-driven prediction methods. The main difference between the two methods lies in whether the design of the controller is based on the system model or only on the I/O data. In other words, whether the design of the controller involves the dynamic model of the system or not. If the system model is involved in the design of the controller, it is a model-based prediction method; otherwise, it is a data-driven prediction method^{##UREF##18##19##}. From this perspective, it can be concluded that certain prediction methods, such as those reliant on neural networks, fuzzy control prediction techniques, and various other intelligent control prediction methods, are founded upon data-driven predictive approaches^{##UREF##19##20##}. Many scholars have conducted extensive research and exploration on quality prediction. Table ##TAB##0##1## summarizes the relevant papers.</p>", "<p id=\"Par7\">The current research on quality prediction methods is mainly divided into two categories: model-based prediction methods and data-driven prediction methods. The main difference between the two methods lies in whether the design of the controller is based on the system model or only on the I/O data. In other words, whether the design of the controller involves the dynamic model of the system or not. If the system model is involved in the design of the controller, it is a model-based prediction method; otherwise, it is a data-driven prediction method^{##UREF##18##19##}. From this perspective, some prediction methods based on neural networks, fuzzy control prediction methods, and many other intelligent control prediction methods are based on data-driven prediction methods^{##UREF##19##20##}. Many scholars have conducted extensive research and exploration on quality prediction. Table ##TAB##0##1## summarizes the relevant papers.</p>", "<p id=\"Par8\">From the existing research perspective, improving data acquisition and processing capabilities provides a foundation for data-driven quality control. It provides research ideas for the analysis of equipment operating data. This includes a model identification algorithm, proposing a multi-degree-of-freedom torsional vibration model for transmission systems, serving as a digital twin model for monitoring the remaining useful life of transmission system components^{##UREF##28##30##}. Additionally, a method for predicting the quality of purifier carrier products is developed based on improved principal component analysis (PCA) and enhanced support vector machine (SVM). Other researchers have studied the mixed manifold learning and support vector machine algorithm based on optimized kernel functions (KML-SVM). They use support vector machines to classify and predict low-dimensional embedded data and optimize the kernel function of the support vector machine to maximize classification accuracy^{##UREF##29##31##}. Using random forests for dimensionality reduction and analyzing key quality characteristics^{##UREF##30##32##}. The principle of quality improvement in mechanical product development based on the Bayesian network can be used for the principle-empirical (P-E) model of quality improvement. It provides a method for learning the structure of the P-E model, and the quality characteristic (QC) relationship is determined by empirical data^{##UREF##30##32##,##UREF##31##33##}. By analyzing the relationship between manufacturing resources and product quality status^{##UREF##32##34##}, proposed a real-time quality control system (RTQCS) based on manufacturing process data, establishing the relationship between real-time product quality status and machining task processes^{##REF##29966374##35##}. A single-board computer and sensors were used to construct an edge device that can collect, process, store, and analyze data. Based on this, they developed a machine fault detection model using long short-term memory recurrent neural networks. Additionally, it is crucial to consider a real-time selection of the best model. In many cases, a simple probabilistic model can outperform more complex ones. Beruvides and colleagues achieved good drilling quality measurement and control results by employing the wavelet packet analysis method and fitting a statistical regression model^{##UREF##33##36##}. Cruz and others proposed a two-step machine learning method for dynamic model selection, achieving favorable outcomes in predicting surface roughness during micro-machining processes and addressing complex cutting phenomena^{##UREF##34##37##}.</p>", "<p id=\"Par9\">These scholars have significantly contributed to quality prediction, but there are also some issues. Firstly, on a stable production line, the quantity of qualified products far exceeds the number of faulty products (imbalanced product quality labels). Therefore, the quality prediction problem becomes an imbalanced data classification issue. Secondly, the equipment environment during the production process is complex, with numerous equipment process parameters affecting the quality characteristics of the processed products. Selecting equipment process parameters helps reduce the dimensionality of prediction models. Thirdly, some cloud-based quality prediction methods may result in issues such as delay, high broadband load, and resource limitations. To overcome these shortcomings, this paper initially introduces edge computing into product quality prediction to ensure shorter response times and higher reliability. Then, a method for selecting quality-correlated parameters is designed. Finally, addressing imbalanced data classification problems is achieved by employing the Synthetic Minority Oversampling Technique (SMOTE) and Extreme Gradient Boosting (XGBoost). The scientific-technical contribution of this article:<list list-type=\"order\"><list-item><p id=\"Par10\">Explored an edge computing-based framework for predicting the manufacturing quality of industrial products, offering guidance for flexible handling of industrial data.</p></list-item><list-item><p id=\"Par11\">The proposed is an active control method for quality prediction using SMOTE-XGBoost based on joint optimization of hyperparameters, applied in predicting manufacturing quality for industrial products to address the imbalanced data classification issue within product quality prediction. The experimental results validated the superiority of the proposed method.</p></list-item><list-item><p id=\"Par12\">Based on this paper’s proposed active control method for quality prediction, a selection and analysis of equipment process parameters for the brake disc production line was conducted using quality-correlated parameter selection, providing guidance and reference for the actual production and processing of brake disc products.</p></list-item></list></p>", "<p id=\"Par13\">For modern manufacturing, ensuring reliable industrial product quality has always been crucial in enterprise manufacturing process control. Guided by data-driven proactive quality control, modern manufacturing enterprises can gather vast amounts of industrial product manufacturing process data and apply it across various models. However, these models must operate at sufficiently high processing speeds to meet the practical production needs. Hence, the introduction of edge computing technology plays a pivotal role. Deploying models to the edge of the production line according to the actual industrial environment and establishing an edge-side IoT platform allows for more effective processing and application of.</p>", "<p id=\"Par14\">The remaining sections of this paper are organized as follows: section “<xref rid=\"Sec3\" ref-type=\"sec\">Industrial product manufacturing quality prediction frame work</xref>” presents an edge computing-based framework for industrial product quality prediction. Section “<xref rid=\"Sec10\" ref-type=\"sec\">Active control method for quality prediction</xref>” introduces a SMOTE-XGboost quality prediction active control method based on joint optimization hyperparameters. Following this, in section “<xref rid=\"Sec17\" ref-type=\"sec\">Case study</xref>”, an experimental analysis of the processing quality of the brake disc production line is conducted based on the proposed quality prediction method, confirming the superiority of this approach and providing guidance and reference for actual brake disc production. Finally, section “<xref rid=\"Sec23\" ref-type=\"sec\">Conclusion</xref>” provides a conclusion.</p>", "<title>Industrial product manufacturing quality prediction frame work</title>", "<p id=\"Par15\">This section constructed an edge-computing architecture for the industrial Internet of Things and analyzed the application methods of existing architectures. This explains the necessity of deploying industrial product quality prediction models using edge computing methods and introduces the quality prediction method proposed in this study.</p>", "<title>Industrial internet of things for industrial production lines</title>", "<p id=\"Par16\">To better manage the production line’s equipment operation status and product quality of the production line, and achieve real-timeproduct quality prediction, an industrial IoT architecture for the production line is established, as shown in Fig. ##FIG##1##2##. This is the basis for implementing industrial intelligence services.</p>", "<p id=\"Par17\">This framework consists of four layers: perception layer, edge layer, central layer, and application layer.</p>", "<title>Quality prediction activecontrol method</title>", "<p id=\"Par18\">Figure ##FIG##2##3## shows an example of industrial product manufacturing quality prediction based on edge computing. The data from the equipment side includes historical and real-time data and analyzes and describes its specific applications, while also analyzing the process parameter data during equipment operation.</p>", "<title>Historical data</title>", "<p id=\"Par19\">Historical data is mainly used for training the prediction model. The collected data is uploaded to the central layer through the perception layer for training the quality prediction model using machine learning algorithms. However, as the production process of products advances, the operating state changes over time. Therefore, the quality diagnosis and prediction model based on historical data is difficult to adapt to current production requirements. Some articles have also studied the update mechanism of predictive models^{##UREF##32##34##,##UREF##35##38##}.</p>", "<p id=\"Par20\">The complexity of manufacturing systems has led to the development of prediction methods that combine historical data and real-time measurement data, which are in line with the characteristics of edge computing technology.</p>", "<title>Real-time data</title>", "<p id=\"Par21\">Real-time data collected by the perception layer is transmitted to the edge layer, which undergoes preprocessing operations on the real-time data. Filtering the collected real-time data based on quality characteristic and then using the prediction model deployed on the edge device to make real-time judgments on product quality.</p>", "<p id=\"Par22\">Simultaneously, the preprocessed data from edge devices is transmitted to the cloud center through the perception layer. As the data volume is reduced after preprocessing, it alleviates the bandwidth pressure and accelerates the transmission speed. For the received data, the central layer can update existing quality prediction models over time using incremental learning methods, addressing the issue of database updates in a time series.</p>", "<title>Equipment process parameters</title>", "<p id=\"Par23\">In existing research^{##UREF##14##15##}, divided equipment process parameters into static process data, direct dynamic process data, and indirect dynamic process data based on their impact on the quality characteristics of the processed products to facilitate the application of equipment process parameters. Among them, static equipment process data refers to the type of equipment process data that generally does not change during the product processing process; direct dynamic process data refers to the equipment process data that changes dynamically during the product processing process, and the numerical changes directly reflect the product quality characteristics; Indirect dynamic process data refers to the equipment process data that changes dynamically during the processing process, but its changes do not directly reflect the product quality characteristics. Table ##TAB##1##2## presents an example classification result of equipment process data^{##UREF##14##15##}. Indirect dynamic equipment process data is the focus of this study.</p>", "<title>Introduction to the method</title>", "<p id=\"Par24\">A proactive control method for quality prediction based on historical data is proposed, comprising two components: quality prediction and proactive control. The Active control methods refer to calculating the difference between the actual qualified rate of the produced product and the predicted qualified rate of products. If this difference exceeds a certain threshold, the edge computing layer will generate corresponding process adjustment control instructions and send them to the relevant processing equipment.</p>", "<p id=\"Par25\">Figure ##FIG##3##4## presents the workflow of this method. Firstly, indirect dynamic process data from production equipment is collected, and crucial quality-related parameters are computed using mutual information. These parameters are then selected based on their importance, followed by splitting the dataset into training and testing sets using stratified sampling. Subsequently, the SMOTE algorithm obtains a balanced dataset fed into the eXtreme Gradient Boosting (XGboost) for quality classification. Furthermore, a grid search method is applied for joint optimization of the hyperparameters of SMOTE and XGboost. Ultimately, the optimal quality prediction model is derived and utilized for product quality prediction. The details of this method are described in section “<xref rid=\"Sec10\" ref-type=\"sec\">Active control method for quality prediction</xref>”.</p>", "<title>Case study</title>", "<p id=\"Par63\">This section takes the brake disc production line as an example to verify the practicality and effectiveness of the proposed method. The experiment consists of two parts: the selection of quality-related parameters and the classification results of the proposed method. Finally, the experimental results were analyzed.</p>", "<title>Experimental background</title>", "<p id=\"Par64\">The data was obtained from a brake disc production line in a certain enterprise, which is mainly used to provide high-quality brake disc products for CRH (China Railway High-speed), urban rail transit, locomotives, and world-leading railway trains. In recent years, with the demand for low energy consumption and lightweight trains, the brake disc production line has undertaken the trial production tasks of new aluminum-based silicon carbide brake discs and carbon-ceramic composite brake discs, realizing the flexible switching between mass production and trial processing to adapt to R&amp;D innovation and new market demands.</p>", "<p id=\"Par65\">The brake disc is a component of the brake system that generates braking force to hinder the movement or motion trend of the vehicle. The surface of the brake disc requires high precision and must meet the qualified performance standards. The final quality inspection of the brake disc is tested by specialized magnetic particle inspection equipment and dynamic balancing equipment to determine whether it is qualified or not. This process takes a long time and the equipment is expensive. Therefore, using data-driven methods to predict the quality of brake disc products has the potential to replace specialized equipment, which can save equipment costs and inspection time.</p>", "<p id=\"Par66\">The entire production line of brake disc machining includes production and processing equipment, inspection equipment, and each equipment is equipped with a data collection gateway, which collects data to the edge-side server for data storage and computing power. The historical data stored in the edge server is uploaded to the private cloud, and the proposed quality prediction model is trained in the private cloud center. The trained model is then deployed on the edge server, and real-time unmarked data is transmitted to the edge server via protocols such as OLE for Process Control Unified Architecture (OPC UA). The data is preprocessed on the edge server, such as removing abnormal values, and then the quality label is obtained through the quality prediction model on an industrial computer.</p>" ]

[ "<title>Author contributions</title>", "<p>Formal analysis, L.L.; data curation, M.C.; writing original draftpreparat-ion, M.C.; supervision, Z.W.; format adjustment K.Z. All authors have read and agreed to the publi-shed version of the manuscript.</p>", "<title>Funding</title>", "<p>The support of this work by the National Natural Science Foundation of China (No. 51975386), Liaoning Province “Unveiling and Commanding” technology projects (2022020630-JH1/108), and Science and Technology Research and Development Program of China National Railway Group Corporation (N2022J014) are gratefully acknowledged.</p>", "<title>Data availability</title>", "<p>The datasets generated during and/or analysed during the current study are not publicly available due to [Information related to product processing] but are available from the corresponding author on reasonable request.</p>", "<title>Competing interests</title>", "<p id=\"Par85\">The authors declare no competing interests.</p>" ]

[ "<fig id=\"Fig1\"><label>Figure 1</label><caption><p>Research directions for equipment management.</p></caption></fig>", "<fig id=\"Fig2\"><label>Figure 2</label><caption><p>Industrial internet of things architecture for industrial production line.</p></caption></fig>", "<fig id=\"Fig3\"><label>Figure 3</label><caption><p>Quality prediction method supported by edge computing.</p></caption></fig>", "<fig id=\"Fig4\"><label>Figure 4</label><caption><p>Shows the flowchart of the prediction method.</p></caption></fig>", "<fig id=\"Fig5\"><label>Figure 5</label><caption><p>Manufacturing processing unit.</p></caption></fig>", "<fig id=\"Fig6\"><label>Figure 6</label><caption><p>SMOTE oversampling algorithm principles schematic.</p></caption></fig>", "<fig id=\"Fig7\"><label>Figure 7</label><caption><p>Importance of each quality characteristic.</p></caption></fig>", "<fig id=\"Fig8\"><label>Figure 8</label><caption><p>Predicted results of some test set data.</p></caption></fig>", "<fig id=\"Fig9\"><label>Figure 9</label><caption><p>The ROC curve plot for the classification algorithms.</p></caption></fig>", "<fig id=\"Fig10\"><label>Figure 10</label><caption><p>Comparison of ROC curves for hyperparameter optimization.</p></caption></fig>" ]

[ "<table-wrap id=\"Tab1\"><label>Table 1</label><caption><p>Research of related papers.</p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th align=\"left\">Type</th><th align=\"left\">Problem-oriented</th><th align=\"left\">Author</th><th align=\"left\">Main work</th></tr></thead><tbody><tr><td align=\"left\" rowspan=\"4\">Model-based prediction methods</td><td align=\"left\" rowspan=\"4\"><p>Accurate mathematical model is available;</p><p>Mathematical model is inaccurate and involves uncertainties;</p></td><td align=\"left\">Wu et al.^{##UREF##20##21##}</td><td align=\"left\">Develop tool wear assessment and life prediction models for real-time monitoring of drill bit wear</td></tr><tr><td align=\"left\">Lin et al.^{##UREF##21##22##}</td><td align=\"left\">Proposed a novel model-based approach for monitoring and predicting rotor-bearing system imbalance</td></tr><tr><td align=\"left\">Marei et al.^{##UREF##22##23##}</td><td align=\"left\">Introduced a transfer learning mechanism to design a hybrid CNN-LSTM model, enhancing the accuracy of predictions</td></tr><tr><td align=\"left\">He et al.^{##UREF##23##24##}</td><td align=\"left\">Calculating the cumulative variation in the assembly process using finite element method</td></tr><tr><td align=\"left\" rowspan=\"5\">Data-driven prediction methods</td><td align=\"left\" rowspan=\"5\"><p>Mathematical model is inaccurate and involves uncertainties;</p><p>Mathematical model is complicated with too high order or too much nonlinearity;</p><p>Mathematical model is difficult to establish or unavailable;</p></td><td align=\"left\">Li et al.^{##UREF##24##25##}</td><td align=\"left\">Implementing multi-source fusion of measured data and model data</td></tr><tr><td align=\"left\">Liu et al.^{##UREF##25##26##}</td><td align=\"left\">Develop a unified product quality prediction framework QTD based on end-to-end time series analysis</td></tr><tr><td align=\"left\">Lee et al.^{##REF##30587841##27##}</td><td align=\"left\">Decision tree, random forest, support vector machine and other algorithms are used for quality prediction in casting processes</td></tr><tr><td align=\"left\">Dong and Fen^{##UREF##26##28##}</td><td align=\"left\">Using the XGBoost intelligent prediction model, the problem of precision prediction and control in the vehicle body assembly process has been solved</td></tr><tr><td align=\"left\">Yu and Zhao^{##UREF##27##29##}</td><td align=\"left\">An ordinal regression network model was applied to an actual industrial process</td></tr></tbody></table></table-wrap>", "<table-wrap id=\"Tab2\"><label>Table 2</label><caption><p>Classification results of equipment process data.</p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th align=\"left\">Type of equipment process data</th><th align=\"left\">Equipment process parameters</th></tr></thead><tbody><tr><td align=\"left\">Static process data</td><td align=\"left\">Operator’s technical level, name, gender, age, etc. equipment type, power-on status, tool number, life, work piece count, etc. item code, item name, process name, etc</td></tr><tr><td align=\"left\">Direct dynamic process data</td><td align=\"left\">tool coordinate location, fault, etc</td></tr><tr><td align=\"left\">Indirect dynamic process data</td><td align=\"left\">spindle power, spindle current, spindle speed, feed rate, etc</td></tr></tbody></table></table-wrap>", "<table-wrap id=\"Tab3\"><label>Table 3</label><caption><p>The detailed description of quality characteristic.</p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th align=\"left\">Quality characteristics</th><th align=\"left\">Sample1</th><th align=\"left\">Sample2</th><th align=\"left\">Sample3</th><th align=\"left\">Sample4</th><th align=\"left\">Sample5</th><th align=\"left\">Sample6</th><th align=\"left\">Sample7</th></tr></thead><tbody><tr><td align=\"left\">Including spindle power(P)/KW</td><td align=\"left\">7.34</td><td align=\"left\">7.25</td><td align=\"left\">7.15</td><td align=\"left\">7.35</td><td align=\"left\">7.28</td><td align=\"left\">7.17</td><td align=\"left\">7.15</td></tr><tr><td align=\"left\"><p>Spindle speed(S)</p><p>r/min</p></td><td align=\"left\">2041</td><td align=\"left\">2051</td><td align=\"left\">1807</td><td align=\"left\">2012</td><td align=\"left\">2037</td><td align=\"left\">2037</td><td align=\"left\">1794</td></tr><tr><td align=\"left\">Spindle current(I)/A</td><td align=\"left\">19.74</td><td align=\"left\">19.64</td><td align=\"left\">20.16</td><td align=\"left\">19.53</td><td align=\"left\">19.69</td><td align=\"left\">19.42</td><td align=\"left\">20.35</td></tr><tr><td align=\"left\"><p>Feed speed(F)</p><p>mm/min</p></td><td align=\"left\">1014</td><td align=\"left\">1014</td><td align=\"left\">1053</td><td align=\"left\">1007</td><td align=\"left\">1007</td><td align=\"left\">1026</td><td align=\"left\">1045</td></tr><tr><td align=\"left\">Clamping force(N) (kN)</td><td align=\"left\">21.43</td><td align=\"left\">21.43</td><td align=\"left\">21.37</td><td align=\"left\">21.39</td><td align=\"left\">21.48</td><td align=\"left\">22.53</td><td align=\"left\">22.13</td></tr><tr><td align=\"left\">Final quality</td><td align=\"left\">Qualified</td><td align=\"left\">Qualified</td><td align=\"left\">Fault</td><td align=\"left\">Qualified</td><td align=\"left\">Qualified</td><td align=\"left\">Qualified</td><td align=\"left\">Fault</td></tr></tbody></table></table-wrap>", "<table-wrap id=\"Tab4\"><label>Table 4</label><caption><p>Description of each dataset.</p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th align=\"left\">Data set</th><th align=\"left\">Qualified product</th><th align=\"left\">Fault product</th><th align=\"left\">Total</th></tr></thead><tbody><tr><td align=\"left\">Training set</td><td align=\"left\">1240</td><td align=\"left\">46</td><td align=\"left\">1286</td></tr><tr><td align=\"left\">Testing set</td><td align=\"left\">538</td><td align=\"left\">20</td><td align=\"left\">558</td></tr><tr><td align=\"left\">Total</td><td align=\"left\">1778</td><td align=\"left\">66</td><td align=\"left\">1844</td></tr></tbody></table></table-wrap>", "<table-wrap id=\"Tab5\"><label>Table 5</label><caption><p>Description of each dataset after SMOTE.</p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th align=\"left\">Data set</th><th align=\"left\">Qualified product</th><th align=\"left\">Fault product</th><th align=\"left\">Total</th></tr></thead><tbody><tr><td align=\"left\">Training set</td><td align=\"left\">1240</td><td align=\"left\">1240</td><td align=\"left\">2480</td></tr><tr><td align=\"left\">Testing set</td><td align=\"left\">538</td><td align=\"left\">20</td><td align=\"left\">558</td></tr><tr><td align=\"left\">Total</td><td align=\"left\">1778</td><td align=\"left\">1260</td><td align=\"left\">3038</td></tr></tbody></table></table-wrap>", "<table-wrap id=\"Tab6\"><label>Table 6</label><caption><p>Comparison experiment of different classification methods.</p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th align=\"left\">Training model</th><th align=\"left\"><italic>R</italic>²</th><th align=\"left\">AUC</th></tr></thead><tbody><tr><td align=\"left\">SMOTE-XGboost_t</td><td char=\".\" align=\"char\">0.897</td><td char=\".\" align=\"char\">0.916</td></tr><tr><td align=\"left\">SMOTE-LR</td><td char=\".\" align=\"char\">0.777</td><td char=\".\" align=\"char\">0.788</td></tr><tr><td align=\"left\">SMOTE-DT</td><td char=\".\" align=\"char\">0.885</td><td char=\".\" align=\"char\">0.808</td></tr><tr><td align=\"left\">SMOTE-RF</td><td char=\".\" align=\"char\">0.891</td><td char=\".\" align=\"char\">0.833</td></tr><tr><td align=\"left\">SMOTE-SVM</td><td char=\".\" align=\"char\">0.882</td><td char=\".\" align=\"char\">0.891</td></tr></tbody></table></table-wrap>", "<table-wrap id=\"Tab7\"><label>Table 7</label><caption><p>The hyperparameters and optimization interval of models.</p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th align=\"left\">Model</th><th align=\"left\">k</th><th align=\"left\">e</th><th align=\"left\">T</th><th align=\"left\">Optimization interval</th></tr></thead><tbody><tr><td align=\"left\">SMOTE-XGboost_t</td><td align=\"left\">6</td><td align=\"left\">100</td><td align=\"left\">3</td><td align=\"left\">k ∈ [3, 10] e ∈ [40,110] T ∈ [2,5]</td></tr><tr><td align=\"left\">SMOTE-XGboost_s</td><td align=\"left\">7</td><td align=\"left\">90</td><td align=\"left\">2</td><td align=\"left\">k ∈ [3,10], e = 90,T = 2</td></tr><tr><td align=\"left\">SMOTE-XGboost_x</td><td align=\"left\">5</td><td align=\"left\">80</td><td align=\"left\">4</td><td align=\"left\">k ∈ 5, e ∈ [40,110],T ∈ [2,5] </td></tr><tr><td align=\"left\">SMOTE-XGboost</td><td align=\"left\">5</td><td align=\"left\">90</td><td align=\"left\">2</td><td align=\"left\">k = 5, e = 90 T = 2</td></tr></tbody></table></table-wrap>", "<table-wrap id=\"Tab8\"><label>Table 8</label><caption><p>Influence of hyperparameters optimization in models.</p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th align=\"left\">Model</th><th align=\"left\">R²</th><th align=\"left\">AUC</th></tr></thead><tbody><tr><td align=\"left\">SMOTE-XGboost_t</td><td align=\"left\">0.897</td><td align=\"left\">0.916</td></tr><tr><td align=\"left\">SMOTE-XGboost_s</td><td align=\"left\">0.863</td><td align=\"left\">0.881</td></tr><tr><td align=\"left\">SMOTE-XGboost_x</td><td align=\"left\">0.881</td><td align=\"left\">0.901</td></tr><tr><td align=\"left\">SMOTE-XGboost</td><td align=\"left\">0.847</td><td align=\"left\">0.868</td></tr></tbody></table></table-wrap>" ]

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id=\"M27\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${X}_{i-1}$$\\end{document}</tex-math><mml:math id=\"M28\"><mml:msub><mml:mi>X</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq15\"><alternatives><tex-math id=\"M29\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$i$$\\end{document}</tex-math><mml:math id=\"M30\"><mml:mi>i</mml:mi></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ1\"><label>1</label><alternatives><tex-math id=\"M31\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$M\\_Data=\\left({m}_{1},{m}_{2}\\cdots {m}_{i}\\right)$$\\end{document}</tex-math><mml:math id=\"M32\" display=\"block\"><mml:mrow><mml:mi>M</mml:mi><mml:mi>_</mml:mi><mml:mi>D</mml:mi><mml:mi>a</mml:mi><mml:mi>t</mml:mi><mml:mi>a</mml:mi><mml:mo>=</mml:mo><mml:mfenced close=\")\" open=\"(\"><mml:msub><mml:mi>m</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>m</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mo>⋯</mml:mo><mml:msub><mml:mi>m</mml:mi><mml:mi>i</mml:mi></mml:msub></mml:mfenced></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq16\"><alternatives><tex-math id=\"M33\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$M\\_data$$\\end{document}</tex-math><mml:math id=\"M34\"><mml:mrow><mml:mi>M</mml:mi><mml:mi>_</mml:mi><mml:mi>d</mml:mi><mml:mi>a</mml:mi><mml:mi>t</mml:mi><mml:mi>a</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq17\"><alternatives><tex-math id=\"M35\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${m}_{i}$$\\end{document}</tex-math><mml:math id=\"M36\"><mml:msub><mml:mi>m</mml:mi><mml:mi>i</mml:mi></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq18\"><alternatives><tex-math id=\"M37\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$k$$\\end{document}</tex-math><mml:math id=\"M38\"><mml:mi>k</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq19\"><alternatives><tex-math id=\"M39\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$M\\_data$$\\end{document}</tex-math><mml:math id=\"M40\"><mml:mrow><mml:mi>M</mml:mi><mml:mi>_</mml:mi><mml:mi>d</mml:mi><mml:mi>a</mml:mi><mml:mi>t</mml:mi><mml:mi>a</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ2\"><label>2</label><alternatives><tex-math id=\"M41\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{array}{c}M\\_Data=\\left|\\begin{array}{cccc}{m}_{11}&amp; {m}_{21}&amp; \\cdots &amp; {m}_{b1}\\\\ {m}_{12}&amp; {m}_{22}&amp; \\cdots &amp; {m}_{b2}\\\\ {m}_{13}&amp; {m}_{23}&amp; \\cdots &amp; {m}_{b3}\\\\ \\vdots &amp; \\vdots &amp; &amp; \\vdots \\\\ {m}_{1a}&amp; {m}_{2a}&amp; {m}_{1a}&amp; {m}_{ba}\\end{array}\\right|\\end{array}$$\\end{document}</tex-math><mml:math id=\"M42\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mrow><mml:mi>M</mml:mi><mml:mi>_</mml:mi><mml:mi>D</mml:mi><mml:mi>a</mml:mi><mml:mi>t</mml:mi><mml:mi>a</mml:mi><mml:mo>=</mml:mo><mml:mfenced close=\"|\" open=\"|\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:msub><mml:mi>m</mml:mi><mml:mn>11</mml:mn></mml:msub></mml:mtd><mml:mtd><mml:msub><mml:mi>m</mml:mi><mml:mn>21</mml:mn></mml:msub></mml:mtd><mml:mtd><mml:mo>⋯</mml:mo></mml:mtd><mml:mtd><mml:msub><mml:mi>m</mml:mi><mml:mrow><mml:mi>b</mml:mi><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mrow/><mml:msub><mml:mi>m</mml:mi><mml:mn>12</mml:mn></mml:msub></mml:mrow></mml:mtd><mml:mtd><mml:msub><mml:mi>m</mml:mi><mml:mn>22</mml:mn></mml:msub></mml:mtd><mml:mtd><mml:mo>⋯</mml:mo></mml:mtd><mml:mtd><mml:msub><mml:mi>m</mml:mi><mml:mrow><mml:mi>b</mml:mi><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mrow/><mml:msub><mml:mi>m</mml:mi><mml:mn>13</mml:mn></mml:msub></mml:mrow></mml:mtd><mml:mtd><mml:msub><mml:mi>m</mml:mi><mml:mn>23</mml:mn></mml:msub></mml:mtd><mml:mtd><mml:mo>⋯</mml:mo></mml:mtd><mml:mtd><mml:msub><mml:mi>m</mml:mi><mml:mrow><mml:mi>b</mml:mi><mml:mn>3</mml:mn></mml:mrow></mml:msub></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mrow/><mml:mo>⋮</mml:mo></mml:mrow></mml:mtd><mml:mtd><mml:mo>⋮</mml:mo></mml:mtd><mml:mtd/><mml:mtd><mml:mo>⋮</mml:mo></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mrow/><mml:msub><mml:mi>m</mml:mi><mml:mrow><mml:mn>1</mml:mn><mml:mi>a</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:mtd><mml:mtd><mml:msub><mml:mi>m</mml:mi><mml:mrow><mml:mn>2</mml:mn><mml:mi>a</mml:mi></mml:mrow></mml:msub></mml:mtd><mml:mtd><mml:msub><mml:mi>m</mml:mi><mml:mrow><mml:mn>1</mml:mn><mml:mi>a</mml:mi></mml:mrow></mml:msub></mml:mtd><mml:mtd><mml:msub><mml:mi>m</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">ba</mml:mi></mml:mrow></mml:msub></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq20\"><alternatives><tex-math id=\"M43\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$I\\left({X}_{i}\\right)$$\\end{document}</tex-math><mml:math id=\"M44\"><mml:mrow><mml:mi>I</mml:mi><mml:mfenced close=\")\" open=\"(\"><mml:msub><mml:mi>X</mml:mi><mml:mi>i</mml:mi></mml:msub></mml:mfenced></mml:mrow></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ3\"><label>3</label><alternatives><tex-math id=\"M45\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$I\\left({X}_{i}\\right)={\\omega }_{0}{I}_{0}\\left({X}_{i},{Y}_{0}\\right)+{\\omega }_{1}{I}_{1}({X}_{i},{Y}_{1})$$\\end{document}</tex-math><mml:math id=\"M46\" display=\"block\"><mml:mrow><mml:mi>I</mml:mi><mml:mfenced close=\")\" open=\"(\"><mml:msub><mml:mi>X</mml:mi><mml:mi>i</mml:mi></mml:msub></mml:mfenced><mml:mo>=</mml:mo><mml:msub><mml:mi>ω</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:msub><mml:mi>I</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mfenced close=\")\" open=\"(\"><mml:msub><mml:mi>X</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>Y</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:mfenced><mml:mo>+</mml:mo><mml:msub><mml:mi>ω</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:msub><mml:mi>I</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>X</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>Y</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ4\"><label>4</label><alternatives><tex-math id=\"M47\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${I}_{0}\\left({X}_{i},{Y}_{0}\\right)=\\sum_{x\\in {X}_{i}}p\\left(x,{Y}_{0}\\right)log\\frac{p(x,{Y}_{0})}{p\\left(x\\right)p({Y}_{0})}$$\\end{document}</tex-math><mml:math id=\"M48\" display=\"block\"><mml:mrow><mml:msub><mml:mi>I</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mfenced close=\")\" open=\"(\"><mml:msub><mml:mi>X</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>Y</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:mfenced><mml:mo>=</mml:mo><mml:munder><mml:mo>∑</mml:mo><mml:mrow><mml:mi>x</mml:mi><mml:mo>∈</mml:mo><mml:msub><mml:mi>X</mml:mi><mml:mi>i</mml:mi></mml:msub></mml:mrow></mml:munder><mml:mi>p</mml:mi><mml:mfenced close=\")\" open=\"(\"><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:msub><mml:mi>Y</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:mfenced><mml:mi>l</mml:mi><mml:mi>o</mml:mi><mml:mi>g</mml:mi><mml:mfrac><mml:mrow><mml:mi>p</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:msub><mml:mi>Y</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mrow><mml:mi>p</mml:mi><mml:mfenced close=\")\" open=\"(\"><mml:mi>x</mml:mi></mml:mfenced><mml:mi>p</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>Y</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:mfrac></mml:mrow></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ5\"><label>5</label><alternatives><tex-math id=\"M49\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${I}_{1}\\left({X}_{i},{Y}_{1}\\right)=\\sum_{x\\in {X}_{i}}p\\left(x,{Y}_{1}\\right)log\\frac{p(x,{Y}_{1})}{p\\left(x\\right)p({Y}_{1})}$$\\end{document}</tex-math><mml:math id=\"M50\" display=\"block\"><mml:mrow><mml:msub><mml:mi>I</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mfenced close=\")\" open=\"(\"><mml:msub><mml:mi>X</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>Y</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:mfenced><mml:mo>=</mml:mo><mml:munder><mml:mo>∑</mml:mo><mml:mrow><mml:mi>x</mml:mi><mml:mo>∈</mml:mo><mml:msub><mml:mi>X</mml:mi><mml:mi>i</mml:mi></mml:msub></mml:mrow></mml:munder><mml:mi>p</mml:mi><mml:mfenced close=\")\" open=\"(\"><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:msub><mml:mi>Y</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:mfenced><mml:mi>l</mml:mi><mml:mi>o</mml:mi><mml:mi>g</mml:mi><mml:mfrac><mml:mrow><mml:mi>p</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:msub><mml:mi>Y</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mrow><mml:mi>p</mml:mi><mml:mfenced close=\")\" open=\"(\"><mml:mi>x</mml:mi></mml:mfenced><mml:mi>p</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>Y</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:mfrac></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq21\"><alternatives><tex-math id=\"M51\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${X}_{i}$$\\end{document}</tex-math><mml:math id=\"M52\"><mml:msub><mml:mi>X</mml:mi><mml:mi>i</mml:mi></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq22\"><alternatives><tex-math id=\"M53\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$i$$\\end{document}</tex-math><mml:math id=\"M54\"><mml:mi>i</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq23\"><alternatives><tex-math id=\"M55\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${Y}_{0}$$\\end{document}</tex-math><mml:math id=\"M56\"><mml:msub><mml:mi>Y</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq24\"><alternatives><tex-math id=\"M57\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${{\\text{Y}}}_{1}$$\\end{document}</tex-math><mml:math id=\"M58\"><mml:msub><mml:mtext>Y</mml:mtext><mml:mn>1</mml:mn></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq25\"><alternatives><tex-math id=\"M59\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$p(x,{Y}_{0})$$\\end{document}</tex-math><mml:math id=\"M60\"><mml:mrow><mml:mi>p</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:msub><mml:mi>Y</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq26\"><alternatives><tex-math id=\"M61\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${X}_{i}$$\\end{document}</tex-math><mml:math id=\"M62\"><mml:msub><mml:mi>X</mml:mi><mml:mi>i</mml:mi></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq27\"><alternatives><tex-math id=\"M63\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${Y}_{0}.$$\\end{document}</tex-math><mml:math id=\"M64\"><mml:mrow><mml:msub><mml:mi>Y</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo>.</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq28\"><alternatives><tex-math id=\"M65\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$p(x,{Y}_{1})$$\\end{document}</tex-math><mml:math id=\"M66\"><mml:mrow><mml:mi>p</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:msub><mml:mi>Y</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq29\"><alternatives><tex-math id=\"M67\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${X}_{i}$$\\end{document}</tex-math><mml:math id=\"M68\"><mml:msub><mml:mi>X</mml:mi><mml:mi>i</mml:mi></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq30\"><alternatives><tex-math id=\"M69\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${Y}_{1}$$\\end{document}</tex-math><mml:math id=\"M70\"><mml:msub><mml:mi>Y</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq31\"><alternatives><tex-math id=\"M71\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$p\\left(x\\right),p\\left({Y}_{0}\\right)$$\\end{document}</tex-math><mml:math id=\"M72\"><mml:mrow><mml:mi>p</mml:mi><mml:mfenced close=\")\" open=\"(\"><mml:mi>x</mml:mi></mml:mfenced><mml:mo>,</mml:mo><mml:mi>p</mml:mi><mml:mfenced close=\")\" open=\"(\"><mml:msub><mml:mi>Y</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:mfenced></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq32\"><alternatives><tex-math id=\"M73\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$p\\left({Y}_{1}\\right)$$\\end{document}</tex-math><mml:math id=\"M74\"><mml:mrow><mml:mi>p</mml:mi><mml:mfenced close=\")\" open=\"(\"><mml:msub><mml:mi>Y</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:mfenced></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq33\"><alternatives><tex-math id=\"M75\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${X}_{i}$$\\end{document}</tex-math><mml:math id=\"M76\"><mml:msub><mml:mi>X</mml:mi><mml:mi>i</mml:mi></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq34\"><alternatives><tex-math id=\"M77\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${Y}_{0}$$\\end{document}</tex-math><mml:math id=\"M78\"><mml:msub><mml:mi>Y</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq35\"><alternatives><tex-math id=\"M79\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${Y}_{1}$$\\end{document}</tex-math><mml:math id=\"M80\"><mml:msub><mml:mi>Y</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq36\"><alternatives><tex-math id=\"M81\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\omega }_{0}$$\\end{document}</tex-math><mml:math id=\"M82\"><mml:msub><mml:mi>ω</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq37\"><alternatives><tex-math id=\"M83\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\omega }_{1}$$\\end{document}</tex-math><mml:math id=\"M84\"><mml:msub><mml:mi>ω</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq38\"><alternatives><tex-math id=\"M85\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$I\\left({X}_{i}\\right)$$\\end{document}</tex-math><mml:math id=\"M86\"><mml:mrow><mml:mi>I</mml:mi><mml:mfenced close=\")\" open=\"(\"><mml:msub><mml:mi>X</mml:mi><mml:mi>i</mml:mi></mml:msub></mml:mfenced></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq39\"><alternatives><tex-math id=\"M87\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$C$$\\end{document}</tex-math><mml:math id=\"M88\"><mml:mi>C</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq40\"><alternatives><tex-math id=\"M89\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${m}_{n}$$\\end{document}</tex-math><mml:math id=\"M90\"><mml:msub><mml:mi>m</mml:mi><mml:mi>n</mml:mi></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq41\"><alternatives><tex-math id=\"M91\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$n$$\\end{document}</tex-math><mml:math id=\"M92\"><mml:mi>n</mml:mi></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ6\"><label>6</label><alternatives><tex-math id=\"M93\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$C=\\left\\{{m}_{a}{m}_{b}\\cdots \\right.\\left.{m}_{n}\\right\\}$$\\end{document}</tex-math><mml:math id=\"M94\" display=\"block\"><mml:mrow><mml:mi>C</mml:mi><mml:mo>=</mml:mo><mml:mfenced open=\"{\"><mml:msub><mml:mi>m</mml:mi><mml:mi>a</mml:mi></mml:msub><mml:msub><mml:mi>m</mml:mi><mml:mi>b</mml:mi></mml:msub><mml:mo>⋯</mml:mo></mml:mfenced><mml:mfenced close=\"}\"><mml:msub><mml:mi>m</mml:mi><mml:mi>n</mml:mi></mml:msub></mml:mfenced></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq42\"><alternatives><tex-math id=\"M95\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${N}_{min}$$\\end{document}</tex-math><mml:math id=\"M96\"><mml:msub><mml:mi>N</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">min</mml:mi></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq43\"><alternatives><tex-math id=\"M97\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${N}_{max}$$\\end{document}</tex-math><mml:math id=\"M98\"><mml:msub><mml:mi>N</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">max</mml:mi></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq44\"><alternatives><tex-math id=\"M99\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$N$$\\end{document}</tex-math><mml:math id=\"M100\"><mml:mi>N</mml:mi></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ7\"><label>7</label><alternatives><tex-math id=\"M101\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$N=\\frac{{N}_{max}}{{N}_{min}}-1$$\\end{document}</tex-math><mml:math id=\"M102\" display=\"block\"><mml:mrow><mml:mi>N</mml:mi><mml:mo>=</mml:mo><mml:mfrac><mml:msub><mml:mi>N</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">max</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mi>N</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">min</mml:mi></mml:mrow></mml:msub></mml:mfrac><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq45\"><alternatives><tex-math id=\"M103\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${X}_{j}$$\\end{document}</tex-math><mml:math id=\"M104\"><mml:msub><mml:mi>X</mml:mi><mml:mi>j</mml:mi></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq46\"><alternatives><tex-math id=\"M105\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${X}_{j}{\\in N}_{min}$$\\end{document}</tex-math><mml:math id=\"M106\"><mml:mrow><mml:msub><mml:mi>X</mml:mi><mml:mi>j</mml:mi></mml:msub><mml:msub><mml:mrow><mml:mo>∈</mml:mo><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"italic\">min</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq47\"><alternatives><tex-math id=\"M107\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$k$$\\end{document}</tex-math><mml:math id=\"M108\"><mml:mi>k</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq48\"><alternatives><tex-math id=\"M109\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$k$$\\end{document}</tex-math><mml:math id=\"M110\"><mml:mi>k</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq49\"><alternatives><tex-math id=\"M111\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${X}_{j}$$\\end{document}</tex-math><mml:math id=\"M112\"><mml:msub><mml:mi>X</mml:mi><mml:mi>j</mml:mi></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq50\"><alternatives><tex-math id=\"M113\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${X}_{K}$$\\end{document}</tex-math><mml:math id=\"M114\"><mml:msub><mml:mi>X</mml:mi><mml:mi>K</mml:mi></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq51\"><alternatives><tex-math id=\"M115\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${X}_{new}$$\\end{document}</tex-math><mml:math id=\"M116\"><mml:msub><mml:mi>X</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">new</mml:mi></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ8\"><label>8</label><alternatives><tex-math id=\"M117\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${X}_{new}={X}_{k}+rand\\left(\\mathrm{0,1}\\right)\\times \\left({X}_{j}-{X}_{K}\\right)$$\\end{document}</tex-math><mml:math id=\"M118\" display=\"block\"><mml:mrow><mml:msub><mml:mi>X</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">new</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:msub><mml:mi>X</mml:mi><mml:mi>k</mml:mi></mml:msub><mml:mo>+</mml:mo><mml:mi>r</mml:mi><mml:mi>a</mml:mi><mml:mi>n</mml:mi><mml:mi>d</mml:mi><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:mfenced><mml:mo>×</mml:mo><mml:mfenced close=\")\" open=\"(\"><mml:msub><mml:mi>X</mml:mi><mml:mi>j</mml:mi></mml:msub><mml:mo>-</mml:mo><mml:msub><mml:mi>X</mml:mi><mml:mi>K</mml:mi></mml:msub></mml:mfenced></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq52\"><alternatives><tex-math id=\"M119\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$rand\\left(\\mathrm{0,1}\\right)$$\\end{document}</tex-math><mml:math id=\"M120\"><mml:mrow><mml:mi>r</mml:mi><mml:mi>a</mml:mi><mml:mi>n</mml:mi><mml:mi>d</mml:mi><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:mfenced></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq53\"><alternatives><tex-math id=\"M121\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$N{*N}_{min}$$\\end{document}</tex-math><mml:math id=\"M122\"><mml:mrow><mml:mi>N</mml:mi><mml:msub><mml:mrow><mml:mrow/><mml:mo>∗</mml:mo><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"italic\">min</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq54\"><alternatives><tex-math id=\"M123\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\left\\{\\left({x}_{1}{y}_{1}\\right),\\left({x}_{2}{y}_{2}\\right),\\cdots ,\\right.\\left.\\left({x}_{n}{y}_{n}\\right)\\right\\}$$\\end{document}</tex-math><mml:math id=\"M124\"><mml:mrow><mml:mfenced open=\"{\"><mml:mfenced close=\")\" open=\"(\"><mml:msub><mml:mi>x</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:msub><mml:mi>y</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:mfenced><mml:mo>,</mml:mo><mml:mfenced close=\")\" open=\"(\"><mml:msub><mml:mi>x</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:msub><mml:mi>y</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:mfenced><mml:mo>,</mml:mo><mml:mo>⋯</mml:mo><mml:mo>,</mml:mo></mml:mfenced><mml:mfenced close=\"}\"><mml:mfenced close=\")\" open=\"(\"><mml:msub><mml:mi>x</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:msub><mml:mi>y</mml:mi><mml:mi>n</mml:mi></mml:msub></mml:mfenced></mml:mfenced></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq55\"><alternatives><tex-math id=\"M125\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$K$$\\end{document}</tex-math><mml:math id=\"M126\"><mml:mi>K</mml:mi></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ9\"><label>9</label><alternatives><tex-math id=\"M127\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\widehat{y}}_{i}=\\sum_{k=1}^{K}{f}_{k}\\left({x}_{i}\\right)$$\\end{document}</tex-math><mml:math id=\"M128\" display=\"block\"><mml:mrow><mml:msub><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"true\">^</mml:mo></mml:mover><mml:mi>i</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:munderover><mml:mo>∑</mml:mo><mml:mrow><mml:mi>k</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mi>K</mml:mi></mml:munderover><mml:msub><mml:mi>f</mml:mi><mml:mi>k</mml:mi></mml:msub><mml:mfenced close=\")\" open=\"(\"><mml:msub><mml:mi>x</mml:mi><mml:mi>i</mml:mi></mml:msub></mml:mfenced></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq56\"><alternatives><tex-math id=\"M129\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${f}_{k}\\left({x}_{i}\\right)$$\\end{document}</tex-math><mml:math id=\"M130\"><mml:mrow><mml:msub><mml:mi>f</mml:mi><mml:mi>k</mml:mi></mml:msub><mml:mfenced close=\")\" open=\"(\"><mml:msub><mml:mi>x</mml:mi><mml:mi>i</mml:mi></mml:msub></mml:mfenced></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq57\"><alternatives><tex-math id=\"M131\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$k$$\\end{document}</tex-math><mml:math id=\"M132\"><mml:mi>k</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq58\"><alternatives><tex-math id=\"M133\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$L$$\\end{document}</tex-math><mml:math id=\"M134\"><mml:mi>L</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq59\"><alternatives><tex-math id=\"M135\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\Omega $$\\end{document}</tex-math><mml:math id=\"M136\"><mml:mi mathvariant=\"normal\">Ω</mml:mi></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ10\"><label>10</label><alternatives><tex-math id=\"M137\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$L=\\sum_{i}l\\left({y}_{i}{\\widehat{,y}}_{i}\\right)+\\sum_{k}\\Omega \\left({f}_{k}\\right)$$\\end{document}</tex-math><mml:math id=\"M138\" display=\"block\"><mml:mrow><mml:mi>L</mml:mi><mml:mo>=</mml:mo><mml:munder><mml:mo>∑</mml:mo><mml:mi>i</mml:mi></mml:munder><mml:mi>l</mml:mi><mml:mfenced close=\")\" open=\"(\"><mml:msub><mml:mi>y</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:msub><mml:mover accent=\"true\"><mml:mrow><mml:mo>,</mml:mo><mml:mi>y</mml:mi></mml:mrow><mml:mo stretchy=\"true\">^</mml:mo></mml:mover><mml:mi>i</mml:mi></mml:msub></mml:mfenced><mml:mo>+</mml:mo><mml:munder><mml:mo>∑</mml:mo><mml:mi>k</mml:mi></mml:munder><mml:mi mathvariant=\"normal\">Ω</mml:mi><mml:mfenced close=\")\" open=\"(\"><mml:msub><mml:mi>f</mml:mi><mml:mi>k</mml:mi></mml:msub></mml:mfenced></mml:mrow></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ11\"><label>11</label><alternatives><tex-math id=\"M139\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\Omega \\left({f}_{k}\\right)=\\gamma T+\\frac{1}{2}\\lambda {\\parallel \\omega \\parallel }^{2}$$\\end{document}</tex-math><mml:math id=\"M140\" display=\"block\"><mml:mrow><mml:mi mathvariant=\"normal\">Ω</mml:mi><mml:mfenced close=\")\" open=\"(\"><mml:msub><mml:mi>f</mml:mi><mml:mi>k</mml:mi></mml:msub></mml:mfenced><mml:mo>=</mml:mo><mml:mi>γ</mml:mi><mml:mi>T</mml:mi><mml:mo>+</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:mn>2</mml:mn></mml:mfrac><mml:mi>λ</mml:mi><mml:msup><mml:mrow><mml:mo stretchy=\"false\">‖</mml:mo><mml:mi>ω</mml:mi><mml:mo stretchy=\"false\">‖</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:msup></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq60\"><alternatives><tex-math id=\"M141\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$L$$\\end{document}</tex-math><mml:math id=\"M142\"><mml:mi>L</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq61\"><alternatives><tex-math id=\"M143\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\Omega \\left({f}_{k}\\right)$$\\end{document}</tex-math><mml:math id=\"M144\"><mml:mrow><mml:mi mathvariant=\"normal\">Ω</mml:mi><mml:mfenced close=\")\" open=\"(\"><mml:msub><mml:mi>f</mml:mi><mml:mi>k</mml:mi></mml:msub></mml:mfenced></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq62\"><alternatives><tex-math id=\"M145\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$T$$\\end{document}</tex-math><mml:math id=\"M146\"><mml:mi>T</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq63\"><alternatives><tex-math id=\"M147\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\omega $$\\end{document}</tex-math><mml:math id=\"M148\"><mml:mi>ω</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq64\"><alternatives><tex-math id=\"M149\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$t$$\\end{document}</tex-math><mml:math id=\"M150\"><mml:mi>t</mml:mi></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ12\"><label>12</label><alternatives><tex-math id=\"M151\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\widehat{y}}_{i}^{\\left(t\\right)}={\\widehat{y}}_{i}^{\\left(t-1\\right)}+{f}_{t}\\left({x}_{i}\\right)$$\\end{document}</tex-math><mml:math id=\"M152\" display=\"block\"><mml:mrow><mml:msubsup><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"true\">^</mml:mo></mml:mover><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mfenced close=\")\" open=\"(\"><mml:mi>t</mml:mi></mml:mfenced></mml:msubsup><mml:mo>=</mml:mo><mml:msubsup><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"true\">^</mml:mo></mml:mover><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mfenced close=\")\" open=\"(\"><mml:mi>t</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mfenced></mml:msubsup><mml:mo>+</mml:mo><mml:msub><mml:mi>f</mml:mi><mml:mi>t</mml:mi></mml:msub><mml:mfenced close=\")\" open=\"(\"><mml:msub><mml:mi>x</mml:mi><mml:mi>i</mml:mi></mml:msub></mml:mfenced></mml:mrow></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ13\"><label>13</label><alternatives><tex-math id=\"M153\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${L}^{\\left(t\\right)}=\\sum_{i}^{n}l\\left({y}_{i ,}{\\widehat{y}}_{i}^{\\left(t-1\\right)}+{f}_{t}\\left({x}_{i}\\right)\\right)+\\Omega \\left({f}_{t}\\right)$$\\end{document}</tex-math><mml:math id=\"M154\" display=\"block\"><mml:mrow><mml:msup><mml:mrow><mml:mi>L</mml:mi></mml:mrow><mml:mfenced close=\")\" open=\"(\"><mml:mi>t</mml:mi></mml:mfenced></mml:msup><mml:mo>=</mml:mo><mml:munderover><mml:mo>∑</mml:mo><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mi>n</mml:mi></mml:munderover><mml:mi>l</mml:mi><mml:mfenced close=\")\" open=\"(\"><mml:msub><mml:mi>y</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mo>,</mml:mo></mml:mrow></mml:msub><mml:msubsup><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"true\">^</mml:mo></mml:mover><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mfenced close=\")\" open=\"(\"><mml:mi>t</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mfenced></mml:msubsup><mml:mo>+</mml:mo><mml:msub><mml:mi>f</mml:mi><mml:mi>t</mml:mi></mml:msub><mml:mfenced close=\")\" open=\"(\"><mml:msub><mml:mi>x</mml:mi><mml:mi>i</mml:mi></mml:msub></mml:mfenced></mml:mfenced><mml:mo>+</mml:mo><mml:mi mathvariant=\"normal\">Ω</mml:mi><mml:mfenced close=\")\" open=\"(\"><mml:msub><mml:mi>f</mml:mi><mml:mi>t</mml:mi></mml:msub></mml:mfenced></mml:mrow></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ14\"><label>14</label><alternatives><tex-math id=\"M155\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${L}^{\\left(t\\right)}=\\sum_{i}^{n}l\\left[{{\\text{g}}}_{i}{f}_{t}\\left({x}_{i}\\right)+\\frac{1}{2}{h}_{i}{f}_{t}^{2}\\left({x}_{i}\\right)\\right]\\left({y}_{i ,}{\\widehat{y}}_{i}^{\\left(t-1\\right)}+{f}_{t}\\left({x}_{i}\\right)\\right)+\\Omega \\left({f}_{t}\\right)$$\\end{document}</tex-math><mml:math id=\"M156\" display=\"block\"><mml:mrow><mml:msup><mml:mrow><mml:mi>L</mml:mi></mml:mrow><mml:mfenced close=\")\" open=\"(\"><mml:mi>t</mml:mi></mml:mfenced></mml:msup><mml:mo>=</mml:mo><mml:munderover><mml:mo>∑</mml:mo><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mi>n</mml:mi></mml:munderover><mml:mi>l</mml:mi><mml:mfenced close=\"]\" open=\"[\"><mml:msub><mml:mtext>g</mml:mtext><mml:mi>i</mml:mi></mml:msub><mml:msub><mml:mi>f</mml:mi><mml:mi>t</mml:mi></mml:msub><mml:mfenced close=\")\" open=\"(\"><mml:msub><mml:mi>x</mml:mi><mml:mi>i</mml:mi></mml:msub></mml:mfenced><mml:mo>+</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:mn>2</mml:mn></mml:mfrac><mml:msub><mml:mi>h</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:msubsup><mml:mi>f</mml:mi><mml:mrow><mml:mi>t</mml:mi></mml:mrow><mml:mn>2</mml:mn></mml:msubsup><mml:mfenced close=\")\" open=\"(\"><mml:msub><mml:mi>x</mml:mi><mml:mi>i</mml:mi></mml:msub></mml:mfenced></mml:mfenced><mml:mfenced close=\")\" open=\"(\"><mml:msub><mml:mi>y</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mo>,</mml:mo></mml:mrow></mml:msub><mml:msubsup><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"true\">^</mml:mo></mml:mover><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mfenced close=\")\" open=\"(\"><mml:mi>t</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mfenced></mml:msubsup><mml:mo>+</mml:mo><mml:msub><mml:mi>f</mml:mi><mml:mi>t</mml:mi></mml:msub><mml:mfenced close=\")\" open=\"(\"><mml:msub><mml:mi>x</mml:mi><mml:mi>i</mml:mi></mml:msub></mml:mfenced></mml:mfenced><mml:mo>+</mml:mo><mml:mi mathvariant=\"normal\">Ω</mml:mi><mml:mfenced close=\")\" open=\"(\"><mml:msub><mml:mi>f</mml:mi><mml:mi>t</mml:mi></mml:msub></mml:mfenced></mml:mrow></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ15\"><label>15</label><alternatives><tex-math id=\"M157\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${{\\text{g}}}_{i}={\\partial }_{{\\widehat{y}}^{\\left(t-1\\right)}}l\\left({y}_{i ,}{\\widehat{y}}_{i}^{\\left(t-1\\right)}\\right)$$\\end{document}</tex-math><mml:math id=\"M158\" display=\"block\"><mml:mrow><mml:msub><mml:mtext>g</mml:mtext><mml:mi>i</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:msub><mml:mi>∂</mml:mi><mml:msup><mml:mrow><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"true\">^</mml:mo></mml:mover></mml:mrow><mml:mfenced close=\")\" open=\"(\"><mml:mi>t</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mfenced></mml:msup></mml:msub><mml:mi>l</mml:mi><mml:mfenced close=\")\" open=\"(\"><mml:msub><mml:mi>y</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mo>,</mml:mo></mml:mrow></mml:msub><mml:msubsup><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"true\">^</mml:mo></mml:mover><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mfenced close=\")\" open=\"(\"><mml:mi>t</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mfenced></mml:msubsup></mml:mfenced></mml:mrow></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ16\"><label>16</label><alternatives><tex-math id=\"M159\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${h}_{i}={\\partial }_{{\\widehat{y}}^{\\left(t-1\\right)}}^{2}l\\left({y}_{i ,}{\\widehat{y}}_{i}^{\\left(t-1\\right)}\\right)$$\\end{document}</tex-math><mml:math id=\"M160\" display=\"block\"><mml:mrow><mml:msub><mml:mi>h</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:msubsup><mml:mi>∂</mml:mi><mml:mrow><mml:msup><mml:mrow><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"true\">^</mml:mo></mml:mover></mml:mrow><mml:mfenced close=\")\" open=\"(\"><mml:mi>t</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mfenced></mml:msup></mml:mrow><mml:mn>2</mml:mn></mml:msubsup><mml:mi>l</mml:mi><mml:mfenced close=\")\" open=\"(\"><mml:msub><mml:mi>y</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mo>,</mml:mo></mml:mrow></mml:msub><mml:msubsup><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"true\">^</mml:mo></mml:mover><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mfenced close=\")\" open=\"(\"><mml:mi>t</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mfenced></mml:msubsup></mml:mfenced></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq65\"><alternatives><tex-math id=\"M161\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${{\\text{g}}}_{i}$$\\end{document}</tex-math><mml:math id=\"M162\"><mml:msub><mml:mtext>g</mml:mtext><mml:mi>i</mml:mi></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq66\"><alternatives><tex-math id=\"M163\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${h}_{i}$$\\end{document}</tex-math><mml:math id=\"M164\"><mml:msub><mml:mi>h</mml:mi><mml:mi>i</mml:mi></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq67\"><alternatives><tex-math id=\"M165\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$k$$\\end{document}</tex-math><mml:math id=\"M166\"><mml:mi>k</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq68\"><alternatives><tex-math id=\"M167\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$e$$\\end{document}</tex-math><mml:math id=\"M168\"><mml:mi>e</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq69\"><alternatives><tex-math id=\"M169\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$T$$\\end{document}</tex-math><mml:math id=\"M170\"><mml:mi>T</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq70\"><alternatives><tex-math id=\"M171\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\text{AUC}}$$\\end{document}</tex-math><mml:math id=\"M172\"><mml:mtext>AUC</mml:mtext></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ17\"><label>17</label><alternatives><tex-math id=\"M173\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$M=SMOTE-XGboost(k,e,{\\text{T}})$$\\end{document}</tex-math><mml:math id=\"M174\" display=\"block\"><mml:mrow><mml:mi>M</mml:mi><mml:mo>=</mml:mo><mml:mi>S</mml:mi><mml:mi>M</mml:mi><mml:mi>O</mml:mi><mml:mi>T</mml:mi><mml:mi>E</mml:mi><mml:mo>-</mml:mo><mml:mi>X</mml:mi><mml:mi>G</mml:mi><mml:mi>b</mml:mi><mml:mi>o</mml:mi><mml:mi>o</mml:mi><mml:mi>s</mml:mi><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>k</mml:mi><mml:mo>,</mml:mo><mml:mi>e</mml:mi><mml:mo>,</mml:mo><mml:mtext>T</mml:mtext><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq71\"><alternatives><tex-math id=\"M175\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$k$$\\end{document}</tex-math><mml:math id=\"M176\"><mml:mi>k</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq72\"><alternatives><tex-math id=\"M177\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$e$$\\end{document}</tex-math><mml:math id=\"M178\"><mml:mi>e</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq73\"><alternatives><tex-math id=\"M179\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$T$$\\end{document}</tex-math><mml:math id=\"M180\"><mml:mi>T</mml:mi></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ18\"><label>18</label><alternatives><tex-math id=\"M181\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$ \\begin{aligned} f = &amp; \\max \\left[ {\\mathop \\sum \\limits_{i = 1}^{t} L\\left( {y_{i} ,\\hat{y}_{i} } \\right)} \\right] \\\\ = &amp; {\\text{max}}\\left\\{ {\\mathop \\sum \\limits_{i = 1}^{t} L\\left[ {y_{i} ,SMOTE - XGboost(k,e,T|D_{1:t} )} \\right]} \\right\\} \\\\ = &amp; G(k,e,T|D_{1:t} ) \\\\ \\end{aligned} $$\\end{document}</tex-math><mml:math id=\"M182\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:mi>f</mml:mi><mml:mo>=</mml:mo></mml:mrow></mml:mtd><mml:mtd columnalign=\"left\"><mml:mrow><mml:mo movablelimits=\"true\">max</mml:mo><mml:mfenced close=\"]\" open=\"[\"><mml:mrow><mml:munderover><mml:mo movablelimits=\"false\">∑</mml:mo><mml:mrow><mml:mi>i</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mi>t</mml:mi></mml:munderover><mml:mi>L</mml:mi><mml:mfenced close=\")\" open=\"(\"><mml:mrow><mml:msub><mml:mi>y</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"false\">^</mml:mo></mml:mover><mml:mi>i</mml:mi></mml:msub></mml:mrow></mml:mfenced></mml:mrow></mml:mfenced></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:mrow/><mml:mo>=</mml:mo></mml:mrow></mml:mtd><mml:mtd columnalign=\"left\"><mml:mrow><mml:mtext>max</mml:mtext><mml:mfenced close=\"}\" open=\"{\"><mml:mrow><mml:munderover><mml:mo movablelimits=\"false\">∑</mml:mo><mml:mrow><mml:mi>i</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mi>t</mml:mi></mml:munderover><mml:mi>L</mml:mi><mml:mfenced close=\"]\" open=\"[\"><mml:mrow><mml:msub><mml:mi>y</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>,</mml:mo><mml:mi>S</mml:mi><mml:mi>M</mml:mi><mml:mi>O</mml:mi><mml:mi>T</mml:mi><mml:mi>E</mml:mi><mml:mo>-</mml:mo><mml:mi>X</mml:mi><mml:mi>G</mml:mi><mml:mi>b</mml:mi><mml:mi>o</mml:mi><mml:mi>o</mml:mi><mml:mi>s</mml:mi><mml:mi>t</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>k</mml:mi><mml:mo>,</mml:mo><mml:mi>e</mml:mi><mml:mo>,</mml:mo><mml:mi>T</mml:mi><mml:mo stretchy=\"false\">|</mml:mo><mml:msub><mml:mi>D</mml:mi><mml:mrow><mml:mn>1</mml:mn><mml:mo>:</mml:mo><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:mfenced></mml:mrow></mml:mfenced></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:mrow/><mml:mo>=</mml:mo></mml:mrow></mml:mtd><mml:mtd columnalign=\"left\"><mml:mrow><mml:mi>G</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>k</mml:mi><mml:mo>,</mml:mo><mml:mi>e</mml:mi><mml:mo>,</mml:mo><mml:mi>T</mml:mi><mml:mo stretchy=\"false\">|</mml:mo><mml:msub><mml:mi>D</mml:mi><mml:mrow><mml:mn>1</mml:mn><mml:mo>:</mml:mo><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow/></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq74\"><alternatives><tex-math id=\"M183\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${y}_{i}$$\\end{document}</tex-math><mml:math id=\"M184\"><mml:msub><mml:mi>y</mml:mi><mml:mi>i</mml:mi></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq75\"><alternatives><tex-math id=\"M185\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\widehat{y}}_{i}$$\\end{document}</tex-math><mml:math id=\"M186\"><mml:msub><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"true\">^</mml:mo></mml:mover><mml:mi>i</mml:mi></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq76\"><alternatives><tex-math id=\"M187\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\left[{y}_{i},SMOTE-XGboost(k,e,T|{D}_{1:t})\\right]$$\\end{document}</tex-math><mml:math id=\"M188\"><mml:mfenced close=\"]\" open=\"[\"><mml:msub><mml:mi>y</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>,</mml:mo><mml:mi>S</mml:mi><mml:mi>M</mml:mi><mml:mi>O</mml:mi><mml:mi>T</mml:mi><mml:mi>E</mml:mi><mml:mo>-</mml:mo><mml:mi>X</mml:mi><mml:mi>G</mml:mi><mml:mi>b</mml:mi><mml:mi>o</mml:mi><mml:mi>o</mml:mi><mml:mi>s</mml:mi><mml:mi>t</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>k</mml:mi><mml:mo>,</mml:mo><mml:mi>e</mml:mi><mml:mo>,</mml:mo><mml:mi>T</mml:mi><mml:mo stretchy=\"false\">|</mml:mo><mml:msub><mml:mi>D</mml:mi><mml:mrow><mml:mn>1</mml:mn><mml:mo>:</mml:mo><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mfenced></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq77\"><alternatives><tex-math id=\"M189\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$G(k,e,T|{D}_{1:t})$$\\end{document}</tex-math><mml:math id=\"M190\"><mml:mrow><mml:mi>G</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>k</mml:mi><mml:mo>,</mml:mo><mml:mi>e</mml:mi><mml:mo>,</mml:mo><mml:mi>T</mml:mi><mml:mo stretchy=\"false\">|</mml:mo><mml:msub><mml:mi>D</mml:mi><mml:mrow><mml:mn>1</mml:mn><mml:mo>:</mml:mo><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq78\"><alternatives><tex-math id=\"M191\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$k,e,T$$\\end{document}</tex-math><mml:math id=\"M192\"><mml:mrow><mml:mi>k</mml:mi><mml:mo>,</mml:mo><mml:mi>e</mml:mi><mml:mo>,</mml:mo><mml:mi>T</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq79\"><alternatives><tex-math id=\"M193\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\text{L}}$$\\end{document}</tex-math><mml:math id=\"M194\"><mml:mtext>L</mml:mtext></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq80\"><alternatives><tex-math id=\"M195\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\text{AUC}}$$\\end{document}</tex-math><mml:math id=\"M196\"><mml:mtext>AUC</mml:mtext></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq81\"><alternatives><tex-math id=\"M197\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${D}_{1:t}$$\\end{document}</tex-math><mml:math id=\"M198\"><mml:msub><mml:mi>D</mml:mi><mml:mrow><mml:mn>1</mml:mn><mml:mo>:</mml:mo><mml:mi>t</mml:mi></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq82\"><alternatives><tex-math id=\"M199\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$t$$\\end{document}</tex-math><mml:math id=\"M200\"><mml:mi>t</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq83\"><alternatives><tex-math id=\"M201\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${R}^{2}$$\\end{document}</tex-math><mml:math id=\"M202\"><mml:msup><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mn>2</mml:mn></mml:msup></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq84\"><alternatives><tex-math id=\"M203\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${R}^{2}$$\\end{document}</tex-math><mml:math id=\"M204\"><mml:msup><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mn>2</mml:mn></mml:msup></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ19\"><label>18</label><alternatives><tex-math id=\"M205\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${R}^{2}=1-\\frac{{\\sum }_{i=1}^{N}\\left({y}_{i},{\\widehat{y}}_{i}\\right)}{{\\sum }_{i=1}^{N}{\\left({y}_{i},\\overline{y }\\right)}^{2}}$$\\end{document}</tex-math><mml:math id=\"M206\" display=\"block\"><mml:mrow><mml:msup><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mn>2</mml:mn></mml:msup><mml:mo>=</mml:mo><mml:mn>1</mml:mn><mml:mo>-</mml:mo><mml:mfrac><mml:mrow><mml:msubsup><mml:mo>∑</mml:mo><mml:mrow><mml:mi>i</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mi>N</mml:mi></mml:msubsup><mml:mfenced close=\")\" open=\"(\"><mml:msub><mml:mi>y</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"true\">^</mml:mo></mml:mover><mml:mi>i</mml:mi></mml:msub></mml:mfenced></mml:mrow><mml:mrow><mml:msubsup><mml:mo>∑</mml:mo><mml:mrow><mml:mi>i</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mi>N</mml:mi></mml:msubsup><mml:msup><mml:mrow><mml:mfenced close=\")\" open=\"(\"><mml:msub><mml:mi>y</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>,</mml:mo><mml:mover><mml:mi>y</mml:mi><mml:mo>¯</mml:mo></mml:mover></mml:mfenced></mml:mrow><mml:mn>2</mml:mn></mml:msup></mml:mrow></mml:mfrac></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq85\"><alternatives><tex-math id=\"M207\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${y}_{i}$$\\end{document}</tex-math><mml:math id=\"M208\"><mml:msub><mml:mi>y</mml:mi><mml:mi>i</mml:mi></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq86\"><alternatives><tex-math id=\"M209\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\widehat{y}}_{i}$$\\end{document}</tex-math><mml:math id=\"M210\"><mml:msub><mml:mover accent=\"true\"><mml:mi>y</mml:mi><mml:mo stretchy=\"true\">^</mml:mo></mml:mover><mml:mi>i</mml:mi></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq87\"><alternatives><tex-math id=\"M211\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\overline{y }$$\\end{document}</tex-math><mml:math id=\"M212\"><mml:mover><mml:mi>y</mml:mi><mml:mo>¯</mml:mo></mml:mover></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq88\"><alternatives><tex-math id=\"M213\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$N$$\\end{document}</tex-math><mml:math id=\"M214\"><mml:mi>N</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq89\"><alternatives><tex-math id=\"M215\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${R}^{2}$$\\end{document}</tex-math><mml:math id=\"M216\"><mml:msup><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mn>2</mml:mn></mml:msup></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq90\"><alternatives><tex-math id=\"M217\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${R}^{2}$$\\end{document}</tex-math><mml:math id=\"M218\"><mml:msup><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mn>2</mml:mn></mml:msup></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq91\"><alternatives><tex-math id=\"M219\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${\\text{AUC}}$$\\end{document}</tex-math><mml:math id=\"M220\"><mml:mtext>AUC</mml:mtext></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ20\"><label>20</label><alternatives><tex-math id=\"M221\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$AUC=\\left[\\sum_{i\\in \\,Positive\\, sample\\, se{\\text{t}}}r(i)-\\frac{M(M+1)}{2}\\right]/M(N-M)$$\\end{document}</tex-math><mml:math id=\"M222\" display=\"block\"><mml:mrow><mml:mi>A</mml:mi><mml:mi>U</mml:mi><mml:mi>C</mml:mi><mml:mo>=</mml:mo><mml:mfenced close=\"]\" open=\"[\"><mml:munder><mml:mo>∑</mml:mo><mml:mrow><mml:mi>i</mml:mi><mml:mo>∈</mml:mo><mml:mspace width=\"0.166667em\"/><mml:mi>P</mml:mi><mml:mi>o</mml:mi><mml:mi>s</mml:mi><mml:mi>i</mml:mi><mml:mi>t</mml:mi><mml:mi>i</mml:mi><mml:mi>v</mml:mi><mml:mi>e</mml:mi><mml:mspace width=\"0.166667em\"/><mml:mi>s</mml:mi><mml:mi>a</mml:mi><mml:mi>m</mml:mi><mml:mi>p</mml:mi><mml:mi>l</mml:mi><mml:mi>e</mml:mi><mml:mspace width=\"0.166667em\"/><mml:mi>s</mml:mi><mml:mi>e</mml:mi><mml:mtext>t</mml:mtext></mml:mrow></mml:munder><mml:mi>r</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>i</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>-</mml:mo><mml:mfrac><mml:mrow><mml:mi>M</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>M</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:mfrac></mml:mfenced><mml:mo stretchy=\"false\">/</mml:mo><mml:mi>M</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>N</mml:mi><mml:mo>-</mml:mo><mml:mi>M</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq92\"><alternatives><tex-math id=\"M223\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$r(i)$$\\end{document}</tex-math><mml:math id=\"M224\"><mml:mrow><mml:mi>r</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>i</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq93\"><alternatives><tex-math id=\"M225\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$M$$\\end{document}</tex-math><mml:math id=\"M226\"><mml:mi>M</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq94\"><alternatives><tex-math id=\"M227\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$N$$\\end{document}</tex-math><mml:math id=\"M228\"><mml:mi>N</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq95\"><alternatives><tex-math id=\"M229\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\Delta Q$$\\end{document}</tex-math><mml:math id=\"M230\"><mml:mrow><mml:mi mathvariant=\"normal\">Δ</mml:mi><mml:mi>Q</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ21\"><label>21</label><alternatives><tex-math id=\"M231\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\Delta Q=\\frac{{q}_{1}}{{Q}_{1}}-\\frac{{q}_{2 }}{{Q}_{2}}$$\\end{document}</tex-math><mml:math id=\"M232\" display=\"block\"><mml:mrow><mml:mi mathvariant=\"normal\">Δ</mml:mi><mml:mi>Q</mml:mi><mml:mo>=</mml:mo><mml:mfrac><mml:msub><mml:mi>q</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:msub><mml:mi>Q</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:mfrac><mml:mo>-</mml:mo><mml:mfrac><mml:msub><mml:mi>q</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:msub><mml:mi>Q</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:mfrac></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq96\"><alternatives><tex-math id=\"M233\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${q}_{1}$$\\end{document}</tex-math><mml:math id=\"M234\"><mml:msub><mml:mi>q</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq97\"><alternatives><tex-math id=\"M235\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${q}_{2}$$\\end{document}</tex-math><mml:math id=\"M236\"><mml:msub><mml:mi>q</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq98\"><alternatives><tex-math id=\"M237\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${Q}_{1}$$\\end{document}</tex-math><mml:math id=\"M238\"><mml:msub><mml:mi>Q</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq99\"><alternatives><tex-math id=\"M239\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${Q}_{2}$$\\end{document}</tex-math><mml:math id=\"M240\"><mml:msub><mml:mi>Q</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq100\"><alternatives><tex-math id=\"M241\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${R}^{2}$$\\end{document}</tex-math><mml:math id=\"M242\"><mml:msup><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mn>2</mml:mn></mml:msup></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq101\"><alternatives><tex-math id=\"M243\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${R}^{2}$$\\end{document}</tex-math><mml:math id=\"M244\"><mml:msup><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mn>2</mml:mn></mml:msup></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq102\"><alternatives><tex-math id=\"M245\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$k$$\\end{document}</tex-math><mml:math id=\"M246\"><mml:mi>k</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq103\"><alternatives><tex-math id=\"M247\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$e$$\\end{document}</tex-math><mml:math id=\"M248\"><mml:mi>e</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq104\"><alternatives><tex-math id=\"M249\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$T$$\\end{document}</tex-math><mml:math id=\"M250\"><mml:mi>T</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq105\"><alternatives><tex-math id=\"M251\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$k$$\\end{document}</tex-math><mml:math id=\"M252\"><mml:mi>k</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq106\"><alternatives><tex-math id=\"M253\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$e$$\\end{document}</tex-math><mml:math id=\"M254\"><mml:mi>e</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq107\"><alternatives><tex-math id=\"M255\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$T$$\\end{document}</tex-math><mml:math id=\"M256\"><mml:mi>T</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq108\"><alternatives><tex-math id=\"M257\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${R}^{2}$$\\end{document}</tex-math><mml:math id=\"M258\"><mml:msup><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mn>2</mml:mn></mml:msup></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq109\"><alternatives><tex-math id=\"M259\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${R}^{2}$$\\end{document}</tex-math><mml:math id=\"M260\"><mml:msup><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mn>2</mml:mn></mml:msup></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq110\"><alternatives><tex-math id=\"M261\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$${R}^{2}$$\\end{document}</tex-math><mml:math id=\"M262\"><mml:msup><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mn>2</mml:mn></mml:msup></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq111\"><alternatives><tex-math id=\"M263\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$k\\hspace{0.17em}$$\\end{document}</tex-math><mml:math id=\"M264\"><mml:mrow><mml:mi>k</mml:mi><mml:mspace width=\"1.69998pt\"/></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq112\"><alternatives><tex-math id=\"M265\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$e\\hspace{0.17em}$$\\end{document}</tex-math><mml:math id=\"M266\"><mml:mrow><mml:mi>e</mml:mi><mml:mspace width=\"1.69998pt\"/></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq113\"><alternatives><tex-math id=\"M267\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$T\\hspace{0.17em}$$\\end{document}</tex-math><mml:math id=\"M268\"><mml:mrow><mml:mi>T</mml:mi><mml:mspace width=\"1.69998pt\"/></mml:mrow></mml:math></alternatives></inline-formula>" ]

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[ "<title>Introduction</title>", "<p id=\"Par2\">Online health consultation is a novel form of doctor-patient communication that has emerged alongside the rapid advancements in Internet technology^{##UREF##0##1##}. Patients can employ mobile phones, computers and other terminal devices to engage in various communication modes, such as pictures, texts, voice or phone consultation on online health consultation platforms. Similarly, physicians can also choose an appropriate method to provide professional advice for patients^{##UREF##1##2##}. In traditional offline health-seeking behavior, patients face limitations in accessing physicians’ information when choosing a healthcare provider^{##REF##11083036##3##}, because beyond readily identifiable information such as the hospital and professional title of physicians, obtaining nuanced information concerning personal characteristics and communication styles of physicians poses a frequent obstacle for patients. While online health consultation platforms can provide a wealth of information, such as physicians’ professional title, service attitude, communication styles, response speed and so on. This multifaceted information help patients form cognitive judgments and affective attitudes towards a physician, so as to make further decisions^{##REF##31647466##4##,##UREF##2##5##}.</p>", "<p id=\"Par3\">As mentioned above, in traditional offline health-seeking behavior, patients have limited access to physicians’ information^{##REF##11083036##3##}. This leads to a scenario where patients’ preferences for physicians tend to converge, resulting in a chaotic pursuit of top-tier hospital renowned physicians irrespective of major or minor ailments^{##REF##23270526##6##}. So now online health consultation platforms can provide comprehensive physicians’ information for patients^{##REF##31647466##4##,##UREF##3##7##}, do patients still exhibit uniform preference for physicians? That is, what specific type of physician information do patients rely on to make decisions when choosing a physician on online health consultation platform? Do they continue to prioritize cognitive stimulation information that can reflect the physician’s professionalism, such as the hospital and professional title of physicians^{##UREF##1##2##}, or do they favor affective stimulation information, which reflects the physician’s care for the patient, such as the physician's attitude and communication skills^{##UREF##1##2##}? Thus, the primary objective of this study is to explore the preferences of patients for different types of physician information during the process of choosing physicians on online health consultation platforms.</p>", "<p id=\"Par4\">Previous studies have shown that the influence of information on individual decision-making behavior is regulated by environmental and contextual factors. These factors encompass the characteristics of users, products or services^{##UREF##4##8##}, of which involvement is an important variable that modulates this influence relationship^{##UREF##5##9##–##UREF##7##11##}. Involvement is originally a psychological concept, referring to the perceived significance and relevance that individuals attribute to specific entities^{##UREF##8##12##}. Diverse manifestations of involvement emerge based on the objects under consideration, including but not limited to product involvement, shopping involvement, brand involvement and advertising involvement^{##UREF##9##13##}. In the field of health decision-making, the importance and relevance of different disease types to patients exhibit distinctions. Therefore, this study defines different disease types as health involvement^{##UREF##10##14##}. Results from previous studies indicate that patients’ behavior is modulated by health involvement, and patients with different disease types exhibit distinct preferences for physicians’ information. For example, Lu and Wu^{##UREF##4##8##} found that the lower the risk of disease, that is, the lower the health involvement, patients’ choice behavior becomes more susceptible to affective stimulus information, such as physicians’ attitude. Thus the first objective of this study is to investigate the moderating effect of health involvement (disease type) on the patients’ preferences for physicians’ information on online health consultation platforms.</p>", "<p id=\"Par5\">However, it is evident that the same disease varying degrees of significance among distinct patients. For example, individuals deeply preoccupied with their health status or even have health anxiety^{##REF##24102547##15##}, they may perceive a common ailment such as a headache as a potentially grave threat to their well-being. This perception could subsequently influence their health information searching behavior and even health decision-making^{##UREF##11##16##,##UREF##12##17##}. Conversely, individuals with lower levels of health anxiety may regard the same symptom as less severe, deeming it necessitating only minor recuperation. Consequently, during the follow-up experiments, deliberate efforts were made to minimize the potential influence of health anxiety on the experimental conditions pertaining to health involvement.</p>", "<p id=\"Par6\">Individual preference or adoption of different types of information is actually a process in which an individual makes a value evaluation on the received information and then makes decision according to the value ranking of different alternatives^{##UREF##13##18##–##REF##30626917##20##}. Hence, the neural mechanism of individual information value ranking process can refer to value-based decision theory. Value-based decision theory posits that the varied preferences and attitudes of individuals towards the same entity are termed subjective value. Subjective value enables individuals to integrate diverse and complex alternatives into a common dimension for comparative evaluation^{##REF##23507394##21##,##REF##34508645##22##}. On the neural level, extensive research has shown that the ventromedial prefrontal cortex (VMPFC) and ventral striatum (VS) are crucial brain regions associated with subjective value. These regions exhibit notable activation during individual evaluations of subjective value^{##REF##30626917##20##–##REF##27516744##23##}. Therefore, the second objective of this study is to reveal the neural mechanism of the moderating effect of health involvement on patients’ preferences for physicians’ information by analyzing the activation differences of VMPFC and VS brain regions when patients process information under different health involvement.</p>", "<p id=\"Par7\">According to the above, to achieve our two objectives in this paper, we first employed behavioral experiment to identify patients’ preferences for different types of physicians’ information (cognitive stimulation information and affective stimulation information) under different health involvement, and then used functional magnetic resonance imaging (fMRI) experiment to analyze the brain activation mechanism of patients under different health involvement from the neural level, thereby providing neural/physiological evidence elucidating the influence of health involvement on the information preferences of patients in online health consultation services.</p>" ]

[ "<title>Methods</title>", "<title>Manipulation check</title>", "<p id=\"Par18\">To divide the abundant physician information provided on the online health consultation platform into two types: cognitive stimulation information and affective stimulation information, this paper referred to the physician-related information provided on an actual online health consultation platform, and extracted seven types of physician attributes: (1) physician rank (indicating both the hospital rank and the physician’s professional title); (2) professional knowledge (signifying the specific expertise of the physician); (3) treatment effect; (4) service attitude; (5) communication skills; (6) response speed; (7) service commitment (the presence of physician on the online health consultation platform means that the physician promises to provide services for users online. The indicator of the number of replies in recent two weeks on the actual platform can reflect whether physicians have enough time and energy to provide services for users online, so we use service commitment to represent the number of replies in recent two weeks). 53 participants (M_age = 28.98 years) were recruited via the sample service provided by the website wjx (<ext-link ext-link-type=\"uri\" xlink:href=\"https://www.wjx.cn/\">https://www.wjx.cn/</ext-link>). The participants were stratified into two groups: the high satisfaction group (experimental condition: physician information + 100% favorable rating, N = 26) and the low satisfaction group (experimental condition: physician information + 75% favorable rating, N = 27) to assess the seven physician attributes mentioned above. Participants were required to assess both cognitive and affective trust perceptions after reviewing a physician’s information. If the information about a physician differed solely in the cognitive trust scores of the participants, without affecting affective trust, it categorizes as cognitive stimulation information; otherwise it is considered affective stimulation information (the measurement scales see ##SUPPL##0##Supplementary 1. Table S2##).</p>", "<title>Behavioral experiment</title>", "<p id=\"Par19\">Given the objective of this study is to identify the physician preferences of patients with different health involvement. 94 participants with prior experience in online health consultations were recruited for a mixed-design experiment using the sample service provided by the website wjx (<ext-link ext-link-type=\"uri\" xlink:href=\"https://www.wjx.cn/\">https://www.wjx.cn/</ext-link>). The participants were required to engage in this experiment by recalling their most recent online health consultation experience. Subsequently, participants were categorized into disease types corresponding to their perception of the severity of their previous consultation illness, distinguishing between mild and acute cases. Specifically, 48 participants were low health involvement in mild diseases, 46 participants were high health involvement in acute diseases. The mean age of the participants was 29.28 ± 7.84 years. During the experiment, participants rated the physician’s willingness to choose according to a combination of a cognitive stimulation information + favorite rating and an affective stimulation information + favorite rating. These sets of information were randomly drawn from the physician’s information selected from the manipulation check. Each participant encountered scenarios involving high cognitive trust &amp; high affective trust (Hh: cognitive stimulation information + 100% favorable rating &amp; affective stimulation information + 100% favorable rating), high cognitive trust &amp; low affective trust (Hl: cognitive stimulation information + 100% favorable rating &amp; affective stimulation information + 75% favorable rating), low cognitive trust &amp; high affective trust (Lh: cognitive stimulation information + 75% favorable rating &amp; affective stimulation information + 100% favorable rating), low cognitive trust &amp; low affective trust (Ll: cognitive stimulation information + 75% favorable rating &amp; affective stimulation information + 75% favorable rating) these four experimental conditions, corresponding to four different physicians. Participants were explicitly informed that the four physicians shared identical conditions, differing only in the ratings assigned to the experimental materials. Participants were then instructed to evaluate the willingness to choose for each physician based on the information provided by the experimental materials (the measurement scales see ##SUPPL##0##Supplementary 1. Table S2##).</p>", "<title>Imaging experiment</title>", "<p id=\"Par20\">The experimental task in the imaging experiment was consistent with the behavioral experimental task. However, to generate a stronger BOLD signal in the imaging experiment, the low satisfaction rating was adjusted from 75% favorable rating to 25% favorable rating. The experimental paradigm employed in this study was rapid-presentation (jittered) event-related design, and the order and timing of events were generated with the help of optseq2 tool^{##REF##10524601##35##}. We recruited 33 undergraduate and graduate students as participants for a within-subject design experiment via online promotional channels, including WeChat groups and campus forums. One participant was excluded due to exceeding the specified head movement range of 2 mm, other 32 participants were included in the final analysis (M_age = 23.97 ± 2.74 years). All the participants were right-handed, physically healthy, devoid of mental related diseases, and exhibited either normal or corrected vision. Upon completion of the experiment, each participant got RMB 150 yuan as a reward for their participation.</p>", "<title>Scanning procedure</title>", "<p id=\"Par21\">The participants first closed their eyes to receive a T1 structural image scan with a duration of about 6 min, and then performed two functional image scans with a duration of about 11 min. Participants were allowed to take a brief respite between each scan at their discretion. The two functional image scans were two tasks, corresponding to the two types of diseases with different health involvement. The imaging experiment was a simulation in which participants were required to hypothetically possess a specific disease. Drawing upon the severity and urgency of the disease, and building on the findings of Li et al.^{##REF##30115610##36##}, this study explicitly defined the disease under low health involvement experimental conditions as a cold, while designating acute abdominal pain for high health involvement conditions. In each task, participants were initially presented with an experimental instruction with a duration of 12 s, followed by a fixation point with a duration of 3 s, and then a formal experimental stimulus with a duration of 7 s. Each task contained 80 trails, including four experimental conditions: Hh, Hl, Lh, and Ll. Each condition repeated 20 times within a single task. Participants were instructed to provide keystroke responses during the duration of the formal experimental stimulus. Because the right-handed single-handed keypresses were used, the participants used the Likert 5-level scale to evaluate the physician’s willingness to choose in response to the experimental stimulus (where 1 indicated very unwilling, 5 indicated very willing). The experimental stimulus presentation program is E-prime, and the presentation procedures is shown in Fig. ##FIG##2##3##.</p>", "<title>Data collection</title>", "<p id=\"Par22\">The imaging data was collected on a Siemens PRISMA 3T fMRI scanner at magnetic resonance imaging research center of Peking University. The scanning parameters of T1 structure images were: Voxel size = 0.5 mm × 0.5 mm × 1.0 m, Field of View = 256 mm, Slice thickness = 1.00 mm, TR = 2530.0 ms, TE = 2.98 ms, Flip angle = 7°, Matrix = 256 × 256. The functional image was scanned by echo planar imaging (EPI), and the scanning parameters were: Voxel size = 2.0 mm × 2.0 mm × 2.0 mm, Slices = 62, Field of View = 224 mm, Slice thickness = 2.0 mm, TR = 2000 ms, TE = 30.0 ms, Flip angle = 90, Matrix = 112 × 112.</p>", "<title>Ethics approval and consent to participate</title>", "<p id=\"Par23\">All study procedures and methods were performed in accordance with the relevant guidelines and regulations and approved by Ethical Review Board of School of Psychological and Cognitive Sciences, Peking University. Written Informed Consent was obtained from all participants before data collection.</p>" ]

[ "<title>Results</title>", "<title>Manipulation check results</title>", "<p id=\"Par8\">Through the manipulation checks, this paper selected four types of physician information from the seven types of physician information extracted on the online health consultation platform for formal experiment. The cognitive stimulation category included: physician rank (p &lt; 0.001) and professional knowledge (p &lt; 0.001), while the affective stimulation category encompassed service attitude (p = 0.012) and communication skills (p = 0.033). The results are shown in Table ##TAB##0##1##.</p>", "<title>Behavioral results</title>", "<p id=\"Par9\">This paper used 2 (cognitive trust high/low) × 2 (affective trust high/low) behavioral experiment to explore the influence of cognitive trust and affective trust on patients’ willingness to choose under two different levels of health involvement (high/low) through two-way repeated ANOVAs, so as to identify patients’ preference to cognitive and affective stimulation information under different levels of health involvement. The results showed that health involvement had a significant impact on patients’ preferences for different types of physicians’ information. When health involvement was low, the interaction between cognitive trust and affective trust on participants’ willingness to choose was not significant (df = 47, F = 0.323, p = 0.572; Partial η² = 0.007). Conversely, under conditions of high health involvement, the interaction between cognitive trust and affective trust on participants’ willingness to choose was significant (df = 45, F = 13.049, p = 0.001; Partial η² = 0.225). The results are shown in Table ##TAB##1##2##.</p>", "<p id=\"Par10\">Further post-hoc test found that (see Table ##TAB##2##3## and Fig. ##FIG##0##1##), when the health involvement was low, both cognitive and affective stimulus information could influence patients' decision-making behavior. The results indicated that an enhancement in either cognitive trust or affective trust significantly heightened patients’ willingness to choose the physician. When health involvement was high, the isolated improvement of either cognitive stimulus information or affective stimulus information did not elicit any discernible impact on patients' decision-making behavior. The outcomes illustrated that enhancing either cognitive trust or affective trust independently did not significantly improve participants’ willingness to choose. Notably, it was only when both aspects were improved simultaneously that participants’ willingness to choose would be significantly improved.</p>", "<title>Imaging results</title>", "<p id=\"Par11\">To examine the activation differences of VMPFC and VS brain regions when patients process information under different health involvement conditions, this paper first used the general linear model (GLM) in SPM12 to generate the design matrix for the individual-level analyses (Specify 1st-level), calculated the task activation regions under four different experimental conditions of Hh, Hl, Lh, Ll for each participant, and then performed group-level analyses (Specify 2nd-level). On the group-level, we computed the contrast “CT [(Hh + Hl)−(Lh + Ll), AT [(Hh + Lh)−Hl + Ll)]” (for additional experimental parameters and design matrix used in the SPM analysis, see ##SUPPL##0##Supplementary 1. Fig. S1##). Within our priori ROIs hypotheses-driven, we focused on VMPFC (6-mm sphere centered at the MNI coordinate: 2/46/− 16)^{##UREF##15##24##} and VS (6-mm sphere centered at the MNI coordinate: − 16/6/− 12)^{##REF##25868676##25##} these two regions of interest (ROIs), extracted the contrast value of these two ROIs to analyze their differences in brain activation under different health involvement conditions. The results are shown in Fig. ##FIG##1##2## (for whole-brain results, see ##SUPPL##0##Supplementary 1. Table S1##), in the condition of low health involvement, activation in the VMPFC differed between CT contrast and AT contrast (t = 3.232, df = 31, p = 0.003; Cohen’s d = 0.572), CT contrast in the VMPFC was significantly activated (small volume FWE corrected at p = 0.001 and k = 26), but not AT contrast. Activation in the VS also differed between CT contrast and AT contrast (t = 2.800, df = 31, p = 0.009; Cohen’s d = 0.498), AT contrast in the VS was significantly activated (small volume FWE corrected at p = 0.003 and k = 17), while CT contrast was not significantly activated. In the condition of high health involvement, activation in the VMPFC (t = 0.762, df = 31, p = 0.452; Cohen’s d = 0.134) and VS (t = 0.699, df = 31, p = 0.489; Cohen’s d = 0.123) was not significantly different between CT contrast and AT contrast, and neither brain region was significantly activated in either contrast.</p>" ]

[ "<title>Discussion</title>", "<p id=\"Par12\">The results of behavioral experiment indicate that the preferences of patients in choosing physician online varied based on their levels of health involvement. When the health involvement was low, physicians with professionalism or good attitude can be preferred by patients. This study observed a noteworthy correlation between the improvement of cognitive trust or affective trust and patients’ willingness to choose physicians, which means that both cognitive stimulation information and affective stimulation information can influence the patient’s decision-making, and the effects of the two can be mutually substituted/compensated. When the health involvement was high, patients exclusively favored physicians possessing both good professionalism and attitude. This suggests that an increase in either cognitive trust or affective trust alone did not substantially improve patients’ willingness to choose. Instead, a significant improvement in the willingness to choose only occurred when both aspects were enhanced simultaneously. This underscores the crucial role of combining cognitive and affective stimulus information in influencing patients’ choice behavior.</p>", "<p id=\"Par13\">The results have also been confirmed at the neural level. Imaging experiment revealed that patients with low health involvement exhibited significant activation in the VMPFC when evaluating cognitive stimulation information and in the VS when evaluating affective stimulation information, and previous studies have demonstrated the close association of both VMPFC and VS with subjective value evaluation ^{##REF##30626917##20##–##REF##27516744##23##}. This implies that when health involvement is low, patients integrate cognitive and affective stimulation information within the same dimension for comparative evaluation to make decisions ^{##REF##23507394##21##,##REF##34508645##22##}. In essence, patients with low health involvement exhibit a preference for both cognitive and affective stimuli information. Conversely, when health involvement was high, the results indicated the absence of activation in related brain regions when patients evaluated cognitive and affective stimulation information. According to previous studies, the activation degree of related brain regions can serve as a reflection of cognitive resources allocated by individuals^{##UREF##16##26##}. Therefore, the results of the imaging experiment may provide a theoretical explanation for the modulation of health involvement on patients’ preferences for different types of physicians’ information.</p>", "<p id=\"Par14\">According to the Elaboration Likelihood Model (ELM), a theoretical framework delineating individual information processing, two distinct routes exist for individuals when processing information, one is the central route and the other is the peripheral route^{##UREF##17##27##}. When an individual uses the central route to process information, the individual's attitude will change after a comprehensive evaluation of the detailed information in a cognitive / rational way, which requires more cognitive resources. When an individual adopts the peripheral route, the individual will engage in a less refined processing of received information, allocating fewer cognitive resources^{##UREF##18##28##–##UREF##20##30##}. This study employed imaging experiment to reveal the information processing processes of patients. The results indicated that when the health involvement was low, patients had a higher activation degree of brain regions, allocated more cognitive resources, which means they preferred to use the central route to process information. When the health involvement was high, patients allocated less cognitive resources and preferred to use peripheral route to process information.</p>", "<p id=\"Par15\">Although the results in this paper indicated that health involvement had an impact on patients' information processing routes, that is, patients with low health involvement used the central route, while those with high health involvement used peripheral route. However, this finding is inconsistent in the field of consumption decision-making. Previous studies have shown that when the product or service is more important to consumers (i.e., high involvement), they allocate more cognitive resources to evaluate information through the central route. Conversely, under low involvement, consumers resort to the peripheral route for information processing^{##UREF##21##31##–##UREF##23##33##}. We posit that this discrepancy arises due to the distinct nature of health decision-making compared to product purchases. In the field of consumption decision-making, individuals freely choose their information processing route without restricting cognitive resources. However, in the field of health decision-making, the priority of the disease itself affects the patient's energy. When the disease is milder, patients possess more energy to comprehensively and carefully evaluate all obtained physician information when choosing physicians. Therefore, in instances of low health involvement, patients are more inclined to adopt the central route for information processing. Conversely, under high health involvement, the overwhelming nature of \"care is chaos\" may deprive patients of sufficient time and energy to comprehensively evaluate all information, leading them to rely on the peripheral route for decision-making.</p>", "<p id=\"Par16\">To sum up, this paper first found the impact of health involvement in the field of online health-seeking on patients' physician preferences through behavioral experiment, and then used imaging experiment to record the brain reaction process of patients’ decision-making to reveal the neural mechanism of this impact, provided neural/physiological evidence to support the derived conclusions. The findings expand the application of individual information processing theories in the field of health decision-making, and establish an effective connection between individual psychological function and decision-making behavior. Furthermore, the relevant conclusions can also provide insights to practical applications. From the behavioral results, it can be found that patients exhibiting diverse disease types employ distinct ways in processing physician information, thereby expressing preferences for physicians of different categories. Consequently, in practical applications, the platform has the potential to formulate a physician recommendation mechanism rooted in these observations. This mechanism could tailor recommendations based on the preferences of patients with specific disease types, fostering a targeted and efficacious alignment between patients and physicians. For example, with respect to patients with mild diseases, the platform can not only recommend physicians demonstrating high professional competence but also those general physicians skilled in cultivating affective trust with patients. This strategic approach mitigates the occurrence of a distorted phenomenon wherein all patients converge toward high-quality medical resources, thereby alleviating the strain on such resources and preventing the idle wastage of ordinary medical resources. Moreover, this approach bolsters the pivotal role of online health consultation platforms in the equitable distribution of doctors and efficient channeling of patient flows. Consequently, it contributes to the optimization of grassroots resource utilization and facilitates the judicious allocation of medical resources.</p>", "<p id=\"Par17\">The study has certain limitations. Firstly, all the experimental methods employed in this study were situational experiments. While situational experiments manage interference from irrelevant variables, a disparity exists between the experimental setting and the real-world scenario. Therefore, future research could employ field experiments or integrate virtual reality technology to enhance the fidelity of the experimental setting, bridging the gap between the experimental and real-world scenarios for more precise results. Secondly, as mentioned earlier, health anxiety can influence patients’ perception of health involvement. Although in the behavioral experiment, participants were categorized into disease types corresponding to their perception of the severity of their previous consultation illness, distinguishing between mild and acute cases. In the imaging experiment, we chose two clearly distinguishable diseases, cold and acute abdominal pain, to represent conditions with low and high health involvement, thereby mitigating potential influences. However, the control of health anxiety constitutes a limitation of this study. Future research should contemplate measuring participants’ health anxiety during the experiment to control this impact. In addition, the participants of the imaging experiment are mainly university students. Despite the predominant users of the online health consultation platform being young people^{##UREF##24##34##}, the significant health demands of the aging population must not be overlooked.. To enhance the effective utilization of online health consultation services among the elderly, future research should prioritize investigating the behavior patterns of this demographic.</p>" ]

[]

[ "<p id=\"Par1\">In traditional offline health-seeking behavior, patients consistently exhibit a preference for similar types of physicians due to limited access to physicians’ information. Nevertheless, with the advent of online health consultation platforms offering comprehensive physicians’ information for patients, raises the question: do patients continue to exhibit uniform preference for physicians? To address this issue, we first employed a behavioral experiment to discern patients’ preferences for different types of physicians’ information under different health involvement, and then conducted a functional magnetic resonance imaging (fMRI) experiment to furnish neural/physiological evidence. The results showed that health involvement modulates patients’ preferences, when health involvement was low, patients had diverse preferences for physicians, that is, different types of physicians’ information could individually impact patients’ choice and could serve as substitutes for each other. When health involvement was high, patients’ preference for physicians were uniform, highlighting that the collective influence of different types of physicians’ information on patients’ choice behavior. From the neural level, an explanation for the results was that the ventromedial prefrontal cortex (VMPFC) and ventral striatum (VS) brain regions, two key brain regions reflecting individual cognitive resource allocation, had different activation levels under different health involvement, indicating that patients allocated different cognitive resources.</p>", "<title>Subject terms</title>" ]

[ "<title>Supplementary Information</title>", "<p>\n</p>" ]

[ "<title>Supplementary Information</title>", "<p>The online version contains supplementary material available at 10.1038/s41598-024-51519-4.</p>", "<title>Acknowledgements</title>", "<p>This work was supported by the Open Project Funding from the Shanghai Key Laboratory of Brain-Machine Intelligence for Information Behavior, Shanghai International Studies University, Shanghai, China (2023KFKT007), National Natural Science Foundation of China (71874018, 71942003), and Science and Technology Research Program of Chongqing Municipal Education Commission (KJQN202300634).</p>", "<title>Author contributions</title>", "<p>Y.Z., Y.W. and H.R. participated in the research design and project implementation. Y.Z. and Y.W. participated in data collection and analysis. Y.W. and H.R. guided manuscript drafting. Y.Z. drafted the manuscript. All of the authors read and approved the final manuscript.</p>", "<title>Data availability</title>", "<p>The datasets used and/or analysed during the current study available from the corresponding author on reasonable request.</p>", "<title>Competing interests</title>", "<p id=\"Par24\">The authors declare no competing interests.</p>" ]

[ "<fig id=\"Fig1\"><label>Figure 1</label><caption><p>Simple effect analysis results of patients' scores of willingness to choose with different levels of health involvement. Note: (<bold>a</bold>) low level health involvement; (<bold>b</bold>) high level health involvement; H, high level of cognitive trust; L, low level of cognitive trust; h, high level of affective trust; l, low level of affective trust; all the scores of willingness to choose ranged between 1 and 9; ns p &gt; 0.05, *p &lt;  = 0.05, **p &lt;  = 0.01, ***p &lt;  = 0.001; error bars: SEM.</p></caption></fig>", "<fig id=\"Fig2\"><label>Figure 2</label><caption><p>Task-related activation in the VMPFC and VS. Note: (<bold>a</bold>) activation for the CT contrast in the VMPFC with low level health involvement; (<bold>b</bold>) activation for the AT contrast in the VS with low level health involvement; (<bold>c</bold>) plots of contrast values in the VMPFC; (<bold>d</bold>) plots of contrast values in the VS; CT = CT contrast, AT = AT contrast; ns p &gt; 0.05, *p &lt;  = 0.05, **p &lt;  = 0.01, ***p &lt;  = 0.001; error bars: SEM.</p></caption></fig>", "<fig id=\"Fig3\"><label>Figure 3</label><caption><p>Illustration of trial procedures. Note: ISI, inter-stimulus interval.</p></caption></fig>" ]

[ "<table-wrap id=\"Tab1\"><label>Table 1</label><caption><p>Physician information manipulation checks results.</p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th align=\"left\" rowspan=\"2\" colspan=\"2\">Physician information</th><th align=\"left\">CT</th><th align=\"left\" rowspan=\"2\">Sig</th><th align=\"left\">AT</th><th align=\"left\" rowspan=\"2\">Sig</th></tr><tr><th align=\"left\">Mean (S.D.)</th><th align=\"left\">Mean (S.D.)</th></tr></thead><tbody><tr><td align=\"left\" rowspan=\"2\">Physician rank</td><td align=\"left\">100% favorable rating</td><td char=\"(\" align=\"char\">7.372 (0.732)</td><td char=\".\" align=\"char\" rowspan=\"2\">0.000</td><td char=\"(\" align=\"char\">6.769 (0.992)</td><td char=\".\" align=\"char\" rowspan=\"2\">0.506</td></tr><tr><td align=\"left\">75% favorable rating</td><td char=\"(\" align=\"char\">6.444 (0.716)</td><td char=\"(\" align=\"char\">6.602 (0.824)</td></tr><tr><td align=\"left\" rowspan=\"2\">Professional knowledge</td><td align=\"left\">100% favorable rating</td><td char=\"(\" align=\"char\">7.513 (0.855)</td><td char=\".\" align=\"char\" rowspan=\"2\">0.000</td><td char=\"(\" align=\"char\">6.769 (1.086)</td><td char=\".\" align=\"char\" rowspan=\"2\">0.973</td></tr><tr><td align=\"left\">75% favorable rating</td><td char=\"(\" align=\"char\">6.679 (0.754)</td><td char=\"(\" align=\"char\">6.778 (0.670)</td></tr><tr><td align=\"left\" rowspan=\"2\">Treatment effect</td><td align=\"left\">100% favorable rating</td><td char=\"(\" align=\"char\">7.641 (1.045)</td><td char=\".\" align=\"char\" rowspan=\"2\">0.000</td><td char=\"(\" align=\"char\">7.346 (0.718)</td><td char=\".\" align=\"char\" rowspan=\"2\">0.022</td></tr><tr><td align=\"left\">75% favorable rating</td><td char=\"(\" align=\"char\">6.531 (0.921)</td><td char=\"(\" align=\"char\">6.889 (0.695)</td></tr><tr><td align=\"left\" rowspan=\"2\">Service attitude</td><td align=\"left\">100% favorable rating</td><td char=\"(\" align=\"char\">7.115 (0.699)</td><td char=\".\" align=\"char\" rowspan=\"2\">0.127</td><td char=\"(\" align=\"char\">7.327 (0.897)</td><td char=\".\" align=\"char\" rowspan=\"2\">0.012</td></tr><tr><td align=\"left\">75% favorable rating</td><td char=\"(\" align=\"char\">6.840 (0.595)</td><td char=\"(\" align=\"char\">6.750 (0.714)</td></tr><tr><td align=\"left\" rowspan=\"2\">Communication skills</td><td align=\"left\">100% favorable rating</td><td char=\"(\" align=\"char\">7.000 (0.993)</td><td char=\".\" align=\"char\" rowspan=\"2\">0.321</td><td char=\"(\" align=\"char\">7.212 (0.836)</td><td char=\".\" align=\"char\" rowspan=\"2\">0.033</td></tr><tr><td align=\"left\">75% favorable rating</td><td char=\"(\" align=\"char\">6.753 (0.793)</td><td char=\"(\" align=\"char\">6.639 (1.050)</td></tr><tr><td align=\"left\" rowspan=\"2\">Response speed</td><td align=\"left\">100% favorable rating</td><td char=\"(\" align=\"char\">7.090 (0.910)</td><td char=\".\" align=\"char\" rowspan=\"2\">0.072</td><td char=\"(\" align=\"char\">7.038 (0.734)</td><td char=\".\" align=\"char\" rowspan=\"2\">0.300</td></tr><tr><td align=\"left\">75% favorable rating</td><td char=\"(\" align=\"char\">6.716 (0.478)</td><td char=\"(\" align=\"char\">6.843 (0.625)</td></tr><tr><td align=\"left\" rowspan=\"2\">Service commitment</td><td align=\"left\">100% favorable rating</td><td char=\"(\" align=\"char\">7.205 (1.360)</td><td char=\".\" align=\"char\" rowspan=\"2\">0.087</td><td char=\"(\" align=\"char\">7.231 (1.317)</td><td char=\".\" align=\"char\" rowspan=\"2\">0.105</td></tr><tr><td align=\"left\">75% favorable rating</td><td char=\"(\" align=\"char\">6.593 (1.192)</td><td char=\"(\" align=\"char\">6.778 (0.543)</td></tr></tbody></table></table-wrap>", "<table-wrap id=\"Tab2\"><label>Table 2</label><caption><p>Two-way multivariate ANOVA results of patients’ scores of willingness to choose with different levels of health involvement.</p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th align=\"left\">Health involvement</th><th align=\"left\">Source of variance</th><th align=\"left\">df</th><th align=\"left\">F</th><th align=\"left\">Sig</th><th align=\"left\">Partial η²</th></tr></thead><tbody><tr><td align=\"left\" rowspan=\"4\">Low level health involvement</td><td align=\"left\">CT</td><td align=\"left\">1</td><td char=\".\" align=\"char\">22.528</td><td char=\".\" align=\"char\">0.000</td><td char=\".\" align=\"char\">0.324</td></tr><tr><td align=\"left\">AT</td><td align=\"left\">1</td><td char=\".\" align=\"char\">31.853</td><td char=\".\" align=\"char\">0.000</td><td char=\".\" align=\"char\">0.404</td></tr><tr><td align=\"left\">CT*AT</td><td align=\"left\">1</td><td char=\".\" align=\"char\">0.323</td><td char=\".\" align=\"char\">0.572</td><td char=\".\" align=\"char\">0.007</td></tr><tr><td align=\"left\">Error</td><td align=\"left\">47</td><td char=\".\" align=\"char\" colspan=\"3\"/></tr><tr><td align=\"left\" rowspan=\"4\">High level health involvement</td><td align=\"left\">CT</td><td align=\"left\">1</td><td char=\".\" align=\"char\">28.389</td><td char=\".\" align=\"char\">0.000</td><td char=\".\" align=\"char\">0.387</td></tr><tr><td align=\"left\">AT</td><td align=\"left\">1</td><td char=\".\" align=\"char\">25.968</td><td char=\".\" align=\"char\">0.000</td><td char=\".\" align=\"char\">0.366</td></tr><tr><td align=\"left\">CT*AT</td><td align=\"left\">1</td><td char=\".\" align=\"char\">13.049</td><td char=\".\" align=\"char\">0.001</td><td char=\".\" align=\"char\">0.225</td></tr><tr><td align=\"left\">Error</td><td align=\"left\">45</td><td char=\".\" align=\"char\" colspan=\"3\"/></tr></tbody></table></table-wrap>", "<table-wrap id=\"Tab3\"><label>Table 3</label><caption><p>Simple effect analysis results of patients’ scores of willingness to choose with different levels of health involvement.</p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th align=\"left\" rowspan=\"2\">Health involvement</th><th align=\"left\" rowspan=\"2\" colspan=\"2\">Variance</th><th align=\"left\" rowspan=\"2\">Mean (S.D.)</th><th align=\"left\" colspan=\"2\">95% confidence interval for difference</th><th align=\"left\" rowspan=\"2\">Sig</th></tr><tr><th align=\"left\">2.5% </th><th align=\"left\">97.5%</th></tr></thead><tbody><tr><td align=\"left\" rowspan=\"8\">Low level health involvement</td><td align=\"left\" rowspan=\"2\">H</td><td align=\"left\">h</td><td char=\"(\" align=\"char\">7.674 (1.075)</td><td char=\".\" align=\"char\" rowspan=\"2\">0.578</td><td char=\".\" align=\"char\" rowspan=\"2\">1.269</td><td char=\".\" align=\"char\" rowspan=\"2\">0.000</td></tr><tr><td align=\"left\">l</td><td char=\"(\" align=\"char\">6.750 (1.112)</td></tr><tr><td align=\"left\" rowspan=\"2\">L</td><td align=\"left\">h</td><td char=\"(\" align=\"char\">6.847 (1.138)</td><td char=\".\" align=\"char\" rowspan=\"2\">0.390</td><td char=\".\" align=\"char\" rowspan=\"2\">1.207</td><td char=\".\" align=\"char\" rowspan=\"2\">0.000</td></tr><tr><td align=\"left\">l</td><td char=\"(\" align=\"char\">6.049 (1.645)</td></tr><tr><td align=\"left\" rowspan=\"2\">h</td><td align=\"left\">H</td><td char=\"(\" align=\"char\">7.674 (1.075)</td><td char=\".\" align=\"char\" rowspan=\"2\">0.523</td><td char=\".\" align=\"char\" rowspan=\"2\">1.130</td><td char=\".\" align=\"char\" rowspan=\"2\">0.000</td></tr><tr><td align=\"left\">L</td><td char=\"(\" align=\"char\">6.847 (1.138)</td></tr><tr><td align=\"left\" rowspan=\"2\">l</td><td align=\"left\">H</td><td char=\"(\" align=\"char\">6.750 (1.112)</td><td char=\".\" align=\"char\" rowspan=\"2\">0.237</td><td char=\".\" align=\"char\" rowspan=\"2\">1.165</td><td char=\".\" align=\"char\" rowspan=\"2\">0.004</td></tr><tr><td align=\"left\">L</td><td char=\"(\" align=\"char\">6.049 (1.645)</td></tr><tr><td align=\"left\" rowspan=\"8\">High level health involvement</td><td align=\"left\" rowspan=\"2\">H</td><td align=\"left\">h</td><td char=\"(\" align=\"char\">8.000 (1.035)</td><td char=\".\" align=\"char\" rowspan=\"2\">0.725</td><td char=\".\" align=\"char\" rowspan=\"2\">1.463</td><td char=\".\" align=\"char\" rowspan=\"2\">0.000</td></tr><tr><td align=\"left\">l</td><td char=\"(\" align=\"char\">6.906 (1.340)</td></tr><tr><td align=\"left\" rowspan=\"2\">L</td><td align=\"left\">h</td><td char=\"(\" align=\"char\">6.891 (1.201)</td><td char=\".\" align=\"char\" rowspan=\"2\">− 0.033</td><td char=\".\" align=\"char\" rowspan=\"2\">0.641</td><td char=\".\" align=\"char\" rowspan=\"2\">0.076</td></tr><tr><td align=\"left\">l</td><td char=\"(\" align=\"char\">6.587 (1.416)</td></tr><tr><td align=\"left\" rowspan=\"2\">h</td><td align=\"left\">H</td><td char=\"(\" align=\"char\">8.000 (1.035)</td><td char=\".\" align=\"char\" rowspan=\"2\">0.764</td><td char=\".\" align=\"char\" rowspan=\"2\">1.453</td><td char=\".\" align=\"char\" rowspan=\"2\">0.000</td></tr><tr><td align=\"left\">L</td><td char=\"(\" align=\"char\">6.891 (1.201)</td></tr><tr><td align=\"left\" rowspan=\"2\">l</td><td align=\"left\">H</td><td char=\"(\" align=\"char\">6.906 (1.340)</td><td char=\".\" align=\"char\" rowspan=\"2\">− 0.033</td><td char=\".\" align=\"char\" rowspan=\"2\">0.671</td><td char=\".\" align=\"char\" rowspan=\"2\">0.075</td></tr><tr><td align=\"left\">L</td><td char=\"(\" align=\"char\">6.587 (1.416)</td></tr></tbody></table></table-wrap>" ]

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[ "<supplementary-material content-type=\"local-data\" id=\"MOESM1\"></supplementary-material>" ]

[ "<table-wrap-foot><p>Note: CT, cognitive trust; AT, affective trust; all the cognitive and affective trust scores ranged between 1 to 9.</p></table-wrap-foot>", "<table-wrap-foot><p>Note: CT, cognitive trust; AT, affective trust.</p></table-wrap-foot>", "<table-wrap-foot><p>Note: H, high level of cognitive trust; L, low level of cognitive trust; h, high level of affective trust; l, low level of affective trust; all the scores of willingness to choose ranged between 1 and 9.</p></table-wrap-foot>", "<fn-group><fn><p><bold>Publisher's note</bold></p><p>Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.</p></fn></fn-group>" ]

[ "<graphic xlink:href=\"41598_2024_51519_Fig1_HTML\" id=\"MO1\"/>", "<graphic xlink:href=\"41598_2024_51519_Fig2_HTML\" id=\"MO2\"/>", "<graphic xlink:href=\"41598_2024_51519_Fig3_HTML\" id=\"MO3\"/>" ]

[ "<media xlink:href=\"41598_2024_51519_MOESM1_ESM.pdf\"><caption><p>Supplementary Information.</p></caption></media>" ]

{ "acronym": [], "definition": [] }

[ "<title>Introduction</title>", "<p id=\"Par2\">Advances in computational materials science have enabled the accurate prediction of novel materials possessing exceptional properties. Remarkably, these computational advancements have facilitated the successful experimental synthesis of materials that exhibit the anticipated properties. Some predicted materials, such as near-room-temperature superconductors, have been successfully synthesized under high-pressure conditions, with their superconducting temperatures in accordance with density functional theory (DFT) calculations^{##UREF##0##1##,##UREF##1##2##}. To achieve accurate predictions, <italic>a priori</italic> knowledge of plausible molecular and crystal structures play a vital role in both theoretical and experimental studies. Several algorithms, such as evolutionary algorithms, swarm particle optimization, random sampling method, and etc., have been employed for structure prediction^{##REF##16821993##3##–##REF##21406903##5##}. These algorithms rely on identifying local minima on the potential energy landscape obtained from, for example, DFT calculations and machine learning-driven methods^{##UREF##3##6##–##UREF##5##8##}. In the case of crystal structures, where atoms are arranged in a three-dimensional space with periodic boundaries, additional criteria are necessary to enforce crystal symmetry constraints^{##REF##21406903##5##}.</p>", "<p id=\"Par3\">Recent approach for structure prediction employs denoising diffusion models to perform probabilistic inference. These models sample molecular and crystal structures from a probability distribution of atomic coordinates and types^{##UREF##6##9##–##UREF##9##12##}, bypassing the computationally intensive DFT calculation that tediously determines the potential energy landscape. By leveraging sufficiently large datasets containing various compounds, this method enables the generation of diverse compositions and combinations of elements simultaneously. Furthermore, the models allow for the control of desired physical properties of the generated structures through conditional probability sampling^{##REF##30016587##13##–##UREF##10##15##}. These machine learning-based algorithms also hold promise for solving inverse problem efficiently, resolving structures from experimental characterizations, e.g., x-ray absorption spectroscopy and other techniques, a challenging problem in materials science^{##UREF##11##16##–##UREF##12##18##}.</p>", "<p id=\"Par4\">There are two primary types of denoising diffusion models: score matching approach and denoising diffusion probabilistic models (DDPM)^{##UREF##13##19##–##UREF##15##21##}. These two models can denoise (reverse) a normal distribution such that the distribution gradually transforms into the data distribution of interest. The score matching approach estimates the score function of the perturbed data directing the normal distribution toward the data distribution and employing large step sizes of variance. In contrast, DDPM gradually denoises the random noise through a joint distribution of data perturbed at different scales of variance. Both approaches have been utilized for generating molecular structures^{##UREF##7##10##–##UREF##9##12##}. However, models based on DDPM tend to sample molecules with higher diversity and energy closer to the ground truth than models based on the score matching approach^{##UREF##8##11##}.</p>", "<p id=\"Par5\">Since atomic positions in crystal structures are periodic and can be invariant under some rotation groups depending on their crystal symmetry, the core neural networks should favourably possess roto-translational equivariance^{##UREF##16##22##–##UREF##18##24##}. Xie et al.^{##UREF##6##9##} has proposed a model for crystal prediction by a combination between variational autoencoder (VAE)^{##UREF##19##25##} and the denoising diffusion model, called crystal diffusion VAE (CDVAE). The model employs the score matching approach with (annealed) Langevin dynamics to generate new crystal structures from random coordinates^{##UREF##13##19##}. The neural networks for an encoder and the diffusion model are roto-translationally equivariant. As a result, CDVAE can generate crystal structures with realistic bond lengths and respect crystal symmetry.</p>", "<p id=\"Par6\">Because of the periodic boundary condition imposed on the unit cell, gradually injecting sufficiently strong noises (in the forward process) to the fractional coordinates can lead to the uniform distribution of atomic positions at late times, the consequence of ergodicity in statistical mechanical sense. Rather than beginning with a Gaussian distribution and denoising it as in the original CDVAE formulation, Jiao et al.^{##UREF##20##26##} perturbed and sampled atomic positions beginning with a wrapped normal distribution which satisfies the periodic boundary condition. With this approach, the reconstruction performance has been significantly improved. Other circular (periodic) distributions, e.g., the wrapped normal and von Mises distributions, are not natural for DDPM framework since there is no known analytical method to explicitly incorporate such distributions into the framework. There, one needs to resort to an additional sampling procedure to construct the DDPM^{##UREF##21##27##}.</p>", "<p id=\"Par7\">In this work, we introduce a crystal generation framework called diffusion probabilistic CDVAE (DP-CDVAE). Similar to the original CDVAE, our model consists of two parts: the VAE part and the diffusion part. The purpose of the VAE part is to predict the lattice parameters and the number of atoms in the unit cell of crystal structures. On the other hand, the diffusion part utilizes the diffusion probabilistic approach to denoise fractional coordinates and predict atomic coordinates. By employing the DDPM instead of the score matching approach, the DP-CDVAE model shows reconstruction and generation task performances that are statistically comparable to those obtained from original CDVAE. Importantly, we demonstrate the significantly higher ground-state generation performance of DP-CDVAE, through the distance comparison between generated structures and those optimized using the DFT method. We also analyze the changes in energy and volume after relaxation to gain further insights into models’ capabilities.</p>" ]

[ "<title>Methods</title>", "<title>Diffusion probabilistic model</title>", "<p id=\"Par23\">In the diffusion probabilistic model, the data distribution is gradually perturbed by noise in the forward process until it becomes a normal distribution at late times. In this study, the distribution of the fractional coordinate () is considered since their values of every crystal structures distribute over the same range ,i.e., . The Markov process is assumed for the forward diffusion such that the joint distribution is a product of the conditional distributions conditioned on the knowledge of the fractional coordinate at the previous time step:where the data distribution of the fractional coordinate, <italic>t</italic> is the discretized diffusion time step, <italic>T</italic> is the final diffusion time, is a noise schedule with a sigmoid scheduler^{##UREF##37##44##}, and the conditional is a Gaussian kernel due to the Markov diffusion process assumption. Then can be expressed in the Langevin’s form through the reparameterization trick aswhere , and . This update rule does not necessitate to remain in ; however, we can impose the periodic boundary condition for the fractional coordinate so thatThen, .</p>", "<p id=\"Par24\">In the reverse diffusion process, if the consecutive discretized time step is small compared to the diffusion timescale, the reverse coordinate trajectories can be approximately sampled also from the product of Gaussian diffusion kernels aswhereThe reverse conditional distribution can be trained by minimizing the Kullback-Leibler divergence between and , the posterior of the corresponding forward process^{##UREF##15##21##}. We use GemNetT for the diffusion network to train the parametrized noise ^{##UREF##38##45##}. Then, the coordinate in the earlier time can be sampled from , whose corresponding reverse Langevin’s dynamics readswhere . Crucially, we empirically found that the final reconstruction performance is considerably improved when we impose the periodic boundary condition on the fractional coordinate at every time step such that and in the first term of Eq. (##FORMU##113##6##) is replaced by . Namely, in our modified reverse process, the coordinate is sampled fromAn illustration of denoising atomic coordinates with Eq. (##FORMU##118##7##) is demonstrated in Fig. ##FIG##1##2##. The model performance using Eq. (##FORMU##113##6##) is shown in Table ##SUPPL##0##S1## in SI, whereas the performance using Eq. (##FORMU##118##7##) is shown in Table ##TAB##0##1##.</p>", "<title>Graph neural networks</title>", "<p id=\"Par25\">Graph neural networks architecture facilitate machine learning of crystal graphs , graph representations of crystal structures. and are sets of nodes and edges, respectively, defined aswhere <italic>n</italic> and <italic>m</italic> are indices of atoms in a crystal structure, is a vector of <italic>M</italic> features of an atom in the unit cell, is a translation vector, and is a lattice matrix that converts a fractional coordinate into its atomic Cartesian coordinate . The atomic features, fractional coordinates, and atomic Cartesian coordinates of the crystal structure are vectorized (concatenated) as , , and . Three graph neural networks implemented in this work are DimeNet^{##UREF##22##28##}, GINE^{##UREF##23##29##}, and GemNetT^{##UREF##38##45##}. DimeNet and GINE are employed for encoders, and GemNetT is used for a diffusion network. DimeNet and GemNetT, whose based architecture concerns geometry of the graphs, are rotationally equivariant. GemNetT has been devised by incorporating the polar angles between four atoms into DimeNet. This development grants GemNetT a higher degree of expressive power compared to DimeNet^{##UREF##39##46##}. Furthermore, GINE has been developed to distinguish a graph isomorphism, but not graph geometry nor the distance between nodes, which is important for our study. Thus we supplement the edge attributes into GINE with the distances between atoms, i.e. .</p>", "<title>DP-CDVAE’s architecture</title>", "<p id=\"Par26\">The forward process of DP-CDVAE model is illustrated in Fig. ##FIG##0##1##. The model is a combination of two generative models, which are VAE and diffusion probabilistic model. The pristine crystal structures consist of the fractional coordinate (), the lattice matrix (), ground-truth atomic type (<italic>Z</italic>), and the number of atoms in a unit cell (). For crystal graphs of the encoders, their node features are . The number of atoms in a unit cell is encoded through multilayer perceptron before concatenated with the latent features from other graph encoders. They are encoded to train and where is a learnable parameter of the encoders. The latent variables () can be obtained bywhere . Then, will be decoded to compute the lattice lengths and angles, which then yield the lattice matrix (), , and . In the original CDVAE, is the probability vector indicating the fraction of each atomic type in the compound and is used to perturb <italic>Z</italic> bywhere is a multinomial distribution, is a one-hot vector of ground-truth atomic type <italic>Z</italic>, and is the variance for perturbing atomic types at time <italic>t</italic>, which is distinct from used for perturbing the atomic coordinates. Similar to the original CDVAE, is selected from the range of [0.01, 5].</p>", "<p id=\"Par27\">For the diffusion network, the input structures are constructed from , , and where the (Cartesian) atomic coordinates at time <italic>t</italic> are computed by . These are then utilized by the crystal graphs for the diffusion network, whose node features are where is a Fourier embedding feature of (see SI). As proposed by Ho et al.^{##UREF##15##21##}, we use the simple loss to train the model such thatSince the diffusion model is trained to predict both and , so the loss of the diffusion network iswhere is the cross entropy loss, is a loss scaling factor, and where . In this work, <italic>t</italic> is randomly chosen for each crystal graph and randomly reinitialized for each epoch in the training process. The total loss in the trainig process is shown in Eq. ##SUPPL##0##S1## in SI.</p>", "<p id=\"Par28\">In the reverse diffusion process, we measure the model performance of two tasks: reconstruction and generation tasks. For the former task, is obtained from Eq. (##FORMU##148##8##) by using the ground-truth structure as an input of the encoders. For the latter task, , which is then used to predict , , , and concatenate with the node feature of the crystal graph in the diffusion network. At the initial step, , is sampled from the highest probability of , and the final-time coordinate is obtained from sampling a Gaussian distribution, i.e. The coordinates can be denoised using Eq. (##FORMU##118##7##), and the predicted atomic types are updated in each reversed time step by .</p>", "<title>DFT calculations</title>", "<p id=\"Par29\">The Vienna <italic>ab initio</italic> Simulation Package (VASP) was employed for structural relaxations and energy calculations based on DFT^{##UREF##40##47##,##UREF##41##48##}. The calculations were conducted under the generalized gradient approximation (GGA), which is Perdew-Burke-Ernzerhof (PBE) exchange-correlation functional, and the project augmented wave (PAW) method^{##UREF##42##49##,##UREF##43##50##}. The thresholds for energy and force convergence were set to eV and eV/Å, respectively. The plane-wave energy cutoff was set to 800 eV, and the Brillouin zone integration was carried out on a k-point mesh of created by the Monkhorst-Pack method^{##UREF##44##51##,##UREF##45##52##}.</p>" ]

[ "<title>Results</title>", "<p id=\"Par8\">The performances of DP-CDVAE models are herein presented. There are four DP-CDVAE models, differed by the choice of the encoder (see Fig. ##SUPPL##0##S1## in Supplemental Information (SI)). DimeNet has been employed for the main encoder for every DP-CDVAE models^{##UREF##22##28##}. We then modify the encoder of DP-CDVAE to encode the crystal structure by two additional neural networks: a multilayer perceptron that takes the number of atoms in the unit cell () as an input, and a graph isomorphism network (GINE)^{##UREF##23##29##}. Their latent features are combined with the latent features from DimeNet through another multilayer perceptron. The is encoded such that the model can decode the accurately, and GINE encoder is inspired by GeoDiff^{##UREF##8##11##} whose model is a combination of SchNet^{##UREF##24##30##} and GINE which yields better performance.</p>", "<p id=\"Par9\">Three datasets, Perov-5^{##UREF##25##31##,##UREF##26##32##}, Carbon-24^{##UREF##27##33##}, and MP-20^{##UREF##28##34##}, were selected to evaluate the performance of the model. The Perov-5 dataset consists of perovskite materials with cubic structures, but with variations in the combinations of elements within the structures. The Carbon-24 dataset comprises carbon materials, where the data consists of carbon element with various crystal systems obtained from <italic>ab initio</italic> random structure searching algorithm at pressure of 10 GPa^{##UREF##27##33##}. The MP-20 dataset encompasses a wide range of compounds and structure types.</p>", "<title>Reconstruction performance</title>", "<p id=\"Par10\">The reconstruction performance is determined by the similarity between reconstructed and ground-truth structures. The similarity can be evaluated using StructureMatcher algorithm from pymatgen library^{##REF##14691322##35##}. The algorithm takes a pair of crystal structures and performs Niggli reduction to reduce their cells^{##UREF##29##36##,##UREF##30##37##}. They are then compared by determining an average displacement between the two structures. If it falls within the error tolerence, the two structures are matched. The reconstructed and ground-truth structures are similar if they pass the criteria of StructureMatcher which are stol=0.5, angle_tol=10, ltol=0.3. <italic>Match rate</italic> is the percentage of those structures passed the criteria. If the reconstructed and ground-truth structures are similar under the criteria, the root-mean-square distance between their atomic positions is computed and then normalized by , where <italic>V</italic> is the unit-cell volume, and is the number of atoms in the unit cell. An average of the distances of every pair of structures (), computed using the StructureMatcher algorithm, is used as the performance metric.</p>", "<p id=\"Par11\">Table ##TAB##0##1## presents the reconstruction performance of different models for three different datasets: Perov-5, Carbon-24, and MP-20. Note that Fourier-transformed crystal properties (FTCP) model is presented as a baseline model, which is based on VAE. It encodes and decodes both the real-space and reciprocal-space features of crystal structures^{##UREF##31##38##}. For the Perov-5 dataset, the DP-CDVAE model achieves a match rate of 90.04%, indicating its ability to reconstruct a significant portion of the ground-truth structures. This performance is 7.48% lower than the CDVAE model but still demonstrates the effectiveness of our model. In terms of , the DP-CDVAE model achieves a value of 0.0212, comparable to the FTCP model^{##UREF##31##38##}, but slightly higher than the CDVAE model. Similarly, for the Carbon-24 and MP-20 datasets, the DP-CDVAE model performs well in terms of both match rate and . It achieves match rates of 45.57% and 32.42% for Carbon-24 and MP-20, respectively. The corresponding values for Carbon-24 and MP-20 are 0.1513 and 0.0383, respectively, comparable to the CDVAE model.</p>", "<p id=\"Par12\">Regarding the DP-CDVAE+ model, the additional encoding of into the model leads to improved match rates for all datasets, with an increase of 2–5%. This enhancement can be attributed to the accurate prediction of . However, in terms of , only the Perov-5 dataset shows an improvement, with a value of 0.0149. On the other hand, for the Carbon-24 and MP-20 datasets, the values are higher compared to the DP-CDVAE model.</p>", "<p id=\"Par13\">For the DP-CDVAE+GINE and DP-CDVAE++GINE models, the additional encoding of GINE into the models leads to a substantial drop in match rates compared to the DP-CDVAE model, particularly for the Perov-5 and Carbon-24 datasets. In contrast, there is a moderate increase in the match rates for the MP-20 dataset. The values for the Perov-5 and Carbon-24 datasets are comparable to those of the DP-CDVAE model. However, for the MP-20 dataset, the is noticeably higher in the models with GINE encoder compared to the DP-CDVAE model.</p>", "<p id=\"Par14\">Overall, while the reconstruction performance of the DP-CDVAE model may be lower than the CDVAE model in terms of match rate, it still demonstrates competitive performance with relatively low . The match rate can be enhanced by additionally encoding the , but the performance is traded off by the increase in .</p>", "<title>Generation performance</title>", "<p id=\"Par15\">We follow the CDVAE model that used three metrics to determine the generation performance of the models^{##UREF##6##9##}. The first metric is the <italic>Validity</italic> percentage, which encompasses two sub-metrics: <italic>Structural Validity</italic> (Struc.) with a criterion that ensures the distances between every pair of atoms are larger than 0.5 Å, and <italic>Compositional Validity</italic> (Comp.) with a criterion that maintains a neutral total charge in the unit cell. The second metric is called <italic>coverage</italic> (COV), which utilizes structure and composition fingerprints to evaluate the similarity between the generated and ground-truth structures. COV-R (Recall) represents the percentage of ground-truth structures covered by the generated structures. COV-P (Precision) represents the percentage of generated structures that are similar to the ground-truth structures, indicating the quality of the generation. The third metric is the Wasserstein distance between property distributions of generated and ground-truth structures. Three property statistics are density (), which is total atomic mass per volume (unit g/cm), formation energy (, unit eV/atom), and the number of elements in the unit cell (# elem.). A separated and pre-trained neural network is employed to predict <italic>E</italic> of the structures where the detail of the pre-training can be found in Ref.^{##UREF##6##9##}. The first and second metrics are computed over 10,240 generated structures, and 1000 structures are randomly chosen from the generated structures that pass the validity tests to compute the third metric. The ground-truth structures used to evaluate the generation performance are from the test set.</p>", "<p id=\"Par16\">In Table ##TAB##1##2##, the DP-CDVAE model achieves a validity rate of 100% for the Perov-5 dataset and close to 100% for the Carbon-24 and MP-20 datasets in terms of structure. The validity rate for composition is comparable to that of the CDVAE model. The DP-CDVAE model also demonstrates comparable COV-R values to the CDVAE model across all three datasets. Furthermore, the DP-CDVAE models with and/or GINE encoders exhibit similar Validity and COV-R metrics to those of the DP-CDVAE model. However, for COV-P, all DP-CDVAE models yield lower values compared to CDVAE.</p>", "<p id=\"Par17\">On the other hand, our models show significant improvements in property statistics. In the case of the MP-20 dataset, the DP-CDVAE models, particularly those with the GINE encoder, yield substantially smaller Wasserstein distances for , , and the number of elements compared to other models. For the Carbon-24 dataset, our models also exhibit a smaller Wasserstein distance for compared to the CDVAE model.</p>", "<title>Ground-state performance</title>", "<p id=\"Par18\">Another objective of the structure generator is to generate novel structures that also are close to the ground state. To verify that, the generated structures are relaxed using the DFT calculation where the <italic>relaxed structures</italic> exhibit balanced internal stresses with external pressures and reside in local energy minima. These relaxed structures are then compared with the generated structures to evaluate their similarity. In this study, we have chosen a set of 100 generated structures from each of CDVAE, CDVAE+Fourier, and DP-CDVAE models for relaxation where CDVAE+Fourier model is CDVAE model with Fourier embedding features of the perturbed coordinates. However, relaxation procedures for multi-element compounds can be computationally intensive. To address this, we have specifically selected materials composed solely of carbon atoms, using the model trained on Carbon-24 dataset. This selection ensures a convergence of the self-consistent field in DFT calculation. Moreover, in the relaxation, we consider the ground state of the relaxed structures at a temperature of 0 K and a pressure of 10 GPa since the carbon structures in the training set are stable at 10 GPa^{##UREF##27##33##}.</p>", "<p id=\"Par19\">We here introduce a ground-state performance presented in Table ##TAB##2##3##. The StructureMatcher with the same criteria as in the reconstruction performance is used to evaluate the similarity between the generated and relaxed structures. The relaxed structure was used as a based structure to determine if the generated structure can be matched. Four metrics used to determine the similarity are 1) match rate, 2) , 3) and 4) . The and represent the root mean square differences in volume and energy, respectively, between the generated structures and the relaxed structures in the dataset.</p>", "<p id=\"Par20\">In Table ##TAB##2##3##, the DP-CDVAE model achieves the highest match rate and the lowest and . Although the CDVAE+Fourier model achieves the lowest , the DP-CDVAE model demonstrates the that is comparable to that of the CDVAE+Fourier model.</p>" ]

[ "<title>Discussion</title>", "<p id=\"Par21\">The DP-CDVAE models significantly enhance the generation performance, particularly in terms of property statistics, while maintaining comparable COVs to those of CDVAE. Specifically, for Carbon-24 and MP-20 datasets, the density distributions between the generated and ground-truth structures from DP-CDVAE models exhibit substantially smaller Wasserstein distance compared those of the CDVAE model (see Table ##TAB##1##2##). The of the DP-CDVAE model presented in Table ##TAB##2##3## is significantly lower than that of the original CDVAE. This is corresponding to smaller Wasserstein distance of shown in Table ##TAB##1##2##. The DP-CDVAE model also demonstrates significantly smaller than the original CDVAE. These suggest that our lattice generation closely approximates the relaxed lattice, while also achieving atomic positions that closely resemble the ground-state configuration. This could be an attribute of the DP approach that gradually learns perturbed coordinates, which in turn enhances the quality of sampled coordinates during the reverse process, much like its successful applications in image and molecular structure generation^{##UREF##8##11##,##UREF##15##21##,##UREF##32##39##}. Additionally, the distribution of the number of elements in the unit cells is relatively similar to that of the data in the test set, particularly in the results from the models with GINE encoder. This could be attributed to the capability of GINE to search for graph isomorphism^{##UREF##33##40##}. For the MP-20 dataset, the Wasserstein distances of the values generated by our models are notably lower. This suggests that the crystal structures we generate are more likely to have values within the specific range we are interested in. Hence, by selecting an appropriate training set, we can concentrate on structures with values falling within the synthesizable candidate range.</p>", "<p id=\"Par22\">Moreover, is the energy difference between the generated structures and their corresponding relaxed structures. The ground-state energy represents a local minimum that the generated structure is relaxed towards. A value of close to zero indicates that the generated structure is in close proximity to the ground state. In Table ##TAB##2##3##, it can be observed that our model achieves the value of 400.7 meV/atom which is about 68.1 meV/atom lower than the of CDVAE. The mode of of our model is 64–128 meV/atom, which is lower than its root-mean-square value (see Fig. ##SUPPL##0##S2## in SI). Nevertheless, both the and the mode of exhibit relatively high values. In many cases, the formation energy of synthesized compounds is reported to be above the convex hull less than 36 meV/atom^{##UREF##34##41##–##UREF##36##43##}. To obviate the need for time-consuming DFT relaxation, it is essential for the generated structures to be even closer to the ground state. Therefore, achieving lower values remains a milestone for future work.</p>" ]

[]

[ "<p id=\"Par1\">The crystal diffusion variational autoencoder (CDVAE) is a machine learning model that leverages score matching to generate realistic crystal structures that preserve crystal symmetry. In this study, we leverage novel diffusion probabilistic (DP) models to denoise atomic coordinates rather than adopting the standard score matching approach in CDVAE. Our proposed DP-CDVAE model can reconstruct and generate crystal structures whose qualities are statistically comparable to those of the original CDVAE. Furthermore, notably, when comparing the carbon structures generated by the DP-CDVAE model with relaxed structures obtained from density functional theory calculations, we find that the DP-CDVAE generated structures are remarkably closer to their respective ground states. The energy differences between these structures and the true ground states are, on average, 68.1 meV/atom lower than those generated by the original CDVAE. This significant improvement in the energy accuracy highlights the effectiveness of the DP-CDVAE model in generating crystal structures that better represent their ground-state configurations.</p>", "<title>Subject terms</title>" ]

[ "<title>Supplementary Information</title>", "<p>\n</p>" ]

[ "<title>Supplementary Information</title>", "<p>The online version contains supplementary material available at 10.1038/s41598-024-51400-4.</p>", "<title>Acknowledgements</title>", "<p>This research project is supported by the Second Century Fund (C2F), Chulalongkorn University. This Research is funded by Thailand Science Research and Innovation Fund Chulalongkorn University and National Research Council of Thailand (NRCT): (NRCT5-RSA63001-04). T.C. acknowledges funding support from the NSRF via the Program Management Unit for Human Resources and Institutional Development, Research and Innovation [grant number B05F650024]. The authors acknowledge high performance computing resources including NVIDIA A100 GPUs from Chula Intelligent and Complex Systems Lab, Faculty of Science, and from the Center for AI in Medicine (CU-AIM), Faculty of Medicine, Chulalongkorn University, Thailand. We acknowledge the supporting computing infrastructure provided by NSTDA, CU, CUAASC, NSRF via PMUB [B05F650021, B37G660013] (Thailand). URL:www.e-science.in.th. The Computational Materials Physics (CMP) Project, SLRI, Thailand, is acknowledged for providing computational resource.</p>", "<title>Author contributions</title>", "<p>T.P and N.C. implemented the methodology into the code. T.P., N.C., R.A., and C.A. designed the project and contributed to discussion of the results. S.K. performed the DFT calculations. T.P., N.C., R.A., and T.C. contributed to discussion of the theory and the algorithm. T.P. and T.C. wrote the paper. T.B. supervised all aspects of the project.</p>", "<title>Data availability</title>", "<p>The code and datasets generated and/or analysed during the current study are available at <ext-link ext-link-type=\"uri\" xlink:href=\"https://github.com/trachote/dp-cdvae\">https://github.com/trachote/dp-cdvae</ext-link>.</p>", "<title>Competing interests</title>", "<p id=\"Par30\">The authors declare no competing interests.</p>" ]

[ "<fig id=\"Fig1\"><label>Figure 1</label><caption><p>The schematic summarizing the architecture for training the DP-CDVAE model. Multiple sub-networks are trained to minimize the total loss function. The encoder () compresses input pristine crystal structures into the latent feature (). The predicted lattice parameters (), the predicted number of atoms (), and are decoded from . Here, enables the sampling of atomic types (), and all the decoded features enable the reconstruction of crystal structures. The input fractional coordinates undergo perturbation (dash-dotted line) at time step <italic>t</italic> and then are transformed by to satisfy the periodic boundary condition (dotted line), serving as the coordinates for the reconstructed crystal structures. These reconstructed structures, , are subsequently fed into the diffusion network (), where is a node feature composing of , , and <italic>t</italic>. The diffusion network predicts the noise added to the fractional coordinates () as well as the one-hot vector of atomic types (). Dashed-line boxes represent the unit cells of the crystal structures.</p></caption></fig>", "<fig id=\"Fig2\"><label>Figure 2</label><caption><p>The schematic depicting the reverse diffusion process of the DP-CDVAE model. Initially, atomic coordinates are sampled from a normal distribution and subsequently mapped into the unit cell (dashed-line box) using the periodic boundary-imposing function . White circles outside the unit cell depict the atomic coordinates prior to the periodic boundary condition is imposed, while colored circles represent atoms that are inside the unit cell of interest. The action of on the atoms outside the unit cell is represented by an arrow that translates the white circles into colored circles in the unit cell. Left to right show the reverse direction of the arrow of time, depicting the reverse diffusion process.</p></caption></fig>" ]

[ "<table-wrap id=\"Tab1\"><label>Table 1</label><caption><p>Reconstruction performance.</p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th align=\"left\" rowspan=\"2\">Models</th><th align=\"left\" colspan=\"3\">Match rate (%) </th><th align=\"left\" colspan=\"3\">\n</th></tr><tr><th align=\"left\">Perov-5</th><th align=\"left\">Carbon-24</th><th align=\"left\">MP-20</th><th align=\"left\">Perov-5</th><th align=\"left\">Carbon-24</th><th align=\"left\">MP-20</th></tr></thead><tbody><tr><td align=\"left\">FTCP^{##UREF##6##9##}</td><td align=\"left\"><bold>99.34</bold></td><td align=\"left\"><bold>62.28</bold></td><td align=\"left\"><bold>69.89</bold></td><td align=\"left\">0.0259</td><td align=\"left\">0.2563</td><td align=\"left\">0.1593</td></tr><tr><td align=\"left\">CDVAE^{##UREF##6##9##}</td><td align=\"left\">97.52</td><td align=\"left\">55.22</td><td align=\"left\">45.43</td><td align=\"left\">0.0156</td><td align=\"left\"><bold>0.1251</bold></td><td align=\"left\"><bold>0.0356</bold></td></tr><tr><td align=\"left\">DP-CDVAE</td><td align=\"left\">90.04</td><td align=\"left\">45.57</td><td align=\"left\">32.42</td><td align=\"left\">0.0212</td><td align=\"left\">0.1513</td><td align=\"left\">0.0383</td></tr><tr><td align=\"left\">DP-CDVAE+</td><td align=\"left\">91.86</td><td align=\"left\">50.99</td><td align=\"left\">36.17</td><td align=\"left\"><bold>0.0149</bold></td><td align=\"left\">0.1612</td><td align=\"left\">0.0560</td></tr><tr><td align=\"left\">DP-CDVAE+GINE</td><td align=\"left\">80.50</td><td align=\"left\">49.02</td><td align=\"left\">34.08</td><td align=\"left\">0.0214</td><td align=\"left\">0.1599</td><td align=\"left\">0.0455</td></tr><tr><td align=\"left\">DP-CDVAE++GINE</td><td align=\"left\">88.30</td><td align=\"left\">38.28</td><td align=\"left\">37.44</td><td align=\"left\">0.0180</td><td align=\"left\">0.1921</td><td align=\"left\">0.0525</td></tr></tbody></table></table-wrap>", "<table-wrap id=\"Tab2\"><label>Table 2</label><caption><p>Generation performance.</p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th align=\"left\" rowspan=\"2\">Datasets</th><th align=\"left\" rowspan=\"2\">Models</th><th align=\"left\" colspan=\"2\">Validity (%) </th><th align=\"left\" colspan=\"2\">COV (%) </th><th align=\"left\" colspan=\"3\">Property statistics </th></tr><tr><th align=\"left\">Struc.</th><th align=\"left\">Comp.</th><th align=\"left\">R.</th><th align=\"left\">P.</th><th align=\"left\"></th><th align=\"left\"></th><th align=\"left\"># elem.</th></tr></thead><tbody><tr><td align=\"left\" rowspan=\"7\">Perov-5</td><td align=\"left\">G-SchNet^{##UREF##6##9##}</td><td align=\"left\">99.92</td><td align=\"left\">98.79</td><td align=\"left\">0.18</td><td align=\"left\">0.23</td><td align=\"left\">1.625</td><td align=\"left\">4.746</td><td align=\"left\">0.0368</td></tr><tr><td align=\"left\">P-G-SchNet^{##UREF##6##9##}</td><td align=\"left\">79.63</td><td align=\"left\"><bold>99.13</bold></td><td align=\"left\">0.37</td><td align=\"left\">0.25</td><td align=\"left\">0.2755</td><td align=\"left\">1.388</td><td align=\"left\">0.4552</td></tr><tr><td align=\"left\">CDVAE^{##UREF##6##9##}</td><td align=\"left\"><bold>100</bold></td><td align=\"left\">98.59</td><td align=\"left\">99.45</td><td align=\"left\"><bold>98.46</bold></td><td align=\"left\">0.1258</td><td align=\"left\"><bold>0.0264</bold></td><td align=\"left\">0.0628</td></tr><tr><td align=\"left\">DP-CDVAE</td><td align=\"left\"><bold>100</bold></td><td align=\"left\">98.07</td><td align=\"left\">99.52</td><td align=\"left\">98.39</td><td align=\"left\">0.1807</td><td align=\"left\">0.0713</td><td align=\"left\">0.0767</td></tr><tr><td align=\"left\">DP-CDVAE+</td><td align=\"left\">99.99</td><td align=\"left\">97.34</td><td align=\"left\"><bold>99.55</bold></td><td align=\"left\">97.22</td><td align=\"left\"><bold>0.1027</bold></td><td align=\"left\">0.0287</td><td align=\"left\">0.0437</td></tr><tr><td align=\"left\">DP-CDVAE+GINE</td><td align=\"left\"><bold>100</bold></td><td align=\"left\">96.11</td><td align=\"left\">98.94</td><td align=\"left\">95.63</td><td align=\"left\">0.2114</td><td align=\"left\">0.0832</td><td align=\"left\">0.0498</td></tr><tr><td align=\"left\">DP-CDVAE++GINE</td><td align=\"left\"><bold>100</bold></td><td align=\"left\">97.09</td><td align=\"left\">99.52</td><td align=\"left\">96.73</td><td align=\"left\">0.1368</td><td align=\"left\">0.0425</td><td align=\"left\"><bold>0.0210</bold></td></tr><tr><td align=\"left\" rowspan=\"7\">Carbon-24</td><td align=\"left\">G-SchNet^{##UREF##6##9##}</td><td align=\"left\">99.94</td><td align=\"left\">–</td><td align=\"left\">0.00</td><td align=\"left\">0.00</td><td align=\"left\">0.9427</td><td align=\"left\">1.320</td><td align=\"left\">–</td></tr><tr><td align=\"left\">P-G-SchNet^{##UREF##6##9##}</td><td align=\"left\">48.39</td><td align=\"left\">–</td><td align=\"left\">0.00</td><td align=\"left\">0.00</td><td align=\"left\">1.533</td><td align=\"left\">134.7</td><td align=\"left\">–</td></tr><tr><td align=\"left\">CDVAE^{##UREF##6##9##}</td><td align=\"left\"><bold>100</bold></td><td align=\"left\">–</td><td align=\"left\">99.80</td><td align=\"left\"><bold>83.08</bold></td><td align=\"left\">0.1407</td><td align=\"left\">0.2850</td><td align=\"left\">–</td></tr><tr><td align=\"left\">DP-CDVAE</td><td align=\"left\">99.92</td><td align=\"left\">–</td><td align=\"left\">99.56</td><td align=\"left\">77.98</td><td align=\"left\">0.1109</td><td align=\"left\"><bold>0.2596</bold></td><td align=\"left\">–</td></tr><tr><td align=\"left\">DP-CDVAE+</td><td align=\"left\">99.73</td><td align=\"left\">–</td><td align=\"left\">99.61</td><td align=\"left\">72.29</td><td align=\"left\">0.1080</td><td align=\"left\">0.3030</td><td align=\"left\">–</td></tr><tr><td align=\"left\">DP-CDVAE+GINE</td><td align=\"left\">99.50</td><td align=\"left\">–</td><td align=\"left\"><bold>100</bold></td><td align=\"left\">68.13</td><td align=\"left\"><bold>0.0977</bold></td><td align=\"left\">0.3623</td><td align=\"left\">–</td></tr><tr><td align=\"left\">DP-CDVAE++GINE</td><td align=\"left\">98.61</td><td align=\"left\">–</td><td align=\"left\">99.21</td><td align=\"left\">65.13</td><td align=\"left\">0.1267</td><td align=\"left\">0.4136</td><td align=\"left\">–</td></tr><tr><td align=\"left\" rowspan=\"7\">MP-20</td><td align=\"left\">G-SchNet^{##UREF##6##9##}</td><td align=\"left\">99.65</td><td align=\"left\">75.96</td><td align=\"left\">38.33</td><td align=\"left\">99.57</td><td align=\"left\">3.034</td><td align=\"left\">42.09</td><td align=\"left\">0.6411</td></tr><tr><td align=\"left\">P-G-SchNet^{##UREF##6##9##}</td><td align=\"left\">77.51</td><td align=\"left\">76.40</td><td align=\"left\">41.93</td><td align=\"left\"><bold>99.74</bold></td><td align=\"left\">4.04</td><td align=\"left\">2.448</td><td align=\"left\">0.6234</td></tr><tr><td align=\"left\">CDVAE^{##UREF##6##9##}</td><td align=\"left\"><bold>100</bold></td><td align=\"left\"><bold>86.70</bold></td><td align=\"left\">99.15</td><td align=\"left\">99.49</td><td align=\"left\">0.6875</td><td align=\"left\">0.2778</td><td align=\"left\">1.432</td></tr><tr><td align=\"left\">DP-CDVAE</td><td align=\"left\">99.59</td><td align=\"left\">85.44</td><td align=\"left\">98.93</td><td align=\"left\">98.96</td><td align=\"left\">0.4037</td><td align=\"left\">0.1547</td><td align=\"left\">0.9179</td></tr><tr><td align=\"left\">DP-CDVAE+</td><td align=\"left\">99.81</td><td align=\"left\">84.95</td><td align=\"left\">99.36</td><td align=\"left\">99.33</td><td align=\"left\">0.4889</td><td align=\"left\">0.1800</td><td align=\"left\">1.053</td></tr><tr><td align=\"left\">DP-CDVAE+GINE</td><td align=\"left\">99.82</td><td align=\"left\">81.92</td><td align=\"left\"><bold>99.48</bold></td><td align=\"left\">99.00</td><td align=\"left\">0.2785</td><td align=\"left\">0.0603</td><td align=\"left\"><bold>0.5679</bold></td></tr><tr><td align=\"left\">DP-CDVAE++GINE</td><td align=\"left\">99.90</td><td align=\"left\">83.89</td><td align=\"left\">95.51</td><td align=\"left\">99.27</td><td align=\"left\"><bold>0.1790</bold></td><td align=\"left\"><bold>0.0522</bold></td><td align=\"left\">0.6909</td></tr></tbody></table></table-wrap>", "<table-wrap id=\"Tab3\"><label>Table 3</label><caption><p>Ground-state performance.</p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th align=\"left\">Model</th><th align=\"left\">Match rate (%) </th><th align=\"left\">\n</th><th align=\"left\"> (Å/atom) </th><th align=\"left\"> (meV/atom) </th></tr></thead><tbody><tr><td align=\"left\">CDVAE</td><td align=\"left\">63</td><td align=\"left\">0.0321</td><td align=\"left\">0.0227</td><td align=\"left\">468.8</td></tr><tr><td align=\"left\">CDVAE+Fourier</td><td align=\"left\">62</td><td align=\"left\">0.0216</td><td align=\"left\"><bold>0.0157</bold></td><td align=\"left\">494.4</td></tr><tr><td align=\"left\">DP-CDVAE</td><td align=\"left\"><bold>64</bold></td><td align=\"left\"><bold>0.0141</bold></td><td align=\"left\">0.0158</td><td align=\"left\"><bold>400.7</bold></td></tr></tbody></table></table-wrap>" ]

[ "<inline-formula id=\"IEq1\"><alternatives><tex-math id=\"M1\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$G_{\\phi }(\\textbf{L}\\textbf{r}_f, Z, N_a)$$\\end{document}</tex-math><mml:math id=\"M2\"><mml:mrow><mml:msub><mml:mi>G</mml:mi><mml:mi>ϕ</mml:mi></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi mathvariant=\"bold\">L</mml:mi><mml:msub><mml:mi mathvariant=\"bold\">r</mml:mi><mml:mi>f</mml:mi></mml:msub><mml:mo>,</mml:mo><mml:mi>Z</mml:mi><mml:mo>,</mml:mo><mml:msub><mml:mi>N</mml:mi><mml:mi>a</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq2\"><alternatives><tex-math id=\"M3\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\textbf{z}$$\\end{document}</tex-math><mml:math id=\"M4\"><mml:mi mathvariant=\"bold\">z</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq3\"><alternatives><tex-math id=\"M5\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\textbf{L}_\\textbf{z}$$\\end{document}</tex-math><mml:math id=\"M6\"><mml:msub><mml:mi mathvariant=\"bold\">L</mml:mi><mml:mi 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id=\"M13\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\textbf{A}_\\textbf{z}$$\\end{document}</tex-math><mml:math id=\"M14\"><mml:msub><mml:mi mathvariant=\"bold\">A</mml:mi><mml:mi mathvariant=\"bold\">z</mml:mi></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq8\"><alternatives><tex-math id=\"M15\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$Z_t$$\\end{document}</tex-math><mml:math 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\n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$^{++}$$\\end{document}</tex-math><mml:math id=\"M40\"><mml:msup><mml:mrow/><mml:mrow><mml:mo>+</mml:mo><mml:mo>+</mml:mo></mml:mrow></mml:msup></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq21\"><alternatives><tex-math id=\"M41\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$N_a$$\\end{document}</tex-math><mml:math id=\"M42\"><mml:msub><mml:mi>N</mml:mi><mml:mi>a</mml:mi></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq22\"><alternatives><tex-math id=\"M43\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$N_a$$\\end{document}</tex-math><mml:math id=\"M44\"><mml:msub><mml:mi>N</mml:mi><mml:mi>a</mml:mi></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq23\"><alternatives><tex-math id=\"M45\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\root 3 \\of {V/N_a}$$\\end{document}</tex-math><mml:math id=\"M46\"><mml:mroot><mml:mrow><mml:mi>V</mml:mi><mml:mo stretchy=\"false\">/</mml:mo><mml:msub><mml:mi>N</mml:mi><mml:mi>a</mml:mi></mml:msub></mml:mrow><mml:mn>3</mml:mn></mml:mroot></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq24\"><alternatives><tex-math id=\"M47\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$N_a$$\\end{document}</tex-math><mml:math id=\"M48\"><mml:msub><mml:mi>N</mml:mi><mml:mi>a</mml:mi></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq25\"><alternatives><tex-math id=\"M49\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\langle \\delta _{\\text {rms}}\\rangle$$\\end{document}</tex-math><mml:math id=\"M50\"><mml:mrow><mml:mo stretchy=\"false\">⟨</mml:mo><mml:msub><mml:mi>δ</mml:mi><mml:mtext>rms</mml:mtext></mml:msub><mml:mo stretchy=\"false\">⟩</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq26\"><alternatives><tex-math id=\"M51\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\langle \\delta _{\\text {rms}}\\rangle$$\\end{document}</tex-math><mml:math id=\"M52\"><mml:mrow><mml:mo stretchy=\"false\">⟨</mml:mo><mml:msub><mml:mi>δ</mml:mi><mml:mtext>rms</mml:mtext></mml:msub><mml:mo stretchy=\"false\">⟩</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq27\"><alternatives><tex-math id=\"M53\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\langle \\delta _{\\text {rms}}\\rangle$$\\end{document}</tex-math><mml:math id=\"M54\"><mml:mrow><mml:mo stretchy=\"false\">⟨</mml:mo><mml:msub><mml:mi>δ</mml:mi><mml:mtext>rms</mml:mtext></mml:msub><mml:mo stretchy=\"false\">⟩</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq28\"><alternatives><tex-math id=\"M55\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\langle \\delta _{\\text {rms}}\\rangle$$\\end{document}</tex-math><mml:math id=\"M56\"><mml:mrow><mml:mo stretchy=\"false\">⟨</mml:mo><mml:msub><mml:mi>δ</mml:mi><mml:mtext>rms</mml:mtext></mml:msub><mml:mo stretchy=\"false\">⟩</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq29\"><alternatives><tex-math id=\"M57\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$N_a$$\\end{document}</tex-math><mml:math id=\"M58\"><mml:msub><mml:mi>N</mml:mi><mml:mi>a</mml:mi></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq30\"><alternatives><tex-math id=\"M59\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$N_a$$\\end{document}</tex-math><mml:math id=\"M60\"><mml:msub><mml:mi>N</mml:mi><mml:mi>a</mml:mi></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq31\"><alternatives><tex-math id=\"M61\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$N_a$$\\end{document}</tex-math><mml:math id=\"M62\"><mml:msub><mml:mi>N</mml:mi><mml:mi>a</mml:mi></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq32\"><alternatives><tex-math id=\"M63\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\langle \\delta _{\\text {rms}}\\rangle$$\\end{document}</tex-math><mml:math id=\"M64\"><mml:mrow><mml:mo stretchy=\"false\">⟨</mml:mo><mml:msub><mml:mi>δ</mml:mi><mml:mtext>rms</mml:mtext></mml:msub><mml:mo stretchy=\"false\">⟩</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq33\"><alternatives><tex-math id=\"M65\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\langle \\delta _{\\text {rms}}\\rangle$$\\end{document}</tex-math><mml:math id=\"M66\"><mml:mrow><mml:mo stretchy=\"false\">⟨</mml:mo><mml:msub><mml:mi>δ</mml:mi><mml:mtext>rms</mml:mtext></mml:msub><mml:mo stretchy=\"false\">⟩</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq34\"><alternatives><tex-math id=\"M67\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$N_a$$\\end{document}</tex-math><mml:math id=\"M68\"><mml:msub><mml:mi>N</mml:mi><mml:mi>a</mml:mi></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq35\"><alternatives><tex-math id=\"M69\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\langle \\delta _{\\text {rms}}\\rangle$$\\end{document}</tex-math><mml:math id=\"M70\"><mml:mrow><mml:mo stretchy=\"false\">⟨</mml:mo><mml:msub><mml:mi>δ</mml:mi><mml:mtext>rms</mml:mtext></mml:msub><mml:mo stretchy=\"false\">⟩</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq36\"><alternatives><tex-math id=\"M71\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\langle \\delta _{\\text {rms}}\\rangle$$\\end{document}</tex-math><mml:math id=\"M72\"><mml:mrow><mml:mo stretchy=\"false\">⟨</mml:mo><mml:msub><mml:mi>δ</mml:mi><mml:mtext>rms</mml:mtext></mml:msub><mml:mo stretchy=\"false\">⟩</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq37\"><alternatives><tex-math id=\"M73\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\langle \\delta _{\\text {rms}}\\rangle$$\\end{document}</tex-math><mml:math id=\"M74\"><mml:mrow><mml:mo stretchy=\"false\">⟨</mml:mo><mml:msub><mml:mi>δ</mml:mi><mml:mtext>rms</mml:mtext></mml:msub><mml:mo stretchy=\"false\">⟩</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq38\"><alternatives><tex-math id=\"M75\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$N_a$$\\end{document}</tex-math><mml:math id=\"M76\"><mml:msub><mml:mi>N</mml:mi><mml:mi>a</mml:mi></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq39\"><alternatives><tex-math id=\"M77\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\langle \\delta _{\\text {rms}}\\rangle$$\\end{document}</tex-math><mml:math id=\"M78\"><mml:mrow><mml:mo stretchy=\"false\">⟨</mml:mo><mml:msub><mml:mi>δ</mml:mi><mml:mtext>rms</mml:mtext></mml:msub><mml:mo stretchy=\"false\">⟩</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq40\"><alternatives><tex-math id=\"M79\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\uparrow$$\\end{document}</tex-math><mml:math id=\"M80\"><mml:mo stretchy=\"false\">↑</mml:mo></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq41\"><alternatives><tex-math id=\"M81\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\langle \\delta _{\\text {rms}}\\rangle$$\\end{document}</tex-math><mml:math id=\"M82\"><mml:mrow><mml:mo stretchy=\"false\">⟨</mml:mo><mml:msub><mml:mi>δ</mml:mi><mml:mtext>rms</mml:mtext></mml:msub><mml:mo stretchy=\"false\">⟩</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq42\"><alternatives><tex-math id=\"M83\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\downarrow$$\\end{document}</tex-math><mml:math id=\"M84\"><mml:mo stretchy=\"false\">↓</mml:mo></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq43\"><alternatives><tex-math id=\"M85\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$N_a$$\\end{document}</tex-math><mml:math id=\"M86\"><mml:msub><mml:mi>N</mml:mi><mml:mi>a</mml:mi></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq44\"><alternatives><tex-math id=\"M87\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$N_a$$\\end{document}</tex-math><mml:math id=\"M88\"><mml:msub><mml:mi>N</mml:mi><mml:mi>a</mml:mi></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq45\"><alternatives><tex-math id=\"M89\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\rho$$\\end{document}</tex-math><mml:math id=\"M90\"><mml:mi>ρ</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq46\"><alternatives><tex-math id=\"M91\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$^3$$\\end{document}</tex-math><mml:math id=\"M92\"><mml:msup><mml:mrow/><mml:mn>3</mml:mn></mml:msup></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq47\"><alternatives><tex-math id=\"M93\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$E_{form}$$\\end{document}</tex-math><mml:math id=\"M94\"><mml:msub><mml:mi>E</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">form</mml:mi></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq48\"><alternatives><tex-math id=\"M95\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$N_a$$\\end{document}</tex-math><mml:math id=\"M96\"><mml:msub><mml:mi>N</mml:mi><mml:mi>a</mml:mi></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq49\"><alternatives><tex-math id=\"M97\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\rho$$\\end{document}</tex-math><mml:math id=\"M98\"><mml:mi>ρ</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq50\"><alternatives><tex-math id=\"M99\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$E_{form}$$\\end{document}</tex-math><mml:math id=\"M100\"><mml:msub><mml:mi>E</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">form</mml:mi></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq51\"><alternatives><tex-math id=\"M101\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\rho$$\\end{document}</tex-math><mml:math id=\"M102\"><mml:mi>ρ</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq52\"><alternatives><tex-math id=\"M103\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\langle \\delta _{\\text {rms}}\\rangle$$\\end{document}</tex-math><mml:math id=\"M104\"><mml:mrow><mml:mo stretchy=\"false\">⟨</mml:mo><mml:msub><mml:mi>δ</mml:mi><mml:mtext>rms</mml:mtext></mml:msub><mml:mo stretchy=\"false\">⟩</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq53\"><alternatives><tex-math id=\"M105\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\Delta V_{\\text {rms}}$$\\end{document}</tex-math><mml:math id=\"M106\"><mml:mrow><mml:mi mathvariant=\"normal\">Δ</mml:mi><mml:msub><mml:mi>V</mml:mi><mml:mtext>rms</mml:mtext></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq54\"><alternatives><tex-math id=\"M107\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\Delta E_{\\text {rms}}$$\\end{document}</tex-math><mml:math id=\"M108\"><mml:mrow><mml:mi mathvariant=\"normal\">Δ</mml:mi><mml:msub><mml:mi>E</mml:mi><mml:mtext>rms</mml:mtext></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq55\"><alternatives><tex-math id=\"M109\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\Delta V_{\\text {rms}}$$\\end{document}</tex-math><mml:math id=\"M110\"><mml:mrow><mml:mi mathvariant=\"normal\">Δ</mml:mi><mml:msub><mml:mi>V</mml:mi><mml:mtext>rms</mml:mtext></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq56\"><alternatives><tex-math id=\"M111\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\Delta E_{\\text {rms}}$$\\end{document}</tex-math><mml:math id=\"M112\"><mml:mrow><mml:mi mathvariant=\"normal\">Δ</mml:mi><mml:msub><mml:mi>E</mml:mi><mml:mtext>rms</mml:mtext></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq57\"><alternatives><tex-math id=\"M113\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\langle \\delta _{\\text {rms}}\\rangle$$\\end{document}</tex-math><mml:math id=\"M114\"><mml:mrow><mml:mo stretchy=\"false\">⟨</mml:mo><mml:msub><mml:mi>δ</mml:mi><mml:mtext>rms</mml:mtext></mml:msub><mml:mo stretchy=\"false\">⟩</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq58\"><alternatives><tex-math id=\"M115\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\Delta E_{\\text {rms}}$$\\end{document}</tex-math><mml:math id=\"M116\"><mml:mrow><mml:mi mathvariant=\"normal\">Δ</mml:mi><mml:msub><mml:mi>E</mml:mi><mml:mtext>rms</mml:mtext></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq59\"><alternatives><tex-math id=\"M117\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\Delta V_{\\text {rms}}$$\\end{document}</tex-math><mml:math id=\"M118\"><mml:mrow><mml:mi mathvariant=\"normal\">Δ</mml:mi><mml:msub><mml:mi>V</mml:mi><mml:mtext>rms</mml:mtext></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq60\"><alternatives><tex-math id=\"M119\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\Delta V_{\\text {rms}}$$\\end{document}</tex-math><mml:math id=\"M120\"><mml:mrow><mml:mi mathvariant=\"normal\">Δ</mml:mi><mml:msub><mml:mi>V</mml:mi><mml:mtext>rms</mml:mtext></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq61\"><alternatives><tex-math id=\"M121\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\uparrow$$\\end{document}</tex-math><mml:math id=\"M122\"><mml:mo stretchy=\"false\">↑</mml:mo></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq62\"><alternatives><tex-math id=\"M123\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\uparrow$$\\end{document}</tex-math><mml:math id=\"M124\"><mml:mo stretchy=\"false\">↑</mml:mo></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq63\"><alternatives><tex-math id=\"M125\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\downarrow$$\\end{document}</tex-math><mml:math id=\"M126\"><mml:mo stretchy=\"false\">↓</mml:mo></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq64\"><alternatives><tex-math id=\"M127\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\rho$$\\end{document}</tex-math><mml:math id=\"M128\"><mml:mi>ρ</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq65\"><alternatives><tex-math id=\"M129\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$E_{form}$$\\end{document}</tex-math><mml:math id=\"M130\"><mml:msub><mml:mi>E</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">form</mml:mi></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq66\"><alternatives><tex-math id=\"M131\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$N_a$$\\end{document}</tex-math><mml:math id=\"M132\"><mml:msub><mml:mi>N</mml:mi><mml:mi>a</mml:mi></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq67\"><alternatives><tex-math id=\"M133\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$N_a$$\\end{document}</tex-math><mml:math id=\"M134\"><mml:msub><mml:mi>N</mml:mi><mml:mi>a</mml:mi></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq68\"><alternatives><tex-math id=\"M135\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$N_a$$\\end{document}</tex-math><mml:math id=\"M136\"><mml:msub><mml:mi>N</mml:mi><mml:mi>a</mml:mi></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq69\"><alternatives><tex-math id=\"M137\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$N_a$$\\end{document}</tex-math><mml:math id=\"M138\"><mml:msub><mml:mi>N</mml:mi><mml:mi>a</mml:mi></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq70\"><alternatives><tex-math id=\"M139\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$N_a$$\\end{document}</tex-math><mml:math id=\"M140\"><mml:msub><mml:mi>N</mml:mi><mml:mi>a</mml:mi></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq71\"><alternatives><tex-math id=\"M141\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$N_a$$\\end{document}</tex-math><mml:math id=\"M142\"><mml:msub><mml:mi>N</mml:mi><mml:mi>a</mml:mi></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq72\"><alternatives><tex-math id=\"M143\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\uparrow$$\\end{document}</tex-math><mml:math id=\"M144\"><mml:mo stretchy=\"false\">↑</mml:mo></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq73\"><alternatives><tex-math id=\"M145\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\langle \\delta _{\\text {rms}}\\rangle$$\\end{document}</tex-math><mml:math id=\"M146\"><mml:mrow><mml:mo stretchy=\"false\">⟨</mml:mo><mml:msub><mml:mi>δ</mml:mi><mml:mtext>rms</mml:mtext></mml:msub><mml:mo stretchy=\"false\">⟩</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq74\"><alternatives><tex-math id=\"M147\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\downarrow$$\\end{document}</tex-math><mml:math id=\"M148\"><mml:mo stretchy=\"false\">↓</mml:mo></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq75\"><alternatives><tex-math id=\"M149\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\Delta V_{\\text {rms}}$$\\end{document}</tex-math><mml:math id=\"M150\"><mml:mrow><mml:mi mathvariant=\"normal\">Δ</mml:mi><mml:msub><mml:mi>V</mml:mi><mml:mtext>rms</mml:mtext></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq76\"><alternatives><tex-math id=\"M151\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$^3$$\\end{document}</tex-math><mml:math id=\"M152\"><mml:msup><mml:mrow/><mml:mn>3</mml:mn></mml:msup></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq77\"><alternatives><tex-math id=\"M153\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\downarrow$$\\end{document}</tex-math><mml:math id=\"M154\"><mml:mo stretchy=\"false\">↓</mml:mo></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq78\"><alternatives><tex-math id=\"M155\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\Delta E_{\\text {rms}}$$\\end{document}</tex-math><mml:math id=\"M156\"><mml:mrow><mml:mi mathvariant=\"normal\">Δ</mml:mi><mml:msub><mml:mi>E</mml:mi><mml:mtext>rms</mml:mtext></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq79\"><alternatives><tex-math id=\"M157\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\downarrow$$\\end{document}</tex-math><mml:math id=\"M158\"><mml:mo stretchy=\"false\">↓</mml:mo></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq80\"><alternatives><tex-math id=\"M159\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\Delta V_{\\text {rms}}$$\\end{document}</tex-math><mml:math id=\"M160\"><mml:mrow><mml:mi mathvariant=\"normal\">Δ</mml:mi><mml:msub><mml:mi>V</mml:mi><mml:mtext>rms</mml:mtext></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq81\"><alternatives><tex-math id=\"M161\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\rho$$\\end{document}</tex-math><mml:math id=\"M162\"><mml:mi>ρ</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq82\"><alternatives><tex-math id=\"M163\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\langle \\delta _{\\text {rms}}\\rangle$$\\end{document}</tex-math><mml:math id=\"M164\"><mml:mrow><mml:mo stretchy=\"false\">⟨</mml:mo><mml:msub><mml:mi>δ</mml:mi><mml:mtext>rms</mml:mtext></mml:msub><mml:mo stretchy=\"false\">⟩</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq83\"><alternatives><tex-math id=\"M165\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$E_{form}$$\\end{document}</tex-math><mml:math id=\"M166\"><mml:msub><mml:mi>E</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">form</mml:mi></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq84\"><alternatives><tex-math id=\"M167\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$E_{form}$$\\end{document}</tex-math><mml:math id=\"M168\"><mml:msub><mml:mi>E</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">form</mml:mi></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq85\"><alternatives><tex-math id=\"M169\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$E_{form}$$\\end{document}</tex-math><mml:math id=\"M170\"><mml:msub><mml:mi>E</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">form</mml:mi></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq86\"><alternatives><tex-math id=\"M171\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\Delta E$$\\end{document}</tex-math><mml:math id=\"M172\"><mml:mrow><mml:mi mathvariant=\"normal\">Δ</mml:mi><mml:mi>E</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq87\"><alternatives><tex-math id=\"M173\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\Delta E$$\\end{document}</tex-math><mml:math id=\"M174\"><mml:mrow><mml:mi mathvariant=\"normal\">Δ</mml:mi><mml:mi>E</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq88\"><alternatives><tex-math id=\"M175\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\Delta E_{\\text {rms}}$$\\end{document}</tex-math><mml:math id=\"M176\"><mml:mrow><mml:mi mathvariant=\"normal\">Δ</mml:mi><mml:msub><mml:mi>E</mml:mi><mml:mtext>rms</mml:mtext></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq89\"><alternatives><tex-math id=\"M177\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\Delta E_{\\text {rms}}$$\\end{document}</tex-math><mml:math id=\"M178\"><mml:mrow><mml:mi mathvariant=\"normal\">Δ</mml:mi><mml:msub><mml:mi>E</mml:mi><mml:mtext>rms</mml:mtext></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq90\"><alternatives><tex-math id=\"M179\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\Delta E$$\\end{document}</tex-math><mml:math id=\"M180\"><mml:mrow><mml:mi mathvariant=\"normal\">Δ</mml:mi><mml:mi>E</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq91\"><alternatives><tex-math id=\"M181\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\Delta E_{\\text {rms}}$$\\end{document}</tex-math><mml:math id=\"M182\"><mml:mrow><mml:mi mathvariant=\"normal\">Δ</mml:mi><mml:msub><mml:mi>E</mml:mi><mml:mtext>rms</mml:mtext></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq92\"><alternatives><tex-math id=\"M183\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\Delta E$$\\end{document}</tex-math><mml:math id=\"M184\"><mml:mrow><mml:mi mathvariant=\"normal\">Δ</mml:mi><mml:mi>E</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq93\"><alternatives><tex-math id=\"M185\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\Delta E_{\\text {rms}}$$\\end{document}</tex-math><mml:math id=\"M186\"><mml:mrow><mml:mi mathvariant=\"normal\">Δ</mml:mi><mml:msub><mml:mi>E</mml:mi><mml:mtext>rms</mml:mtext></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq94\"><alternatives><tex-math id=\"M187\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\textbf{r}_f$$\\end{document}</tex-math><mml:math id=\"M188\"><mml:msub><mml:mi mathvariant=\"bold\">r</mml:mi><mml:mi>f</mml:mi></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq95\"><alternatives><tex-math id=\"M189\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\textbf{r}_f \\in [0, 1)^3$$\\end{document}</tex-math><mml:math id=\"M190\"><mml:mrow><mml:msub><mml:mi mathvariant=\"bold\">r</mml:mi><mml:mi>f</mml:mi></mml:msub><mml:mo>∈</mml:mo><mml:msup><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mn>3</mml:mn></mml:msup></mml:mrow></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ1\"><label>1</label><alternatives><tex-math id=\"M191\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} \\begin{aligned} q(\\textbf{r}_{1:T}|\\textbf{r}_0)&amp;= \\prod _{t=1}^T q(\\textbf{r}_t|\\textbf{r}_{t-1}), \\\\ q(\\textbf{r}_t|\\textbf{r}_{t-1})&amp;= \\mathcal {N}(\\textbf{r}_t; \\sqrt{\\alpha _t}\\textbf{r}_{t-1}, (1-\\alpha _t)\\textbf{I}), \\end{aligned} \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M192\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:mi>q</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi mathvariant=\"bold\">r</mml:mi><mml:mrow><mml:mn>1</mml:mn><mml:mo>:</mml:mo><mml:mi>T</mml:mi></mml:mrow></mml:msub><mml:mo stretchy=\"false\">|</mml:mo><mml:msub><mml:mi mathvariant=\"bold\">r</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mtd><mml:mtd columnalign=\"left\"><mml:mrow><mml:mo>=</mml:mo><mml:munderover><mml:mo>∏</mml:mo><mml:mrow><mml:mi>t</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mi>T</mml:mi></mml:munderover><mml:mi>q</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi mathvariant=\"bold\">r</mml:mi><mml:mi>t</mml:mi></mml:msub><mml:mo stretchy=\"false\">|</mml:mo><mml:msub><mml:mi mathvariant=\"bold\">r</mml:mi><mml:mrow><mml:mi>t</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>,</mml:mo></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:mrow/><mml:mi>q</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi mathvariant=\"bold\">r</mml:mi><mml:mi>t</mml:mi></mml:msub><mml:mo stretchy=\"false\">|</mml:mo><mml:msub><mml:mi mathvariant=\"bold\">r</mml:mi><mml:mrow><mml:mi>t</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mtd><mml:mtd columnalign=\"left\"><mml:mrow><mml:mo>=</mml:mo><mml:mi mathvariant=\"script\">N</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi mathvariant=\"bold\">r</mml:mi><mml:mi>t</mml:mi></mml:msub><mml:mo>;</mml:mo><mml:msqrt><mml:msub><mml:mi>α</mml:mi><mml:mi>t</mml:mi></mml:msub></mml:msqrt><mml:msub><mml:mi mathvariant=\"bold\">r</mml:mi><mml:mrow><mml:mi>t</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>1</mml:mn><mml:mo>-</mml:mo><mml:msub><mml:mi>α</mml:mi><mml:mi>t</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mi mathvariant=\"bold\">I</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>,</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq96\"><alternatives><tex-math id=\"M193\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\textbf{r}_0 \\sim q(\\textbf{r}_f)$$\\end{document}</tex-math><mml:math id=\"M194\"><mml:mrow><mml:msub><mml:mi mathvariant=\"bold\">r</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo>∼</mml:mo><mml:mi>q</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi mathvariant=\"bold\">r</mml:mi><mml:mi>f</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq97\"><alternatives><tex-math id=\"M195\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\alpha _t$$\\end{document}</tex-math><mml:math id=\"M196\"><mml:msub><mml:mi>α</mml:mi><mml:mi>t</mml:mi></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq98\"><alternatives><tex-math id=\"M197\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$q(\\cdot | \\cdot )$$\\end{document}</tex-math><mml:math id=\"M198\"><mml:mrow><mml:mi>q</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mo>·</mml:mo><mml:mo stretchy=\"false\">|</mml:mo><mml:mo>·</mml:mo><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq99\"><alternatives><tex-math id=\"M199\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\textbf{r}_t$$\\end{document}</tex-math><mml:math id=\"M200\"><mml:msub><mml:mi mathvariant=\"bold\">r</mml:mi><mml:mi>t</mml:mi></mml:msub></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ2\"><label>2</label><alternatives><tex-math id=\"M201\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} \\textbf{r}_t = \\sqrt{\\bar{\\alpha }_t}\\textbf{r}_0 + \\sqrt{1 - \\bar{\\alpha }_t}\\varvec{\\epsilon }, \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M202\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:msub><mml:mi mathvariant=\"bold\">r</mml:mi><mml:mi>t</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:msqrt><mml:msub><mml:mover accent=\"true\"><mml:mrow><mml:mi>α</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">¯</mml:mo></mml:mrow></mml:mover><mml:mi>t</mml:mi></mml:msub></mml:msqrt><mml:msub><mml:mi mathvariant=\"bold\">r</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo>+</mml:mo><mml:msqrt><mml:mrow><mml:mn>1</mml:mn><mml:mo>-</mml:mo><mml:msub><mml:mover accent=\"true\"><mml:mrow><mml:mi>α</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">¯</mml:mo></mml:mrow></mml:mover><mml:mi>t</mml:mi></mml:msub></mml:mrow></mml:msqrt><mml:mrow><mml:mi mathvariant=\"bold-italic\">ϵ</mml:mi></mml:mrow><mml:mo>,</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq100\"><alternatives><tex-math id=\"M203\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\varvec{\\epsilon } \\sim \\mathcal {N}(0,\\textbf{I})$$\\end{document}</tex-math><mml:math id=\"M204\"><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">ϵ</mml:mi></mml:mrow><mml:mo>∼</mml:mo><mml:mi mathvariant=\"script\">N</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mi mathvariant=\"bold\">I</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq101\"><alternatives><tex-math id=\"M205\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\bar{\\alpha }_t = \\prod _{i=1}^t\\alpha _i$$\\end{document}</tex-math><mml:math id=\"M206\"><mml:mrow><mml:msub><mml:mover accent=\"true\"><mml:mrow><mml:mi>α</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">¯</mml:mo></mml:mrow></mml:mover><mml:mi>t</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:msubsup><mml:mo>∏</mml:mo><mml:mrow><mml:mi>i</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mi>t</mml:mi></mml:msubsup><mml:msub><mml:mi>α</mml:mi><mml:mi>i</mml:mi></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq102\"><alternatives><tex-math id=\"M207\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\textbf{r}_t$$\\end{document}</tex-math><mml:math id=\"M208\"><mml:msub><mml:mi mathvariant=\"bold\">r</mml:mi><mml:mi>t</mml:mi></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq103\"><alternatives><tex-math id=\"M209\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$[0,1)^3$$\\end{document}</tex-math><mml:math id=\"M210\"><mml:msup><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mn>3</mml:mn></mml:msup></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ3\"><label>3</label><alternatives><tex-math id=\"M211\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} \\textbf{r}_{f_t} = \\varvec{\\pi }(\\textbf{r}_t) :=\\textbf{r}_t - \\lfloor \\textbf{r}_t \\rfloor . \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M212\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:msub><mml:mi mathvariant=\"bold\">r</mml:mi><mml:msub><mml:mi>f</mml:mi><mml:mi>t</mml:mi></mml:msub></mml:msub><mml:mo>=</mml:mo><mml:mrow><mml:mi mathvariant=\"bold-italic\">π</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi mathvariant=\"bold\">r</mml:mi><mml:mi>t</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>:</mml:mo><mml:mo>=</mml:mo><mml:msub><mml:mi mathvariant=\"bold\">r</mml:mi><mml:mi>t</mml:mi></mml:msub><mml:mo>-</mml:mo><mml:mrow><mml:mo>⌊</mml:mo><mml:msub><mml:mi mathvariant=\"bold\">r</mml:mi><mml:mi>t</mml:mi></mml:msub><mml:mo>⌋</mml:mo></mml:mrow><mml:mo>.</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq104\"><alternatives><tex-math id=\"M213\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\textbf{r}_{f_t} \\in [0, 1)^3$$\\end{document}</tex-math><mml:math id=\"M214\"><mml:mrow><mml:msub><mml:mi mathvariant=\"bold\">r</mml:mi><mml:msub><mml:mi>f</mml:mi><mml:mi>t</mml:mi></mml:msub></mml:msub><mml:mo>∈</mml:mo><mml:msup><mml:mrow><mml:mo stretchy=\"false\">[</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mn>3</mml:mn></mml:msup></mml:mrow></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ4\"><label>4</label><alternatives><tex-math id=\"M215\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} \\begin{aligned} p_{\\theta }(\\textbf{r}_{0:T})&amp;= p(\\textbf{r}_T)\\prod _{t=1}^Tp_{\\theta }(\\textbf{r}_{t-1}|\\textbf{r}_t), \\\\ p_{\\theta }(\\textbf{r}_{t-1}|\\textbf{r}_t)&amp;= \\mathcal {N}(\\textbf{r}_{t-1};\\varvec{\\mu }_{\\theta },\\sigma ^2_t\\textbf{I}), \\end{aligned} \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M216\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:msub><mml:mi>p</mml:mi><mml:mi>θ</mml:mi></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi mathvariant=\"bold\">r</mml:mi><mml:mrow><mml:mn>0</mml:mn><mml:mo>:</mml:mo><mml:mi>T</mml:mi></mml:mrow></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:mtd><mml:mtd columnalign=\"left\"><mml:mrow><mml:mo>=</mml:mo><mml:mi>p</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi mathvariant=\"bold\">r</mml:mi><mml:mi>T</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:munderover><mml:mo>∏</mml:mo><mml:mrow><mml:mi>t</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mi>T</mml:mi></mml:munderover><mml:msub><mml:mi>p</mml:mi><mml:mi>θ</mml:mi></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi mathvariant=\"bold\">r</mml:mi><mml:mrow><mml:mi>t</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo stretchy=\"false\">|</mml:mo><mml:msub><mml:mi mathvariant=\"bold\">r</mml:mi><mml:mi>t</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>,</mml:mo></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:mrow/><mml:msub><mml:mi>p</mml:mi><mml:mi>θ</mml:mi></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi mathvariant=\"bold\">r</mml:mi><mml:mrow><mml:mi>t</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo stretchy=\"false\">|</mml:mo><mml:msub><mml:mi mathvariant=\"bold\">r</mml:mi><mml:mi>t</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:mtd><mml:mtd columnalign=\"left\"><mml:mrow><mml:mo>=</mml:mo><mml:mi mathvariant=\"script\">N</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi mathvariant=\"bold\">r</mml:mi><mml:mrow><mml:mi>t</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>;</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant=\"bold-italic\">μ</mml:mi></mml:mrow><mml:mi>θ</mml:mi></mml:msub><mml:mo>,</mml:mo><mml:msubsup><mml:mi>σ</mml:mi><mml:mi>t</mml:mi><mml:mn>2</mml:mn></mml:msubsup><mml:mi mathvariant=\"bold\">I</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>,</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<disp-formula id=\"Equ5\"><label>5</label><alternatives><tex-math id=\"M217\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} \\begin{aligned} \\varvec{\\mu }_{\\theta }&amp;= \\frac{1}{\\sqrt{\\alpha _t}}\\Big (\\textbf{r}_t -\\frac{1-\\alpha _t}{\\sqrt{1-\\bar{\\alpha }_t}}\\varvec{\\epsilon }_{\\theta }\\Big ), \\\\ \\sigma _t^2&amp;= \\frac{(1 - \\bar{\\alpha }_{t-1})(1 - \\alpha _t)}{1 - \\bar{\\alpha }_t}. \\end{aligned} \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M218\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:msub><mml:mrow><mml:mi mathvariant=\"bold-italic\">μ</mml:mi></mml:mrow><mml:mi>θ</mml:mi></mml:msub></mml:mtd><mml:mtd columnalign=\"left\"><mml:mrow><mml:mo>=</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:msqrt><mml:msub><mml:mi>α</mml:mi><mml:mi>t</mml:mi></mml:msub></mml:msqrt></mml:mfrac><mml:mrow><mml:mo maxsize=\"1.623em\" minsize=\"1.623em\" stretchy=\"true\">(</mml:mo></mml:mrow><mml:msub><mml:mi mathvariant=\"bold\">r</mml:mi><mml:mi>t</mml:mi></mml:msub><mml:mo>-</mml:mo><mml:mfrac><mml:mrow><mml:mn>1</mml:mn><mml:mo>-</mml:mo><mml:msub><mml:mi>α</mml:mi><mml:mi>t</mml:mi></mml:msub></mml:mrow><mml:msqrt><mml:mrow><mml:mn>1</mml:mn><mml:mo>-</mml:mo><mml:msub><mml:mover accent=\"true\"><mml:mrow><mml:mi>α</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">¯</mml:mo></mml:mrow></mml:mover><mml:mi>t</mml:mi></mml:msub></mml:mrow></mml:msqrt></mml:mfrac><mml:msub><mml:mrow><mml:mi mathvariant=\"bold-italic\">ϵ</mml:mi></mml:mrow><mml:mi>θ</mml:mi></mml:msub><mml:mrow><mml:mo maxsize=\"1.623em\" minsize=\"1.623em\" stretchy=\"true\">)</mml:mo></mml:mrow><mml:mo>,</mml:mo></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:mrow/><mml:msubsup><mml:mi>σ</mml:mi><mml:mi>t</mml:mi><mml:mn>2</mml:mn></mml:msubsup></mml:mrow></mml:mtd><mml:mtd columnalign=\"left\"><mml:mrow><mml:mo>=</mml:mo><mml:mfrac><mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>1</mml:mn><mml:mo>-</mml:mo><mml:msub><mml:mover accent=\"true\"><mml:mrow><mml:mi>α</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">¯</mml:mo></mml:mrow></mml:mover><mml:mrow><mml:mi>t</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>1</mml:mn><mml:mo>-</mml:mo><mml:msub><mml:mi>α</mml:mi><mml:mi>t</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow><mml:mrow><mml:mn>1</mml:mn><mml:mo>-</mml:mo><mml:msub><mml:mover accent=\"true\"><mml:mrow><mml:mi>α</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">¯</mml:mo></mml:mrow></mml:mover><mml:mi>t</mml:mi></mml:msub></mml:mrow></mml:mfrac><mml:mo>.</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq105\"><alternatives><tex-math id=\"M219\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$p_{\\theta }(\\textbf{r}_{t-1}|\\textbf{r}_t)$$\\end{document}</tex-math><mml:math id=\"M220\"><mml:mrow><mml:msub><mml:mi>p</mml:mi><mml:mi>θ</mml:mi></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi mathvariant=\"bold\">r</mml:mi><mml:mrow><mml:mi>t</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo stretchy=\"false\">|</mml:mo><mml:msub><mml:mi mathvariant=\"bold\">r</mml:mi><mml:mi>t</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq106\"><alternatives><tex-math id=\"M221\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$q(\\textbf{r}_{t-1}| \\textbf{r}_t, \\textbf{r}_0)$$\\end{document}</tex-math><mml:math id=\"M222\"><mml:mrow><mml:mi>q</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi mathvariant=\"bold\">r</mml:mi><mml:mrow><mml:mi>t</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo stretchy=\"false\">|</mml:mo><mml:msub><mml:mi mathvariant=\"bold\">r</mml:mi><mml:mi>t</mml:mi></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi mathvariant=\"bold\">r</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq107\"><alternatives><tex-math id=\"M223\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\varvec{\\epsilon }_{\\theta }$$\\end{document}</tex-math><mml:math id=\"M224\"><mml:msub><mml:mrow><mml:mi mathvariant=\"bold-italic\">ϵ</mml:mi></mml:mrow><mml:mi>θ</mml:mi></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq108\"><alternatives><tex-math id=\"M225\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\textbf{r}_{t-1} \\sim p_{\\theta }(\\textbf{r}_{t-1}|\\textbf{r}_t)$$\\end{document}</tex-math><mml:math id=\"M226\"><mml:mrow><mml:msub><mml:mi mathvariant=\"bold\">r</mml:mi><mml:mrow><mml:mi>t</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>∼</mml:mo><mml:msub><mml:mi>p</mml:mi><mml:mi>θ</mml:mi></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi mathvariant=\"bold\">r</mml:mi><mml:mrow><mml:mi>t</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo stretchy=\"false\">|</mml:mo><mml:msub><mml:mi mathvariant=\"bold\">r</mml:mi><mml:mi>t</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ6\"><label>6</label><alternatives><tex-math id=\"M227\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} \\textbf{r}_{t-1} = \\frac{1}{\\sqrt{\\alpha _t}}\\Big (\\textbf{r}_t -\\frac{1-\\alpha _t}{\\sqrt{1-\\bar{\\alpha }_t}}\\varvec{\\epsilon }_{\\theta }\\Big ) + \\sigma _t\\varvec{\\epsilon }^{\\prime }, \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M228\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:msub><mml:mi mathvariant=\"bold\">r</mml:mi><mml:mrow><mml:mi>t</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:msqrt><mml:msub><mml:mi>α</mml:mi><mml:mi>t</mml:mi></mml:msub></mml:msqrt></mml:mfrac><mml:mrow><mml:mo maxsize=\"1.623em\" minsize=\"1.623em\" stretchy=\"true\">(</mml:mo></mml:mrow><mml:msub><mml:mi mathvariant=\"bold\">r</mml:mi><mml:mi>t</mml:mi></mml:msub><mml:mo>-</mml:mo><mml:mfrac><mml:mrow><mml:mn>1</mml:mn><mml:mo>-</mml:mo><mml:msub><mml:mi>α</mml:mi><mml:mi>t</mml:mi></mml:msub></mml:mrow><mml:msqrt><mml:mrow><mml:mn>1</mml:mn><mml:mo>-</mml:mo><mml:msub><mml:mover accent=\"true\"><mml:mrow><mml:mi>α</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">¯</mml:mo></mml:mrow></mml:mover><mml:mi>t</mml:mi></mml:msub></mml:mrow></mml:msqrt></mml:mfrac><mml:msub><mml:mrow><mml:mi mathvariant=\"bold-italic\">ϵ</mml:mi></mml:mrow><mml:mi>θ</mml:mi></mml:msub><mml:mrow><mml:mo maxsize=\"1.623em\" minsize=\"1.623em\" stretchy=\"true\">)</mml:mo></mml:mrow><mml:mo>+</mml:mo><mml:msub><mml:mi>σ</mml:mi><mml:mi>t</mml:mi></mml:msub><mml:msup><mml:mrow><mml:mi mathvariant=\"bold-italic\">ϵ</mml:mi></mml:mrow><mml:mo>′</mml:mo></mml:msup><mml:mo>,</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq109\"><alternatives><tex-math id=\"M229\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\varvec{\\epsilon }^{\\prime } \\sim \\mathcal {N}(0,\\textbf{I})$$\\end{document}</tex-math><mml:math id=\"M230\"><mml:mrow><mml:msup><mml:mrow><mml:mi mathvariant=\"bold-italic\">ϵ</mml:mi></mml:mrow><mml:mo>′</mml:mo></mml:msup><mml:mo>∼</mml:mo><mml:mi mathvariant=\"script\">N</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mi mathvariant=\"bold\">I</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq110\"><alternatives><tex-math id=\"M231\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\textbf{r}_{t-1} \\sim p_{\\theta }(\\textbf{r}_{t-1}|\\textbf{r}_{f_t})$$\\end{document}</tex-math><mml:math id=\"M232\"><mml:mrow><mml:msub><mml:mi mathvariant=\"bold\">r</mml:mi><mml:mrow><mml:mi>t</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>∼</mml:mo><mml:msub><mml:mi>p</mml:mi><mml:mi>θ</mml:mi></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi mathvariant=\"bold\">r</mml:mi><mml:mrow><mml:mi>t</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo stretchy=\"false\">|</mml:mo><mml:msub><mml:mi mathvariant=\"bold\">r</mml:mi><mml:msub><mml:mi>f</mml:mi><mml:mi>t</mml:mi></mml:msub></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq111\"><alternatives><tex-math id=\"M233\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\alpha _t$$\\end{document}</tex-math><mml:math id=\"M234\"><mml:msub><mml:mi>α</mml:mi><mml:mi>t</mml:mi></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq112\"><alternatives><tex-math id=\"M235\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\bar{\\alpha }_t$$\\end{document}</tex-math><mml:math id=\"M236\"><mml:msub><mml:mover accent=\"true\"><mml:mrow><mml:mi>α</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">¯</mml:mo></mml:mrow></mml:mover><mml:mi>t</mml:mi></mml:msub></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ7\"><label>7</label><alternatives><tex-math id=\"M237\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} \\begin{aligned} \\textbf{r}_{t-1}&amp;= \\frac{1}{\\sqrt{\\bar{\\alpha _t}}}\\Big (\\textbf{r}_{f_t} -\\sqrt{1-\\bar{\\alpha }_t}\\varvec{\\epsilon }_{\\theta }\\Big ) + \\sigma _t\\varvec{\\epsilon }^{\\prime }, \\\\ \\textbf{r}_{f_{t}}&amp;= \\varvec{\\pi }(\\textbf{r}_{t}). \\end{aligned} \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M238\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:msub><mml:mi mathvariant=\"bold\">r</mml:mi><mml:mrow><mml:mi>t</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:mtd><mml:mtd columnalign=\"left\"><mml:mrow><mml:mo>=</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:msqrt><mml:mover accent=\"true\"><mml:mrow><mml:msub><mml:mi>α</mml:mi><mml:mi>t</mml:mi></mml:msub></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">¯</mml:mo></mml:mrow></mml:mover></mml:msqrt></mml:mfrac><mml:mrow><mml:mo maxsize=\"1.623em\" minsize=\"1.623em\" stretchy=\"true\">(</mml:mo></mml:mrow><mml:msub><mml:mi mathvariant=\"bold\">r</mml:mi><mml:msub><mml:mi>f</mml:mi><mml:mi>t</mml:mi></mml:msub></mml:msub><mml:mo>-</mml:mo><mml:msqrt><mml:mrow><mml:mn>1</mml:mn><mml:mo>-</mml:mo><mml:msub><mml:mover accent=\"true\"><mml:mrow><mml:mi>α</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">¯</mml:mo></mml:mrow></mml:mover><mml:mi>t</mml:mi></mml:msub></mml:mrow></mml:msqrt><mml:msub><mml:mrow><mml:mi mathvariant=\"bold-italic\">ϵ</mml:mi></mml:mrow><mml:mi>θ</mml:mi></mml:msub><mml:mrow><mml:mo maxsize=\"1.623em\" minsize=\"1.623em\" stretchy=\"true\">)</mml:mo></mml:mrow><mml:mo>+</mml:mo><mml:msub><mml:mi>σ</mml:mi><mml:mi>t</mml:mi></mml:msub><mml:msup><mml:mrow><mml:mi mathvariant=\"bold-italic\">ϵ</mml:mi></mml:mrow><mml:mo>′</mml:mo></mml:msup><mml:mo>,</mml:mo></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:mrow/><mml:msub><mml:mi mathvariant=\"bold\">r</mml:mi><mml:msub><mml:mi>f</mml:mi><mml:mi>t</mml:mi></mml:msub></mml:msub></mml:mrow></mml:mtd><mml:mtd columnalign=\"left\"><mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mi mathvariant=\"bold-italic\">π</mml:mi></mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi mathvariant=\"bold\">r</mml:mi><mml:mi>t</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo><mml:mo>.</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq113\"><alternatives><tex-math id=\"M239\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\varvec{\\pi }(\\cdot )$$\\end{document}</tex-math><mml:math id=\"M240\"><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">π</mml:mi></mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mo>·</mml:mo><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq114\"><alternatives><tex-math id=\"M241\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\varvec{\\pi }(\\cdot )$$\\end{document}</tex-math><mml:math id=\"M242\"><mml:mrow><mml:mrow><mml:mi mathvariant=\"bold-italic\">π</mml:mi></mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mo>·</mml:mo><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq115\"><alternatives><tex-math id=\"M243\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\mathcal {G} =(\\mathcal {V},\\mathcal {E})$$\\end{document}</tex-math><mml:math id=\"M244\"><mml:mrow><mml:mi mathvariant=\"script\">G</mml:mi><mml:mo>=</mml:mo><mml:mo stretchy=\"false\">(</mml:mo><mml:mi mathvariant=\"script\">V</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant=\"script\">E</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq116\"><alternatives><tex-math id=\"M245\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\mathcal {V}$$\\end{document}</tex-math><mml:math id=\"M246\"><mml:mi mathvariant=\"script\">V</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq117\"><alternatives><tex-math id=\"M247\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\mathcal {E}$$\\end{document}</tex-math><mml:math id=\"M248\"><mml:mi mathvariant=\"script\">E</mml:mi></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ12\"><alternatives><tex-math id=\"M249\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned}&amp;\\mathcal {V} = \\{(\\textbf{f}_{n}, \\textbf{r}_{c_{n}}) \\; | \\; \\textbf{f}_{n} \\in \\mathbb {R}^{M}, \\; \\textbf{r}_{c_{n}} = \\textbf{L}\\textbf{r}_{f_{n}} \\in \\mathbb {R}^{3} \\}, \\\\&amp;\\mathcal {E} = \\{\\Delta \\textbf{r}_{c_{mn}}^{(\\textbf{T})} \\; | \\; \\Delta \\textbf{r}_{c_{mn}}^{(\\textbf{T})} = \\textbf{r}_{c_{m}} - \\textbf{r}_{c_{n}} + \\textbf{T}; \\ \\textbf{r}_{c_{m}}, \\textbf{r}_{c_{n}} \\in \\mathbb {R}^{3} \\}, \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M250\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd/><mml:mtd columnalign=\"left\"><mml:mrow><mml:mi mathvariant=\"script\">V</mml:mi><mml:mo>=</mml:mo><mml:mo stretchy=\"false\">{</mml:mo><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi mathvariant=\"bold\">f</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi mathvariant=\"bold\">r</mml:mi><mml:msub><mml:mi>c</mml:mi><mml:mi>n</mml:mi></mml:msub></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mspace width=\"0.277778em\"/><mml:mo stretchy=\"false\">|</mml:mo><mml:mspace width=\"0.277778em\"/><mml:msub><mml:mi mathvariant=\"bold\">f</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mo>∈</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant=\"double-struck\">R</mml:mi></mml:mrow><mml:mi>M</mml:mi></mml:msup><mml:mo>,</mml:mo><mml:mspace width=\"0.277778em\"/><mml:msub><mml:mi mathvariant=\"bold\">r</mml:mi><mml:msub><mml:mi>c</mml:mi><mml:mi>n</mml:mi></mml:msub></mml:msub><mml:mo>=</mml:mo><mml:mi mathvariant=\"bold\">L</mml:mi><mml:msub><mml:mi mathvariant=\"bold\">r</mml:mi><mml:msub><mml:mi>f</mml:mi><mml:mi>n</mml:mi></mml:msub></mml:msub><mml:mo>∈</mml:mo><mml:msup><mml:mrow><mml:mi 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mathvariant=\"italic\">mn</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi mathvariant=\"bold\">T</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:msubsup><mml:mo>=</mml:mo><mml:msub><mml:mi mathvariant=\"bold\">r</mml:mi><mml:msub><mml:mi>c</mml:mi><mml:mi>m</mml:mi></mml:msub></mml:msub><mml:mo>-</mml:mo><mml:msub><mml:mi mathvariant=\"bold\">r</mml:mi><mml:msub><mml:mi>c</mml:mi><mml:mi>n</mml:mi></mml:msub></mml:msub><mml:mo>+</mml:mo><mml:mi mathvariant=\"bold\">T</mml:mi><mml:mo>;</mml:mo><mml:mspace width=\"4pt\"/><mml:msub><mml:mi mathvariant=\"bold\">r</mml:mi><mml:msub><mml:mi>c</mml:mi><mml:mi>m</mml:mi></mml:msub></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi mathvariant=\"bold\">r</mml:mi><mml:msub><mml:mi>c</mml:mi><mml:mi>n</mml:mi></mml:msub></mml:msub><mml:mo>∈</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant=\"double-struck\">R</mml:mi></mml:mrow><mml:mn>3</mml:mn></mml:msup><mml:mo stretchy=\"false\">}</mml:mo><mml:mo>,</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq118\"><alternatives><tex-math id=\"M251\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\textbf{f}_n$$\\end{document}</tex-math><mml:math id=\"M252\"><mml:msub><mml:mi mathvariant=\"bold\">f</mml:mi><mml:mi>n</mml:mi></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq119\"><alternatives><tex-math id=\"M253\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\textbf{T}$$\\end{document}</tex-math><mml:math id=\"M254\"><mml:mi mathvariant=\"bold\">T</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq120\"><alternatives><tex-math id=\"M255\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\textbf{L}$$\\end{document}</tex-math><mml:math id=\"M256\"><mml:mi mathvariant=\"bold\">L</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq121\"><alternatives><tex-math id=\"M257\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\textbf{r}_{f_n}$$\\end{document}</tex-math><mml:math id=\"M258\"><mml:msub><mml:mi mathvariant=\"bold\">r</mml:mi><mml:msub><mml:mi>f</mml:mi><mml:mi>n</mml:mi></mml:msub></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq122\"><alternatives><tex-math id=\"M259\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\textbf{r}_{c_n}$$\\end{document}</tex-math><mml:math id=\"M260\"><mml:msub><mml:mi mathvariant=\"bold\">r</mml:mi><mml:msub><mml:mi>c</mml:mi><mml:mi>n</mml:mi></mml:msub></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq123\"><alternatives><tex-math id=\"M261\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\textbf{f} = (\\textbf{f}_1, \\dots , \\textbf{f}_{N_a}) \\in \\mathbb {R}^{N_a\\times M}$$\\end{document}</tex-math><mml:math id=\"M262\"><mml:mrow><mml:mi mathvariant=\"bold\">f</mml:mi><mml:mo>=</mml:mo><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi mathvariant=\"bold\">f</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>,</mml:mo><mml:mo>⋯</mml:mo><mml:mo>,</mml:mo><mml:msub><mml:mi mathvariant=\"bold\">f</mml:mi><mml:msub><mml:mi>N</mml:mi><mml:mi>a</mml:mi></mml:msub></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>∈</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant=\"double-struck\">R</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mi>N</mml:mi><mml:mi>a</mml:mi></mml:msub><mml:mo>×</mml:mo><mml:mi>M</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq124\"><alternatives><tex-math id=\"M263\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\textbf{r}_f = (\\textbf{r}_{f_1}, \\dots , \\textbf{r}_{f_{N_a}}) \\in \\mathbb {R}^{N_a\\times 3}$$\\end{document}</tex-math><mml:math id=\"M264\"><mml:mrow><mml:msub><mml:mi mathvariant=\"bold\">r</mml:mi><mml:mi>f</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi mathvariant=\"bold\">r</mml:mi><mml:msub><mml:mi>f</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:msub><mml:mo>,</mml:mo><mml:mo>⋯</mml:mo><mml:mo>,</mml:mo><mml:msub><mml:mi mathvariant=\"bold\">r</mml:mi><mml:msub><mml:mi>f</mml:mi><mml:msub><mml:mi>N</mml:mi><mml:mi>a</mml:mi></mml:msub></mml:msub></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>∈</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant=\"double-struck\">R</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mi>N</mml:mi><mml:mi>a</mml:mi></mml:msub><mml:mo>×</mml:mo><mml:mn>3</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq125\"><alternatives><tex-math id=\"M265\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\textbf{r}_c = (\\textbf{r}_{c_1}, \\dots , \\textbf{r}_{c_{N_a}}) \\in \\mathbb {R}^{N_a\\times 3}$$\\end{document}</tex-math><mml:math id=\"M266\"><mml:mrow><mml:msub><mml:mi mathvariant=\"bold\">r</mml:mi><mml:mi>c</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi mathvariant=\"bold\">r</mml:mi><mml:msub><mml:mi>c</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:msub><mml:mo>,</mml:mo><mml:mo>⋯</mml:mo><mml:mo>,</mml:mo><mml:msub><mml:mi mathvariant=\"bold\">r</mml:mi><mml:msub><mml:mi>c</mml:mi><mml:msub><mml:mi>N</mml:mi><mml:mi>a</mml:mi></mml:msub></mml:msub></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>∈</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant=\"double-struck\">R</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mi>N</mml:mi><mml:mi>a</mml:mi></mml:msub><mml:mo>×</mml:mo><mml:mn>3</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq126\"><alternatives><tex-math id=\"M267\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$^{++}$$\\end{document}</tex-math><mml:math id=\"M268\"><mml:msup><mml:mrow/><mml:mrow><mml:mo>+</mml:mo><mml:mo>+</mml:mo></mml:mrow></mml:msup></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq127\"><alternatives><tex-math id=\"M269\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$^{++}$$\\end{document}</tex-math><mml:math id=\"M270\"><mml:msup><mml:mrow/><mml:mrow><mml:mo>+</mml:mo><mml:mo>+</mml:mo></mml:mrow></mml:msup></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq128\"><alternatives><tex-math id=\"M271\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$^{++}$$\\end{document}</tex-math><mml:math id=\"M272\"><mml:msup><mml:mrow/><mml:mrow><mml:mo>+</mml:mo><mml:mo>+</mml:mo></mml:mrow></mml:msup></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq129\"><alternatives><tex-math id=\"M273\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$^{++}$$\\end{document}</tex-math><mml:math id=\"M274\"><mml:msup><mml:mrow/><mml:mrow><mml:mo>+</mml:mo><mml:mo>+</mml:mo></mml:mrow></mml:msup></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq130\"><alternatives><tex-math id=\"M275\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$^{++}$$\\end{document}</tex-math><mml:math id=\"M276\"><mml:msup><mml:mrow/><mml:mrow><mml:mo>+</mml:mo><mml:mo>+</mml:mo></mml:mrow></mml:msup></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq131\"><alternatives><tex-math id=\"M277\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\mathcal {E} = \\{||\\Delta \\textbf{r}_{c_{mn}}^{(\\textbf{T})}|| \\}$$\\end{document}</tex-math><mml:math id=\"M278\"><mml:mrow><mml:mi mathvariant=\"script\">E</mml:mi><mml:mo>=</mml:mo><mml:mo stretchy=\"false\">{</mml:mo><mml:mo stretchy=\"false\">|</mml:mo><mml:mo stretchy=\"false\">|</mml:mo><mml:mi mathvariant=\"normal\">Δ</mml:mi><mml:msubsup><mml:mi mathvariant=\"bold\">r</mml:mi><mml:mrow><mml:msub><mml:mi>c</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">mn</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi mathvariant=\"bold\">T</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:msubsup><mml:mo stretchy=\"false\">|</mml:mo><mml:mo stretchy=\"false\">|</mml:mo><mml:mo stretchy=\"false\">}</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq132\"><alternatives><tex-math id=\"M279\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\textbf{r}_f$$\\end{document}</tex-math><mml:math id=\"M280\"><mml:msub><mml:mi mathvariant=\"bold\">r</mml:mi><mml:mi>f</mml:mi></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq133\"><alternatives><tex-math id=\"M281\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\textbf{L}$$\\end{document}</tex-math><mml:math id=\"M282\"><mml:mi mathvariant=\"bold\">L</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq134\"><alternatives><tex-math id=\"M283\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$N_a$$\\end{document}</tex-math><mml:math id=\"M284\"><mml:msub><mml:mi>N</mml:mi><mml:mi>a</mml:mi></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq135\"><alternatives><tex-math id=\"M285\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\textbf{f}=Z$$\\end{document}</tex-math><mml:math id=\"M286\"><mml:mrow><mml:mi mathvariant=\"bold\">f</mml:mi><mml:mo>=</mml:mo><mml:mi>Z</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq136\"><alternatives><tex-math id=\"M287\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$N_a$$\\end{document}</tex-math><mml:math id=\"M288\"><mml:msub><mml:mi>N</mml:mi><mml:mi>a</mml:mi></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq137\"><alternatives><tex-math id=\"M289\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\varvec{\\mu }_{\\phi }$$\\end{document}</tex-math><mml:math id=\"M290\"><mml:msub><mml:mrow><mml:mi mathvariant=\"bold-italic\">μ</mml:mi></mml:mrow><mml:mi>ϕ</mml:mi></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq138\"><alternatives><tex-math id=\"M291\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$logvar_{\\phi }$$\\end{document}</tex-math><mml:math id=\"M292\"><mml:mrow><mml:mi>l</mml:mi><mml:mi>o</mml:mi><mml:mi>g</mml:mi><mml:mi>v</mml:mi><mml:mi>a</mml:mi><mml:msub><mml:mi>r</mml:mi><mml:mi>ϕ</mml:mi></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq139\"><alternatives><tex-math id=\"M293\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\phi$$\\end{document}</tex-math><mml:math id=\"M294\"><mml:mi>ϕ</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq140\"><alternatives><tex-math id=\"M295\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\textbf{z}$$\\end{document}</tex-math><mml:math id=\"M296\"><mml:mi mathvariant=\"bold\">z</mml:mi></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ8\"><label>8</label><alternatives><tex-math id=\"M297\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} \\textbf{z} = \\varvec{\\mu }_{\\phi } + e^{logvar_{\\phi }}\\varvec{\\epsilon }^{\\prime \\prime }, \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M298\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:mi mathvariant=\"bold\">z</mml:mi><mml:mo>=</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant=\"bold-italic\">μ</mml:mi></mml:mrow><mml:mi>ϕ</mml:mi></mml:msub><mml:mo>+</mml:mo><mml:msup><mml:mi>e</mml:mi><mml:mrow><mml:mi>l</mml:mi><mml:mi>o</mml:mi><mml:mi>g</mml:mi><mml:mi>v</mml:mi><mml:mi>a</mml:mi><mml:msub><mml:mi>r</mml:mi><mml:mi>ϕ</mml:mi></mml:msub></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mi mathvariant=\"bold-italic\">ϵ</mml:mi></mml:mrow><mml:mo>″</mml:mo></mml:msup><mml:mo>,</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq141\"><alternatives><tex-math id=\"M299\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\varvec{\\epsilon }^{\\prime \\prime } \\sim \\mathcal {N}(0,\\textbf{I})$$\\end{document}</tex-math><mml:math id=\"M300\"><mml:mrow><mml:msup><mml:mrow><mml:mi mathvariant=\"bold-italic\">ϵ</mml:mi></mml:mrow><mml:mo>″</mml:mo></mml:msup><mml:mo>∼</mml:mo><mml:mi mathvariant=\"script\">N</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mi mathvariant=\"bold\">I</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq142\"><alternatives><tex-math id=\"M301\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\textbf{z}$$\\end{document}</tex-math><mml:math id=\"M302\"><mml:mi mathvariant=\"bold\">z</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq143\"><alternatives><tex-math id=\"M303\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\textbf{L}_\\textbf{z}$$\\end{document}</tex-math><mml:math id=\"M304\"><mml:msub><mml:mi mathvariant=\"bold\">L</mml:mi><mml:mi mathvariant=\"bold\">z</mml:mi></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq144\"><alternatives><tex-math id=\"M305\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$N_a$$\\end{document}</tex-math><mml:math id=\"M306\"><mml:msub><mml:mi>N</mml:mi><mml:mi>a</mml:mi></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq145\"><alternatives><tex-math id=\"M307\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\textbf{A}_\\textbf{z}$$\\end{document}</tex-math><mml:math id=\"M308\"><mml:msub><mml:mi mathvariant=\"bold\">A</mml:mi><mml:mi mathvariant=\"bold\">z</mml:mi></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq146\"><alternatives><tex-math id=\"M309\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\textbf{A}_\\textbf{z}$$\\end{document}</tex-math><mml:math id=\"M310\"><mml:msub><mml:mi mathvariant=\"bold\">A</mml:mi><mml:mi mathvariant=\"bold\">z</mml:mi></mml:msub></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ9\"><label>9</label><alternatives><tex-math id=\"M311\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} Z_t \\sim \\mathcal {M}(\\text {softmax}(\\textbf{A} + \\sigma _t^{\\prime }\\textbf{A}_\\textbf{z})) \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M312\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:msub><mml:mi>Z</mml:mi><mml:mi>t</mml:mi></mml:msub><mml:mo>∼</mml:mo><mml:mi mathvariant=\"script\">M</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mtext>softmax</mml:mtext><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi mathvariant=\"bold\">A</mml:mi><mml:mo>+</mml:mo><mml:msubsup><mml:mi>σ</mml:mi><mml:mi>t</mml:mi><mml:mo>′</mml:mo></mml:msubsup><mml:msub><mml:mi mathvariant=\"bold\">A</mml:mi><mml:mi mathvariant=\"bold\">z</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq147\"><alternatives><tex-math id=\"M313\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\mathcal {M}$$\\end{document}</tex-math><mml:math id=\"M314\"><mml:mi mathvariant=\"script\">M</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq148\"><alternatives><tex-math id=\"M315\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\textbf{A}$$\\end{document}</tex-math><mml:math id=\"M316\"><mml:mi mathvariant=\"bold\">A</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq149\"><alternatives><tex-math id=\"M317\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\sigma _t^{\\prime }$$\\end{document}</tex-math><mml:math id=\"M318\"><mml:msubsup><mml:mi>σ</mml:mi><mml:mi>t</mml:mi><mml:mo>′</mml:mo></mml:msubsup></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq150\"><alternatives><tex-math id=\"M319\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\sigma _t$$\\end{document}</tex-math><mml:math id=\"M320\"><mml:msub><mml:mi>σ</mml:mi><mml:mi>t</mml:mi></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq151\"><alternatives><tex-math id=\"M321\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\sigma _t^{\\prime }$$\\end{document}</tex-math><mml:math id=\"M322\"><mml:msubsup><mml:mi>σ</mml:mi><mml:mi>t</mml:mi><mml:mo>′</mml:mo></mml:msubsup></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq152\"><alternatives><tex-math id=\"M323\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\textbf{r}_{f_t}$$\\end{document}</tex-math><mml:math id=\"M324\"><mml:msub><mml:mi mathvariant=\"bold\">r</mml:mi><mml:msub><mml:mi>f</mml:mi><mml:mi>t</mml:mi></mml:msub></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq153\"><alternatives><tex-math id=\"M325\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$Z_t$$\\end{document}</tex-math><mml:math id=\"M326\"><mml:msub><mml:mi>Z</mml:mi><mml:mi>t</mml:mi></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq154\"><alternatives><tex-math id=\"M327\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\textbf{L}_\\textbf{z}$$\\end{document}</tex-math><mml:math id=\"M328\"><mml:msub><mml:mi mathvariant=\"bold\">L</mml:mi><mml:mi mathvariant=\"bold\">z</mml:mi></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq155\"><alternatives><tex-math id=\"M329\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\textbf{r}_{c_t} = \\textbf{L}_\\textbf{z}\\textbf{r}_{f_t}$$\\end{document}</tex-math><mml:math id=\"M330\"><mml:mrow><mml:msub><mml:mi mathvariant=\"bold\">r</mml:mi><mml:msub><mml:mi>c</mml:mi><mml:mi>t</mml:mi></mml:msub></mml:msub><mml:mo>=</mml:mo><mml:msub><mml:mi mathvariant=\"bold\">L</mml:mi><mml:mi mathvariant=\"bold\">z</mml:mi></mml:msub><mml:msub><mml:mi mathvariant=\"bold\">r</mml:mi><mml:msub><mml:mi>f</mml:mi><mml:mi>t</mml:mi></mml:msub></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq156\"><alternatives><tex-math id=\"M331\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\textbf{f}_{t} = (Z_t, \\textbf{F}_t, \\textbf{z}, t)$$\\end{document}</tex-math><mml:math id=\"M332\"><mml:mrow><mml:msub><mml:mi mathvariant=\"bold\">f</mml:mi><mml:mi>t</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>Z</mml:mi><mml:mi>t</mml:mi></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi mathvariant=\"bold\">F</mml:mi><mml:mi>t</mml:mi></mml:msub><mml:mo>,</mml:mo><mml:mi mathvariant=\"bold\">z</mml:mi><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq157\"><alternatives><tex-math id=\"M333\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\textbf{F}_t$$\\end{document}</tex-math><mml:math id=\"M334\"><mml:msub><mml:mi mathvariant=\"bold\">F</mml:mi><mml:mi>t</mml:mi></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq158\"><alternatives><tex-math id=\"M335\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\textbf{r}_t$$\\end{document}</tex-math><mml:math id=\"M336\"><mml:msub><mml:mi mathvariant=\"bold\">r</mml:mi><mml:mi>t</mml:mi></mml:msub></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ10\"><label>10</label><alternatives><tex-math id=\"M337\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} \\mathcal {L}_{simple} = \\Vert \\varvec{\\epsilon } - \\varvec{\\epsilon }_{\\theta }(\\textbf{r}_{c_t}, \\textbf{f}_{t})\\Vert ^2. \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M338\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:msub><mml:mi mathvariant=\"script\">L</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">simple</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:msup><mml:mrow><mml:mo stretchy=\"false\">‖</mml:mo><mml:mrow><mml:mi mathvariant=\"bold-italic\">ϵ</mml:mi></mml:mrow><mml:mo>-</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant=\"bold-italic\">ϵ</mml:mi></mml:mrow><mml:mi>θ</mml:mi></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi mathvariant=\"bold\">r</mml:mi><mml:msub><mml:mi>c</mml:mi><mml:mi>t</mml:mi></mml:msub></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi mathvariant=\"bold\">f</mml:mi><mml:mi>t</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo stretchy=\"false\">‖</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:msup><mml:mo>.</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq159\"><alternatives><tex-math id=\"M339\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\varvec{\\epsilon }$$\\end{document}</tex-math><mml:math id=\"M340\"><mml:mrow><mml:mi mathvariant=\"bold-italic\">ϵ</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq160\"><alternatives><tex-math id=\"M341\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\textbf{A}$$\\end{document}</tex-math><mml:math id=\"M342\"><mml:mi mathvariant=\"bold\">A</mml:mi></mml:math></alternatives></inline-formula>", "<disp-formula id=\"Equ11\"><label>11</label><alternatives><tex-math id=\"M343\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} \\mathcal {L}_{diff} = \\mathcal {L}_{simple} + \\lambda \\mathcal {L}_{CE}(\\textbf{A},\\textbf{A}_{\\theta }(\\textbf{r}_{c_t}, \\textbf{f}_{t})), \\end{aligned}$$\\end{document}</tex-math><mml:math id=\"M344\" display=\"block\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd columnalign=\"right\"><mml:mrow><mml:msub><mml:mi mathvariant=\"script\">L</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">diff</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:msub><mml:mi mathvariant=\"script\">L</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">simple</mml:mi></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:mi>λ</mml:mi><mml:msub><mml:mi mathvariant=\"script\">L</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">CE</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi mathvariant=\"bold\">A</mml:mi><mml:mo>,</mml:mo><mml:msub><mml:mi mathvariant=\"bold\">A</mml:mi><mml:mi>θ</mml:mi></mml:msub><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi mathvariant=\"bold\">r</mml:mi><mml:msub><mml:mi>c</mml:mi><mml:mi>t</mml:mi></mml:msub></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi mathvariant=\"bold\">f</mml:mi><mml:mi>t</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>,</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></alternatives></disp-formula>", "<inline-formula id=\"IEq161\"><alternatives><tex-math id=\"M345\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\mathcal {L}_{CE}$$\\end{document}</tex-math><mml:math id=\"M346\"><mml:msub><mml:mi mathvariant=\"script\">L</mml:mi><mml:mrow><mml:mi mathvariant=\"italic\">CE</mml:mi></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq162\"><alternatives><tex-math id=\"M347\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\lambda$$\\end{document}</tex-math><mml:math id=\"M348\"><mml:mi>λ</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq163\"><alternatives><tex-math id=\"M349\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$t \\in \\{1, \\ldots , T\\}$$\\end{document}</tex-math><mml:math id=\"M350\"><mml:mrow><mml:mi>t</mml:mi><mml:mo>∈</mml:mo><mml:mo stretchy=\"false\">{</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mo>…</mml:mo><mml:mo>,</mml:mo><mml:mi>T</mml:mi><mml:mo stretchy=\"false\">}</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq164\"><alternatives><tex-math id=\"M351\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$T=1000$$\\end{document}</tex-math><mml:math id=\"M352\"><mml:mrow><mml:mi>T</mml:mi><mml:mo>=</mml:mo><mml:mn>1000</mml:mn></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq165\"><alternatives><tex-math id=\"M353\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\textbf{z}$$\\end{document}</tex-math><mml:math id=\"M354\"><mml:mi mathvariant=\"bold\">z</mml:mi></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq166\"><alternatives><tex-math id=\"M355\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\textbf{z} \\sim \\mathcal {N}(0, \\textbf{I})$$\\end{document}</tex-math><mml:math id=\"M356\"><mml:mrow><mml:mi mathvariant=\"bold\">z</mml:mi><mml:mo>∼</mml:mo><mml:mi mathvariant=\"script\">N</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mi mathvariant=\"bold\">I</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq167\"><alternatives><tex-math id=\"M357\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$N_a$$\\end{document}</tex-math><mml:math id=\"M358\"><mml:msub><mml:mi>N</mml:mi><mml:mi>a</mml:mi></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq168\"><alternatives><tex-math id=\"M359\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\textbf{L}_\\textbf{z}$$\\end{document}</tex-math><mml:math id=\"M360\"><mml:msub><mml:mi mathvariant=\"bold\">L</mml:mi><mml:mi mathvariant=\"bold\">z</mml:mi></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq169\"><alternatives><tex-math id=\"M361\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\textbf{A}_\\textbf{z}$$\\end{document}</tex-math><mml:math id=\"M362\"><mml:msub><mml:mi mathvariant=\"bold\">A</mml:mi><mml:mi mathvariant=\"bold\">z</mml:mi></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq170\"><alternatives><tex-math id=\"M363\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$t=T$$\\end{document}</tex-math><mml:math id=\"M364\"><mml:mrow><mml:mi>t</mml:mi><mml:mo>=</mml:mo><mml:mi>T</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq171\"><alternatives><tex-math id=\"M365\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$Z_T$$\\end{document}</tex-math><mml:math id=\"M366\"><mml:msub><mml:mi>Z</mml:mi><mml:mi>T</mml:mi></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq172\"><alternatives><tex-math id=\"M367\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\textbf{A}_\\textbf{z}$$\\end{document}</tex-math><mml:math id=\"M368\"><mml:msub><mml:mi mathvariant=\"bold\">A</mml:mi><mml:mi mathvariant=\"bold\">z</mml:mi></mml:msub></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq173\"><alternatives><tex-math id=\"M369\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\textbf{r}_T \\sim \\mathcal {N}(0, \\textbf{I}).$$\\end{document}</tex-math><mml:math id=\"M370\"><mml:mrow><mml:msub><mml:mi mathvariant=\"bold\">r</mml:mi><mml:mi>T</mml:mi></mml:msub><mml:mo>∼</mml:mo><mml:mi mathvariant=\"script\">N</mml:mi><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mi mathvariant=\"bold\">I</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow><mml:mo>.</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq174\"><alternatives><tex-math id=\"M371\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\text {argmax}(\\textbf{A}_{\\theta })$$\\end{document}</tex-math><mml:math id=\"M372\"><mml:mrow><mml:mtext>argmax</mml:mtext><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi mathvariant=\"bold\">A</mml:mi><mml:mi>θ</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq175\"><alternatives><tex-math id=\"M373\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$10^{-5}$$\\end{document}</tex-math><mml:math id=\"M374\"><mml:msup><mml:mn>10</mml:mn><mml:mrow><mml:mo>-</mml:mo><mml:mn>5</mml:mn></mml:mrow></mml:msup></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq176\"><alternatives><tex-math id=\"M375\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$10^{-5}$$\\end{document}</tex-math><mml:math id=\"M376\"><mml:msup><mml:mn>10</mml:mn><mml:mrow><mml:mo>-</mml:mo><mml:mn>5</mml:mn></mml:mrow></mml:msup></mml:math></alternatives></inline-formula>", "<inline-formula id=\"IEq177\"><alternatives><tex-math id=\"M377\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$5 \\times 5 \\times 5$$\\end{document}</tex-math><mml:math id=\"M378\"><mml:mrow><mml:mn>5</mml:mn><mml:mo>×</mml:mo><mml:mn>5</mml:mn><mml:mo>×</mml:mo><mml:mn>5</mml:mn></mml:mrow></mml:math></alternatives></inline-formula>" ]

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[ "<supplementary-material content-type=\"local-data\" id=\"MOESM1\"></supplementary-material>" ]

[ "<table-wrap-foot><p>Significant values are in [bold].</p></table-wrap-foot>", "<table-wrap-foot><p>Significant values are in [bold].</p></table-wrap-foot>", "<table-wrap-foot><p>Significant values are in [bold].</p></table-wrap-foot>", "<fn-group><fn><p><bold>Publisher's note</bold></p><p>Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.</p></fn></fn-group>" ]

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[ "<media xlink:href=\"41598_2024_51400_MOESM1_ESM.pdf\"><caption><p>Supplementary Information.</p></caption></media>" ]

{ "acronym": [], "definition": [] }

[ "<title>Introduction</title>", "<p id=\"Par2\">With the continuous improvement of the global economy, the prevalence of chronic kidney disease (CKD) continues to elevate which has emerged as a severe public health problem, ≈15% of people suffer from CKD^{##REF##34571064##1##,##REF##31205024##2##}. CKD is typically characterized by renal interstitial fibrosis and excessive extracellular matrix deposition contributing to the progressive loss of renal function^{##REF##35788561##3##}. Unfortunately, existing effective clinical treatments only alleviate the process of renal fibrosis but do not complete the cure for the disease^{##REF##35641620##4##}. Therefore, developing a method for anti-renal fibrosis has very important clinical significance^{##REF##35779658##5##}.</p>", "<p id=\"Par3\">Numerous studies have reported that Smad2/3 positively regulates cell fibrosis-like change and inflammation in renal interstitial fibrosis^{##REF##33418880##6##,##REF##31917288##7##}. In addition, expression of Smad2/3 in the nucleus can aggregates in the renal tubular cell and accelerate CKD progression^{##REF##34774561##8##,##UREF##0##9##}. Therefore, the key to anti-fibrosis is to regulate the expression of Smad2/3 protein. A recent study revealed that carboxyl terminus of Hsp70 interacting protein (CHIP) as an E3 ubiquitin ligase, may directly participate in regulating the stability of Smad2/3 protein and inhibiting tumorigenesis^{##REF##14701756##10##,##REF##31414713##11##}. This finding suggests that increasing the activity of CHIP in the renal tubular cells to promote the clearance of Smad2/3 may be important for RIF therapy. However, CHIP ligase was susceptible to biochemical and physical instability, which may induce to inhibit Smad2/3 activity weakness and lead to decrease the therapeutic efficacy^{##REF##33082516##12##}. Moreover, the low enrichment in the renal for CHIP-mediated ubiquitin degradation significantly limits their research and the future clinical transformation applications for RIF therapy^{##UREF##1##13##}. The key to solving problems is to improve CHIP activity and selectively rich in the injured kidney tissue to promote Smad2/3 ubiquitination and degradation.</p>", "<p id=\"Par4\">Extracellular vesicles (EVs) such as microvesicles, are double membrane microparticles (60–1000 nm in size) secreted by cells in a constitutive or inducible manner^{##REF##32029601##14##,##REF##36185597##15##}. The released EVs naturally function as intercellular messengers by selecting transporting nucleic acids and proteins to distal or nearby recipient cells^{##REF##29290805##16##}. A large number of studies have shown the potential of using EVs as mighty and feasible nanocarriers for drug delivery vector in various situations, from tumor therapy to gene regeneration therapy^{##UREF##2##17##,##REF##28529644##18##}. Compared with current delivery systems, EVs have a unique advantage in the natural origin, which enables them to escape phagocytosis, extend the half-life of therapeutic agents, and low immunogenicity^{##REF##30126052##19##}. In addition, magnetic targeted therapy has become a novel research hotspot in recent years, and magnetic nanoparticles have small size and magnetic guided properties^{##REF##34578651##20##}, and targeted migration of tissue damage sites under the action of magnetic fields has broad application prospects^{##REF##31803890##21##}. Engineered EVs have shown great potential as a promising therapeutic platform and have attracted considerable attention in tissue regeneration^{##REF##36193769##22##}. EVs could be engineered to overexpress various proteins in targeting tissues and active signals pathway for the regulation of fibrosis like cells^{##REF##32952493##23##}. Therefore, we use superparamagnetic iron oxide nanoparticles to modify engineering EVs loaded CHIP delivery system to enhance anti-fibrosis therapeutic effect for the effectively treatment of CKD.</p>", "<p id=\"Par5\">Here, we provide the valid method for SPION decorated CHIP high-expressing MSC-EVs in CKD renal fibrosis treatment. We found that SPION-EVs-CHIP showed great ability to target injury renal in unilateral ureteral obstruction (UUO) rat. More importantly, SPION-EVs-CHIP significantly reversed collagen deposition by inducing Smad2/3 ubiquitination and degradation of renal tubular cells and inhibiting tubular damage-mediated inflammatory responses as compared to MSC-EVs. Our EV-engineering technology demonstrated that SPION-EVs with CHIP overexpression provide a potential platform for effective renal interstitial fibrosis therapy by inducing Smad2/3 degradation.</p>" ]

[ "<title>Methods</title>", "<title>Ethics statement for human samples and animal models</title>", "<p id=\"Par23\">Written informed consent was obtained from healthy donors, and discarded human umbilical cord tissues were acquired according to the approved protocol of the Institutional Review Board (IRB) at the Affiliated Hospital of Jiangsu University. And conducted in accordance with ethical principles of the World Medical Association Declaration of Helsinki. All animal experimental protocols were approved by the Animal Care and Use Committee of Jiangsu University (Approval number: 2020280). All animal experiments were performed in accordance with the associated relevant guidelines and regulations for working with live vertebrate animals.</p>", "<title>Animals</title>", "<p id=\"Par24\">Sprague-Dawley rat (male, 6–8 weeks old, 180–200 g) were purchased from the Experimental Animal Center of the Jiangsu University. And db/db mice (female, 6–8 weeks old, 18–22 g) were purchased from the Cavens Experimental Animal Co., Ltd (Changzhou). They were provided free access to pellet food and kept water in plastic cages at 20 ± 2 °C and kept on a 12 h light/dark cycle in specific pathogen-free facilities. The experimental endpoints the animals were anesthetized euthanasia with pentobarbital sodium (Sprague-Dawley rat: 50 mg/kg, db/db mice: 30 mg/kg, Sigma-Aldrich) by intraperitoneal injection. Kidneys were fixed with 4% formaldehyde, embedded in paraffin, and sectioned to 4 μm thickness.</p>", "<title>Cell culture</title>", "<p id=\"Par25\">The rat renal tubular cells line NRK-52E was purchased from National Collection of Authenticated Cell Cultures. NRK-52E cells were maintained in DMEM (GIBCO, USA) containing 10% fetal bovine serum, 100 U mL⁻¹ penicillin, and 100 mg mL⁻¹ streptomycin in 5% CO₂ at 37 °C.</p>", "<p id=\"Par26\">Mesenchymal stem cells were isolated from human umbilical cord according to a previously described method. In brief, take the umbilical cord from a full-term newborn, wash with PBS and cut off the arteries and veins. Cut it into 2 mm sized tissue blocks and stick them on a 6-well plate with an interval of 5 mm. Add basic DMEM (GIBCO, USA) medium containing 10% fetal bovine serum and 100 U mL⁻¹ penicillin, and 100 mg mL⁻¹ streptomycin in 5% CO₂ at 37 °C. Thereafter, the medium was refreshed every 3 days and continuous culture until the purity of mesenchymal stem cells reached 85%.</p>", "<title>Extraction and Purification of MSC-EVs-CHIP</title>", "<p id=\"Par27\">A lentivirus vector encoding murine CHIP was purchased from genepharma (Suzhou, China). Mesenchymal stem cells were transfected with lentivirus encoding CHIP gene. When the transfection efficiency reached 80% measured by western blotting and flow cytometry analysis. The EVs in medium and FBS were depleted by ultracentrifugation. The medium was replaced and collected after 48 h. MSC-EVs were isolated and purified by ultracentrifugation^{##UREF##5##33##}. In brief, the medium was centrifuged at 800 g for 10 min, 2000g for 10 min, 10 000 g for 30 min, 100,000 g for 70 min at 4 °C, to obtain MSC-EVs. The specific marker expression of CD63, CD9, CD81, and the CHIP expression was measured by western blotting.</p>", "<title>Biological characterization analysis of MSC-EVs-CHIP</title>", "<p id=\"Par28\">Prepared MSC-EVs and MSC-EVs-CHIP samples were pipetted onto a 300 meshes copper and incubated for 3 min. 20 µL of 1% uranyl acetate was pipetted onto the mesh copper and incubated for 1 min, washed and dry for TEM. The images were acquired using transmission electron microscope. MSC-EVs size determination by dynamic light scattering (DLS), in brief the volume of MSC-EVs and MSC-EVs-CHIP samples was increased to 1 mL with PBS and loaded into a quartz cuvette. The size distribution of EVs was measured by dynamic light scattering at 25 °C. Nanometer tracking analyzer detected its zeta potential and nanoparticle size.</p>", "<title>Renal interstitial fibrosis model construction and treatment</title>", "<p id=\"Par29\">RIF was established in Sprague-Dawley rats, anesthetize with inhalation of 2% isoflurane, expose the left ureter through a lateral incision, and ligated with double straps and surgical thread. The rats were randomly assigned into five experimental groups: Control, UUO, MSC-EVs, and MSC-EVs-CHIP, SPION-EVs. Rats were ureteral obstruction for 2 weeks, and intravenously injected with MSC-EVs, MSC-EVs-CHIP and SPION-EVs (2 mg per rat/every three days). After 2 weeks of treatment, the renal and blood serum were collected and analyzed renal function. The biodistribution of MSC-EVs-CHIP and SPION-EVs was performed a PerkinElmer IVIS Lumina II. In brief, MSC-EVs-CHIP and SPION-EVs was labeled with CM-DIR, and the free CM-DIR was washed thrice with cold PBS by ultracentrifugation for 3 times with cold PBS. Then, the CM-DIR-labeled EVs were injected intravenously into rat for 12 h and the CM-DIR signal in various tissues by PerkinElmer IVIS Lumina II.</p>", "<title>RNA extraction and RT-qPCR</title>", "<p id=\"Par30\">Trizol reagent was applied to extract total RNA from cultured cells or tissues in strict accordance with the instructions provided on the kit, followed by the determination of RNA concentration. mRNA was reverse-transcribed to cDNA and subjected to quantitative PCR, which was performed with the BioRad CFX96 Real-Time PCR Detection System (BioRad, USA). mRNA expression was compared using the 2^−ΔΔCt relative quantification method, GAPDH was used as the endogenous control.</p>", "<title>Western blot analysis</title>", "<p id=\"Par31\">Renal tubular cells and kidney tissues were lysed using lysis buffer and the obtained protein lysates were quantitated by BCA assay, then degenerated at 100 °C for 5 min. Proteins were resolved by sodium dodecyl sulfate-polyacrylamide gel electrophoresis, transferred onto polyvinylidene fluoride membranes and blocked with 5% bovine serum albumin. Membranes were immunoblotted with primary rabbit polyclonal antibodies to alpha-smooth muscle actin (α-SMA; 1:500, AB32575), Mothers against decapentaplegic homolog 2 (Smad2/3; 1:1000, CST3102), Fibronectin (FN, 1:1000, CST26836), Collagen I (Collagen I, 1:1000, CST72026) and incubated overnight at 4 °C with constant shaking. Next, the primary antibody-incubated membranes were washed five times with washing buffer and incubated with HRP-coupled secondary antibody for 1 h at room temperature. Protein bands were visualized by western blotting. Band intensity was quantified using the Image J software. The relative expression of the target protein was normalized to the band intensity of β-actin. We have included original western blot chemiluminescent images with corresponding light micrographs showing molecular weight markers for all western blots in Supplementary Fig. ##SUPPL##0##10##.</p>", "<title>Histologic analysis</title>", "<p id=\"Par32\">Renal tissues samples were fixed in 10% formalin solution, embedded in paraffin and sliced into 5μm thick sections. After deparaffinization and rehydration, sections were stained with hematoxylin and eosin (HE), Sirius red and Masson. The area at the junction of the cortex and medulla in the renal section was selected for histological analysis. HE staining was used to assess tubular injury, Sirius red and Masson staining was performed to evaluate the deposition of collagenous fibers in the renal interstitium. The positive area (red) of Sirius red and (blue) Masson staining was quantified using Image J.</p>", "<title>Immunohistochemistry and Immunofluorescence</title>", "<p id=\"Par33\">Paraffin-embedded tissues were heat-fixed, deparaffinized, rehydrated, antigen retrieved, and subsequent process. First, renal tissue sections were heat-fixed for 2 h at 60 °C and deparaffinized in xylene for 30 min, then rehydrated in 100%, 75%, and 50% ethanol respectively for 10 min. Antigen retrieval was performed using 10 × 10⁻³ M citrate antigen retrieval solution under high pressure for 30 min. The sections were incubated with 3% H₂O₂ for 10 min in the dark, blocked with 10% BSA for 1 h and stained with Smad2/3 and Fibronectin antibodies overnight at 4 °C, and washed three times with PBS. 50 µL HRP-labeled goat anti-mouse/rabbit Ig mixture was added to sections for 30 min and washed with PBST thrice. 50 µL DAB reagent was added for color reaction and restained with hematoxylin for 30 s. Finally, the sections were soaked in 50%, 75%, 100% ethanol, and xylene for 10 min respectively to dehydrate. The images of samples were acquired using Leica fluorescence optical microscope.</p>", "<p id=\"Par34\">The renal sections after deparaffinized and rehydrated, then stained with Smad2/3, α-SMA, CD66, and CHIP, Slc5a12 antibodies overnight at 4 °C, and washed thrice times with PBS. The sections were stained with DAPI for confocal laser scanning microscopy (GE, USA). NRK-52E cells were washed twice with cold PBS. Then cells were fixed in 4% paraformaldehyde (PFA) for 10 min, blocked with 10% BSA for 1 h at room temperature, and stained with CHIP, Smad2/3, and α-SMA antibodies overnight at 4 °C. The cells were then washed thrice times with cold PBS, and stained with DAPI for confocal laser scanning microscopy (GE, USA).</p>", "<title>Statistical analysis</title>", "<p id=\"Par35\">All the experiments were randomized and blinded. All studies were performed in at least three independent experiments with each experiment including triplicate sets in vitro, or six animals per group in vivo. All data are presented as the mean ± SEM. GraphPad Prism 8.0 software (San Diego, CA, USA) was used for the statistical analysis. Western blotting and immunofluorescence analyses were performed using the Image J software. Data among multiple groups were compared using one-way analysis of variance, followed by a post hoc correction using Tukey’s test. a student’s t-test was used to test the difference between two groups. A value of *<italic>p</italic> &lt; 0.05 was indicative of a statistically significant difference.</p>" ]

[ "<title>Results</title>", "<title>Characteristics of MSC derived extracellular vesicles with CHIP high expression</title>", "<p id=\"Par6\">Umbilical derived MSCs with high purity were isolated from rat according to previous protocols (Supplementary Fig. ##SUPPL##0##1##). We next infected MSCs with lentivirus encoding the CHIP gene and selected with puromycin to obtain CHIP high-expressing MSCs, and confocal graph shows that CHIP is mainly expressed in the cytoplasm (Fig. ##FIG##0##1a, b##). The morphology of MSCs was not affected by high CHIP expression levels (Fig. ##FIG##0##1c##), lipogenesis and osteogenesis experiment slightly induced stem-cell differentiation compared to non-transfected MSCs (Supplementary Fig. ##SUPPL##0##2##). Then MSCs derived EVs were then isolated and purified according to previously described protocols. Transmission electron microscopy (TEM) imaging shows then cup shaped vesicle morphology of MSC-EVs-CHIP (Fig. ##FIG##0##1d##). And the average particle diameter of MSC-EVs-CHIP measured by dynamic light scattering (DLS) is 108 ± 10.6 nm (Fig. ##FIG##0##1e##). The zeta potential of MSC-EVs-CHIP was ≈−15 mV (Fig. ##FIG##0##1f##). Finally, we confirmed the expression of CHIP in MSCs-EVs by western blotting assay. The results showed that CHIP was also highly expressed in engineered EVs containing EVs-associated proteins, such as CD63, CD9, Alix and Calnexin was expressed negatively (Fig. ##FIG##0##1g##), and GFP signals was also found in EVs-CHIP. These results suggested that the genetically engineered MSC-EVs with high CHIP expression were successfully prepared.</p>", "<title>MSC-EVs-CHIP reduced fibrotic change by inducing Smad2/3 degradation in renal tubular cells</title>", "<p id=\"Par7\">Evidence suggests that activation of the Smad2/3 pathway and inflammatory infiltration exacerbate fibrosis in CKD. To test whether MSC-EVs-CHIP induces Smad2/3 degradation and reduce renal interstitial fibrosis, MSC-EVs with GFP-tagged CHIP high expression were incubated with renal tubular cells NRK-52E. As shown by confocal imaging, a high GFP fluorescence signal was observed in NRK-52E (Fig. ##FIG##1##2a##). Imaging flow cytometry showed that GFP MSC-EVs-CHIP can be internalized by NE-52E cells (Fig. ##FIG##1##2b##). Western blot experiment confirmed high expression of CHIP in NRK-52E (Fig. ##FIG##1##2c##). Furthermore, the GFP signal was colocalized with Smad2/3 (Red) in the cells, suggesting that CHIP shifted into the cytoplasm with MSC-EVs (Fig. ##FIG##1##2d##).</p>", "<p id=\"Par8\">Next, we used fibrosis stimulating factor TGF-β₁ (10 nM) to induce fibrosis-like change in NRK-52E cells and establish in vitro RIF model. In our experiment, it was found that TGF-β₁ stimulates α-SMA overexpress and transform into myofibroblasts cells (Supplementary Fig. ##SUPPL##0##3a##), MSC-EVs-CHIP significantly inhibited Smad2/3 and fibrotic of TGF-β₁ incubated cells, as shown by the confocal microphage and western blot (Fig. ##FIG##1##2e, f##). Additionally, Co-immunoprecipitation observed that MSC-EVs-CHIP promoted ubiquitin molecules expression and decreased the expression of Smad2/3 in TGF-β₁-incubated cells (Fig. ##FIG##1##2g##). In order to simulate the mechanical pressure environment in <italic>vivo</italic> by using stiff gel (30Kpa) to stimulate NRK-52E (Fig. ##FIG##1##2h##, Supplementary Fig. ##SUPPL##0##3b##), confocal microscopy also showed that the number of Smad2/3 in nucleus was decreased obviously in NK-52E cells treated with MSCEVs-CHIP relative to that in controls (Fig. ##FIG##1##2i##), and statistical analysis confirms the results (Fig. ##FIG##1##2j##). These results demonstrated that MSC-EVs-CHIP could alleviate TGF-β₁ and stiff gel induced renal tubular cell fibrotic like change by promoting Smad2/3 ubiquitination degradation.</p>", "<title>MSC-EVs-CHIP migrated to the kidney and ameliorated inflammatory infiltration by reduced Smad2/3 accumulation</title>", "<p id=\"Par9\">MSC-EVs-CHIP have been shown to promote Smad2/3 degradation, anti-fibrotic-like change of renal tubular cell, all of which play a critical role in the development and procession of RIF. We further evaluated the therapeutic efficacy of MSC-EVs-CHIP in RIF rat model established by unilateral UUO rat model. First, we assessed the organizational distribution of MSC-EVs-CHIP. CM-DIR labeled MSC-EVs and MSC-EVs-CHIP were injected into rat intravenously for 48 h, then the organ fluorescence was evaluated by PerkinElmer IVIS Lumina II ex vivo imaging. As shown in Figure ##SUPPL##0##S4##, MSC-EVs and MSC-EVs-CHIP was enriched in liver, hardly enriched in the kidney of control groups, while accumulated in kidney of UUO treated rat (Supplementary Fig. ##SUPPL##0##4##). In addition, the fluorescence imaging of the kidney tissue of RIF-rat receiving MSC-EVs-CHIP exhibited significant CM-DIR signals relative to the control (Supplementary Fig. ##SUPPL##0##5a##). Western blotting analysis further showed that MSC-EVs treatment resulted in a high CHIP level in kidney tissue (Supplementary Fig. ##SUPPL##0##5b##). These data suggested that under RIF pathological conditions, MSC-EVs-CHIP have an ability to target the injury kidney.</p>", "<p id=\"Par10\">We next investigated whether MSC-EVs-CHIP promote smad2/3 degradation and inhibits inflammation in the kidney of RIF model. Rats were raised for 7 days after unilateral ureteral obstruction, and were then intravenously treated with MSC-EVs or MSC-EVs-CHIP (2 mg per rat/every three days). After one week, we performed immunofluorescence analysis of the renal tissue. In UUO rats, Smad2/3 is highly expressed in the nucleus of renal tubular epithelial cells, and fibrosis-related proteins (Collagen I, α-SMA) are significantly increased (Supplementary Fig. ##SUPPL##0##6##). While the renal of rat receiving MSC-EVs-CHIP, we observed a significantly reduced expression of Smad2/3 in renal tubular nucleus compared to MSC-EVs treatment, suggesting the downregulation of Smad2/3 in the renal tissue of RIF rat (Fig. ##FIG##2##3a##). In addition, the fibrosis related indicators α-SMA and Fibronectin in the kidneys of RIF rats receiving MSC-EVs-CHIP were significantly reduced (Fig. ##FIG##2##3b##). We further observed the reduced Smad2/3 accumulation and tubular cells fibrotic like change in the renal of MSC-EVs-CHIP-treated rat by immunofluorescence analysis, suggesting that MSC-EVs-CHIP can significantly rescue RIF pathology by promoting Smad2/3 degradation and recovering normal homeostatic processes (Fig. ##FIG##2##3c, d##).</p>", "<p id=\"Par11\">Given the critical role of inflammation in the progression of RIF, its effect on inflammation in renal was also evaluated after MSC-EVs-CHIP intervene. Persistent mechanical damage can stimulate the inflammatory response in tubular cells, especially the CD66 positive neutrophil accumulation, as shown in the HE staining and immunofluorescence (Fig. ##FIG##2##3e–g##). Encouragingly, the induced inflammation and cytokine (IL-6, IL-1β, TNF-α) release was remarkably inhibited by MSC-EVs-CHIP treatment (Fig. ##FIG##2##3h##). These data suggested that CHIP overexpression mediated by MSC-EVs inhibited Smad2/3 and decreased fibrosis index, further remitted the inflammation in RIF-rat.</p>", "<title>MSC-EVs-CHIP reduced interstitial collagen deposition in UUO model</title>", "<p id=\"Par12\">Inject the prepared MSC-EVs-CHIP and MSC-EVs into the tail vein of the UUO model, and collect large mouse notebooks from each group on the 15th day. The biochemical analyzer detected renal function indicators such as creatinine and urea nitrogen, and found that the MSC-EVs-CHIP group significantly downregulated creatinine and urea nitrogen compared to the MSC-EVs group, significantly improving renal function (Fig. ##FIG##3##4a, b##). The appearance of kidneys in the UUO group significantly larger and lighter in color, while the kidney tissue significantly returned to control after MSC-EVs-CHIP treatment (Fig. ##FIG##3##4c##). Further through the HE staining results of renal tissue pathology showed that the UUO injury group had an increase in tubular vacuolar degeneration, a significant thickening of the glomerular basement membrane, and a tissue damage score of 70%. Compared with the MSC-EVs group, the MSC-EVs-CHIP group restored normal renal tissue structure after intervention, and reduced tubular vacuolar degeneration. The damage score was reduced to 18% (Fig. ##FIG##3##4d, e##). The results of Sirius red staining and Masson staining showed significant fibrosis in UUO renal tissue. The change area reached 64%, while the MSC-EVs-CHIP group showed a significant reduction in renal interstitial collagen fibers and a significant decrease in fibrosis degree after treatment, which was better than the MSC-EVs treated group alone (Fig. ##FIG##3##4f-h##). Extract kidney tissue proteins from each group for Western blot detection, displaying fibrosis related proteins Fibronectin, Collagen I, and α-SMA were significantly reduced (Fig. ##FIG##3##4i##). The above research results indicate that high expression of CHIP enhances MSC-EVs in improving renal function and reducing renal interstitial collagen.</p>", "<title>In vivo renal-targeting and antifibrotic therapeutic effect of SPION-EVs</title>", "<p id=\"Par13\">In order to further enhance the anti-fibrotic effect of MSC-EVs, we used the SPION with surface modified transferrin (Tf) to combine with the transferrin receptor on the surface of MSC EVs CHIP for engineering modification to enhance its ability to target the injured site. Scanning electron microscope observation shows that the size of the nanospheres is about 5 nm, and dynamic light scattering shows that the size of the nanoparticle is about 10 nm (Fig. ##FIG##4##5a##). The superparamagnetic modification of transferrin binds to the transferrin receptor on the surface of the MSC-EVs-CHIP membrane after co incubation at 4 °C for 4 h, and SPION-EVs are collected and dissolved in PBS by external magnetic field adsorption (Fig. ##FIG##4##5b##). The elliptical vesicle structure combined with multiple SPION was observed by transmission electron microscopy (Fig. ##FIG##4##5c##). Analyze the stability and magnetic characteristics of SPION-EVs in aqueous solutions using hysteresis loops and PDI polymer dispersion index (Fig. ##FIG##4##5d, e##). The average particle size of SPION-EVs measured by dynamic light scattering (DLS) is 115 ± 11.3 nm, compared to MSC-EVs-CHIP, the average particle diameter increases by about 7 nm, indicating that SPION not obvious change the size of extracellular vesicles (Fig. ##FIG##4##5f##). The zeta potential of SPION-EVs was ≈−16 mV (Supplementary Fig. ##SUPPL##0##7##). Western blot was used to detect the expression of surface positive marker proteins CD9, CD81, Alix and negative marker Calnexin in SPION-EVs, and strong positive expression of CHIP and transferrin receptor TfR (Fig. ##FIG##4##5g##). This indicates that we have successfully prepared engineered SPION-EVs.</p>", "<p id=\"Par14\">Construct the renal fibrosis treatment model using prepared engineered SPION-EVs under external magnetic field intervention (Fig. ##FIG##4##5h##). By using a small animal live imaging system, it was shown that under the action of an external magnetic field, the signal intensity of DIR labeled SPION-EVs was enriched by targeting the liver to the kidneys (Fig. ##FIG##4##5i##). The pathological tissue showed a decrease in tubular vacuolar degeneration and a return to normal structure in the SPION-EVs intervention group. Sirius red and Masson staining confirmed that compared to the MSC-EVs -CHIP group, the SPION-EVs group showed a decrease in interstitial collagen fibers and a significant decrease in fibrotic areas (Fig. ##FIG##4##5j##). Immunofluorescence showed that SPION-EVs significantly inhibited the nuclear uptake of Smad2/3 in renal tubular cells and decreased α-SMA was superior to the MSC-EVs-CHIP group (Fig. ##FIG##4##5k##). The above results indicate that the targeted enrichment of engineered SPION-EVs under magnetic field significantly enhances the anti-fibrotic effect.</p>", "<title>SPION-EVs attenuated renal interstitial fibrosis in the DKD model</title>", "<p id=\"Par15\">To further prove that SPION-EVs is a promising nanoplatform for the delivery of CHIP and treatment of renal fibrosis, we constructed another renal interstitial fibrosis model in diabetic kidney disease (DKD) rat. The DKD injury model was established by feeding high-fat combined with streptozotocin (STZ) for 18 weeks, intervened by tail vein injection of SPION-EVs. Firstly, we assessed the alleviation of SPION-EVs against interstitial fibrosis in DKD rat. As expected, histological analysis of kidney sections at 24 weeks after STZ injection revealed significant tubulointerstitial damage, including tubular atrophy, cast formation, infiltrates of leukocytes, and fibrosis, all of which were markedly ameliorated by SPION-EVs treatment (Fig. ##FIG##5##6a, b##). To fully assess the efficacy of SPION-EVs, kidney sections were subjected to Masson and Sirius red staining to determine the fibrosis index and stained with anti-α-SMA/Fibronectin antibody for the detection of fibrosis-like tubules. The results demonstrated that the renal fibrosis was reduced in DKD rat treated with SPION-EVs (Fig. ##FIG##5##6c–f##), and reduction of α-SMA, Collagen I, and Fibronectin deposition in renal tissues of SPION-EVs treated DKD rat (Fig. ##FIG##5##6g–j##). Moreover, immunofluorescence shows that SPION-EVs can significantly inhibit the nuclear expression of Smad2/3 in renal tubules stimulated by high glucose environment and reduce α-SMA (Fig. ##FIG##5##6k##). These data suggested that SPION-EVs could attenuate renal fibrosis and retarding the progress of DKD (Fig. ##FIG##6##7##).</p>", "<title>Assessment of toxicity and inflammation in SPION-EVs–treated rat model</title>", "<p id=\"Par16\">We next examined the concentrations of hepatic enzyme such as alanine aminotransferase and aspartate aminotransferase in the serum of SPION-EVs treated rats showed no obvious differences from those of phosphate-buffered saline (PBS) treated rats (Supplementary Fig. ##SUPPL##0##8a##). Similarly, SPION-EVs and MSC-EVs-CHIP group toxicity on the major organs, including the heart, liver, spleen and lung. The histological analysis showed no evidence of in vivo toxicity of SPION-EVs (Supplementary Fig. ##SUPPL##0##8b##). Consistent with previous studies that engineered cell-derived EVs are biocompatible and not significantly toxic.</p>" ]

[ "<title>Discussion</title>", "<p id=\"Par17\">In this study, we reported a method for manufacturing engineered SPION-EVs and evaluated the feasibility, safety, and effectiveness of MSC-EVs with CHIP overexpressed in treating renal interstitial fibrosis in UUO and DKD models. Impressively, provide packed CHIP protein into MSC-derived EVs by genetic engineering with lentivirus transfection (Fig. ##FIG##0##1##), then through nanomaterials SPION decorated MSC-EVs with CHIP overexpressing target and enrich the damaged kidneys, resulting in notable renal protective effects, specifically by promoting Smad2/3 ubiquitination and degradation to reverse the transformation of renal tubular cells into myofibroblast. Our findings not only demonstrate the role of engineered MSC extracellular vesicles in the anti-fibrotic activity, but also as a new renal fibrosis therapeutic agent and target delivery nanoplatform. In a word, engineered SPION-EVs constitute an effective nanotherapeutic for the renal fibrosis treatment.</p>", "<p id=\"Par18\">Over the past several years, the clinical studies of RIF therapies have concentrated on antifibrotic, including anti-fibrosis antibodies and YAP inhibitors^{##REF##32382019##24##,##REF##37328872##25##}. Unfortunately, these treatment strategies have not achieved good results and were not significantly reduce fibrosis in clinical tests. Furthermore, the pathogenesis of RIF is complex and includes inflammation, tubular injury, oxidative stress and metabolic dysfunction^{##REF##28388958##26##}. There is increasing evidence that Smad2/3 ubiquitination and degradation in renal interstitial is a promising treatment method for CKD therapy^{##UREF##3##27##}. we developed an SPION decorated engineered MSC-EVs platform by CHIP expression to reduced renal fibrosis and inflammatory infiltration in RIF rats. In vitro, we observed that MSC-EVs-CHIP inhibit Smad2/3 expression in rat renal tubular epithelial cells and relieved cells myofibroblast change induced by TGF-β₁, and inhibited inflammasome activation. In UUO and DKD rat model, MSC-EVs-CHIP promoted Smad2/3 ubiquitination and degradation, further alleviated tubular cells fibrosis and inflammation.</p>", "<p id=\"Par19\">CHIP as an E3 Ubiquitin ligase, that induces protein folding and degradation balance by upregulating ligase expression or activation^{##REF##35020437##28##}. And which is an important component of the protein quality control system, preventing and treating diseases related to protein metabolism disorders^{##REF##32494669##29##}. CHIP protein could be used as a biological drug for the treatment of RIF. However, due to the distribution of metabolism in <italic>vivo</italic>, and poor renal target efficacy, the poor stability, lead to the CHIP protein can hardly be used in clinical practice. Moreover, we verified the low enrichment of CHIP protein in kidney tissue by intravenous injection in the RIF rat model. According to our previous works on genetically engineering decorated MSC-EVs for delivery of the key protein molecules, which have packed CHIP protein into MSC-EVs by genetic engineering with lentivirus transfection, then use SPION decorated MSC-EVs-CHIP that resolved the challenges for enrichment and renal-target.</p>", "<p id=\"Par20\">Mesenchymal stem cell derived extracellular vesicles as cell-free long-distance delivery systems have displayed apparent advantages, for instance high injury tissue target ability, tissue repairment capacity and low immunogenicity, which have been applicated in clinical therapies^{##REF##35787632##30##}. It has reported that MSC-EVs have strong tissues target capacity to realize drug delivery^{##UREF##4##31##}. Our previous research confirmed that MSC-EVs attenuate renal interstitial fibrosis through the kinase ubiquitin system CK1δ/β-TRCP mediated YAP ubiquitination and degradation^{##REF##32382019##24##}. MSC-EVs could to some extent reach the site of damaged kidney tissue and exert the effective on tissues fibrosis, however, the fluorescence statistical distribution of MSC-EVs labeled with DIR membrane dyes, we found that MSC-EVs were mainly concentrated in the liver and lungs. This has prompted us to think deeply about the need to enhance the targeted enrichment of extracellular vesicles through engineering transformation.</p>", "<p id=\"Par21\">Superparamagnetic iron oxide nanoparticles (SPION) have been widely used in disease therapy for excellent superparamagnetism property. The conjugation of SPIONs with extracellular vesicles has many advantages, including magnetic targeted functionalization, magnetic thermotherapy, and delivery of anti-fibrosis agents^{##REF##36858264##32##}. Furthermore, in order to reduce the uptake of MSC-EVs in the liver and lungs after tail vein injection, nanomaterials SPION modified MSC-EVs were used to enhances the penetration and enrichment of MSC-EVs to renal under external magnetic field.</p>", "<p id=\"Par22\">In conclusion, we have exploited a therapeutic strategy for renal fibrosis using nanomaterials SPION decorated MSC-EVs-CHIP to increase target the concentration of CHIP in renal tissue. We have constructed a platform for targeted delivery of CHIP by using MSC-EVs and highlighted the potential of SPION-EVs as a promising nanotherapeutic for renal interstitial fibrosis treatment. Our platform may provide potential for clinical application in the future owing to the advantages of engineered MSC-EVs and the novel inducing Smad2/3 protein degradation strategy in RIF treatment.</p>" ]

[]

[ "<p id=\"Par1\">Renal interstitial fibrosis (RIF) is a fundamental pathological feature of chronic kidney disease (CKD). However, toxicity and poor renal enrichment of fibrosis inhibitors limit their further applications. In this study, a platform for CKD therapy is developed using superparamagnetic iron oxide nanoparticles (SPION) decorated mesenchymal stem cells derived extracellular vesicles with carboxyl terminus of Hsc70-interacting protein (CHIP) high expression (SPION-EVs) to achieve higher renal-targeting antifibrotic therapeutic effect. SPION-EVs selectively accumulate at the injury renal sites under an external magnetic field. Moreover, SPION-EVs deliver CHIP to induce Smad2/3 degradation in renal tubular cells which alleviates Smad2/3 activation-mediated fibrosis-like changes and collagen deposition. The extracellular vesicle engineering technology provides a potential nanoplatform for RIF therapy through CHIP-mediated Smad2/3 degradation.</p>", "<title>Subject terms</title>" ]

[ "<title>Supplementary information</title>", "<p>\n\n\n</p>" ]

[ "<title>Supplementary information</title>", "<p>The online version contains supplementary material available at 10.1038/s41536-024-00348-0.</p>", "<title>Acknowledgements</title>", "<p>C.J. and J.Z. contributed equally to this work. The authors thank the members of Qian lab for helpful discussion and paper preparation. This work was supported by the National Natural Science Foundation of China (82172102, 81871496), the Jiangsu Province’s Major Project in Research and Development (BE2021689), the Natural Science Foundation of Jiangsu Province (BK20220527), China Postdoctoral Science Foundation (2023M731376), Zhenjiang Key Laboratory of High Technology Research on Exosomes Foundation and Transformation Application (Grant ss2018003).</p>", "<title>Author contributions</title>", "<p>C.J. performed experimental design, tissue procurement, data generation pathological assessments, data analysis and interpretation, and manuscript preparation; J.Z. performed experimental design, data generation, and data analysis; H.S. and L.S. performed tissue procurement, data generation, interpretation, and intellectual contribution. W.X., J.J. provided intellectual contribution and critically appraised the manuscript; H.Q. conceived the study, designed experiments, interpreted data, and prepared the manuscript.</p>", "<title>Data availability</title>", "<p>All data needed to evaluate the conclusions in the paper are present in the paper and/or the Supplementary Materials. Additional data related to this paper may be requested from the authors.</p>", "<title>Competing interests</title>", "<p id=\"Par36\">The authors declare no competing interests.</p>" ]

[ "<fig id=\"Fig1\"><label>Fig. 1</label><caption><title>Characteristics of MSC-derived extracellular vesicles with CHIP high-expression.</title><p><bold>a</bold> MSCs were transfected with lentivirus encoding the CHIP gene. The expression of CHIP (green) with GFP-tagged in MSCs was determined by immunofluorescence (left). Nuclei were stained with DAPI (Blue). Scale bar: 50 µm. Representative quantification of the mean fluorescence intensity (MFI) of CHIP (<italic>n</italic> = 3) (right). <bold>b</bold> Flow cytometry analysis of CHIP expression in MSCs (left) and representative quantification of the mean fluorescence intensity (MFI) of CHIP (<italic>n</italic> = 3) (right). <bold>c</bold> Morphology of MSCs with lentivirus transfection or non-transfection. Scale bar: 50 µm. <bold>d</bold> Representative transmission electron microscope (TEM) images of MSC-EVs and MSC-EVs- CHIP. Scale bar: 100 nm. <bold>e</bold> The size distribution of MSC-EVs and MSC-EVs-CHIP was measured by dynamic light scattering (DLS). <bold>f</bold> Surface zeta potential of MSC-EVs and MSC-EVs-CHIP (<italic>n</italic> = 3). <bold>g</bold> The expression of CHIP, CD73, CD63, and CD9 in MSCs and MSC-EVs with lentivirus was determined by western blotting (left). Relative values of CHIP/GAPDH and CHIP/CD9 stipe gray (<italic>n</italic> = 3) (right). Data are represented as the mean ± standard error of the mean (SEM). Statistical significance was calculated by a student’s t test. ns non-significance, **<italic>P</italic> &lt; 0.01, ***<italic>P</italic> &lt; 0.001.</p></caption></fig>", "<fig id=\"Fig2\"><label>Fig. 2</label><caption><title>MSC-EVs-CHIP induced Smad2/3 degradation in renal tubular cells.</title><p><bold>a</bold> MSC-EVs and MSC-EVs-CHIP were incubated with NRK-52E cells for 12 h. Confocal observed the expression of GFP-CHIP in cells. Nuclei were stained with DAPI (Blue). Scale bar: 25 µm. <bold>b</bold> Flow cytometry shows that NRK-52E cells internalize GFP-CHIP high expression of MSC-EVs. <bold>c</bold> The expression of CHIP in NRK-52E cells was determined by western blotting (left). Representative quantification of CHIP (right) (n = 3). <bold>d</bold> Confocal observed the co-localization of CHIP (Green) with Smad2/3 (Red) of NRK-52E cells. Nuclei were stained with DAPI (Blue). Scale bar: 25 µm. <bold>e</bold> The expression of Smad2/3 in NRK-52E was determined by western blotting. <bold>f</bold> The renal tubular cells (NRK-52E cell line) were pretreated with 10 nM TGF-β₁ protein in the absence or presence of MSC-EVs and MSC-EVs-CHIP for 24 h. The fibrosis index α-SMA (Red) was observed. Nuclei were stained with DAPI (Blue). Scale bar: 25 µm. <bold>g</bold> Determination of ubiquitin binding on Smad2/3 protein by CO-IP after MSC-EVs and MSC-EVs-CHIP treatment. <bold>h</bold> Diagrammatic of NRK-52E cells were stimulated with 30Kpa stiff gel in the presence of MSC-EVs and MSC-EVs-CHIP for 24 h. <bold>i</bold> The co-localized of CHIP (Green, stained by GFP) with Smad2/3 (Red, stained by Cy3) was observed. Nuclei were stained with DAPI (Blue). Scale bar: 25 µm. <bold>j</bold> Representative quantification of nuclear cytoplasmic ratio of Smad2/3. Data are represented as the mean ± SEM. Statistical significance was calculated by a student’s t-test. ns non-significance, *<italic>P</italic> &lt; 0.05, **<italic>P</italic> &lt; 0.01.</p></caption></fig>", "<fig id=\"Fig3\"><label>Fig. 3</label><caption><title>MSC-EVs-CHIP inhibited Smad2/3 and reduced inflammatory infiltration.</title><p><bold>a</bold> RIF was established by ureteral ligation of SD rat for 14 days, and intravenously injected with MSC-EVs and MSC-EVs-CHIP for three times in 7 days. Representative images of Smad2/3 in renal interstitial were demonstrated by immunohistochemistry (<italic>n</italic> = 3). Scale bar: 100 µm. <bold>b</bold> The localized of fibrosis marker α-SMA (Red) with renal tubular cell marker Slc5a1 (Green) in renal was showed by immunofluorescence (left). Nuclei were stained with DAPI (Blue). And representative images of Fibronectin were demonstrated by immunohistochemistry. Scale bar: 100 µm. <bold>c</bold> The co-localized of Smad2/3 (Green) with α-SMA (Red) in renal was showed by immunofluorescence (left). Nuclei were stained with DAPI (Blue). Scale bar: 100 µm. <bold>d</bold> The expression of Smad2/3, α-SMA and β-actin in kidney was determined by western blotting (left). Relative values of Smad2/3 and α-SMA stipe gray (<italic>n</italic> = 3) (right). <bold>e</bold> Representative images of HE staining of renal inflammatory infiltration. Scale bars: 100 µm. <bold>f</bold> The representative quantitation of Inflammatory infiltration was demonstrated by CD66 (Red) immunofluorescence staining. Nuclei were stained with DAPI (Blue). Scale bar: 100 µm. <bold>g</bold> Representative number of CD66 (Red) positive cell (<italic>n</italic> = 6). Nuclei were stained with DAPI (Blue). Scale bar: 100 µm. <bold>h</bold> The production of cytokines (IL-6, IL-1β, TNF-α) in the rat serum was measured using enzyme-linked immunosorbent assay (ELISA) kits (<italic>n</italic> = 3). Data are represented as the mean ± SEM. Statistical significance was calculated by a student’s t test. ns non-significance, *<italic>P</italic> &lt; 0.05, **<italic>P</italic> &lt; 0.01, ***<italic>P</italic> &lt; 0.001.</p></caption></fig>", "<fig id=\"Fig4\"><label>Fig. 4</label><caption><title>MSC-EVs-CHIP reduced collagen deposition in UUO model.</title><p><bold>a</bold> Effects of MSC-EVs-CHIP on serum urea nitrogen and <bold>b)</bold> creatinine. <bold>c</bold> Representative images of renal tissues established by UUO rats (<italic>n</italic> = 6 per group) that received different treatments including sham, PBS, MSC-EVs (10 mg/kg of body weight), MSC-EVs-CHIP (10 mg/kg of body weight). <bold>d</bold> Representative images of HE staining in MSC-EVs and MSC-EVs-CHIP group (<italic>n</italic> = 3). Scale bars: 100 µm. <bold>e</bold> The quantification of tubular injury based on HE staining (<italic>n</italic> = 5). <bold>f</bold> Representative images of Sirius red staining (<italic>n</italic> = 3). The bottom was a higher magnification of the boxed region. Scale bars: 1 mm (top) and 100 µm (bottom). <bold>g</bold> The quantification of fibrotic area based on Sirius red staining (<italic>n</italic> = 5). <bold>h</bold> Representative images of Masson trichrome staining on renal tissues sections (<italic>n</italic> = 3). Scale bars: 100 µm. <bold>i</bold> Western blotting analysis of Fibronectin, α-SMA, and Collagen I in kidney tissues.</p></caption></fig>", "<fig id=\"Fig5\"><label>Fig. 5</label><caption><title>In <italic>vivo</italic> renal-targeting and antifibrotic therapeutic effect of SPION-EVs.</title><p><bold>a</bold> TEM and dynamic light scattering (DLS) detect the graphic and nanoparticle size of SPION. Scale bar: 500 nm. <bold>b</bold> Separation and preparation of SPION-EVs flowchart. <bold>c</bold> Representative TEM images of MSC-EVs-CHIP and SPION-EVs. Scale bar: 100 nm. <bold>D</bold> Stability analysis of SPION-EVs using polymer dispersibility index (PDI). <bold>e</bold> Hysteresis loop analysis of SPION-EVs. <bold>f</bold> DLS detect the nanoparticle size of SPION-EVs. <bold>g</bold> The expression of CHIP, TfR, Alix, CD63, and CD9 in MSC-EVs-CHIP and SPION-EVs was determined by western blotting. <bold>h</bold> Schematic diagram of the experimental design. Briefly, rats were concurrently treated with MSC-EVs-CHIP and SPION-EVs (2 mg) every 3 days after renal UUO injury 7 days and were euthanized at 7 days after disease induction. <bold>i</bold> Intravenously injected with CM-DIR-labeled MSC-EVs-CHIP and SPION-EVs for 24, 48,72 h in UUO model. Ex vivo fluorescence images of CM-DIR-labeled EVs in major organs (left). Fluorescence intensity per gram of tissue in heart, liver, spleen, lung and kidney (right) (n = 3). <bold>j</bold> The role of SPION EVs in anti-fibrotic were demonstrated by HE and Masson staining, Sirius red staining (n = 3). Scale bar: 100 µm. <bold>k</bold> The expression of Smad2/3 (Green) with α-SMA (Red) after MSC-EVs-CHIP and SPION-EVs treatment in renal was showed by immunofluorescence. Nuclei were stained with DAPI (Blue). Scale bar: 100 µm. Data are represented as the mean ± S.E.M. Statistical significance was calculated by a student’s t test. ns, non-significance, *<italic>P</italic> &lt; 0.05, **<italic>P</italic> &lt; 0.01, ***<italic>P</italic> &lt; 0.001.</p></caption></fig>", "<fig id=\"Fig6\"><label>Fig. 6</label><caption><title>The anti-fibrotic effect of SPION-EVs in DKD model.</title><p><bold>a</bold> Representative image of HE staining with MSC-EVs-CHIP and SPION-EVs treatment in DKD model (<italic>n</italic> = 3). The bottom was a higher magnification of the boxed region. Scale bars: 1 mm (top) and 100 µm (bottom). <bold>b</bold> The quantification of tubular injury score based on HE staining (<italic>n</italic> = 5). <bold>c</bold> Representative images of Sirius red staining (<italic>n</italic> = 3). Scale bars: 100 µm. <bold>d</bold> The quantification of Masson positive area based on Masson staining (n = 5). <bold>e</bold> Representative images of Sirius red staining (<italic>n</italic> = 3). Scale bars: 100 µm. <bold>f</bold> The quantification of fibrotic area based on Sirius red staining (<italic>n</italic> = 5). <bold>g</bold> The expression of Smad2/3, α-SMA, Fibronectin and Collagen I in DKD model was determined by western blotting (left). <bold>h</bold> Relative values of Smad2/3, α-SMA, Fibronectin and Collagen I stipe gray (<italic>n</italic> = 3) (right). <bold>i</bold> Representative images of α-SMA were demonstrated by immunohistochemistry. Scale bar: 100 µm. <bold>j</bold> Representative images of Fibronectin in DKD model were demonstrated by immunohistochemistry. Scale bar: 100 µm. <bold>k</bold> The expression of Smad2/3 (Green) with α-SMA (Red) after MSC-EVs-CHIP and SPION-EVs treatment in DKD model was showed by immunofluorescence. Nuclei were stained with DAPI (Blue). Scale bar: 100 µm. Data are represented as the mean ± SEM. Statistical significance was calculated by a student’s t test. ns, non-significance, *<italic>P</italic> &lt; 0.05, **<italic>P</italic> &lt; 0.01, ***<italic>P</italic> &lt; 0.001.</p></caption></fig>", "<fig id=\"Fig7\"><label>Fig. 7</label><caption><title>Schematic illustrations of engineered extracellular vesicle as novel nanotherapeutics against renal fibrosis.</title><p>A platform for RIF therapy using SPION decorate nanosized mesenchymal stem cells-derived extracellular vesicles with carboxyl terminus of Hsc70-interacting protein (CHIP) high-expression (SPION-EVs) is developed. SPION-EVs significantly enrichment of renal injury sites and induces Smad2/3 degradation of renal tubular cells, which alleviates fibrosis and inflammation, accompanying with rescued collagen deposition in RIF.</p></caption></fig>" ]

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[ "<supplementary-material content-type=\"local-data\" id=\"MOESM1\"></supplementary-material>", "<supplementary-material content-type=\"local-data\" id=\"MOESM2\"></supplementary-material>" ]

[ "<fn-group><fn><p><bold>Publisher’s note</bold> Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.</p></fn><fn><p>These authors contributed equally: Cheng Ji, Jiahui Zhang, Linru Shi.</p></fn></fn-group>" ]

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[ "<media xlink:href=\"41536_2024_348_MOESM1_ESM.pdf\"><caption><p>Supplementary material</p></caption></media>", "<media xlink:href=\"41536_2024_348_MOESM2_ESM.pdf\"><caption><p>Reporting-summary</p></caption></media>" ]

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[ "<title>Introduction</title>", "<p>Diarrhea is one of the top three infectious diseases responsible for ~8%-10% of total deaths among children below the age of five years; 90% of these deaths are estimated to occur in South Asia and sub-Saharan Africa [##UREF##0##1##, ##UREF##1##2##]. Rotavirus and diarrheagenic <italic>Escherichia coli</italic> (<italic>E. coli</italic>) are the two most common pathogens responsible for acute diarrhea in children below the age of five years. <italic>Shigella</italic>, <italic>Cryptosporidium</italic>, <italic>Campylobacter spp.</italic>, and <italic>Vibrio cholerae</italic> are a few other pathogens frequently associated with diarrhea in children under five [##REF##15825143##3##], with variable region-specific incidence.</p>", "<p>The Global Enteric Multicentric Study (GEMS), a large case-control study, determined the etiology of childhood diarrhea in South Asia and sub-Saharan Africa. Rotavirus, <italic>Cryptosporidium</italic>, <italic>E. coli</italic> producing heat-stable toxin (ST-ETEC), and <italic>Shigella </italic>were identified as major attributable organisms for the cause of diarrhea [##REF##23680352##4##]. Later, stool samples from GEMS were re-analyzed using quantitative polymerase chain reaction (PCR) and revised pathogen-specific burdens were calculated. Pathogen-attributable burden increased significantly, and <italic>Shigella </italic>was reported as the most common causative agent [##REF##27673470##5##]. Customarily, <italic>Shigella </italic>infection is associated with bloody diarrhea, but this re-analysis using PCR techniques documented that most such children have only loose stools without blood. Despite <italic>Shigella </italic>being reported as the most common cause of acute diarrhea in GEMS re-analysis, there is a paucity of studies evaluating <italic>Shigella </italic>by molecular methods and comparing the isolation rate with conventional culture techniques. Also, the clinical characteristics and outcomes of children with non-bloody <italic>Shigella </italic>diarrhea managed as per current WHO diarrhea treatment guidelines are not known.</p>", "<p>In this study, we determined the frequency of <italic>Shigella </italic>detected by molecular and conventional methods in under-five children with diarrhea and the clinical profile and outcome of children with <italic>Shigella </italic>diarrhea over the next three months.</p>" ]

[ "<title>Materials and methods</title>", "<p>This hospital-based prospective observational study was conducted between November 2017 and April 2019 at the University College of Medical Sciences, Delhi, an urban hospital catering mainly to the low-income population of eastern Delhi and adjoining areas of the state of Uttar Pradesh in India. Approval from the Institutional Ethics Committee for Human Research (IEC-HR) of the institute was obtained before enrolling the study participants (approval number: IEC-HR/2017/32/88). Informed consent was obtained from the parents or caregivers of all the participants enrolled in the study.</p>", "<p>Participants</p>", "<p>All consecutive children aged between one month and five years, presenting with acute diarrhea to the outpatient department or pediatric emergency services of the hospital and requiring hospitalization due to any reason (such as dehydration, severe malnutrition, septicemia, etc.), were eligible for inclusion. Acute diarrhea was defined as diarrhea of &lt; seven days duration with or without blood in stools [##UREF##2##6##]. Children who received antibiotic therapy seven days prior and those with any other known cause of blood in stools (rectal polyp, bleeding diathesis, inflammatory bowel disease, etc.) were excluded.</p>", "<p>At the time of enrollment, detailed clinical history, including the treatment prescribed (oral rehydration solution (ORS), zinc, paracetamol, etc.) during the current diarrheal episode, was elicited from the parents or guardian. A detailed clinical examination, including an assessment of hydration status, was performed. Anthropometry was performed on all children after the correction of dehydration. All findings were recorded in a case record form.</p>", "<p>Collection and processing of stool samples</p>", "<p>A freshly passed stool sample was collected in a wide-mouthed, clean container. A part of the stool sample was concentrated by the formal ether sedimentation method for wet-mount microscopy for ova, parasites, and cysts. The second part of the sample was enriched in selenite F broth (SFB) and alkaline peptone water (APW) for four to six hours at 37º C and cultured on MacConkey’s medium, xylose lysine deoxycholate (XLD) medium, and bile salt agar (BSA) medium for the isolation of pathogenic enteric bacteria by standard laboratory methods. Antibiotic sensitivity testing of isolates was performed as per Clinical and Laboratory Standards Institute (CLSI) guidelines [##UREF##3##7##]. The third part of the sample was stored at -80 °C for PCR. The DNA was extracted from a stool sample using the QIAamp DNA Stool Mini Kit (QIAGEN, Hilden, Germany) and stored at -20 °C. The extracted DNA was subjected to PCR amplification using specific primers for the invasion plasmid antigen H (ipaH) gene at 424 bp for <italic>Shigella </italic>[##REF##11478669##8##]. Forward 5’- GCTGGAAAAACTCAGTGCCT-3’ and reverse 5’- CCAGTCCTGTAAATTCATTCT-3’ amplification was performed using a reaction mixture of Taq polymerase, deoxynucleotide triphosphate (dNTP), and buffer (QIAGEN). Thirty-five amplification cycles were performed with the following steps: Initial denaturation at 95 °C for five minutes, cycling denaturation at 95 °C for 30 seconds, annealing at 56 °C for 30 seconds, extension at 72 °C for 40 seconds, and final extension at 72 °C for 10 minutes. The PCR amplicons were loaded onto a 2% agarose gel in a specific order for electrophoresis and run at 150 volts for 20-25 minutes. After electrophoresis, the 0.5 mM ethidium bromide-stained gel was visualized and interpreted for specific amplification patterns on the UV transilluminator.</p>", "<p>Patient management</p>", "<p>Hydration status was assessed, classified, and managed with ORS and intravenous fluids as per WHO guidelines [##UREF##4##9##]. Oral zinc was administered (10 mg/day for two- to six-month-olds, 20 mg/day for six- to 59-month-olds) for a total duration of 14 days. All children with visible blood in their stools received antimicrobial therapy as per WHO and Indian Academy of Pediatrics (IAP) guidelines [##UREF##4##9##, ##UREF##5##10##]. Children with bloody diarrhea, aged more than one year, with no risk factors (no malnutrition, no dehydration, preserved appetite) received oral antibiotics (cefixime), while those with malnutrition, severe dehydration, or poor oral acceptance received injectable antibiotics. Enrolled children were evaluated for comorbidities like hypoglycemia, hypothermia, and electrolyte imbalance and managed accordingly. Children with severe acute malnutrition (SAM) were stabilized and managed as per WHO guidelines [##UREF##6##11##].</p>", "<p>Outcome variables</p>", "<p>Outcomes were assessed as a percentage of children with a diagnosis of <italic>Shigella </italic>infection by molecular (PCR) and conventional methods (stool microscopy, culture). Children were discharged once the dehydration was corrected, stool frequency was reduced, and oral acceptance improved. After discharge, follow-up for persistence or recurrence of diarrhea was done on days three and seven, and then fortnightly for three months with simultaneous evaluation for growth faltering. Any recurrence of diarrhea or dysentery over three months after discharge was noted. The need for follow-up was emphasized by personal communication at each visit or by telephonic communication to minimize the attrition rate during follow-up.</p>", "<p>Sample size</p>", "<p>In an earlier study [##REF##27061990##12##], <italic>Shigella </italic>infection was diagnosed by molecular methods in 11 out of 60 children with acute watery diarrhea (18.33%) and 34 out of 60 children with dysentery (56.66%). Assuming 10% of our enrolled cases to have dysentery (visible blood in stools), the estimated proportion of <italic>Shigella </italic>positivity by molecular methods was 22.2%. With an estimated proportion of 22.2%, the calculated sample size was 150 children with a relative precision of 30% (absolute precision of 6.66%) at a 95% confidence level. Thus, we planned to enroll 150 children with acute diarrhea.</p>", "<p>Statistical analysis</p>", "<p>Data collected were entered in a Microsoft Excel spreadsheet (Microsoft Corp., Redmond, WA) and analyzed using IBM Statistical Package for the Social Sciences (SPSS) software version 25 (IBM Corp., Armonk, NY). Outcome parameters such as frequency of isolation, duration of diarrhea, and recurrence of diarrhea were measured using descriptive statistics. Categorical variables were compared between <italic>Shigella</italic>-positive and <italic>Shigella</italic>-negative cases by the chi-square test to identify clinical characteristics and the outcome of <italic>Shigella </italic>infection. For normally distributed data, continuous variables between the two groups were compared using the Student's t-test, and for data that were not normally distributed, the Mann-Whitney U test was used. Odds ratios and 95% CI were calculated to determine risk factors (e.g., age &lt; one month, socioeconomic status, malnutrition status) and clinical predictors (e.g., blood in stools, vomiting, dehydration) by univariate analysis.</p>" ]

[ "<title>Results</title>", "<p>Enrollment and baseline characteristics</p>", "<p>During the study period, 209 children aged between one month and five years with acute diarrhea were hospitalized, out of which 150 were included in the present study. Figure ##FIG##0##1## depicts the flow of participants in the study, along with the reasons for the exclusion of cases not enrolled in the study.</p>", "<p>The median (interquartile range (IQR)) age of enrolled children (72 (48%) males and 78 (52%) females) was 10 (5-23) months; 87 (58%) were infants (age &lt;1 year). Only 94 (63%) were completely immunized for age. The majority (142, 94.7%) of the children’s families belonged to the lower middle, upper lower, or lower socioeconomic class on the Kuppuswamy scale [##UREF##7##13##]. The median (IQR) duration of diarrhea at hospitalization was two (one to three) days, and the number of stools passed in the previous 24 hours was 15 (10-20). Out of 150 children, 141 (94%) had watery, mucoid, or loose stools, and nine (6%) had blood in their stools. The most common associated symptom with diarrhea was vomiting (85%), followed by decreased oral acceptance (80%), and fever (40%).</p>", "<p>Clinical characteristics</p>", "<p>Undernutrition was prevalent in the enrolled children; 58 (38.7%) were stunted (height-for-age Z-score (HAZ) &lt;−2SD) and 64 (42.7%) were wasted (weight-for-height Z-score (WHZ) &lt;−2SD). Among these, 28 (19%) were severely stunted (HAZ &lt;−3SD), while 40 (26.7%) children had SAM (weight for height Z-score &lt;-3SD or mid-upper arm circumference &lt;11.5 cm for children aged between six months and five years). At admission, dehydration was present in 138 (92%) children; 89 (59%) were severely dehydrated, and 15 (10%) had features suggestive of circulatory shock. All enrolled children received ORS during the hospitalization with a mean (SD) duration of 89.0 (31.3) hours, while IV fluids were administered to 95% of children for a mean (SD) duration of 44.1 (31.6) hours. Antibiotics were used in 101 (67.3%) cases for a mean (SD) of 5.3 (2.8) days. The presence of SAM (37%) and suspected or confirmed sepsis (33%) were the most common indications for the use of antibiotics in children admitted with acute diarrhea, followed by clinically suspected cholera (22%), dysentery (9%), pneumonia (5%), and meningitis (2%).</p>", "<p>Stool microscopy</p>", "<p><italic>Shigella sonnei</italic> was diagnosed by stool culture in only one child. <italic>Escherichia coli</italic> was cultured from 67 stool samples, whereas <italic>Klebsiella pneumonia</italic>, trophozoites of <italic>Giardia lamblia</italic> and ova of <italic>Ascaris lumbricoides</italic>, and motile <italic>Vibrio cholera</italic> in hanging drop were detected in one stool sample each. <italic>Shigella </italic>was detected in 13 (8.7%) out of 150 stool samples by PCR amplification using the ipaH gene. The only culture-positive case was also detected by PCR. The sensitivity of stool culture was 7.7% against PCR for the diagnosis of <italic>Shigella </italic>infection. Out of 13 children in whom the <italic>Shigella </italic>PCR was positive, 11 (84.6%) had non-bloody diarrhea, and only two (15.4%) had dysentery. In 11 children with <italic>Shigella </italic>PCR positivity and non-bloody diarrhea, the characteristics of the stool did not change to bloody diarrhea until the resolution of the diarrhea episode, but eight of these children had received antibiotics due to associated comorbidities. Among three children with PCR positivity and non-bloody diarrhea managed without antibiotics, two had vomiting and poor oral acceptance at presentation, whereas none of them had fever or abdominal pain. All three children presented with some or severe dehydration, and none of these children developed bloody diarrhea or persistent diarrhea, despite no antibiotic usage.</p>", "<p>Comparison of the characteristics and outcomes of children who were PCR-positive vs. PCR-negative for <italic>Shigella</italic>\n</p>", "<p>Table ##TAB##0##1## compares the clinical and anthropometric characteristics of children with stool PCR positivity for <italic>Shigella </italic>with those who had a negative test result on PCR.</p>", "<p>There was no statistically significant difference between the two groups. The mean (SD) duration of diarrhea, as monitored after enrolment, was not significantly different (P=0.94) between PCR-positive participants (2.2 (1.54) days) and PCR-negative participants (2.3 (1.45) days). On univariate analysis, none of the evaluated risk factors or clinical predictors had a significant association with the <italic>Shigella </italic>infection (Table ##TAB##1##2##).</p>", "<p>Follow-up</p>", "<p>Out of 150 enrolled children, 23 were lost to follow-up, whereas 43 had a recurrence of diarrhea over the next three months. Among these, 14 patients required hospitalization, and antibiotics were used in 12 patients. During follow-up, only two patients had dysentery, and one patient developed persistent diarrhea. The recurrence of diarrhea episodes over three months was seen slightly more often among the children who had initial PCR (5, 45.4%) positivity than in PCR-negative participants (38, 37.8%), but this was not statistically significant (P=0.507). Also, there were no statistically significant differences in outcome in terms of growth faltering, recurrence of diarrhea, or hospitalization (Table ##TAB##2##3##).</p>" ]

[ "<title>Discussion</title>", "<p>In this hospital-based study on under-five children with acute diarrhea, we documented that stool culture's diagnostic yield was very low compared to stool PCR amplification using a specific primer for the ipaH gene for the diagnosis of <italic>Shigella </italic>infection. The majority of <italic>Shigella </italic>infections presented with watery diarrhea rather than bloody diarrhea, and a history of blood in stools was a poor marker for the diagnosis of shigellosis. We documented that the molecular diagnostic approach performed better than conventional culture methods for the diagnosis of shigellosis. We could not find any significant clinical predictors associated with <italic>Shigella </italic>infection. There was no statistically significant difference in the overall outcome between PCR-positive and PCR-negative children at three months of follow-up.</p>", "<p>The conventional methods of detection of <italic>Shigella </italic>infection rely on the stool culture, which turns out positive in a small fraction of actual cases of shigellosis due to multiple reasons like low bacterial load, loss of bacterial viability due to changes in ambient temperature and pH during specimen transport and storage, and use of antibiotics before specimen collection, transport, and storage [##REF##1184731##14##]. Molecular diagnostic methods seem to overcome some of the shortcomings of conventional diagnostic methods, thus improving the diagnosis of <italic>Shigella </italic>infection. The use of PCR assays based on the amplification of the ipaH gene sequence is one such molecular method for diagnosis in cases of dysentery, as it is carried by all four species of <italic>Shigella </italic>[##REF##8421181##15##].</p>", "<p>Recent data from other settings also suggest that conventional culture yield is poor in episodes of <italic>Shigella</italic> diarrhea in young children [##UREF##8##16##]. The PCR-derived incidence for <italic>Shigella </italic>surpassed the original estimates by two-fold in the GEMS re-analysis, thus making it the most attributable pathogen of diarrhea among children under five [##REF##27673470##5##]. In a re-analysis by PCR of stool samples from the multi-country Malnutrition and Enteric Disease (MAL-ED) Study, <italic>Shigella</italic>-attributable incidence increased from 4% to 41.3% in children aged between 12-24 months [##UREF##9##17##]. Earlier, Dutta et al. from Kolkata, India, reported a <italic>Shigella </italic>detection rate of 15.3% (46 of 300) by stool PCR as against 7.7% (23 of 300) by stool culture [##REF##11478669##8##]. The positivity rate in the study by Dutta et al. was higher than that seen in our study, which is likely to be due to the inclusion of a very high proportion (42%; 126 out of 300) of children with dysentery. Similarly, Aggarwal et al. reported that the prevalence of <italic>Shigella </italic>increased from 8% (17 of 207) to 18.3% (11 of 60) in diarrhea and from 33% (39 of 118) to 56.7% (34 of 60) in dysentery when stool samples were analyzed by culture and PCR, respectively [##REF##27061990##12##].</p>", "<p>In two large population-based surveillance studies from Vietnam and China, an ipaH gene amplification-based PCR assay was used to detect <italic>Shigella spp</italic>. from the rectal swab specimens of cases presenting with dysentery. Results showed the presence of the ipaH gene in approximately 93%-97% of randomly selected <italic>Shigella </italic>culture-positive specimens and 46%-58% of randomly selected <italic>Shigella </italic>culture-negative specimens [##REF##15131166##18##, ##REF##20951728##19##]. In a study by Von Seidlein et al. on the analysis of pooled data from studies from six Southeast Asian countries, i.e., Bangladesh, China, Pakistan, Indonesia, Vietnam, and Thailand, <italic>Shigella </italic>was isolated from 2,927 (5%) of 56,958 diarrhea episodes, and more than half of these were in children under the age of five, and PCR detected the ipaH gene in 33% of samples of culture-negative stool specimens [##UREF##10##20##]. In a study from Bangladesh, Islam et al. documented that a PCR assay based on an ipaH probe improved the rate of detection of <italic>Shigella </italic>in stool samples to 61% from 44% by the conventional culture method [##REF##10453122##21##]. They also tested 123 strains of <italic>E. coli</italic> by PCR to identify enteroinvasive <italic>E. coli</italic>, but none yielded any positive results. Though these studies have included variable age groups, the high detection rates of the ipaH gene in culture-negative stool specimens suggest that earlier estimates of shigellosis burden measured by conventional culture may have underestimated the true disease burden.</p>", "<p>Blood in stools has been used as a surrogate marker for <italic>Shigella </italic>diarrhea. The WHO's diarrhea management guidelines recommend antibiotics effective against <italic>Shigella </italic>only when visible blood is present in stools. In our study, 84.6% (11 of 13) of the children with stool PCR positivity for <italic>Shigella </italic>presented with non-bloody diarrhea. In the GEMS re-analysis, <italic>Shigella spp</italic>. was the second highest cause of watery diarrhea, and overall, 40.3% (~527 of 1,310) of cases due to S<italic>higella spp.</italic> were non-dysenteric [##REF##27673470##5##]. Likewise, 86.2% of the attributable incidence for <italic>Shigella </italic>was non-dysenteric in the re-analysis of the MAL-ED cohort study [##UREF##9##17##]. The results of a recent meta-analysis showed that the proportion of <italic>Shigella </italic>infections presenting as dysentery has been decreasing [##UREF##11##22##]. This may be related to the diagnosis of such cases by molecular methods or early treatment with antibiotics. However, it is unclear whether <italic>Shigella </italic>infection without blood in stools needs therapy on similar lines. In our series of cases, antibiotics were administered for other reasons in cases where <italic>Shigella </italic>was later detected by molecular methods, except in three children in whom the infection resolved without any use of antibiotics.</p>", "<p>The strength of our study was the prospective nature of enrollment, with a focus on the clinical presentation and outcome of <italic>Shigella </italic>infection detected with molecular methods. Our study also attempted to evaluate the clinical profile (resolution of diarrhea, change to bloody diarrhea in those with watery diarrhea, and recurrence) during a three-month follow-up period for children with <italic>Shigella </italic>infection managed as per current diarrhea treatment guidelines, which has not been evaluated in any of the studies comparing detection rates of <italic>Shigella </italic>between conventional and molecular methods. However, because of antibiotic use for other indications, we were unable to validly comment on the outcome of PCR-positive cases when they are managed without antibiotics. A relatively low <italic>Shigella </italic>prevalence in our series precluded the identification of any significant clinical marker of <italic>Shigella </italic>infection. Another limitation of our study was the use of the ipaH gene as a marker for <italic>Shigella </italic>detection, which may also be found in enteroinvasive <italic>E. coli</italic>. We enrolled hospitalized patients who primarily had moderate-to-severe diarrhea and concurrent illnesses; therefore, results from our study may not be generalizable to community settings with less severe diarrheal illnesses.</p>" ]

[ "<title>Conclusions</title>", "<p>The present study concludes that <italic>Shigella </italic>is an important cause of diarrhea in children under five, often missed by conventional laboratory methods. Blood in stools as a syndromic indicator for <italic>Shigella </italic>infection needs to be reconsidered as the majority of <italic>Shigella </italic>diarrhea cases have non-bloody stools. As the current management guidelines for childhood diarrhea recommend antibiotics for only bloody diarrhea, studies evaluating other clinical predictors of <italic>Shigella </italic>infection and faster and more cost-effective techniques for its molecular diagnosis are required. Large, community-based longitudinal studies are needed to evaluate the outcome of non-dysenteric <italic>Shigella </italic>infections diagnosed by molecular methods so that management guidelines for such infections can be formulated.</p>" ]

[ "<p>Background and objectives: <italic>Shigella </italic>is an important cause of diarrhea in children under five, often missed by conventional laboratory methods. Blood in stools has always been a syndromic indicator for <italic>Shigella </italic>diarrhea, but most cases present with watery diarrhea without blood. This study aimed to determine the frequency of <italic>Shigella </italic>detected by molecular and conventional methods in children under five. Additionally, we aimed to study the clinical profile and outcome of children with <italic>Shigella </italic>diarrhea managed as per current diarrhea treatment guidelines.</p>", "<p>Methods: In this hospital-based prospective observational study, stool samples from 150 children (age range: one month to five years) with acute diarrhea (duration &lt; seven days) were subjected to routine microscopic examination, stool culture, and DNA extraction. The extracted DNA from stored stool samples was subjected to polymerase chain reaction (PCR) amplification using a specific primer for the invasion plasmid antigen H gene sequence (ipaH) gene at 424 bp. Results were interpreted in the context of the percentage of isolation of <italic>Shigella </italic>by molecular (PCR) and conventional methods (stool microscopy and culture) and the follow-up outcome in terms of recurrence of diarrhea or dysentery and growth faltering over three months after discharge.</p>", "<p>Results: <italic>Shigella </italic>infection was diagnosed in stool samples by PCR from 13 (8.7%) children, whereas it was isolated by conventional stool culture in only one (0.7%) child. The sensitivity of culture was only 7.7% against PCR for the diagnosis of <italic>Shigella </italic>infection, whereas blood in stools had a sensitivity of 15.4%. The majority of <italic>Shigella </italic>PCR-positive cases (11 out of 13) presented with non-bloody diarrhea. None of the evaluated clinical predictors had a significant association with the <italic>Shigella </italic>infection. No statistically significant difference was found between PCR-positive and PCR-negative children at the end of follow-up (P&gt;0.05).</p>", "<p>Conclusion: The majority of children with <italic>Shigella </italic>infection present with watery diarrhea rather than bloody diarrhea, and a history of blood in stools is a poor marker for the diagnosis of shigellosis. The diagnostic performance of stool culture is also very low compared to stool PCR for the diagnosis of <italic>Shigella </italic>diarrhea.</p>" ]

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[ "<fig position=\"anchor\" fig-type=\"figure\" id=\"FIG1\"><label>Figure 1</label><caption><title>Flowchart depicting the selection of participants for the study</title><p>mo: month; y: year; d: days; TB: tuberculosis</p></caption></fig>" ]

[ "<table-wrap position=\"float\" id=\"TAB1\"><label>Table 1</label><caption><title>Comparison of clinical and anthropometric parameters at presentation between children with or without stool PCR Shigella positivity</title><p>^aData in number (%) unless stated otherwise; ^bn=7 missing cases as the measurement taken for children &gt;6 months; ^cn=136 as the value could not be calculated for children with a length less than 45 cm; ^dn=96 missing cases as the measurement taken for children &gt;6 months</p><p>n: number of patients; IQR: interquartile range; SD: standard deviation; PCR: polymerase chain reaction; SAM: severe acute malnutrition</p></caption><table frame=\"hsides\" rules=\"groups\"><tbody><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Parameter</td><td rowspan=\"1\" colspan=\"1\">Stool PCR-positive  (n=13, 8.7%)^a\n</td><td rowspan=\"1\" colspan=\"1\">Stool PCR-negative  (n=137, 91.3%)^a\n</td><td rowspan=\"1\" colspan=\"1\">P-value</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Diarrhea duration (days); median (IQR)</td><td rowspan=\"1\" colspan=\"1\">2.00 (1.0, 3.0)</td><td rowspan=\"1\" colspan=\"1\">2.00 (1.0,3.0)</td><td rowspan=\"1\" colspan=\"1\">0.908</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">No. of stools in last 24 hours; median (IQR)</td><td rowspan=\"1\" colspan=\"1\">15.00 (8.0, 19.0)</td><td rowspan=\"1\" colspan=\"1\">15.00 (11.0, 20.0)</td><td rowspan=\"1\" colspan=\"1\">0.390</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Blood in stools</td><td rowspan=\"1\" colspan=\"1\">2 (15.4%)</td><td rowspan=\"1\" colspan=\"1\">7 (5.1%)</td><td rowspan=\"1\" colspan=\"1\">0.176</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Vomiting</td><td rowspan=\"1\" colspan=\"1\">11 (84.6%)</td><td rowspan=\"1\" colspan=\"1\">116 (84.7%)</td><td rowspan=\"1\" colspan=\"1\">0.996</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Fever</td><td rowspan=\"1\" colspan=\"1\">6 (46.2%)</td><td rowspan=\"1\" colspan=\"1\">54 (39.4%)</td><td rowspan=\"1\" colspan=\"1\">0.636</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Abdominal pain</td><td rowspan=\"1\" colspan=\"1\">0</td><td rowspan=\"1\" colspan=\"1\">8 (5.8%)</td><td rowspan=\"1\" colspan=\"1\">1.000</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Poor oral acceptance</td><td rowspan=\"1\" colspan=\"1\">10 (76.9%)</td><td rowspan=\"1\" colspan=\"1\">89 (65%)</td><td rowspan=\"1\" colspan=\"1\">0.544</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Convulsions</td><td rowspan=\"1\" colspan=\"1\">1 (7.7%)</td><td rowspan=\"1\" colspan=\"1\">8 (5.8%)</td><td rowspan=\"1\" colspan=\"1\">0.568</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Rash</td><td rowspan=\"1\" colspan=\"1\">0</td><td rowspan=\"1\" colspan=\"1\">5 (3.6%)</td><td rowspan=\"1\" colspan=\"1\">1.000</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Cough</td><td rowspan=\"1\" colspan=\"1\">1 (7.7%)</td><td rowspan=\"1\" colspan=\"1\">18 (13.1%)</td><td rowspan=\"1\" colspan=\"1\">1.000</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Immunized for age</td><td rowspan=\"1\" colspan=\"1\">7 (53.8%)</td><td rowspan=\"1\" colspan=\"1\">87 (63.5%)</td><td rowspan=\"1\" colspan=\"1\">0.679</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Dehydration</td><td rowspan=\"1\" colspan=\"1\">11 (84.6%)</td><td rowspan=\"1\" colspan=\"1\">127 (92.7%)</td><td rowspan=\"1\" colspan=\"1\">0.465</td></tr><tr><td colspan=\"4\" rowspan=\"1\">Anthropometry at enrolment</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Weight for age (WFA) (kg); mean ± SD</td><td rowspan=\"1\" colspan=\"1\">6.62 (2.30)</td><td rowspan=\"1\" colspan=\"1\">6.76 (2.68)</td><td rowspan=\"1\" colspan=\"1\">0.847</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Height for age (HFA) (cm); mean ± SD</td><td rowspan=\"1\" colspan=\"1\">69.10 (12.91)</td><td rowspan=\"1\" colspan=\"1\">71.05 (13.63)</td><td rowspan=\"1\" colspan=\"1\">0.620</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Weight for age Z-score (WAZ); mean ± SD</td><td rowspan=\"1\" colspan=\"1\">-2.62 (1.60)</td><td rowspan=\"1\" colspan=\"1\">-2.98 (1.35)</td><td rowspan=\"1\" colspan=\"1\">0.359</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Height for age Z-score (HAZ); mean ± SD</td><td rowspan=\"1\" colspan=\"1\">-1.62 (1.63)</td><td rowspan=\"1\" colspan=\"1\">-1.79 (1.69)</td><td rowspan=\"1\" colspan=\"1\">0.724</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Weight for height Z-score (WHZ); mean ± SD</td><td rowspan=\"1\" colspan=\"1\">-2.10 (1.97)</td><td rowspan=\"1\" colspan=\"1\">-2.68 (1.58)^c\n</td><td rowspan=\"1\" colspan=\"1\">0.219</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Mid upper arm circumference (MUAC) (cm); mean ± SD</td><td rowspan=\"1\" colspan=\"1\">12.36 (1.53)^b\n</td><td rowspan=\"1\" colspan=\"1\">12.48 (1.21)^d\n</td><td rowspan=\"1\" colspan=\"1\">0.803</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Wasting</td><td rowspan=\"1\" colspan=\"1\">2 (15.4%)</td><td rowspan=\"1\" colspan=\"1\">31 (22.8%)</td><td rowspan=\"1\" colspan=\"1\">0.607</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Severe wasting</td><td rowspan=\"1\" colspan=\"1\">4 (30.8%)</td><td rowspan=\"1\" colspan=\"1\">27 (19.9%) ^c\n</td><td rowspan=\"1\" colspan=\"1\">-</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Stunting</td><td rowspan=\"1\" colspan=\"1\">2 (15.4%)</td><td rowspan=\"1\" colspan=\"1\">28 (19%)</td><td rowspan=\"1\" colspan=\"1\">0.890</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Severe stunting</td><td rowspan=\"1\" colspan=\"1\">3 (23.1%)</td><td rowspan=\"1\" colspan=\"1\">25 (18.3%)</td><td rowspan=\"1\" colspan=\"1\">-</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">SAM</td><td rowspan=\"1\" colspan=\"1\">5 (38.5%)</td><td rowspan=\"1\" colspan=\"1\">35 (25.7%)</td><td rowspan=\"1\" colspan=\"1\">0.336</td></tr></tbody></table></table-wrap>", "<table-wrap position=\"float\" id=\"TAB2\"><label>Table 2</label><caption><title>Analysis of risk factors and clinical predictors of Shigella infection</title><p>n: number of patients; PCR: polymerase chain reaction; SAM: severe acute malnutrition.</p></caption><table frame=\"hsides\" rules=\"groups\"><tbody><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Parameter</td><td rowspan=\"1\" colspan=\"1\">Stool PCR-positive cases (n=13, 8.7%)</td><td rowspan=\"1\" colspan=\"1\">Stool PCR-negative cases (n=137, 91.3%)</td><td rowspan=\"1\" colspan=\"1\">Odds ratio (95% CI)</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Age &lt;1 year</td><td rowspan=\"1\" colspan=\"1\">9 (69.2%)</td><td rowspan=\"1\" colspan=\"1\">78 (56.9%)</td><td rowspan=\"1\" colspan=\"1\">1.70 (0.50-5.80)</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Male gender</td><td rowspan=\"1\" colspan=\"1\">7 (53.8%)</td><td rowspan=\"1\" colspan=\"1\">65 (47.4%)</td><td rowspan=\"1\" colspan=\"1\">1.29 (0.41-4.04)</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Lower socioeconomic status</td><td rowspan=\"1\" colspan=\"1\">12 (92.3%)</td><td rowspan=\"1\" colspan=\"1\">130 (94.9%)</td><td rowspan=\"1\" colspan=\"1\">0.65 (0.07-5.70)</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Blood in stools</td><td rowspan=\"1\" colspan=\"1\">2 (15.4%)</td><td rowspan=\"1\" colspan=\"1\">7 (5.1%)</td><td rowspan=\"1\" colspan=\"1\">3.38 (0.62-18.26)</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Vomiting</td><td rowspan=\"1\" colspan=\"1\">11 (84.6%)</td><td rowspan=\"1\" colspan=\"1\">116 (84.7%)</td><td rowspan=\"1\" colspan=\"1\">1.00 (0.21-4.82)</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Poor oral acceptance</td><td rowspan=\"1\" colspan=\"1\">10 (76.9%)</td><td rowspan=\"1\" colspan=\"1\">89 (65%)</td><td rowspan=\"1\" colspan=\"1\">1.8 (0.47-6.85)</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Convulsion</td><td rowspan=\"1\" colspan=\"1\">1 (7.7%)</td><td rowspan=\"1\" colspan=\"1\">8 (5.8%)</td><td rowspan=\"1\" colspan=\"1\">1.34 (0.15-11.67)</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Cough</td><td rowspan=\"1\" colspan=\"1\">1 (7.7%)</td><td rowspan=\"1\" colspan=\"1\">18 (13.1%)</td><td rowspan=\"1\" colspan=\"1\">0.55 (0.07-4.50)</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Dehydration</td><td rowspan=\"1\" colspan=\"1\">11 (84.6%)</td><td rowspan=\"1\" colspan=\"1\">127 (92.7%)</td><td rowspan=\"1\" colspan=\"1\">0.43 (0.08-2.23)</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Wasting</td><td rowspan=\"1\" colspan=\"1\">6 (46.2%)</td><td rowspan=\"1\" colspan=\"1\">58 (42.3%)</td><td rowspan=\"1\" colspan=\"1\">1.17 (0.37-3.66)</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Stunting</td><td rowspan=\"1\" colspan=\"1\">5 (38.5%)</td><td rowspan=\"1\" colspan=\"1\">51 (37.2%)</td><td rowspan=\"1\" colspan=\"1\">1.05 (0.33-3.40)</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">SAM</td><td rowspan=\"1\" colspan=\"1\">5 (38.5%)</td><td rowspan=\"1\" colspan=\"1\">35 (25.7%)</td><td rowspan=\"1\" colspan=\"1\">1.82 (0.56-5.94)</td></tr></tbody></table></table-wrap>", "<table-wrap position=\"float\" id=\"TAB3\"><label>Table 3</label><caption><title>Outcome of enrolled children with acute diarrhea at the three-month follow-up</title><p>^an=127 (11 stool PCR-positive cases, 116 stool PCR-negative cases); ^bn=43 (5 stool PCR-positive cases, 38 stool PCR-negative cases)</p><p>PCR: polymerase chain reaction</p><p>Binomial analysis was not done for persistent diarrhea, bloody diarrhea, and mortality as none of the PCR cases who completed the three months of follow-up had any of these outcomes.</p></caption><table frame=\"hsides\" rules=\"groups\"><tbody><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Parameter</td><td rowspan=\"1\" colspan=\"1\">Stool PCR-positive cases (n=13, 8.7%)</td><td rowspan=\"1\" colspan=\"1\">Stool PCR-negative cases (n=137, 91.3%)</td><td rowspan=\"1\" colspan=\"1\">Odds ratio (95% CI)</td><td rowspan=\"1\" colspan=\"1\">P-value</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Duration of diarrhea ≥7 days</td><td rowspan=\"1\" colspan=\"1\">4/13 (30.8)</td><td rowspan=\"1\" colspan=\"1\">50 (36.1)</td><td rowspan=\"1\" colspan=\"1\">0.77 (0.23-2.64)</td><td rowspan=\"1\" colspan=\"1\">0.68</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Recurrence in the next 3 months^a\n</td><td rowspan=\"1\" colspan=\"1\">5 (45.5)</td><td rowspan=\"1\" colspan=\"1\">38 (32.8)</td><td rowspan=\"1\" colspan=\"1\">1.71 (0.49-5.96)</td><td rowspan=\"1\" colspan=\"1\">0.51</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Hospitalization in the next 3 months ^b\n</td><td rowspan=\"1\" colspan=\"1\">1 (20)</td><td rowspan=\"1\" colspan=\"1\">13 (34.2)</td><td rowspan=\"1\" colspan=\"1\">0.48 (0.05-4.75)</td><td rowspan=\"1\" colspan=\"1\">1.00</td></tr></tbody></table></table-wrap>" ]

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[ "<fn-group content-type=\"other\"><title>Human Ethics</title><fn fn-type=\"other\"><p>Consent was obtained or waived by all participants in this study. Institutional Ethics Committee for Human Research, University College of Medical Sciences, Delhi issued approval IEC-HR/2017/32/88</p></fn></fn-group>", "<fn-group content-type=\"other\"><title>Animal Ethics</title><fn fn-type=\"other\"><p><bold>Animal subjects:</bold> All authors have confirmed that this study did not involve animal subjects or tissue.</p></fn></fn-group>", "<fn-group content-type=\"competing-interests\"><fn fn-type=\"COI-statement\"><p>The authors have declared that no competing interests exist.</p></fn></fn-group>" ]

[ "<graphic xlink:href=\"cureus-0015-00000050546-i01\" position=\"float\"/>" ]

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{ "acronym": [], "definition": [] }

[ "<title>Introduction</title>", "<p>Mucoid degeneration (MD) of the anterior cruciate ligament (ACL) is a rare condition, with a prevalence rate ranging from 0.2% to 1.2% [##REF##11372618##1##]. The cause of this condition is not well understood and has been the subject of vigorous debate, with only a limited number of reports in the literature detailing its origin [##REF##20811733##2##,##REF##22265045##3##]. It is characterized by the infiltration of a mucoid-like substance within the ACL, which can cause knee pain and restricted motion [##REF##19089409##4##]. Although it was once considered rare, recent reports suggest that it may be more common than previously thought, implying that it is underdiagnosed or misdiagnosed [##REF##18210315##5##]. We describe a successful treatment approach for MD of ACL, which involves the application of arthroscopic debridement.</p>" ]

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[ "<title>Discussion</title>", "<p>MD of ACL is an uncommon condition that leads to knee pain and discomfort, primarily impacting individuals in their fourth decade of life [##REF##18997652##6##]. Kumar et al. [##REF##10204939##7##] initially coined the term mucoid cystic degeneration to describe this condition affecting the ACL. It is characterized by significant disruption of the ligament's normal appearance and can contribute to knee pain. The predominant symptom associated with MD of ACL is knee pain, primarily located in the posterior aspect [##UREF##0##8##,##REF##19288177##9##]. This pain is caused by mechanical impingement on the posterior cruciate ligament and the posterior capsule or bone erosions [##REF##19288177##9##]. Additional clinical manifestations include a mechanical hindrance to extension, varying degrees of swelling, and audible clicking sounds [##REF##14997351##10##].</p>", "<p>MRI serves as a valuable tool for evaluating MD of ACL during preoperative assessment. It shows an ACL that is indistinctly defined, exhibiting an enlarged dimension while retaining a typical orientation, and presenting heightened signal intensity interspersed among visible intact ACL fibers, creating a distinctive \"celery stalk\" appearance [##REF##11465770##11##]. On T1-weighted images, MD of ACL typically appears with intermediate signal intensity, while on T2-weighted images, it shows high signal intensity [##REF##11465770##11##,##REF##14760345##12##]. Hodler et al. correlated MRI appearances with histological findings and found that 29 out of 38 ligaments had focal areas of signal increase, suggesting a correlation between the focal MRI signal changes and the presence of degenerative changes in the ligaments [##REF##1632355##13##].</p>", "<p>Arthroscopic observations suggest that MD of ACL involves a hypertrophied, fibrillated ligament with the presence of yellowish mucinous material interspersed among the fibers. This is often accompanied by the absence of the ligamentum mucosum [##REF##14760345##12##]. Additionally, a lack of smooth synovial lining is typically noted [##REF##21331652##14##]. Arthroscopic surgery is a discretionary treatment option for MD of ACL. This procedure involves the partial removal of lesions within the ACL, leading to rapid pain relief and improved range of motion, without any persistent symptoms of instability. The reduction in volume and tension within the ACL is often attributed to the notable pain relief [##REF##27299081##15##]. Debridement of mucinous substance, along with partial resection of ACL, is a recommended and effective therapy that does not cause instability, according to many authors [##REF##18629461##16##].</p>", "<p>While certain authors have underscored the significance of notchplasty as a supplementary step in the procedure, Motmans and Verheyden have challenged this idea by arguing against the necessity of notchplasty. They contend that a comprehensive debridement alone is sufficient to resolve impingement and effectively address the underlying pathology [##REF##19089409##4##].</p>" ]

[ "<title>Conclusions</title>", "<p>The clinical and radiological characteristics of MD of ACL can be often inconclusive, potentially leading to a misdiagnosis of a torn ACL. Arthroscopists must be cognizant of this uncommon condition, as they may encounter a mucoid ACL during surgical procedures. This awareness is crucial for an accurate diagnosis and appropriate management of the condition.</p>" ]

[ "<p>Mucoid degeneration (MD) is an uncommon pathological phenomenon that specifically affects the anterior cruciate ligament (ACL). This condition arises from the infiltration of yellowish material within the fibers of the ACL, contributing to the clinical presentation characterized by discomfort and limited mobility. MRI has proven to be the foremost diagnostic modality in effectively distinguishing MD of ACL from other potential pathologies. Preoperative recognition of this condition facilitates straightforward diagnosis, particularly via characteristic findings observed during knee arthroscopy. We present a case of MD of ACL, review prior studies about the condition, and outline its clinical features and symptoms, including those observed in our specific case.</p>" ]

[ "<title>Case presentation</title>", "<p>The patient was a 57-year-old female who presented to our hospital with a two-year history of right knee pain and limited flexion following a minor injury. A thorough examination was conducted. Notably, there was no apparent swelling or instability observed. The knee exhibited a restricted range of motion, spanning from 0° to 90°, with pain occurring during terminal flexion. Tenderness was localized to the lateral joint line of the knee, and the McMurray test yielded a positive result. Significantly, no clinical signs or symptoms suggestive of instability, ligamentous laxity, or patellofemoral pathology were evident (Figure ##FIG##0##1##).</p>", "<p>A plain X-ray examination provided valuable insights, revealing subtle degenerative changes, particularly in the medial compartment (Figure ##FIG##1##2##). The patient underwent conservative treatment for several months; however, there was no improvement in the condition.</p>", "<p>MRI of the knee was performed, which revealed specific findings in the right knee. In the coronal view, a clear thickening of the ACL was evident, characterized by preserved fibers, and a consistent increase in signal intensity. Noteworthy findings on the sagittal view revealed an indistinct ACL presenting a distinctive \"celery stalk\" appearance (Figure ##FIG##2##3##).</p>", "<p>Arthroscopy was performed, uncovering femoropatellar arthritis marked by advanced osteocartilaginous lesions and degenerative changes in the lateral and medial meniscus. The continuous fibers, notably the anterolateral bundle, exhibited enlargement and displayed a yellowish degenerative appearance, confirming MD. Additionally, degenerative vertical lesions were evident in the external meniscus. A decision was made to perform debridement of the yellowish lesion of the ACL, and subsequent stability testing through the anterior drawer maneuver yielded satisfactory results (Figure ##FIG##3##4##). At the 10-month follow-up, the patient demonstrated a complete range of knee motion without experiencing pain or instability.</p>" ]

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[ "<fig position=\"anchor\" fig-type=\"figure\" id=\"FIG1\"><label>Figure 1</label><caption><title>Examination of the patient at presentation</title><p>A: Range of flexion of the right knee 90°. B: Range of extension of the right knee 0°</p></caption></fig>", "<fig position=\"anchor\" fig-type=\"figure\" id=\"FIG2\"><label>Figure 2</label><caption><title>Plain radiographs show degenerative changes on the medial aspect of the knee</title><p>A: Anteroposterior radiographs of the right knee. B: Lateral radiographs of the right knee</p></caption></fig>", "<fig position=\"anchor\" fig-type=\"figure\" id=\"FIG3\"><label>Figure 3</label><caption><title>MRI of the knee at presentation</title><p>A: MRI Sagittal T1-weighted image showing the high signal intensity of the anterior cruciate ligament with a \"celery stalk\" appearance. B: MRI Coronal T2-weighted image showing a thickening of the anterior cruciate ligament with meniscus lesions</p><p>MRI: magnetic resonance imaging</p></caption></fig>", "<fig position=\"anchor\" fig-type=\"figure\" id=\"FIG4\"><label>Figure 4</label><caption><title>Arthroscopic images of the anterior cruciate ligament</title><p>A: Enlarged anterior cruciate ligament occupying the intercondylar notch. B: Yellowish material and sclerotic lesions within the anterior cruciate ligament were subsequently removed through debridement</p></caption></fig>" ]

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[ "<fn-group content-type=\"other\"><title>Author Contributions</title><fn fn-type=\"other\"><p><bold>Concept and design:</bold>  Oussama El Alaoui, Adnane Lachkar, Najib Abdeljaouad, Hicham Yacoubi</p><p><bold>Acquisition, analysis, or interpretation of data:</bold>  Oussama El Alaoui, Ousama Jelti, Adnane Lachkar</p><p><bold>Drafting of the manuscript:</bold>  Oussama El Alaoui, Ousama Jelti</p><p><bold>Critical review of the manuscript for important intellectual content:</bold>  Oussama El Alaoui, Ousama Jelti, Adnane Lachkar, Najib Abdeljaouad, Hicham Yacoubi</p><p><bold>Supervision:</bold>  Adnane Lachkar, Najib Abdeljaouad, Hicham Yacoubi</p></fn></fn-group>", "<fn-group content-type=\"other\"><title>Human Ethics</title><fn fn-type=\"other\"><p>Consent was obtained or waived by all participants in this study</p></fn></fn-group>", "<fn-group content-type=\"competing-interests\"><fn fn-type=\"COI-statement\"><p>The authors have declared that no competing interests exist.</p></fn></fn-group>" ]

[ "<graphic xlink:href=\"cureus-0015-00000050545-i01\" position=\"float\"/>", "<graphic xlink:href=\"cureus-0015-00000050545-i02\" position=\"float\"/>", "<graphic xlink:href=\"cureus-0015-00000050545-i03\" position=\"float\"/>", "<graphic xlink:href=\"cureus-0015-00000050545-i04\" position=\"float\"/>" ]

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[ "<p>Approved by: Agustin Ibanez, Latin American Brain Health Institute (BrainLat), Chile</p>" ]

[ "<p>The journal retracts the 18 June 2021 article cited above.</p>", "<p>Following publication, the publisher uncovered evidence that false identities were used in the peer-review process. The assignment of fake reviewers was confirmed by an investigation, conducted in accordance with Frontiers' policies and the Committee on Publication Ethics (COPE) guidelines. Given the concerns, the editors no longer have confidence in the findings presented in the article.</p>", "<p>This retraction was approved by the Chief Editors of Frontiers in Aging Neuroscience and the Chief Executive Editor of Frontiers. The authors received a communication regarding the retraction and had a chance to respond. This communication has been recorded by the publisher.</p>" ]

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[ "<title>Introduction and background</title>", "<p>The development of more efficient, well-tolerated, and easy combination antiretroviral treatment (cART) has significantly enhanced HIV-positive people's (PLWH) life expectancy, even exceeding that of the general population [##REF##27831953##1##]. Their extended survival, however, has resulted in a shift in their health profile, with a greater frequency of age-related comorbidities, which frequently manifest 5 to 10 years sooner than in the general population [##REF##21998278##2##]. Furthermore, certain PLWH are becoming more susceptible to geriatric disorders such as frailty [##REF##26009828##3##]. The goal of this study is to look into the role of frailty in the holistic treatment of older people with HIV (OPLWH).</p>", "<p>Frailty, which was initially used by the insurance industry to assess mortality risks in adults over the age of 65 [##REF##510638##4##], has recently been broadened to include physical, cognitive, social, emotional, and economic components [##REF##28210943##5##]. It denotes vulnerability to unfavorable health impacts as a result of several stresses. While frailty was originally studied in community-living persons over the age of 65, it has subsequently been studied across a wide range of demographics and conditions, indicating prevalence rates ranging from 3% to 5% between the ages of 30 and 60 to 25-30% after the age of 80 in the United Kingdom [##UREF##0##6##]. Frailty has been proven to be a more accurate predictor of survival in COVID-19 individuals than chronological age or comorbidities [##UREF##1##7##]. </p>", "<p>Frailty, cognitive decline, and other geriatric disorders are becoming more widespread and emerging sooner as PLWH continues to age, with a median age presently in the mid-50s in high-income countries [##REF##26009828##3##]. Frailty must be identified since it increases the likelihood of acquiring new chronic diseases, such as falling, cognitive impairment, polypharmacy, hospitalization, loss of independence, and mortality. Because aging rates vary, screening for frailty is a critical initial step in the management of OPLWH [##REF##23860844##8##].</p>", "<p>As a result of these developments, the approach to caring for OPLWH has moved from immunovirologic treatment to a more multidisciplinary one that places an emphasis on tailored services [##REF##31395144##9##]. Specialized programs have been developed to accommodate the particular needs of OPLWH [##UREF##1##7##]. While frailty assessments are not necessary for every OPLWH, they are crucial in identifying at-risk individuals since chronological age alone is not always a reliable predictor of age-related disorders in geriatrics [##REF##23860844##8##].</p>" ]

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[ "<title>Conclusions</title>", "<p>As OPLWH lives longer, healthcare for them is changing. However, there are issues with the growth in frailty and other geriatric problems among OPLWH. Once associated with elderly people, fragility is now a critical factor in determining susceptibility to unfavorable health outcomes. Personalized services and an interdisciplinary approach have been developed for OPLWH. Because frailty is associated with mortality and chronic diseases, it is imperative to identify and treat it early. Research on pharmacotherapeutic alternatives and e-health solutions is still ongoing. To offer complete treatment in this evolving healthcare environment, it is vital to steer clear of ageist healthcare practices and give frailty the attention it demands.</p>" ]

[ "<p>The life expectancy of people living with HIV (PLWH) has greatly increased due to advancements in combination antiretroviral treatment (cART). However, this longer life has also increased the prevalence of age-related comorbidities, such as frailty, which now manifest sooner in this group. Frailty, a term coined by the insurance industry, has been broadened to include physical, cognitive, and emotional elements and has been recognized as a critical predictor of negative health outcomes. With the median age of PLWH now in the mid-50s, treating frailty is critical given its link to chronic diseases, cognitive decline, and even death. Frailty assessment tools, such as the Frailty Phenotype (FP) and the Frailty Index (FI), are used to identify vulnerable people. Understanding the pathophysiology of frailty in PLWH indicates the role of immunological mechanisms. Frailty screening and management in this group have progressed, with specialized clinics and programs concentrating on multidisciplinary care. Potential pharmacotherapeutic solutions, as well as novel e-health programs and sensors, are in the future of frailty treatment, but it is critical to ensure that frailty evaluation is not exploited to perpetuate ageist healthcare practices. This narrative review investigates the changing healthcare environment for older people living with HIV (OPLWH), notably in high-income countries. It emphasizes the significance of identifying and managing frailty as a crucial feature of OPLWH's holistic care and well-being.</p>" ]

[ "<title>Review</title>", "<p>Frailty: concept and evolution</p>", "<p>Derived from the Latin word \"fragilis\" (meaning \"breakable\"), the term frailty indicates a condition of increased sensitivity. Fried devised the Frailty Phenotype (FP) and the Frailty Index (FI) to emphasize the complexity of the idea in contrast to impairments or comorbidities [##UREF##2##10##]. Although the fundamental traits and consequences of frailty are well acknowledged, the absence of a clear operational definition makes diagnosis difficult. There are at least 67 metrics that have been developed; the most often used ones in research are the FP and FI [##REF##26674984##11##].</p>", "<p>The FP diagnoses frailty when three of the five predefined physical criteria - unintentional weight loss, self-reported fatigue, poor hand grip strength, sluggish walking speed, and low physical activity - are met. One or two conditions indicate pre-frailty, whereas none indicate non-frailty. Based on the accumulation of many age-related health problems, the FI computes frailty. Generally speaking, frailty in older adults who live in communities is indicated by a FI greater than 0.25. A prognosis of poor quality and several acute illnesses is indicated by a FI larger than 0.7 [##UREF##3##12##]. There are 30 prerequisites in all. Even though the FP is simpler to use, it does require specialized assessments and qualified personnel. The FI, on the other hand, appears more complicated but provides superior risk discrimination and may be determined using electronic health information [##REF##26944937##13##].</p>", "<p>Models predict comparable results when applied to diverse populations, and they also identify subgroups of frail individuals who do not have impairments or comorbidities. Research on the prevalence of frailty varies, with estimates ranging from 2% to 25% [##REF##31395144##9##]. Other proven methods for diagnosing frailty include the Edmonton Frail Scale (EFS), the Clinical Frailty Scale (CFS), and the Frail questionnaire. Even among those who are not yet diagnosed as frail, a meta-analysis confirmed the association between these markers of frailty and increased rates of death, hospitalization, loss of independence, impairments, falls, fractures, and cognitive decline [##UREF##4##14##].</p>", "<p>A gradual loss of normal homeostatic mechanisms brought on by biological and environmental stresses that affect cells, tissues, organs, and the overall health of a person ultimately leads to frailty. A chronic inflammatory state is shaped by immunological traits and CMV seropositivity, and immunosenescence is a major factor in this process. Numerous processes and serum markers are associated with this chronic inflammation, which is sometimes referred to as inflammaging [##REF##20571864##15##]. Hormonal fluctuations, epigenetic alterations, telomere shortening, genetic control, and changes in body composition, such as abdominal fat and muscle loss, can all have an impact on frailty [##REF##30017798##16##].</p>", "<p>The CFS is a measure of independence in older PLWH that shows favorable correlations with the FP and FI [##REF##16129869##17##]. A multi-domain evaluation called the EFS has earlier been compared to the FP and FI in PLWH [##REF##16757522##18##]. A simple instrument consisting of five parts is the FRAIL Scale [##REF##22836700##19##]. Originally intended to predict death in PLWH, the Veterans Aging Cohort Study Index (VACS-I) is currently utilized as a frailty screening tool with construct validity and predictive validity [##REF##31565954##20##]. These methods aid in identifying persons who, due to their frailty, may require treatments and care.</p>", "<p>Pathophysiology of frailty in PLWH</p>", "<p>Frailty in PLWH has molecular underpinnings associated with age-related dysregulation of several physiological systems. Dysregulated systems include those related to the stress response (neuro-immuno-endocrine), metabolism (insulin and mitochondria), and musculoskeletal systems [##REF##27549318##21##]. When dysregulation escalates to a point where negative outcomes are more likely, frailty is categorized as a syndrome. Other variables are involved, even though the majority of evidence indicates that frailty in OPLWH is comparable to physiologic aging [##REF##30885572##22##].</p>", "<p>Prolonged low-grade HIV replication triggers persistent immunological activation even in suppressed individuals, which results in immunosenescence and inflammation that invariably produce frailty in clinical trials. Additionally, epigenomic dysregulation is at play. Clinical variables such as co-infections, early suppression of HIV replication, CD4/CD8 ratio, and microbiota translocation facilitate the development of frailty. These traits emphasize the role that immunoinflammatory variables play in the development of frailty [##REF##28203694##23##]. Frailty is a common dysregulated condition of physiological functioning and impaired functional reserve that leads to negative outcomes typical of aging, although it may manifest earlier in PLWH [##REF##28386608##24##].</p>", "<p>Studies employing the FP for frailty diagnosis</p>", "<p>In the multicenter AIDS Cohort Study (MACS), frailty was measured using an adapted frailty-related phenotype (aFRP) in untreated, seropositive, college-educated, Caucasian males with a mean age of 55. In the same group of males over 65 who tested negative for HIV, frailty was common in 3.4% of cases [##REF##18000149##25##]. Immuno-virologic traits were associated with frailty, and among PLWH who started cART, frailty was associated with a higher risk of developing AIDS or passing away [##REF##21719610##26##]. Risk factors for frailty included old age, non-Hispanic black ethnicity, and potentially controllable conditions such as AIDS, smoking, hepatitis C infection, depression, diabetes, and renal disease. Frailty also happens in the absence of concomitant comorbidities [##REF##18000149##25##].</p>", "<p>A 12% frailty prevalence was observed in the AIDS Linked to Intravenous Experience (ALIVE) cohort, which consisted primarily of male African-American injectable drug users (median age 49). Risk factors for this cohort included female sex, advanced HIV disease, lower education, depression, and multimorbidity, as well as older age and HIV infection. Frailty has been associated with all-cause hospitalizations [##UREF##5##27##], overall mortality [##REF##27516622##28##], and chronic illnesses such as lung, heart, and mental disorders.</p>", "<p>The HIV Infection, Aging, and Immune Function Long-Term Observational Study (HAILO) cohort examined disability and frailty. The prevalence of pre-frailty and frailty was 37% and 6% of the population, respectively, with little overlap between the two conditions. Premature mortality, type II diabetes, and cardiovascular disease have all been related to frailty [##REF##30590451##29##]. Neurocognitive impairment, obesity, smoking, the first choice of cART (with NNRTI [nucleoside reverse transcriptase inhibitor]-based cART increasing the risk of frailty), and educational level were among the modifiable risk factors. Both moderate alcohol use and physical activity were protective [##REF##28453849##30##].</p>", "<p>Additionally, it was demonstrated that more seropositive women than males had frailty, which is in line with the overall population. 10.0% of uninfected women and 17.3% of seropositive women in the Women's Interagency HIV Study (WIHS) were frail [##REF##26980368##31##]. Concerns regarding economic and healthcare delivery challenges as the PLWH population ages are raised by other research, like those carried out in South Africa and Spain, which revealed a high prevalence of frailty in PLWH [##REF##31373215##32##].</p>", "<p>Screening of frailty in PLWH</p>", "<p>When a senior is diagnosed as frail, it signifies more than simply a condition with a dismal prognosis. For instance, one important risk factor for perioperative issues is frailty. By altering recognized preoperative variables for surgical morbidity, pre-habilitation clinics improve outcomes [##UREF##6##33##]. An integrated geriatric strategy is becoming more and more recommended for certain older PLWH, especially those who have been classified as weak [##REF##28387803##34##].</p>", "<p>Other surrogates, besides frailty, can help identify PLWH who would benefit from a geriatric assessment. For instance, polypharmacy is more common in PLWH compared to controls. Similarly, decreased functional status as determined by gait speed or the whole Short Physical Performance Battery (SPPB) assessment tool is also more common [##REF##24576251##35##]. Frailty, functional status, and deficits all interact in PLWH, despite being distinct illnesses [##REF##27715455##36##]. PLWH frequently experience functional impairments and impairments, especially those who also have concurrent geriatric symptoms [##REF##27715455##36##,##REF##24966138##37##]. Strong PLWH in need of geriatric review may also be indicated by a combination of immunological markers (e.g., a low nadir CD4 count of 200, a plateau CD4 level of 500 on suppressive cART, and a CD4/CD8 ratio of one) [##UREF##7##38##].</p>", "<p>Weakness is a dynamic state. The majority of non-frail and pre-frail individuals retained their status in a 12-month follow-up analysis involving over 300 treated PLWH, but the majority of frail individuals relapsed to pre-frailty [##REF##25495766##39##]. Pre-frailty, which affects 30-60% of PLWH, has to be identified since it is associated with unfavorable outcomes. We have previously discussed the elements that contribute to the shift of PLWH in the MACS to frailty. The only factor associated with a reversal of frailty was age [##REF##24127428##40##]. Guaraldi spent four years studying the MHMC Cohort's frailty transition determinants. The following factors predicted FI at follow-up: baseline FI, female gender, length of HIV infection and cART usage, and history of smoking [##UREF##8##41##].</p>", "<p>There are currently no guidelines for which PLWH to refer, and it is unclear if geriatric referrals are clinically effective. The public should be evaluated for frailty if they are older than 70. It is reasonable to think about screening PLWH above the age of 50 based on data showing PLWH age-advancement. The responsibilities of geriatricians as knowledgeable consultants or involved team members are being established [##REF##23764209##42##].</p>", "<p>Management of frailty in PLWH</p>", "<p>In high-income nations, the primary healthcare approach for OPLWH is based on specialized community-based primary care or tertiary care that is offered in infectious disease clinics. Specialized geriatric or aging-HIV clinics, HIV-metabolic clinics, and HIV-rehabilitation programs have been established to address the wider range of social, emotional, and health concerns experienced by OPLWH as their healthcare needs have expanded to include non-HIV-related illnesses [##UREF##9##43##].</p>", "<p>Numerous of these clinics have adopted care methods designed for senior citizens that are based on geriatric concepts. With an emphasis on modifying the Comprehensive Geriatric Assessment (CGA) to identify the most susceptible OPLWH, these models entail screening for frailty and other geriatric disorders. CGA, a well-researched multidimensional diagnostic procedure, is used to evaluate the functional, psychological, and medical abilities of a subset of elderly people [##UREF##10##44##]. After the age of 70, it is advised to do yearly frailty screenings in the general population [##REF##23764209##42##]. According to recent guidelines, PLWH over 50 should undergo yearly frailty screening using the FRAIL Scale. If the screening findings are positive, the individual should then be sent for a thorough geriatric evaluation [##REF##34652753##45##]. </p>", "<p>To test for frailty, HIV clinics that specialize in OPLWH use a variety of measures, such as gait speed, the FRAIL Scale, and the Clinical Frailty Scale [##REF##34652753##45##]. Many clinics have provided insights into how they manage OPLWH who are frail, using multidisciplinary geriatric evaluations to determine who is most in need [##REF##28387803##34##]. For instance, the CFS is used by the Chelsea and Westminster Clinic in London to screen older people with HIV (OPLWH). Individuals with a CFS score of 5 or more, which indicates at least mild frailty, are sent to a specialized geriatric HIV clinic [##REF##34269603##46##]. Clinics may use several measures to screen for frailty. While some use electronic health records to identify the FP, others use the FI. When electronic health data are accessible, the FI may be applied more easily than the FP, which needs specific equipment and training [##REF##26944937##13##, ##REF##30668637##47##].</p>", "<p>The goals of therapy for OPLWH who have been classified as frail are to prevent and control impairment and comorbidities. Furthermore, geriatric syndrome assessment and treatment are essential elements of this care approach [##REF##33987636##48##]. To control OPLWH, tactics borrowed from research with frail, uninfected older adults are used. Maintaining the quality of life and possibly averting cognitive decline also depend on acknowledging the increased health risks linked to social isolation. This concept aims to actively involve older people with HIV/AIDS within their social networks and reintegrate HIV care into primary care while facilitating access to community resources [##REF##33128225##49##].</p>", "<p>Fundamental care principles for frail older HIV-negative individuals include implementing strength-training exercise programs, managing sarcopenia, assessing for polypharmacy, treating weight loss and undernutrition with protein-calorie supplements, and evaluating and treating reversible causes of fatigue such as anemia, depression, hypothyroidism, and B12 deficiency. Furthermore, vitamin D levels should be monitored, and it should be administered if necessary [##REF##29678160##50##,##REF##31641726##51##].</p>", "<p>Programs for preventing frailty that involve many components have shown promise in reducing the pace of frailty development, boosting cognitive performance, and increasing physical function [##UREF##11##52##]. CGA informs care plans that can improve functional abilities, lower the chance of institutionalization, postpone the onset of impairment, cut down on hospital admissions and stays, and increase survival [##UREF##12##53##]. A physiotherapist should ideally be available or included in the multidisciplinary team. Exercise programs of varying intensities have enhanced physical function in OPLWH, while short-duration exercise programs have shown beneficial effects on frail older individuals. Improvements in quality-of-life measures in PLWH have also been associated with participation in yoga programs [##REF##32583707##54##].</p>", "<p>While exercise has been shown to be the most effective intervention for frailty, group-based physiotherapy classes have also demonstrated efficacy in protecting against functional decline. On the other hand, addressing the social determinants of frailty is important. This includes, but is not limited to, isolation, loneliness, depression, or any other psychiatric diagnoses. However, this may not always be feasible, as it often depends on the availability of local professional resources and the presence of a community-based support system [##UREF##13##55##].</p>", "<p>The New Orleans Alcohol Use in HIV (NOAH) study evaluated the body composition, gait speed, and muscle strength of 341 participants living with HIV. Body composition was found to have a modulatory effect on frailty risk among PLWH, which was statistically significant. That is, while obesity was associated with increased risk, greater muscle mass may have had a protective effect, even among individuals who consume alcohol. These findings emphasize the importance of physical activity and weight control in modulating frailty risk [##REF##36124866##56##].</p>", "<p>Furthermore, strict control of the medications PLWH consumes may be beneficial. Polypharmacy may increase the risk of adverse outcomes, and de-prescription is often necessary. For instance, a study demonstrated that frailty was more common in those who used medications with an anticholinergic effect (OR 2.12; 95% CI 0.89-5.0). Therefore, clinicians should be aware of their impact and work to reduce exposure whenever possible [##REF##37644705##57##].</p>", "<p>As mentioned earlier, exercise is a well-established preventive intervention for frailty. However, since many of the OPLWH are already in a frail state, high-intensity training and weight-lifting may be too taxing. Therefore, a study investigated a novel game-based training program, referred to as exergame, and its impact on ameliorating frailty among 10 HIV-infected individuals. The exergame program which was conducted twice weekly for six weeks, incorporated activities such as weight shifting, ankle reaching, and obstacle crossing. After the conclusion of the program, changes in balance, gait, and other parameters were assessed, and improvements were seen in balance and mobility [##REF##27768079##58##].</p>", "<p>E-health systems are being increasingly utilized to provide health information, personalized recommendations, and smartphone reminders to promote healthy habits and help older people preserve and enhance their functional independence. There is much promise in the continuing evaluation of wearable sensors for frailty identification [##REF##32583707##54##]. The Ecological Momentary Assessment, which gathers real-time patient-reported outcome indicators, can help make it easier to incorporate patient-reported outcomes into innovative healthcare models [##UREF##12##53##]. Nonetheless, it is important to maintain vigilance to prevent the possible ageist approach to healthcare delivery from being supported by frailty [##UREF##11##52##].</p>", "<p>Limitations and challenges</p>", "<p>While our review provides insight into frailty pathophysiology, screening, detection, and management for PLWH, it also identifies several limitations and research gaps, the most notable of which is the absence of a standardized operational definition for frailty. This presents a challenge not only for consistent diagnosis in the healthcare setting but also for conducting studies with reliable data and results. Therefore, the need to establish a universally accepted definition is paramount. However, more research is needed to explore the most effective and relevant metrics for frailty while considering the diverse impacted populations, including OPLWH.</p>", "<p>In a comparable context, the variability of estimates of frailty prevalence in the different studies (ranging from 2% to 25%) highlights the importance of understanding any contributing factors and standardizing assessment methods in future studies. On the other hand, it implies that frailty in this population (PLWH) may manifest differently. Furthermore, while our article touches on the integration of geriatric concepts in HIV care, more research is required to assess the feasibility, effectiveness, and scalability of these approaches. Clear guidelines for identifying PLWH who would benefit from frailty assessments and evaluating the clinical effectiveness of geriatric referrals are lacking, and this is an area we aim to address in our ongoing research.</p>", "<p>Lastly, a more thorough exploration of e-health systems and electronic health records used for estimating frailty, especially those employing metrics such as the Frailty Index, would enhance our understanding of technological advancements in frailty assessment and their real-world applicability. Addressing these gaps will contribute to a more comprehensive and nuanced understanding of frailty, improve diagnostic accuracy and inform targeted interventions for vulnerable populations.</p>" ]

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[ "<fn-group content-type=\"other\"><title>Author Contributions</title><fn fn-type=\"other\"><p><bold>Concept and design:</bold>  Mohammad Mansour, Monisha Augustine, Mahendra Kumar, Amna Naveed Butt, Thanmai Reddy Thugu, Parvinder Kaur, Nipakumari J. Patel, Ankit Gaudani, M. Bilal Jahania, Elhama Jami, Mouhammad Sharifa, Rohan Raj</p><p><bold>Acquisition, analysis, or interpretation of data:</bold>  Mohammad Mansour, Monisha Augustine, Mahendra Kumar, Amna Naveed Butt, Thanmai Reddy Thugu, Parvinder Kaur, Nipakumari J. Patel, Ankit Gaudani, M. Bilal Jahania, Elhama Jami, Mouhammad Sharifa, Rohan Raj, Dalia Mehmood</p><p><bold>Drafting of the manuscript:</bold>  Mohammad Mansour, Monisha Augustine, Mahendra Kumar, Amna Naveed Butt, Thanmai Reddy Thugu, Parvinder Kaur, Nipakumari J. Patel, Ankit Gaudani, M. Bilal Jahania, Elhama Jami, Mouhammad Sharifa, Rohan Raj</p><p><bold>Critical review of the manuscript for important intellectual content:</bold>  Mohammad Mansour, Monisha Augustine, Mahendra Kumar, Amna Naveed Butt, Thanmai Reddy Thugu, Parvinder Kaur, Nipakumari J. Patel, Ankit Gaudani, M. Bilal Jahania, Elhama Jami, Mouhammad Sharifa, Rohan Raj, Dalia Mehmood</p><p><bold>Supervision:</bold>  Mohammad Mansour</p></fn></fn-group>", "<fn-group content-type=\"competing-interests\"><fn fn-type=\"COI-statement\"><p>The authors have declared that no competing interests exist.</p></fn></fn-group>" ]

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[ "<title>Introduction</title>", "<p>Autism is a disorder distinguished by significant challenges in social interaction and communication coupled with repetitive and stereotypical patterns of behavior and activities. Deficits in social interaction and language development become apparent before the age of three. In children, this condition is referred to as autism spectrum disorder (ASD). ASD is a disorder that affects a child's neurological system, growth, and general development [##UREF##0##1##]. A child with ASD often has problems communicating; they may have trouble developing social skills. In the Diagnostic and Statistical Manual of Mental Illnesses-5 (DSM-5), ASD is a neurodevelopmental disorder characterized by persistent deficits in social communication and social interaction across multiple contexts, as well as restricted, repetitive patterns of behavior, interests, or activities. The criteria are grouped into two main domains: social communication and behavior. To be diagnosed with ASD, according to the DSM-5, an individual must demonstrate persistent difficulties in social communication and interaction, along with at least two of the four types of restricted, repetitive behaviors. The severity of ASD is specified based on the level of support required: Level 1 (requiring support), Level 2 (requiring substantial support), or Level 3 (requiring very substantial support) [##UREF##1##2##].\nIt is important to note that the field of psychiatry and psychology may evolve, and updates to diagnostic criteria or new editions of diagnostic manuals may occur. Therefore, it is recommended to consult the latest professional literature or experts in the field for the most up-to-date information on the diagnosis of ASD and other mental health conditions. Different approaches, especially psychosocial therapies, are used to address the core symptoms of ASD. These include specific educational programs in special education, speech, occupational, physical, and behavioral analysis, all of which have a favorable impact on the overall efficacy of treatment. Occupational therapy is frequently utilised in managing ASD, although its efficacy can be limited. Consequently, there is considerable anticipation that psychopharmacology may provide additional assistance to these individuals. The CDC states that ASD is a prevalent illness, typically first identified in early childhood [##UREF##2##3##].\nThe signs and symptoms of attention deficit hyperactivity disorder (ADHD), as listed in the DSM-5, include experiencing difficulties with staying focused, paying attention to details, and listening when spoken to directly. Individuals with ADHD often have trouble finishing homework and organizing projects and are reluctant to engage in demanding activities. They regularly lose critical materials needed for assignments, become easily distracted, and are restless or fidgety. Common behaviors include frequently leaving their seat in situations where remaining seated is expected, being unable to play quietly, and exhibiting restlessness as if driven by an inner motor. Additional symptoms are talking excessively, vocalizing thoughts aloud, and showing a lack of self-control or patience [##UREF##3##4##]. The level of impairment in individuals with ASD can vary, affecting perception, cognitive processing, learning, and memory [##UREF##4##5##]. Additionally, genetics plays a significant role in autism, as evidenced by the ASD occurrence rate among twins, which ranges from 36% to 95%. These genetic factors particularly influence the repetitive behavioral aspects associated with ASD [##UREF##5##6##, ####UREF##6##7##, ##UREF##7##8####7##8##].\nClinical assessments using the Indian Scale for Autism Assessment (ISAA) have yielded positive results across the board for symptoms associated with ASD [##UREF##8##9##, ####REF##35996797##10##, ##UREF##9##11####9##11##]. The training intervention was designed to enhance the understanding of managing arousal, preventing annoyance from worsening, and promoting self-regulation skills [##UREF##10##12##]. It is essential to ensure that children on the autism spectrum interventions are supported by research evidence. Given the variety of accessible pharmaceutical therapies, compiling comprehensive research on interventions for children with these conditions is imperative [##REF##36081343##13##]. Physiotherapists are often among the first practitioners to assess children who may be at risk for these disorders [##REF##29971656##14##].\nThe study aims to investigate the efficacy of a multimodal physiotherapy approach in addressing the specific challenges associated with ASD, including speech impairment and attention deficit. The case report will focus on an individual with ASD who presents both speech difficulties and attention deficit symptoms. The multimodal physiotherapy intervention will incorporate a combination of physical therapy techniques, sensory integration strategies, and potentially other therapeutic modalities to enhance both motor and cognitive functions. The study seeks to assess improvements in speech abilities, attention span, and overall functional outcomes in individuals with ASD following the implementation of the multimodal physiotherapy intervention. By exploring the potential benefits of this comprehensive approach, the research aims to contribute valuable insights into the development of effective therapeutic strategies for individuals with ASD who experience speech impairments and attention deficits.</p>" ]

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[ "<title>Discussion</title>", "<p>This study presents intriguing findings that warrant careful consideration and interpretation. The observed improvements in speech abilities and attention deficits following the implementation of the multimodal physiotherapy intervention suggest the potential efficacy of this comprehensive approach. The integration of physical therapy techniques and sensory strategies appears to have a positive impact on both motor and cognitive functions, addressing the unique challenges faced by individuals with ASD who experience concurrent speech impairments and attention deficits. The individualized nature of the treatment plan is a notable strength, emphasizing the importance of tailoring interventions to the specific needs of individuals with ASD.</p>", "<p>In the broader context of interventions for ASD, this study contributes to the growing body of literature exploring novel therapeutic approaches. By incorporating physical therapy and sensory strategies into a multimodal intervention, the research underscores the potential benefits of simultaneously addressing both motor and cognitive aspects. This aligns with the evolving understanding of the interconnected nature of sensory-motor functions and cognitive processes in individuals with ASD. The findings may inform clinicians and therapists working with this population, offering insights into developing more comprehensive and tailored interventions. Despite the promising results, it is essential to approach the study's implications cautiously and recognize the need for continued exploration of multimodal approaches within more extensive and diverse samples, allowing for a more robust understanding of their effectiveness across the spectrum of ASD presentations.\nThe physiotherapy interventions presented in this protocol are a comprehensive and multidisciplinary approach to addressing attention deficit and speech impairment in a three-year-old child with developmental anomalies. Each intervention is tailored to target specific aspects of the child's condition, aiming to improve their overall communication and cognitive abilities. Auditory integration training (AIT) involves exposure to specially filtered and modulated music to enhance auditory processing, which may improve sensory integration and attention in individuals with developmental disorders [##REF##7608035##15##]. Sensory integration therapy (SIT) focuses on improving the brain's processing of sensory information, particularly for children with sensory sensitivities often seen in ASD [##REF##10500857##16##]. Holding therapy, based on the belief that autism is linked to a lack of maternal bonding, has its efficacy and safety debated. Facilitated communication involves a skilled facilitator assisting a child with communication challenges by using an output device for spelling words or phrases [##REF##8050988##17##]. Music therapy is recognized for its potential to enhance interaction and communication skills in individuals with ASD [##REF##19535468##18##]. The Picture Exchange Communication System (PECS) is an established method for children with ASD to communicate, offering a structured approach for non-verbal or minimally verbal individuals to express needs and desires using pictures or symbols [##REF##17501728##19##].\nParent-mediated communication is a crucial component of the child's support system, equipping parents to effectively understand and respond to their child's communication. However, the effectiveness of each intervention can vary from one individual to another, and ongoing evaluation and research are needed to determine the long-term impact and effectiveness of these interventions. ADHD is common in children, and the implementation of therapy in mental health, complemented by interdisciplinary collaboration, is crucial [##UREF##11##20##].</p>", "<p>The pediatric population benefits from the expertise of physiotherapists who specialize in treating children. These professionals possess a deep understanding of the interplay between early childhood development and the body's systems and functions, as well as typical child growth patterns. Physiotherapists use many advanced devices and tools, as well as generic skills, applying their additional training in development and growth to treat infants through teenagers. Physiotherapists use an extensive span of treatment approaches in the pediatrics field. In each case, careful assessment determines the treatment approach and protocol. However, it is crucial to acknowledge the study's limitations, such as the single-case design, which may restrict the generalizability of the findings. Further research with larger sample sizes and controlled designs is warranted to confirm and extend the current results. </p>" ]

[ "<title>Conclusions</title>", "<p>In conclusion, this case report provides valuable insights into a holistic intervention strategy for individuals with ASD facing concurrent challenges of speech impairments and attention deficits. The observed improvements in speech abilities and attention following the multimodal physiotherapy intervention suggest the potential effectiveness of integrating physical therapy techniques and sensory strategies. The individualized nature of the treatment plan highlights the importance of tailoring interventions to the specific needs of each individual with ASD.\nWhile the findings are promising, it is crucial to recognize the study's limitations, including its single-case design, which may limit generalizability. Further research with larger and more diverse samples and controlled designs is essential to validate and extend these preliminary results. The study contributes to the broader understanding of therapeutic approaches for ASD by emphasizing the interconnected nature of sensory-motor functions and cognitive processes. The multimodal physiotherapy approach offers a nuanced perspective for clinicians and therapists, suggesting potential benefits in concurrently addressing both motor and cognitive aspects. As we move forward, ongoing research should continue to explore and refine multimodal interventions, considering their applicability across the diverse spectrum of ASD presentations and informing the development of more comprehensive and tailored therapeutic strategies for individuals with ASD experiencing speech impairments and attention deficits.</p>" ]

[ "<p>Autism is a disorder distinguished by significant challenges in social interaction and communication coupled with repetitive and stereotypical patterns of behavior and activities. Deficits in social interaction and language development become apparent before age three. In children, this condition is referred to as autism spectrum disorder (ASD). Attention Deficit Hyperactivity Disorder (ADHD) is a neurodevelopmental disorder characterized by persistent patterns of inattention, hyperactivity, and impulsivity. Individuals with ADHD may struggle with sustaining attention, organizing tasks, and completing assignments. They may exhibit hyperactive behaviors such as fidgeting, difficulty staying seated, and impulsive actions like interrupting others. ADHD can significantly impact daily functioning and is often diagnosed in childhood, with symptoms potentially persisting into adulthood. The disorder has three main subtypes: predominantly inattentive, predominantly hyperactive-impulsive, and combined presentation. Treatment typically involves a combination of behavioral interventions, psychoeducation, and, in some cases, medication, aiming to provide support and strategies for individuals to manage their symptoms effectively in various aspects of life. Cognitive impairment in ASD varies, meaning it could be at the sensory perception level of cognitive processing, learning, and memory. The goal of the training intervention was to control physiological arousal, enhance awareness, keep annoyance from getting worse, and encourage self-regulation abilities. In this case report, we discuss the approach of multimodal physiotherapy for autism with speech impairment and attention deficit. Furthermore, physiotherapy needs to find a position in the new mental health care paradigm in order to contribute to mental health care.</p>" ]

[ "<title>Case presentation</title>", "<p>Patient information</p>", "<p>We present a case involving a three-year-old male child characterized by a spectrum of developmental anomalies. Predominant concerns encompass intellectual disability, hyperactivity, speech dysfluency, and attentional deficits. The patient was delivered at term following an uneventful gestational period of 9 months and 5 days via a lower segment cesarean section (LSCS). Postnatally, delayed initiation of the birth cry necessitated a brief admission to the neonatal intensive care unit (NICU) for observation. Challenges in breastfeeding ensued due to attentional constraints, resulting in a reliance on artificial feeding. \nAdditionally, the patient was identified as having a calcium deficiency. At birth, he weighed 2.4 kilograms and presented with congenital anomalies, which were first observed by his mother at 2.5 years of age. Parental reports at this age indicated deficits in attention, speech fluency, social integration, and communication skills. His vocalizations remained mostly indiscernible babbling until 36 months, when he produced his inaugural linguistically coherent word. He exhibited challenges in apprehending and executing tasks and directives.\nMotor milestones were accomplished within normative time frames. Ambulation with support was initiated at nine months, and independent ambulation was achieved at 10 months. Equilibrium mastery was attained at 14 months. Regarding ADHD, the patient exhibited a diminished attentional span and proclivity for distractibility. Restlessness and loud vocalizations during play were evident, along with difficulty in maintaining a stationary seated position. A sensitive, temperamental disposition was noted. Modest progress was ascertained by both therapeutic interventionists and parents, with circumscribed strides in attentional endurance and communication acumen. The patient remains reliant on external feeding and continues to necessitate diaper use. Proficiency in toilet training remains unattended.</p>", "<p>Clinical examination</p>", "<p>The child exhibits integration with an examiner but shows less interaction with the surrounding environment. His overall activity level was average, but he exhibited a notable deficit in orientation to time, place, and person. His speech was unimpaired, and his hearing was within normal limits. However, his attention span was observed to be poor, which contributed to difficulties in maintaining focus and engagement. The anthropometric measurements are detailed in Table ##TAB##0##1##.</p>", "<p>Investigation</p>", "<p>The patient underwent a Brainstem Evoked Response Audiometry (BERA) examination, which revealed normal auditory pathway function, indicating no detectable abnormalities in the auditory system (Table ##TAB##1##2##).</p>", "<p>Physiotherapy intervention</p>", "<p>The patient underwent physiotherapy treatment for four weeks (Table ##TAB##2##3##). Throughout the treatment, both the patient and his family participated in counseling sessions and followed an exercise regimen.</p>", "<p>Figure ##FIG##0##1## shows a child undergoing holding therapy under the supervision of a therapist, and Figure ##FIG##1##2## illustrates the Picture Exchange Communication System (PECS) therapy.</p>", "<p>Follow-up and outcome measures</p>", "<p>After four weeks of treatment, outcome measures were assessed. Table ##TAB##3##4## summarizes the findings of the scales after the completion of treatment.</p>" ]

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[ "<fig position=\"anchor\" fig-type=\"figure\" id=\"FIG1\"><label>Figure 1</label><caption><title>Patient undergoing holding therapy.</title></caption></fig>", "<fig position=\"anchor\" fig-type=\"figure\" id=\"FIG2\"><label>Figure 2</label><caption><title>Picture Exchange Communication System (PECS).</title></caption></fig>" ]

[ "<table-wrap position=\"float\" id=\"TAB1\"><label>Table 1</label><caption><title>Physical examination findings.</title><p>cm: Centimeter; kg: Kilogram.</p></caption><table frame=\"hsides\" rules=\"groups\"><tbody><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Anthropometric measures</td><td rowspan=\"1\" colspan=\"1\">At birth</td><td rowspan=\"1\" colspan=\"1\">At present</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Length/Height (in cm)</td><td rowspan=\"1\" colspan=\"1\">10 cm</td><td rowspan=\"1\" colspan=\"1\">94cm</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Weight (in kg)</td><td rowspan=\"1\" colspan=\"1\">2.40 kg</td><td rowspan=\"1\" colspan=\"1\">13.5 kg</td></tr><tr><td rowspan=\"1\" colspan=\"1\">Head circumference (in cm)</td><td rowspan=\"1\" colspan=\"1\">35 cm</td><td rowspan=\"1\" colspan=\"1\">46 cm</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Chest circumference (in cm)</td><td rowspan=\"1\" colspan=\"1\">30-33 cm</td><td rowspan=\"1\" colspan=\"1\">49 cm</td></tr></tbody></table></table-wrap>", "<table-wrap position=\"float\" id=\"TAB2\"><label>Table 2</label><caption><title>BERA test findings.</title><p>The data has been represented as N.</p><p>EP: Evoked Potentials; SPL: Sound Pressure Level; dB: Decibel; ms: Millisecond; Rec: Recorded; Cz-M: Vertex-Mastoid; R: Right; L: Left; BERA: Brainstem Evoked Response Audiometry; NA: Not Applicable.</p></caption><table frame=\"hsides\" rules=\"groups\"><tbody><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">N</td><td rowspan=\"1\" colspan=\"1\">Rec.sites</td><td rowspan=\"1\" colspan=\"1\">I Lat., ms</td><td rowspan=\"1\" colspan=\"1\">Ia Lat., ms</td><td rowspan=\"1\" colspan=\"1\">II Lat., ms</td><td rowspan=\"1\" colspan=\"1\">III Lat., ms</td><td rowspan=\"1\" colspan=\"1\">IIIa Lat., ms</td><td rowspan=\"1\" colspan=\"1\">IV Lat., ms</td><td rowspan=\"1\" colspan=\"1\">V Lat., ms</td><td rowspan=\"1\" colspan=\"1\">I-III Lat., ms</td><td rowspan=\"1\" colspan=\"1\">III-V Lat., ms</td><td rowspan=\"1\" colspan=\"1\">I-V Lat., ms</td><td rowspan=\"1\" colspan=\"1\">Stim.side</td><td rowspan=\"1\" colspan=\"1\">Stimulus</td></tr><tr><td rowspan=\"1\" colspan=\"1\">1</td><td rowspan=\"1\" colspan=\"1\">Cz-M</td><td rowspan=\"1\" colspan=\"1\">1.43</td><td rowspan=\"1\" colspan=\"1\">1.75</td><td rowspan=\"1\" colspan=\"1\">3.25</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">5.15</td><td rowspan=\"1\" colspan=\"1\">6.63</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">5.2</td><td rowspan=\"1\" colspan=\"1\">R</td><td rowspan=\"1\" colspan=\"1\">±110 dB SPL</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">2</td><td rowspan=\"1\" colspan=\"1\">Cz-M</td><td rowspan=\"1\" colspan=\"1\">1.53</td><td rowspan=\"1\" colspan=\"1\">2.85</td><td rowspan=\"1\" colspan=\"1\">3.35</td><td rowspan=\"1\" colspan=\"1\">4.6</td><td rowspan=\"1\" colspan=\"1\">4.85</td><td rowspan=\"1\" colspan=\"1\">5.25</td><td rowspan=\"1\" colspan=\"1\">6.68</td><td rowspan=\"1\" colspan=\"1\">3.08</td><td rowspan=\"1\" colspan=\"1\">2.08</td><td rowspan=\"1\" colspan=\"1\">5.15</td><td rowspan=\"1\" colspan=\"1\">R</td><td rowspan=\"1\" colspan=\"1\">±100 dB SPL</td></tr><tr><td rowspan=\"1\" colspan=\"1\">3</td><td rowspan=\"1\" colspan=\"1\">Cz-M</td><td rowspan=\"1\" colspan=\"1\">2.38</td><td rowspan=\"1\" colspan=\"1\">2.78</td><td rowspan=\"1\" colspan=\"1\">3.5</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">4.6</td><td rowspan=\"1\" colspan=\"1\">5.4</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">R</td><td rowspan=\"1\" colspan=\"1\">±90 dB SPL</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">4</td><td rowspan=\"1\" colspan=\"1\">Cz-M</td><td rowspan=\"1\" colspan=\"1\">2.35</td><td rowspan=\"1\" colspan=\"1\">3.23</td><td rowspan=\"1\" colspan=\"1\">3.55</td><td rowspan=\"1\" colspan=\"1\">3.95</td><td rowspan=\"1\" colspan=\"1\">4.75</td><td rowspan=\"1\" colspan=\"1\">5.5</td><td rowspan=\"1\" colspan=\"1\">5.9</td><td rowspan=\"1\" colspan=\"1\">1.6</td><td rowspan=\"1\" colspan=\"1\">1.95</td><td rowspan=\"1\" colspan=\"1\">3.55</td><td rowspan=\"1\" colspan=\"1\">R</td><td rowspan=\"1\" colspan=\"1\">±80 dB SPL</td></tr><tr><td rowspan=\"1\" colspan=\"1\">5</td><td rowspan=\"1\" colspan=\"1\">Cz-M</td><td rowspan=\"1\" colspan=\"1\">1.73</td><td rowspan=\"1\" colspan=\"1\">2.45</td><td rowspan=\"1\" colspan=\"1\">2.8</td><td rowspan=\"1\" colspan=\"1\">3.75</td><td rowspan=\"1\" colspan=\"1\">4.03</td><td rowspan=\"1\" colspan=\"1\">4.3</td><td rowspan=\"1\" colspan=\"1\">5.83</td><td rowspan=\"1\" colspan=\"1\">2.03</td><td rowspan=\"1\" colspan=\"1\">2.08</td><td rowspan=\"1\" colspan=\"1\">4.1</td><td rowspan=\"1\" colspan=\"1\">R</td><td rowspan=\"1\" colspan=\"1\">±70 dB SPL</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">6</td><td rowspan=\"1\" colspan=\"1\">Cz-M</td><td rowspan=\"1\" colspan=\"1\">1.83</td><td rowspan=\"1\" colspan=\"1\">2.65</td><td rowspan=\"1\" colspan=\"1\">2.88</td><td rowspan=\"1\" colspan=\"1\">3.95</td><td rowspan=\"1\" colspan=\"1\">5.15</td><td rowspan=\"1\" colspan=\"1\">5.35</td><td rowspan=\"1\" colspan=\"1\">5.95</td><td rowspan=\"1\" colspan=\"1\">2.13</td><td rowspan=\"1\" colspan=\"1\">2.0</td><td rowspan=\"1\" colspan=\"1\">4.13</td><td rowspan=\"1\" colspan=\"1\">R</td><td rowspan=\"1\" colspan=\"1\">±60 dB SPL</td></tr><tr><td rowspan=\"1\" colspan=\"1\">7</td><td rowspan=\"1\" colspan=\"1\">Cz-M</td><td rowspan=\"1\" colspan=\"1\">2.2</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">3.75</td><td rowspan=\"1\" colspan=\"1\">4.35</td><td rowspan=\"1\" colspan=\"1\">5.68</td><td rowspan=\"1\" colspan=\"1\">6.63</td><td rowspan=\"1\" colspan=\"1\">1.55</td><td rowspan=\"1\" colspan=\"1\">2.88</td><td rowspan=\"1\" colspan=\"1\">4.43</td><td rowspan=\"1\" colspan=\"1\">R</td><td rowspan=\"1\" colspan=\"1\">±50 dB SPL</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">8</td><td rowspan=\"1\" colspan=\"1\">Cz-M</td><td rowspan=\"1\" colspan=\"1\">1.93</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">3.93</td><td rowspan=\"1\" colspan=\"1\">4.65</td><td rowspan=\"1\" colspan=\"1\">4.88</td><td rowspan=\"1\" colspan=\"1\">5.53</td><td rowspan=\"1\" colspan=\"1\">2.0</td><td rowspan=\"1\" colspan=\"1\">1.6</td><td rowspan=\"1\" colspan=\"1\">3.6</td><td rowspan=\"1\" colspan=\"1\">R</td><td rowspan=\"1\" colspan=\"1\">±40 dB SPL</td></tr><tr><td rowspan=\"1\" colspan=\"1\">9</td><td rowspan=\"1\" colspan=\"1\">Cz-M</td><td rowspan=\"1\" colspan=\"1\">1.35</td><td rowspan=\"1\" colspan=\"1\">2.83</td><td rowspan=\"1\" colspan=\"1\">3.3</td><td rowspan=\"1\" colspan=\"1\">4.4</td><td rowspan=\"1\" colspan=\"1\">4.68</td><td rowspan=\"1\" colspan=\"1\">5.25</td><td rowspan=\"1\" colspan=\"1\">6.78</td><td rowspan=\"1\" colspan=\"1\">3.05</td><td rowspan=\"1\" colspan=\"1\">2.38</td><td rowspan=\"1\" colspan=\"1\">5.43</td><td rowspan=\"1\" colspan=\"1\">L</td><td rowspan=\"1\" colspan=\"1\">±110 dB SPL</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">10</td><td rowspan=\"1\" colspan=\"1\">Cz-M</td><td rowspan=\"1\" colspan=\"1\">1.53</td><td rowspan=\"1\" colspan=\"1\">2.9</td><td rowspan=\"1\" colspan=\"1\">3.35</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">5.2</td><td rowspan=\"1\" colspan=\"1\">6.8</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">5.28</td><td rowspan=\"1\" colspan=\"1\">L</td><td rowspan=\"1\" colspan=\"1\">±100 dB SPL</td></tr><tr><td rowspan=\"1\" colspan=\"1\">11</td><td rowspan=\"1\" colspan=\"1\">Cz-M</td><td rowspan=\"1\" colspan=\"1\">1.6</td><td rowspan=\"1\" colspan=\"1\">2.88</td><td rowspan=\"1\" colspan=\"1\">3.5</td><td rowspan=\"1\" colspan=\"1\">4.65</td><td rowspan=\"1\" colspan=\"1\">4.83</td><td rowspan=\"1\" colspan=\"1\">5.38</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">3.05</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">L</td><td rowspan=\"1\" colspan=\"1\">±90 dB SPL</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">12</td><td rowspan=\"1\" colspan=\"1\">Cz-M</td><td rowspan=\"1\" colspan=\"1\">2.15</td><td rowspan=\"1\" colspan=\"1\">3.0</td><td rowspan=\"1\" colspan=\"1\">3.53</td><td rowspan=\"1\" colspan=\"1\">4.6</td><td rowspan=\"1\" colspan=\"1\">4.93</td><td rowspan=\"1\" colspan=\"1\">5.4</td><td rowspan=\"1\" colspan=\"1\">5.73</td><td rowspan=\"1\" colspan=\"1\">2.45</td><td rowspan=\"1\" colspan=\"1\">1.13</td><td rowspan=\"1\" colspan=\"1\">3.58</td><td rowspan=\"1\" colspan=\"1\">L</td><td rowspan=\"1\" colspan=\"1\">±80 dB SPL</td></tr><tr><td rowspan=\"1\" colspan=\"1\">13</td><td rowspan=\"1\" colspan=\"1\">Cz-M</td><td rowspan=\"1\" colspan=\"1\">2.3</td><td rowspan=\"1\" colspan=\"1\">2.88</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">3.9</td><td rowspan=\"1\" colspan=\"1\">4.8</td><td rowspan=\"1\" colspan=\"1\">5.33</td><td rowspan=\"1\" colspan=\"1\">5.78</td><td rowspan=\"1\" colspan=\"1\">1.6</td><td rowspan=\"1\" colspan=\"1\">1.88</td><td rowspan=\"1\" colspan=\"1\">3.48</td><td rowspan=\"1\" colspan=\"1\">L</td><td rowspan=\"1\" colspan=\"1\">±60 dB SPL</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">14</td><td rowspan=\"1\" colspan=\"1\">Cz-M</td><td rowspan=\"1\" colspan=\"1\">2.08</td><td rowspan=\"1\" colspan=\"1\">3.1</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">4.3</td><td rowspan=\"1\" colspan=\"1\">5.0</td><td rowspan=\"1\" colspan=\"1\">6.05</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">3.98</td><td rowspan=\"1\" colspan=\"1\">L</td><td rowspan=\"1\" colspan=\"1\">±40 dB SPL</td></tr><tr><td rowspan=\"1\" colspan=\"1\">15</td><td rowspan=\"1\" colspan=\"1\">Cz-M</td><td rowspan=\"1\" colspan=\"1\">1.88</td><td rowspan=\"1\" colspan=\"1\">3.28</td><td rowspan=\"1\" colspan=\"1\">3.6</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">4.63</td><td rowspan=\"1\" colspan=\"1\">6.08</td><td rowspan=\"1\" colspan=\"1\">6.53</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">NA</td><td rowspan=\"1\" colspan=\"1\">4.65</td><td rowspan=\"1\" colspan=\"1\">L</td><td rowspan=\"1\" colspan=\"1\">±50 dB SPL</td></tr></tbody></table></table-wrap>", "<table-wrap position=\"float\" id=\"TAB3\"><label>Table 3</label><caption><title>Physiotherapy management for attention deficit and speech impairment.</title><p>ASD: Autism spectrum disorder; AIT: Auditory Integration Training; SIT: Sensory Integration Therapy; PECS: Picture Exchange Communication System; Min: Minute.</p></caption><table frame=\"hsides\" rules=\"groups\"><tbody><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Sr. No.</td><td rowspan=\"1\" colspan=\"1\">Treatment</td><td rowspan=\"1\" colspan=\"1\">Description</td><td rowspan=\"1\" colspan=\"1\">Duration</td></tr><tr><td rowspan=\"1\" colspan=\"1\">1)</td><td rowspan=\"1\" colspan=\"1\">Auditory Integration Training (AIT)</td><td rowspan=\"1\" colspan=\"1\">AIT involves experiencing music that has been filtered and modulated, including varied volume levels and pitches.</td><td rowspan=\"1\" colspan=\"1\">10 min 1 session per day/week</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">2)</td><td rowspan=\"1\" colspan=\"1\">Sensory Integration Therapy (SIT)</td><td rowspan=\"1\" colspan=\"1\">SIT aims to enhance brain processing of information, providing a foundation for advanced skill development.</td><td rowspan=\"1\" colspan=\"1\">10 min 1 session per day/week</td></tr><tr><td rowspan=\"1\" colspan=\"1\">3)</td><td rowspan=\"1\" colspan=\"1\">Holding Therapy</td><td rowspan=\"1\" colspan=\"1\">Based on the theory that autism stems from insufficient bonding with the mother.</td><td rowspan=\"1\" colspan=\"1\">10 min 1 session per day/week</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">4)</td><td rowspan=\"1\" colspan=\"1\">Facilitated Communication</td><td rowspan=\"1\" colspan=\"1\">It involves a method in which a skilled facilitator provides assistance to assist a child in using an output device, like a keyboard, typewriter or similar tool to spell.</td><td rowspan=\"1\" colspan=\"1\">15 min 1 session per day/week</td></tr><tr><td rowspan=\"1\" colspan=\"1\">5)</td><td rowspan=\"1\" colspan=\"1\">Music Therapy</td><td rowspan=\"1\" colspan=\"1\">One of the reasons music therapy is believed to be effective for individuals with ASD is because the processes involved in improvisation can potentially aid in the enhancement of interaction and communication skills.</td><td rowspan=\"1\" colspan=\"1\">5 min 1 session per day/week</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">6)</td><td rowspan=\"1\" colspan=\"1\">Picture Exchange Communication System (PECS)</td><td rowspan=\"1\" colspan=\"1\">Its main goal is to help children who have autism learn to start conversations by giving a picture to someone they want to communicate with in order to communicate their needs.</td><td rowspan=\"1\" colspan=\"1\">10 min 1 session per day/week</td></tr><tr><td rowspan=\"1\" colspan=\"1\">7)</td><td rowspan=\"1\" colspan=\"1\">Parent-mediated Communication</td><td rowspan=\"1\" colspan=\"1\">In this approach, the therapist provides training to parents with the goal of enhancing their ability to understand and respond effectively to their child’s communication.</td><td rowspan=\"1\" colspan=\"1\">10 min 1 session per day/week</td></tr></tbody></table></table-wrap>", "<table-wrap position=\"float\" id=\"TAB4\"><label>Table 4</label><caption><title>Outcome measures.</title><p>MACS: Manual ability classification system; FIM: Functional independence measures; ACS-C: Attentional control scale.</p></caption><table frame=\"hsides\" rules=\"groups\"><tbody><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">Sr. Number</td><td rowspan=\"1\" colspan=\"1\">Pediatrics scale</td><td rowspan=\"1\" colspan=\"1\">Before treatment</td><td rowspan=\"1\" colspan=\"1\">After Treatment</td></tr><tr><td rowspan=\"1\" colspan=\"1\">1)</td><td rowspan=\"1\" colspan=\"1\">MACS Scale </td><td rowspan=\"1\" colspan=\"1\">Level 3</td><td rowspan=\"1\" colspan=\"1\">Level 1</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">2)</td><td rowspan=\"1\" colspan=\"1\">WeeFIM Scale</td><td rowspan=\"1\" colspan=\"1\">Score 1</td><td rowspan=\"1\" colspan=\"1\">Score 5</td></tr><tr><td rowspan=\"1\" colspan=\"1\">3)</td><td rowspan=\"1\" colspan=\"1\">Conners Rating Scale</td><td rowspan=\"1\" colspan=\"1\">Score 4</td><td rowspan=\"1\" colspan=\"1\">Score 1</td></tr><tr style=\"background-color:#ccc\"><td rowspan=\"1\" colspan=\"1\">4)</td><td rowspan=\"1\" colspan=\"1\">Attentional Control Scale</td><td rowspan=\"1\" colspan=\"1\">Score 1</td><td rowspan=\"1\" colspan=\"1\">Score 4</td></tr></tbody></table></table-wrap>" ]

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[ "<fn-group content-type=\"other\"><title>Author Contributions</title><fn fn-type=\"other\"><p><bold>Concept and design:</bold>  Pradhyum D. Kolhe, H V Sharath, Prajyot Ankar, Vaishnavi M. Thakre</p><p><bold>Acquisition, analysis, or interpretation of data:</bold>  Pradhyum D. Kolhe, H V Sharath, Prajyot Ankar, Vaishnavi M. Thakre</p><p><bold>Drafting of the manuscript:</bold>  Pradhyum D. Kolhe, H V Sharath, Prajyot Ankar, Vaishnavi M. Thakre</p><p><bold>Critical review of the manuscript for important intellectual content:</bold>  Pradhyum D. Kolhe, H V Sharath, Prajyot Ankar, Vaishnavi M. Thakre</p><p><bold>Supervision:</bold>  Pradhyum D. Kolhe, H V Sharath, Prajyot Ankar, Vaishnavi M. Thakre</p></fn></fn-group>", "<fn-group content-type=\"other\"><title>Human Ethics</title><fn fn-type=\"other\"><p>Consent was obtained or waived by all participants in this study</p></fn></fn-group>", "<fn-group content-type=\"competing-interests\"><fn fn-type=\"COI-statement\"><p>The authors have declared that no competing interests exist.</p></fn></fn-group>" ]

[ "<graphic xlink:href=\"cureus-0015-00000050547-i01\" position=\"float\"/>", "<graphic xlink:href=\"cureus-0015-00000050547-i02\" position=\"float\"/>" ]

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{ "acronym": [], "definition": [] }

[ "<title>Introduction</title>", "<p>Road accidents are a significant public health concern worldwide, with an estimated 1.35 million deaths caused by road traffic accidents each year.\n^{\n##UREF##0##1##\n} Developing countries, such as India, are disproportionately affected, with over 150,000 fatalities reported annually.\n^{\n##UREF##0##1##\n} Road safety is a major concern in India, with a large number of accidents and fatalities reported each year. According to the Ministry of Road Transport and Highways, there were 449,002 road accidents in India in 2019, resulting in 151,113 deaths and 451,361 injuries.\n^{\n##UREF##1##2##\n}\n</p>", "<p>Road accidents pose a threat to health, leading to approximately 1.35 million deaths globally each year. Countries such as India are particularly affected, experiencing over 150,000 fatalities annually. This alarming situation emphasizes the importance of comprehending the factors that contribute to these accidents in order to develop prevention measures. Our research is motivated by the urgency to improve road safety in high-risk areas, like India, where the number of accidents and fatalities remains distressingly high.</p>", "<p>Our research delves into the modeling of accident severity, a statistical technique in the field of road safety. Traditional statistical models like logit and probit have been used since the 1990s to predict traffic accident severity, but they have limitations, especially when their underlying assumptions are violated.\n^{\n##UREF##2##3##\n}\n^,\n^{\n##REF##17920851##4##\n} On the other hand, Artificial Intelligence (AI) models offer adaptability and can handle complex nonlinear relationships without being constrained by such assumptions. We focus on the Random Forest (RF) algorithm, an advanced machine learning model with distinct advantages over other algorithms for predicting accident severity.\n^{\n##UREF##3##5##\n} Through tuning parameters and preprocessing techniques, we aim to improve the performance of RF even further, making our academic contribution significant in the pursuit of more accurate and reliable predictions for accident severity.</p>", "<p>The modelling process involves analyzing data on past accidents and identifying the factors that contributed to their occurrence and severity.\n^{\n##UREF##4##6##\n} These factors can include road conditions, weather, driver behaviour, and vehicle type, among others. The goal of accident severity modelling is to identify the most important factors contributing to accidents and to develop evidence-based strategies to improve road safety and reduce the number and severity of accidents.\n^{\n##UREF##5##7##\n}\n^–\n^{\n##UREF##7##9##\n}\n</p>", "<p>Statistical models have been widely used for predicting traffic accidents’ severity.\n^{\n##UREF##2##3##\n}\n^,\n^{\n##REF##17920851##4##\n} Artificial intelligence models, in contrast, do not make any assumptions and are more adaptable. They can handle intricate nonlinear relationships and generally offer higher predictive accuracy than statistical approaches.\n^{\n##UREF##3##5##\n} RF, in particular, has been successfully applied in various contexts, and its performance can be further improved by tuning key parameters and careful data preprocessing.\n^{\n##REF##17920851##4##\n}\n^,\n^{\n##UREF##3##5##\n}\n^,\n^{\n##REF##14733994##10##\n}\n^,\n^{\n##UREF##8##11##\n} The performance of the RF algorithm is significantly influenced by the selection of hyperparameters.\n^{\n##UREF##9##12##\n} To optimize its performance, identifying the optimal parameter values is crucial.</p>", "<p>The main objective of our study is to develop a predictive model for the severity of traffic accidents on Indian highways using RF models due to their accuracy and interpretability.</p>", "<p>The findings of our study will be used to develop a predictive model for accident severity that can inform road safety policies and interventions. This model can be used to identify high-risk areas and prioritize resources for accident prevention and mitigation.</p>", "<title>Study Areas</title>", "<p>The study areas selected are the National Highways two stretches as mentioned below\n<list list-type=\"simple\"><list-item><label>1.</label><p>Pune-Sholapur Section of NH-9 in km.144/400 to Km. 249/000 in the State of Maharashtra (\n##FIG##0##Figure 1##).</p></list-item><list-item><label>2.</label><p>Six-Laning of Barwa-Adda-Panagarh Section of NH-2 from km 398.240 to km 521.120 including Panagarh Bypass in the States of Jharkhand and West Bengal (\n##FIG##1##Figure 2##).</p></list-item></list>\n</p>", "<p>The study areas for this research project were selected based on specific criteria. Firstly, the researchers had prior experience of working on one of the stretches, which is the Pune-Sholapur Section of NH-9 in km.144/400 to Km. 249/000 in the State of Maharashtra. This experience could have provided insights and knowledge that could be useful in conducting the study.</p>", "<p>Additionally, data was also provided by the same concessionaire as of the previous stretch on request for another stretch, which is the Six-Laning of Barwa-Adda-Panagarh Section of NH-2 from km 398.240 to km 521.120 including Panagarh Bypass in the States of and West Bengal. This data could have been relevant to the research objectives and could have assisted in achieving the desired outcomes.</p>" ]

[ "<title>Methods</title>", "<p>The proposed methodology for this research involves the following steps for implementing a RF model machine learning technique for accident severity prediction.</p>", "<p>Data Preparation: The first step in implementing a RF model for accident severity prediction is to collect and prepare data. Raw data on road accidents for the selected stretches of the highway can be obtained from secondary sources such as the Ministry of Road Transport and Highways (MoRTH) and the National Highways Authority of India (NHAI).\n^{\n##UREF##1##2##\n} Data wrangling and mining techniques can be used to clean and preprocess the data.</p>", "<p>Feature Selection: Once the data is prepared, selecting appropriate features for the model becomes crucial. Feature selection plays a vital role in reducing the dimensionality of the data and enhancing the model’s accuracy. There are several techniques available for feature selection, such as statistical tests, correlation analysis, and principal component analysis (PCA).\n^{\n##UREF##10##13##\n}\n</p>", "<p>Model Training: In the next step, an RF model can be trained on the preprocessed data. The model can be developed using a machine learning-based framework, as described in Breiman’s work on RF.\n^{\n##UREF##11##14##\n} The RF algorithm involves bagging and random feature selection techniques to create multiple decision trees that are aggregated to form a stronger learner.\n^{\n##UREF##12##15##\n}\n</p>", "<p>RF Algorithm Formulation: The RF algorithm can be represented as:\nwhere X are the input features, B is the number of trees, and T\n_b(X) is the prediction of the b-th individual decision tree.</p>", "<p>Parameter Tuning: To improve the performance of a RF model, it is important to fine-tune its parameters. The three key parameters that significantly impact the tuning performance of the RF model are the total number of trees (n_estimators), the number of features used for each node segmentation (max_feature), and the maximum depth of a tree (max_depth).\n^{\n##REF##20159099##16##\n}\n</p>", "<p>In the construction of the RF model for predicting accident severity, Gini impurity is employed as a criterion to evaluate the significance of different explanatory variables. Gini impurity, a measure utilized within the framework of decision trees (the base learners in a RF), is crucial for the optimal selection of features at each node split. It offers a quantitative metric to discern the effectiveness of a variable in segregating the target classes.</p>", "<p>Mechanism of Gini Impurity: In the context of binary classification, the Gini impurity for a node is calculated as:\nwhere,\n<italic toggle=\"yes\">pk</italic>​ is the proportion of samples classified to class\n<italic toggle=\"yes\">k</italic> at that node, and the summation operates over all classes. A lower Gini impurity score suggests a higher purity of the node, indicating an enhanced classification.</p>", "<p>Gini Importance in RF: In the developed RF model, the Gini impurity plays a dual role:\n<bold>Node Splitting</bold>: It aids in the identification of the most significant variable at each node by evaluating the potential reduction in impurity for each split, and\n<bold>Feature Importance</bold>: Post model training, the average decrease in impurity caused by each feature across all trees is computed, known as Gini importance. This metric offers insights into the relative significance of different features for the prediction task.</p>", "<p>Model Evaluation: After training the RF model and optimizing its parameters, it is important to evaluate the model’s performance. Various evaluation metrics can be used, including accuracy, precision, recall, F1 score, and Area Under the Curve - Receiver Operating Characteristics (AUC-ROC) curve.\n^{\n##UREF##13##17##\n}\n</p>", "<p>Model Implementation: Once the model has been trained and evaluated, it can be deployed for accident severity prediction. The methodology can be designed using Python for building the model and forecasting the severity of road traffic accidents on Indian highways.</p>", "<title>Source data</title>", "<p>Data on road accidents from selected stretches of highways was obtained from the Concessionaires of the National Highways Authority of India (NHAI) for two projects: Pune-Solapur and Bengal (BAEL) Section. For the Pune-Solapur Section of NH-9, which is located between km. 144/400 and km. 249/000 in the state of Maharashtra, accident dates from 2013 to 2018 were used. For the Six-Laning of Barwa-Adda-Panagarh Section of NH-2, which includes Panagarh Bypass and is located in the States of Jharkhand and West Bengal Stretch, accident dates from 2015 to 2019 were used for the stretch between km 398.240 and km 521.120. The raw data was subject to exploratory data analysis, as detailed in the following section.</p>", "<title>Data Preparation</title>", "<p>In this stage, data gathering and exploration is performed using secondary source data. The dataset consists of 3257 observations out of which the 1855 observations are of Bengal (BAEL) Section and 1402 observations are of Pune- Solapur and 32 variables, including the target variable “accident severity.” The 32 attributes and their corresponding mappings are presented in\n##TAB##0##Table 1##.</p>", "<title>Data Modelling</title>", "<p>The RF classification algorithm has been employed in this study to forecast the severity of road traffic accidents in India. This section details the procedure for implementing the model, performance evaluation, and discuss the results obtained. The RF algorithm is written using python programming language.</p>", "<p>The target variable for the random RF is selected as the’Accident Severity’ which has classes as Fatal, Grevious Injury, Minor Injury and No Injury and indexed as 1-Fatal, 2-Grevious Injury, 3-Minor Injury, 4-No Injury.</p>", "<p>The dataset is partitioned into training and testing sets with a ratio of 80% and 20%, respectively. The hyperparameters’n_estimators’ and’max_depth’ are specified, and a grid search is conducted with cross-validation (cv=5) to identify the optimal hyperparameters. The best parameters and scores are obtained. The best estimator is fit on the training data. Predictions are made on the test data and the accuracy of the model is obtained.</p>", "<p>The algorithm and programme for Accident Severity Modelling using RF are written in the Python programming language, and the code is made available to the public for further development. The source code can be\n<ext-link xlink:href=\"http://accessed\" ext-link-type=\"uri\">accessed</ext-link> via the software availability statement.</p>", "<p>Accuracy analysis on test data: Three metrics were employed to evaluate the effectiveness of the algorithms: accuracy, precision, and recall. These metrics are defined as follows:</p>", "<p>Accuracy: The formula for a metric that measures the proportion of correctly predicted observations to the total number of observations is represented as:\n\n</p>", "<p>Precision is a metric that indicates the ratio of correctly predicted positive observations to the total number of predicted positive observations, and is calculated using the formula:\n\n</p>", "<p>Recall is a metric that reflects the ratio of correctly predicted positive observations to the total number of actual positive observations, and is determined using the formula:\n\n</p>" ]

[ "<title>Result and Discussion</title>", "<title>Model Performance</title>", "<p>The classification model used three hyperparameters -’max_depth’: 10,’max_features’:’sqrt’, and’n_estimators’: 100, and the results generated a confusion matrix for the training set. The matrix indicated the number of correctly and incorrectly classified instances for each class. The classification report provided precision, recall, and f1-score for each class, along with support. The model showed high precision and recall for class 1 but low precision and recall for classes 2, 3, and 4, with an overall accuracy of 67% and a weighted average f1-score of 0.64 on the training set. The macro average f1-score, which assigns equal weight to each class, was 0.53.</p>", "<p>The optimal parameters for a RF classifier model were determined through a grid search, with a max depth of 2, n estimators of 5000, and a random state of 0. The model was then applied to the test data, and the predictions were saved in an Excel file called “predicted output3.xls” for further analysis. The accuracy of the model on the test data was determined to be 0.4147, or approximately 41.47%, indicating that it accurately predicted the severity of traffic accidents in about 41.47% of test cases.</p>", "<p>\n<bold>Predicted outputs</bold>\n</p>", "<p>\n<bold>Comparative analysis of observed and predicted accident severity index against dates</bold>\n</p>", "<p>The actual accident severity indices are represented by the observed values, while the predicted values are generated by the RF model using the input features.</p>", "<p>The following is a summary (\n##FIG##2##Figure 3##) of the comparison between the observed and predicted values:</p>", "<p>On dates such as 25-02-2017, 17-04-2017, and 22-04-2017, the RF model accurately predicts the accident severity index.</p>", "<p>In a number of instances, the model predicts a lower accident severity index value than the observed value. 18-02-2017, 23-02-2017, and 27-03-2017, for example.</p>", "<p>Occasionally, the model overestimates the accident severity index by predicting a higher value than the observed value, as on 24-05-2017 and 20-10-2017.</p>", "<p>In general, the model frequently predicts a severity index of 2 for accidents, even when the observed values are distinct. This may indicate a bias in the model, possibly as a result of an imbalance in the training dataset, in which severity index 2 occurs more frequently than other categories.</p>", "<p>\n<bold>Comparative analysis of observed and predicted accident severity index against time</bold>\n</p>", "<p>\n##FIG##3##Figure 4## displays the date, day of the week, and time of the accident, as well as the observed and predicted accident severity indices. The plotted for the 165 rows of predicted data doesn’t fit in A4 sheet hence the data is published and the link is provided in the Tableau graphs visuals availbility [i].</p>", "<p>The dataset contains accident data from February 18, 2017 to December 31, 2017, as determined by Tableau analysis of the plot generated from the provided Excel table.</p>", "<p>The observed accident severity index ranges from 1 to 4, where 1 corresponds to the least severe accident and 4 to the most severe accident.</p>", "<p>The observed severity index for the vast majority of accidents in the dataset is 3, followed by 4. 2 indicates a less severe accident, while 4 indicates a more severe accident.</p>", "<p>The majority of accidents within the dataset have a predicted severity index of 2, followed by an index of 1.</p>", "<p>The analysis of the scatter plot reveals that the predicted severity index is typically lower than the observed severity index. This suggests that the model used to predict the severity of accidents is not always accurate and could be improved.</p>", "<title>Comparative analysis of observed and predicted accident severity index against Location and Chainages- RHS</title>", "<p>The Tableau plot (\n##FIG##4##Figure 5##) presents a detailed visual analysis of accident data on the right-hand side of the road. The data is organized by date and day of the week, displaying the accident location, observed accident severity index, and predicted accident severity index for each incident. The plot effectively illustrates the spatial distribution of accidents and their severity over time, enabling the identification of patterns and trends. The Tableau plot doesn’t fit in A4 sheet hence the data is published and the link is provided in the Tableau graphs visuals availability [ii].</p>", "<p>It is evident from the analysis that the majority of accidents have an observed severity index of 2 or 3, indicating a moderate severity. However, the predicted accident severity index largely remains at 2, indicating that the predictions may be somewhat conservative and do not fully capture the observed severity range.</p>", "<p>In addition, there appears to be no correlation between the day of the week and the frequency or severity of accidents across the different days of the week. This may suggest that external factors, such as traffic patterns or weather conditions, have a greater impact on the occurrence and severity of accidents than the day of the week.</p>", "<title>Comparative analysis of observed and predicted accident severity index against Location and Chainages- LHS</title>", "<p>The graph displays (\n##FIG##5##Figure 6##) the date, day of the week, and accident location on Left Hand Side (LHS) of the road, as well as the observed and predicted accident severity indices. The plotted of predicted data doesn’t fit in A4 sheet hence the data is published and the link is provided in the Tableau graphs visuals availability [iii].</p>", "<p>The scatterplot reveals that the majority of accidents on the left side of the road had a severity index of 2 or 3, with only a few instances of severity index 1 and 4. This indicates that the majority of collisions on the left side of the road were of moderate severity.</p>", "<p>In the majority of cases, the predicted accident severity index was 2, with only a few instances of values 3 and 4. This suggests that the predictive model may be biased towards predicting less severe accidents.</p>", "<p>There was no discernible pattern or trend between the day of the week and the occurrence of accidents. Accidents appeared to occur every day, indicating that the day of the week may not be a significant predictor of accident severity on the left side of the road.</p>", "<p>The accident locations, as measured by Accident Location A Chainage km, were scattered along the roadway at various distances. This suggests that there may not be a particular accident hotspot or concentration on the left-hand side of the road.</p>", "<title>Data Recording and availability</title>", "<p>The recording of road accident data in India must comply with the MoRTH &amp; IRC guidelines, utilizing the Road Accident Recording and Reporting Formats. Despite this, there exists a need for a more advanced data recording system to effectively model road safety. The digital monitoring of road accidents can increase the frequency of data collection and minimize the absence of crucial information. Often, the lack of a system or individual to document the accident leads to the absence of important road accident data. This missing data can be regained through the use of machine learning, thus enhancing the accuracy of road safety modeling.</p>" ]

[ "<title>Result and Discussion</title>", "<title>Model Performance</title>", "<p>The classification model used three hyperparameters -’max_depth’: 10,’max_features’:’sqrt’, and’n_estimators’: 100, and the results generated a confusion matrix for the training set. The matrix indicated the number of correctly and incorrectly classified instances for each class. The classification report provided precision, recall, and f1-score for each class, along with support. The model showed high precision and recall for class 1 but low precision and recall for classes 2, 3, and 4, with an overall accuracy of 67% and a weighted average f1-score of 0.64 on the training set. The macro average f1-score, which assigns equal weight to each class, was 0.53.</p>", "<p>The optimal parameters for a RF classifier model were determined through a grid search, with a max depth of 2, n estimators of 5000, and a random state of 0. The model was then applied to the test data, and the predictions were saved in an Excel file called “predicted output3.xls” for further analysis. The accuracy of the model on the test data was determined to be 0.4147, or approximately 41.47%, indicating that it accurately predicted the severity of traffic accidents in about 41.47% of test cases.</p>", "<p>\n<bold>Predicted outputs</bold>\n</p>", "<p>\n<bold>Comparative analysis of observed and predicted accident severity index against dates</bold>\n</p>", "<p>The actual accident severity indices are represented by the observed values, while the predicted values are generated by the RF model using the input features.</p>", "<p>The following is a summary (\n##FIG##2##Figure 3##) of the comparison between the observed and predicted values:</p>", "<p>On dates such as 25-02-2017, 17-04-2017, and 22-04-2017, the RF model accurately predicts the accident severity index.</p>", "<p>In a number of instances, the model predicts a lower accident severity index value than the observed value. 18-02-2017, 23-02-2017, and 27-03-2017, for example.</p>", "<p>Occasionally, the model overestimates the accident severity index by predicting a higher value than the observed value, as on 24-05-2017 and 20-10-2017.</p>", "<p>In general, the model frequently predicts a severity index of 2 for accidents, even when the observed values are distinct. This may indicate a bias in the model, possibly as a result of an imbalance in the training dataset, in which severity index 2 occurs more frequently than other categories.</p>", "<p>\n<bold>Comparative analysis of observed and predicted accident severity index against time</bold>\n</p>", "<p>\n##FIG##3##Figure 4## displays the date, day of the week, and time of the accident, as well as the observed and predicted accident severity indices. The plotted for the 165 rows of predicted data doesn’t fit in A4 sheet hence the data is published and the link is provided in the Tableau graphs visuals availbility [i].</p>", "<p>The dataset contains accident data from February 18, 2017 to December 31, 2017, as determined by Tableau analysis of the plot generated from the provided Excel table.</p>", "<p>The observed accident severity index ranges from 1 to 4, where 1 corresponds to the least severe accident and 4 to the most severe accident.</p>", "<p>The observed severity index for the vast majority of accidents in the dataset is 3, followed by 4. 2 indicates a less severe accident, while 4 indicates a more severe accident.</p>", "<p>The majority of accidents within the dataset have a predicted severity index of 2, followed by an index of 1.</p>", "<p>The analysis of the scatter plot reveals that the predicted severity index is typically lower than the observed severity index. This suggests that the model used to predict the severity of accidents is not always accurate and could be improved.</p>", "<title>Comparative analysis of observed and predicted accident severity index against Location and Chainages- RHS</title>", "<p>The Tableau plot (\n##FIG##4##Figure 5##) presents a detailed visual analysis of accident data on the right-hand side of the road. The data is organized by date and day of the week, displaying the accident location, observed accident severity index, and predicted accident severity index for each incident. The plot effectively illustrates the spatial distribution of accidents and their severity over time, enabling the identification of patterns and trends. The Tableau plot doesn’t fit in A4 sheet hence the data is published and the link is provided in the Tableau graphs visuals availability [ii].</p>", "<p>It is evident from the analysis that the majority of accidents have an observed severity index of 2 or 3, indicating a moderate severity. However, the predicted accident severity index largely remains at 2, indicating that the predictions may be somewhat conservative and do not fully capture the observed severity range.</p>", "<p>In addition, there appears to be no correlation between the day of the week and the frequency or severity of accidents across the different days of the week. This may suggest that external factors, such as traffic patterns or weather conditions, have a greater impact on the occurrence and severity of accidents than the day of the week.</p>", "<title>Comparative analysis of observed and predicted accident severity index against Location and Chainages- LHS</title>", "<p>The graph displays (\n##FIG##5##Figure 6##) the date, day of the week, and accident location on Left Hand Side (LHS) of the road, as well as the observed and predicted accident severity indices. The plotted of predicted data doesn’t fit in A4 sheet hence the data is published and the link is provided in the Tableau graphs visuals availability [iii].</p>", "<p>The scatterplot reveals that the majority of accidents on the left side of the road had a severity index of 2 or 3, with only a few instances of severity index 1 and 4. This indicates that the majority of collisions on the left side of the road were of moderate severity.</p>", "<p>In the majority of cases, the predicted accident severity index was 2, with only a few instances of values 3 and 4. This suggests that the predictive model may be biased towards predicting less severe accidents.</p>", "<p>There was no discernible pattern or trend between the day of the week and the occurrence of accidents. Accidents appeared to occur every day, indicating that the day of the week may not be a significant predictor of accident severity on the left side of the road.</p>", "<p>The accident locations, as measured by Accident Location A Chainage km, were scattered along the roadway at various distances. This suggests that there may not be a particular accident hotspot or concentration on the left-hand side of the road.</p>", "<title>Data Recording and availability</title>", "<p>The recording of road accident data in India must comply with the MoRTH &amp; IRC guidelines, utilizing the Road Accident Recording and Reporting Formats. Despite this, there exists a need for a more advanced data recording system to effectively model road safety. The digital monitoring of road accidents can increase the frequency of data collection and minimize the absence of crucial information. Often, the lack of a system or individual to document the accident leads to the absence of important road accident data. This missing data can be regained through the use of machine learning, thus enhancing the accuracy of road safety modeling.</p>" ]

[ "<title>Conclusion</title>", "<p>The RF classifier model predicted the severity of traffic accidents with an overall accuracy of 67% on the training set and approximately 41.47% on the test set. Indicating possible bias or imbalance in the training dataset, the model tended to predict a lower severity index than the observed values. There were no discernible relationships between the day of the week and the occurrence or severity of accidents. The performance of the model can be enhanced by correcting the dataset imbalance and refining the model’s hyperparameters.</p>", "<p>The observed and predicted accident severity indices were compared against a number of variables, including dates, times, and locations on both sides of the road. In some instances, the model accurately predicted the accident severity index, but it frequently underestimated accident severity. No discernible patterns or trends were observed in terms of accident location, indicating that external factors may have a greater influence on the occurrence and severity of accidents.</p>", "<p>To improve road safety modelling, it is essential to adopt a more sophisticated data recording system consistent with MoRTH and IRC recommendations. Digital monitoring of road accidents can increase the frequency of data collection and reduce the loss of vital information. Integrating machine learning techniques can contribute to more effective interventions and decision-making in the field of traffic accident prevention and mitigation.</p>", "<p>In the field of accident severity modeling our research stands out for its contributions. We focus on leveraging Artificial Intelligence (AI) models, which excel at capturing the relationships that traditional statistical methods often overlook. Our, in depth study of the Random Forest (RF) algorithm, combined with careful parameter adjustment and data preprocessing highlights its potential in this area. This research addresses not concerns but also specifically tackles India’s road safety challenges providing insights applicable worldwide as well as tailored solutions for the region. A key aspect of our approach is our unwavering dedication to improving accuracy positioning our work as a standard for precise and reliable accident severity predictions. Overall this study makes a contribution to literature, in this field.</p>" ]

[ "<p>No competing interests were disclosed.</p>", "<p>\n<bold>Background:</bold> Road accidents claim around 1.35 million lives annually, with countries like India facing a significant impact. In 2019, India reported 449,002 road accidents, causing 151,113 deaths and 451,361 injuries. Accident severity modeling helps understand contributing factors and develop preventive strategies. AI models, such as random forest, offer adaptability and higher predictive accuracy compared to traditional statistical models. This study aims to develop a predictive model for traffic accident severity on Indian highways using the random forest algorithm.</p>", "<p>\n<bold>Methods:</bold> A multi-step methodology was employed, involving data collection and preparation, feature selection, training a random forest model, tuning parameters, and evaluating the model using accuracy and F1 score. Data sources included MoRTH and NHAI.</p>", "<p>\n<bold>Results:</bold> The classification model had hyperparameters ‘max depth’:  10, ‘max features’: ‘sqrt’, and ‘n estimators’: 100. The model achieved an overall accuracy of 67% and a weighted average F1-score of 0.64 on the training set, with a macro average F1-score of 0.53. Using grid search, a random forest Classifier was fitted with optimal parameters, resulting in 41.47% accuracy on test data.</p>", "<p>\n<bold>Conclusions:</bold> The random forest classifier model predicted traffic accident severity with 67% accuracy on the training set and 41.47% on the test set, suggesting possible bias or imbalance in the dataset. No clear patterns were found between the day of the week and accident occurrence or severity. Performance can be improved by addressing dataset imbalance and refining model hyperparameters. The model often underestimated accident severity, highlighting the influence of external factors. Adopting a sophisticated data recording system in line with MoRTH and IRC guidelines and integrating machine learning techniques can enhance road safety modeling, decision-making, and accident prevention efforts.</p>", "<title>Amendments from Version 1</title>", "<p>We express our gratitude to the reviewers for their constructive comments that greatly enhanced the manuscript's quality. In response to Reviewer-1's insightful observation on the term \"accident\", we have clarified the context and regional usage of the term in the paper, while also acknowledging the global standards. As per suggestions, we've incorporated the Random Forest algorithm formulation in the methods section and discussed the significance of the Gini impurity test in understanding the importance of explanatory variables. Addressing Reviewer-2's feedback, we've emphasized the motivation behind our work and its academic contribution in the Introduction section, ensuring a comprehensive understanding of our study's significance. We revisited and corrected specific literature citations, particularly references 7, 14, and 15. The choice of factors, their contribution, and the transferability of the developed model are now prominently highlighted in a dedicated subsection. Further, to address the concerns about the novelty, we've incorporated a detailed paragraph in the conclusion, elucidating the innovative aspects and unique contributions of our predictive model in comparison to existing literature. We believe these revisions have amplified the clarity, depth, and significance of our research, ensuring alignment with the journal's standards and objectives.</p>" ]

[ "<title>Future Scope</title>", "<p>The study presented provides a good starting point for future research in the field of road safety modeling and accident prevention for Indian highways. However, with the limitations of the present study there opens potential areas for future research as mentioned below which will be taken up in continuation.</p>", "<p>Dataset improvement: The study identified the possibility of dataset bias and imbalance affecting model performance. Future research will focus on improving the quality and quantity of data, reducing bias and improving model performance. This will involve exploring alternative data sources, enhancing data collection methods, and addressing data quality issues.</p>", "<p>Model improvement: The study used the RF algorithm to develop a predictive model for traffic accident severity. In future research, other machine learning algorithms or ensemble models to improve model performance will be explored. Additionally, refining hyperparameters and addressing dataset imbalance will be done to improve model accuracy.</p>", "<p>External factors analysis: The study highlighted the influence of external factors on accident severity prediction. Future research can focus on exploring the impact of external factors such as weather conditions, road infrastructure, and driver behavior on accident severity. This can enhance the accuracy of predictive models and inform decision-making in accident prevention efforts.</p>", "<p>Real-time monitoring: The study highlighted the need for a sophisticated data recording system in line with MoRTH and IRC guidelines. Future research can focus on developing a real-time monitoring system that can capture road safety data in real-time and provide insights for accident prevention efforts.</p>" ]

[ "<title>Acknowledgement</title>", "<p>We are grateful to National Highways Authority of India and IL&amp;FS Engineering and Construction Company for making the raw accident data available.</p>", "<title>Data availability</title>", "<p>Zenodo. Data for Accident Severity Prediction Modelling for Indian Highways Case Study,\n<ext-link xlink:href=\"https://doi.org/10.5281/zenodo.7773156\" ext-link-type=\"uri\">https://doi.org/10.5281/zenodo.7773156</ext-link>.\n^{\n##UREF##14##18##\n}\n</p>", "<p>This project contains the following underlying data:\n<list list-type=\"simple\"><list-item><label>•</label><p>\nAccdataset_hk_PS_BAEL_Combined.csv (The dataset consists of 3257 observations out of which the 1855 observations are of Bengal (BAEL) Section and 1402 observations are of Pune- Solapur.)</p></list-item><list-item><label>•</label><p>\npredicted_output_1.xlsx (This is level-2 processed data derived from raw accident data using prediction modeling. The data has been indexed from 1 to 4 for further analysis, and there are a total of 165 rows in the predicted output observations.\n</p></list-item></list>\n</p>", "<p>Data are available under the terms of the\n<ext-link xlink:href=\"https://creativecommons.org/licenses/by/4.0/legalcode\" ext-link-type=\"uri\">Creative Commons Attribution 4.0 International license</ext-link> (CC-BY 4.0).</p>", "<title>Software availability</title>", "<p>\n<list list-type=\"simple\"><list-item><label>•</label><p>Github:\n<ext-link xlink:href=\"https://github.com/humera-k/RF_Accident_Severity\" ext-link-type=\"uri\">https://github.com/humera-k/RF_Accident_Severity\n</ext-link>\n</p></list-item><list-item><label>•</label><p>\n<ext-link xlink:href=\"https://zenodo.org/badge/latestdoi/616376786\" ext-link-type=\"uri\">https://zenodo.org/badge/latestdoi/616376786</ext-link>\n</p></list-item></list>\n</p>", "<p>Data are available under the terms of the\n<ext-link xlink:href=\"https://creativecommons.org/licenses/by/4.0/legalcode\" ext-link-type=\"uri\">Creative Commons Attribution 4.0 International license</ext-link> (CC-BY 4.0).</p>", "<title>Tableau graphs visuals availability</title>", "<p>\n<list list-type=\"simple\"><list-item><label>1.</label><p>\n<ext-link xlink:href=\"https://public.tableau.com/app/profile/humera.khanum/viz/Accidental_Analysis_1/Sheet52\" ext-link-type=\"uri\">\nAccidental_Analysis_1 | Tableau Public</ext-link> (Comparative analysis of observed and predicted accident severity index against time)</p></list-item><list-item><label>2.</label><p>\n<ext-link xlink:href=\"https://public.tableau.com/app/profile/humera.khanum/viz/Accidental_Analysis_1/Sheet3\" ext-link-type=\"uri\">\nAccidental_Analysis_1\n</ext-link> (Comparative analysis of observed and predicted accident severity index against Location and Chainages-Right hand Side (RHS))</p></list-item><list-item><label>3.</label><p>\n<ext-link xlink:href=\"https://public.tableau.com/views/Accidental_Analysis_1/Sheet4?:language=en-US&amp;publish=yes&amp;:display_count=n&amp;:origin=viz_share_link\" ext-link-type=\"uri\">\nAccidental_Analysis_1\n</ext-link> (Comparative analysis of observed and predicted accident severity index against Location and Chainages-Left Hand Side (LHS))\n</p></list-item></list>\n</p>" ]

[ "<fig position=\"float\" fig-type=\"figure\" id=\"f1\"><label>Figure 1. </label><caption><title>Pune-Sholapur Section of NH-9 in km.144/400 to Km. 249/000 in the State of Maharashtra.</title></caption></fig>", "<fig position=\"float\" fig-type=\"figure\" id=\"f2\"><label>Figure 2. </label><caption><title>Six-Laning of Barwa-Adda-Panagarh Section of NH-2 from km 398.240 to km 521.120 including Panagarh Bypass in the States of Jharkhand and West Bengal.</title></caption></fig>", "<fig position=\"float\" fig-type=\"figure\" id=\"f3\"><label>Figure 3. </label><caption><title>Comparison between Observed and Predicted Accident Severity Index.</title></caption></fig>", "<fig position=\"float\" fig-type=\"figure\" id=\"f4\"><label>Figure 4. </label><caption><title>Comparative analysis of observed and predicted accident severity index against time.</title></caption></fig>", "<fig position=\"float\" fig-type=\"figure\" id=\"f5\"><label>Figure 5. </label><caption><title>Comparative analysis of observed and predicted accident severity index against Location and Chainages-RHS.</title></caption></fig>", "<fig position=\"float\" fig-type=\"figure\" id=\"f6\"><label>Figure 6. </label><caption><title>Comparative analysis of observed and predicted accident severity index against Location and Chainages-LHS.</title></caption></fig>" ]

[ "<table-wrap position=\"float\" id=\"T1\"><label>Table 1. </label><caption><title>Dataset Attributes and Parameters Mapping.</title></caption><table frame=\"hsides\" rules=\"groups\" content-type=\"article-table\"><thead><tr><th align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">Sl No</th><th align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">Attributes</th><th align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">Mapping</th></tr></thead><tbody><tr><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">1</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">Accident Index</td><td colspan=\"1\" rowspan=\"1\"/></tr><tr><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">2</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">Date</td><td colspan=\"1\" rowspan=\"1\"/></tr><tr><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">3</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">Day of Week</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">1-Sunday, 2-Monday, 3-Tuesday, 4-Wednesday, 5- Thursday, 6-Friday, 7-Saturday</td></tr><tr><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">4</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">Time of Accident, Accident Location-A</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">1-Urban, 2-Rural, 3-Unallocated</td></tr><tr><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">5</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">Accident Location-A Chainage-km</td><td colspan=\"1\" rowspan=\"1\"/></tr><tr><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">6</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">Accident Location-A Chainage-km-RoadSide</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">LHS, RHS</td></tr><tr><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">7-9</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">Nature of Accident-B1, Nature of Accident- B2, Nature of Accident-B3</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">1-Overturning, 2-Head on collision, 3-Rear End Collision, 4-Collision Brush/Side Wipe, 5-Right Turn Collision, 6- Skidding, 7a-Others-Hit Cyclist, 7b-Others-Hit Pedestrian, 7c-Others-Hit Parked Vehicle, 7d-Others-Hit Fixed Object, 7e-Others-Wrong Side Driving, 7f-Others-Hit Animal, 7g- Others-Hit Two Wheeler, 7h-Others-Unknown, 7i-Others- Fallen down, 8-Overtaking vehicle, 9-Left Turn Collision</td></tr><tr><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">10</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">Accident Severity -C</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">1-Fatal, 2-Grevious Injury, 3-Minor Injury, 4-No Injury</td></tr><tr><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">11-13</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">Classification of Accident-C1, Classification of Accident-C2, Classification of Accident-C3</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">1-Fatal, 2-Grevious Injury, 3-Minor Injury, 4-Non - Injury (Damage only)</td></tr><tr><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">14-18</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">Causes-D1, Causes-D2, Causes-D3, Causes- D4, Causes-D5</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">1-Drunken, 2-Overspeeding, 3-Vehicle out of control, 4a- Fault of driver of motor vehicle, 4b-Driver of other vehicle, 4c-Cyclist, 4d-Pedestrian, 4e-Passenger, 4f-Animal, 5a- Defect in mechanical condition of motor vehicle, 5b-Road condition</td></tr><tr><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">19</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">Road Feature-E</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">1-Single lane, 2-Two lanes, 3-Three lanes or more without central divider median, 4-Four lanes or more with central divider alongwith carriageway width</td></tr><tr><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">20</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">Road Condition-F</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">1-Straight Road, 2-Slight Curve, 3-Sharp Curve, 4-Flat Road, 5-Gentle incline, 6-Steep incline 7-Hump, 8-Dip</td></tr><tr><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">21</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">Intersection Type-G</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">1-T Junction, 2-’Y Junction, 3-’Four arm junction, 4- Staggered junction, 5-Roundabout, 6-Uncontrolled junction</td></tr><tr><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">22</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">Weather Conditions-H</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">1-Fine, 2-Mist/Fog\n<break/>3-Cloud, 4-Light Rain, 5-Heavy Rain, 6-Hail/sleet, 7- Snow, 8-Strong Wind, 9-Dust Storm\n<break/>10-Very Hot, 11-Very Cold, 12-Other extraordinary weather condition</td></tr><tr><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">23-26</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">Vehicle Type Involved-J-V1, Vehicle Type Involved-J-V2, Vehicle Type Involved-J-V3, Vehicle Type Involved-J-V4</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">1-Car/Jeep/Van, 2-SUV, 3-Bus, 4-Mini Bus, 5-Truck, 6- Two Wheeler, 7-Three Wheeler, 8-Cycle, 9-Pedestrian, 10- Tractor, 11-Unknown, 12-Animal, 13-Objects, 14-LCV, 15- MAV</td></tr><tr><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">27</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">Number of Vehicles</td><td colspan=\"1\" rowspan=\"1\"/></tr><tr><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">28</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">Number of Casualties-Fatal</td><td colspan=\"1\" rowspan=\"1\"/></tr><tr><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">29</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">Number of Casualties-Grievous Injury</td><td colspan=\"1\" rowspan=\"1\"/></tr><tr><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">30</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">Number of Casualties-Minor Injury</td><td colspan=\"1\" rowspan=\"1\"/></tr><tr><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">31</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">Number of Casualties-Non Injured</td><td colspan=\"1\" rowspan=\"1\"/></tr><tr><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">32</td><td align=\"left\" colspan=\"1\" rowspan=\"1\" valign=\"top\">Number of Casualties</td><td colspan=\"1\" rowspan=\"1\"/></tr></tbody></table></table-wrap>" ]

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[ "<fn-group content-type=\"pub-status\"><fn><p>[version 2; peer review: 1 approved</p></fn></fn-group>" ]

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{ "acronym": [], "definition": [] }

[ "<title>Introduction</title>", "<p id=\"Par5\">Breast cancer is the most common malignancy affecting women worldwide [##REF##30207593##1##]. In the United States, approximately 5% of patients with breast cancer have de novo metastatic disease at diagnosis, and at least 20% of those initially diagnosed with early-stage breast cancer subsequently develop metastatic disease [##REF##18695137##2##]. Despite advances in therapies, the prognosis of patients with metastatic breast cancer (MBC) remains poor, with a median time to progression on first-line therapy of 9.7 months for triple-negative breast cancer (TNBC) and 25.3 months for HR + , HER2 negative breast cancer and a median overall survival (OS) of 23 months for TNBC and 63.9 months for HR + , HER2 negative breast cancer [##REF##29718092##3##–##UREF##1##5##]. Patients with metastatic disease develop cumulative toxicity from multiple lines of chemotherapy, as well as chemotherapy resistance, which limits efficacy. In order to improve survival while maintaining quality of life, it is important to identify new treatment regimens for patients with MBC. New chemotherapy combinations may improve the duration of response (DOR) and also may provide a superior chemotherapy backbone for the addition of targeted agents in future studies.</p>", "<p id=\"Par6\">Eribulin mesylate, a nontaxane microtubule dynamics inhibitor, is a structurally simplified, synthetic analog of the natural product Halichondrin B, isolated from the marine sponge <italic>Halichondria okadai</italic> [##UREF##2##6##, ##UREF##3##7##]. Eribulin suppresses polymerization and sequesters tubulin into nonfunctional aggregates [##UREF##2##6##, ##REF##16020666##8##–##REF##18645010##10##] and has other cytotoxic effects, including vascular remodeling, reversal of the epithelial–mesenchymal transition, induction of the differentiation, and suppression of migration and invasion [##REF##27069131##11##, ##REF##25060424##12##]. Eribulin is administered as a 2- to 5-min IV infusion without the need for premedications, so it is easier to administer than other chemotherapy agents [##UREF##4##13##]. Eribulin was approved as monotherapy for the treatment of taxane and anthracycline-resistant MBC based on results from the EMBRACE trial, which reported a 2.5-month improvement in median OS with eribulin compared to treatment of physicians choice (TPC, 13.1 months vs 10.6 months; HR 0.81, 95% CI 0.66–0.99 p = 0.041) in women with heavily pre-treated MBC [##UREF##5##14##]. Response rate (RR) was significantly longer with eribulin, with a non-significant numerical difference in PFS. In another study, eribulin was compared to capecitabine as an earlier line therapy (up to two lines of chemotherapy) and showed no difference in OS or PFS, but a pooled subset analysis suggested improved OS with eribulin in patients with HER2-negative and TNBC [##UREF##6##15##, ##REF##25381136##16##]. In both studies, treatment-related adverse events (AEs) with eribulin included neuropathy and neutropenia, requiring higher rates of growth factor administration [##REF##26567010##17##–##REF##25381136##19##].</p>", "<p id=\"Par7\">Combination chemotherapy with docetaxel and cyclophosphamide (TC) has become a standard treatment option for early-stage lower risk breast cancer based on data from several studies showing improved or similar outcome compared to anthracycline-based regimens [##REF##19204201##20##–##REF##28398846##22##] Compared to doxorubicin and cyclophosphamide (AC), treatment with TC improved OS in patients with up to three positive axillary nodes [##REF##19204201##20##]. Based on encouraging efficacy with TC, we hypothesized that eribulin combined with cyclophosphamide (EC) would be effective in taxane-resistant disease with tolerable toxicity. The aim of this study was to determine the maximum tolerated dose (MTD) of EC, followed by a dose expansion study to estimate the clinical benefit rate (CBR) of EC in patients with advanced breast cancer (ABC). We also performed correlative studies to assess the correlation of circulating tumor cells (CTCs) with response and survival.</p>" ]

[ "<title>Methods</title>", "<title>Patients</title>", "<p id=\"Par8\">Male or female patients ≥ 18 years who had histologically confirmed locally advanced, unresectable or metastatic carcinoma of the breast of all subtypes were eligible to enroll with no limitation on prior lines of therapy. Eligibility included an Eastern Cooperative Oncology Group performance status (ECOG PS) of 0–2, measurable disease, adequate organ and bone marrow functions (neutrophils &gt; 1.0 × 10⁹/L, platelets &gt; 100 × 10⁹/L, hemoglobin &gt; 9 g/dL, total bilirubin &lt; 1.5 × upper limit of normal (ULN), AST and ALT ≤ 3 × ULN or ≤ 5 × ULN in patients with known liver metastasis, creatinine ≤ 1.5 × ULN or ≥ 60 mL/min for patients with creatinine levels &gt; 1.5 × institutional ULN),  ≤ grade 1 peripheral neuropathy, and a life expectancy of at least 3 months. Patients with stable treated brain metastases were also eligible to enroll. Exclusion criteria included known active central nervous system (CNS) metastases and/or carcinomatous meningitis, a corrected QT interval (cQT) &gt; 480 ms, or significant cardiovascular disease within the past 6 months. The study was approved by the University of California San Francisco (UCSF) Comprehensive Cancer Center Protocol Review Committee on Human Research, and written informed consent was obtained from all patients prior to trial enrollment. The trial was registered at ClinicalTrials.gov (NCT01554371).</p>", "<title>Study design</title>", "<p id=\"Par9\">Patients were treated using a 3 + 3 dose confirmation strategy for eribulin with dose expansion at the MTD. Cyclophosphamide was given at a fıxed dose of 600 mg/m² on day 1 of a 21-day cycle; eribulin was given day 1 and 8 every of a 21 day cycle, and was escalated from 1.1 mg/m² at dose level 0 (DL0) to 1.4 mg/m² at dose level 1 (DL1); both drugs were given intravenously. Dose expansion occurred at DL1. The primary objective of dose escalation was to determine the MTD of EC. The highest dose level at which no more than one of six subjects experienced a dose limiting toxicity (DLT) defined the MTD. DLTs were defined as grade 3 or 4 clinically evident non-hematologic toxicity; grade 4 neutropenia, thrombocytopenia lasting &gt; 7 days, febrile neutropenia or any clinically significant toxicity grade 2 or higher that required more than 14 days to resolve occurring within the first 21 days of combination therapy. The primary objective of the dose expansion was CBR at three months; we selected CBR at 3 months rather than 6 months, as these patients were heavily pre-treated so we anticipated shorter responses to therapy in the later line setting. Secondary objectives were RR, DOR, PFS, safety, and correlation of CTCs with clinical benefit.</p>", "<p id=\"Par10\">Drug doses were modified for treatment-related toxicity. These toxicities were neutropenia (absolute neutrophil count [ANC] &lt; 1000/μL), thrombocytopenia, rash, GI toxicity, liver abnormalities and neuropathy. If toxicity occurred in a patient, dose reductions were managed as follows. Dose levels -1 and -2 for eribulin were 1.1 mg/m² and 0.7 mg/m², for cyclophosphamide one dose reduction was allowed to 500 mg/m². Dose reductions were sequential, with the first dose reduction for cyclophosphamide, followed by eribulin to dose level -1 then -2 for persistent toxicity. If toxicity persisted despite these dose reductions and/or if the participant experienced a cycle delay of three or more weeks, study treatment was discontinued (Supplement 1).</p>", "<title>Concomitant medication</title>", "<p id=\"Par11\">Patients received prophylactic antiemetics and premedications according to standard institutional guidelines. Colony stimulating growth factor use was allowed at the discretion of the treating physician. Palliative radiotherapy was permitted to control bone pain as long as the irradiated area was limited in extent. Other investigational agents and potent inhibitors or inducers of CYP3A4 were not permitted.</p>", "<title>Assessments</title>", "<p id=\"Par12\">Baseline evaluations included medical history, a physical examination, Eastern Cooperative Oncology Group (ECOG) Performance Status (PS) tumor imaging with computed tomography (CT), bone scan, laboratory tests (hematology, blood chemistry), a serum pregnancy test for females of child-bearing potential, and an electrocardiogram with QTc measurement. In the dose confirmation cohort (phase Ib), the response to EC was evaluated every 6 weeks until disease progression according to the investigator, based on objective tumor assessments using RECIST version 1.1 criteria. In the dose-expansion cohort (phase II), response to EC was evaluated after study start, then every 9 weeks until end of study therapy.</p>", "<title>Safety/tolerability</title>", "<p id=\"Par13\">Safety evaluations at baseline and subsequent visits included AEs, clinical laboratory tests, physical examination, and vital signs. AEs were assessed and AE severity was graded in accordance with the National Cancer Institute Common Terminology Criteria for Adverse Events (CTCAE version 4.0). Both agents were held for grade 3 or febrile neutropenia (ANC &lt; 1000/μL). Filgrastim or pegylated-filgrastim myeloid growth factor support was encouraged for ANC &lt; 1500/μL and was allowed at the discretion of the treating physician in order to maintain adequate blood counts. Filgrastim was given for ANC &lt; 1000/μL at any time, or as prophylaxis in patients at risk for neutropenia. Neuropathy was assessed using the 10-point Modified Total Neuropathy Score at the start of each cycle and at study termination.</p>", "<title>Correlative studies</title>", "<p id=\"Par14\">Whole blood samples were obtained in fixative-containing tubes (CellSave tubes, Veridex (currently Menarini)) and processed in the laboratory of Dr. John Park at University of California, San Francisco for CTC identification and enumeration using the CellSearch system. Samples with  5 CTCs per 7.5 mLs of blood were considered CTC-positive.</p>", "<title>Statistical analysis</title>", "<p id=\"Par15\">The study followed a standard dose-confirmation schema (phase Ib portion) with three patients per cohort (3 + 3 design) for a total of six patients. A two-stage design was performed in the dose-expansion (phase II portion) with a possible total of 40 patients. An overall RR of 25% was considered clinically meaningful. Using a two-stage Simon’s minimax design, the null and alternative hypothesis were H0: p0 &lt; 10% versus Ha: p1 &gt; 25% for the proportion of patients with complete or partial response (PR) by RECIST criteria. Based on the type I error of 5% and the type II error rate of 20%, p0 = 10% and p1 = 25%. Secondary efficacy variables were analyzed using Kaplan–Meier methods, with a corresponding median and a 95% CI.</p>" ]

[ "<title>Results</title>", "<title>Patient characteristics</title>", "<p id=\"Par16\">Patients with histologically confirmed metastatic or ABC with any number of prior lines of therapy were eligible to enroll in this study, and 44 patients enrolled in total. Baseline patient characteristics are summarized in Table ##TAB##0##1##. The median age was 56 years (range 33–82 years). 31 patients (70.4%) had HR + /HER2- disease, 12 patients (27.3%) had TNBC, and 1 patient (2.3%) had HR + /HER2 + disease. Patients had a median of 1 prior line of hormone therapy (range 0–6) and 2 prior lines of chemotherapy (range 0–7). Most patients (97.7%) had visceral disease. The most common metastatic sites of disease were bone, lymph nodes, liver, and lung (Table ##TAB##0##1##).</p>", "<title>Antitumor activity</title>", "<p id=\"Par17\">The median duration of treatment was 14.7 weeks (1.8–53.3 weeks). The CBR was 79.5% (35/44; 7 PR, 28 SD). The median PFS was 16.4 weeks (95%CI: 13.8–21.1 weeks). (Figure ##FIG##0##1##a). The median DOR was 16.4 weeks (13.8–21.1 weeks). Clinical response to EC therapy is summarized in Table ##TAB##1##2##. Individual patient characteristics of those who had a PFS 24 weeks on EC are summarized in Table ##TAB##2##3##.</p>", "<p id=\"Par18\">The CBR at 3 months in patients with HR + /HER2- disease was 83.9% (n = 31), and 12.9% of patients had a PR. The median PFS was 18.1 weeks. In the 12 patients with TNBC, the CBR was 66.7% with a PR rate of 25%. The median PFS was 10.8 weeks. As expected, PFS was longer in those with HR + disease versus those with TNBC (18.1 vs 10.8 weeks; p = 0.067) (Fig. ##FIG##0##1##b).</p>", "<p id=\"Par19\">The CBR among patients who had received (neo)adjuvant treatment with an anthracycline and/or a taxane (A/T) was 81.8% (18/22), similar to that of the overall study population. There was no difference in PFS among patients who had received prior A/T (n = 24) versus those who had treatment without A/T (n = 20) (21.1 vs.15.1 weeks respectively, p = 0.4251). There was no difference in PFS among patients who received 0–2 prior lines of chemotherapy versus who received lines of chemotherapy (18.1 vs. 14.7 weeks respectively, p = 0.6736). In patients with HR + disease, patients who had received 0–2 prior lines of chemotherapy (n = 17) had a median PFS of 21.1 weeks whereas patients who had received lines of chemotherapy (n = 15) had a median PFS of 14.3 weeks. In patients with TNBC, patients who had received 0–2 prior lines of chemotherapy (n = 9) had a median PFS of 9.8 weeks whereas patients who had received lines of chemotherapy (n = 3) had a median PFS of 19.1 weeks.</p>", "<title>Drug exposure and safety</title>", "<p id=\"Par20\">No DLTs were identified in the dose confirmation (phase Ib) portion of the study. Three patients were treated at DL0 (eribulin 1.1 mg/m²) and 3 were treated at DL1 (eribulin 1.4 mg/m²), the MTD. Thirty-seven patients (84.1%) completed at least three cycles of treatment and 21 (47.7%) received cycles of treatment. The median number of cycles delivered was 5.8 (1.1–17.8); the median exposure was 3.6 weeks and nine patients received treatment for 6 months or longer (20.5%). Seventeen patients (38.6%) had 1 cyclophosphamide dose reduction and 12 patients (27.3%) had 1 eribulin dose reduction.</p>", "<p id=\"Par21\">Adverse events (AEs) are summarized in Table ##TAB##3##4##. Twenty-eight patients (63.6%) experienced a grade 3/4 AE, the most common of which were neutropenia (47.7%, n = 21), fatigue (4.5%, n = 2), dyspnea (4.5%, n = 2), and anemia (2.3%, n = 1). Febrile neutropenia was reported for three patients (6.8%). Most patients (77.3%) received myeloid growth factors. Treatment related AEs led to dose adjustment (interruption/delay or reduction): 26 patients (59.1%) had a dose interruption/delay and 17 patients (38.6%) underwent dose reduction due to a treatment-related AE. Dose reductions due to neutropenia included a decrease of cyclophosphamide to 500 mg/m² (n = 17) and of eribulin to 1.1 mg/m² (n = 12). Five patients discontinued treatment due to fatigue (n = 3) or neutropenia (n = 2).</p>", "<title>Biomarkers</title>", "<p id=\"Par22\">CTCs in blood were enumerated at baseline and during treatment to explore the correlation between CTC levels and response to EC. Of the 44 evaluable patients, 26 had baseline CTC data. Of these, 14 were CTC-positive (53.8%). There was no significant difference in the mean CTCs/7.5 mLs of blood at baseline between subtypes (t-test p = 0.6435). There was no significant association between CTC status at baseline and treatment response at first scan (Fisher p = 0.5901). 18 of the 26 patients had paired CTC data at baseline and follow-up, 2 of whom were CTC positive at baseline and turned CTC-negative on treatment. The change in CTC status was not significantly associated with response to treatment (Fisher p = 0.3355).</p>", "<p id=\"Par23\">The median PFS was significantly shorter in patients who were CTC-positive at baseline compared to those who were CTC-negative (13.1 vs. 30.6 weeks, p = 0.011). Eighteen patients had on-treatment CTC data available (either during treatment or at the end of study). Of these, nine were CTC-positive (50%). The median PFS was significantly shorter in patients who were CTC-positive during treatment compared to those who were CTC-negative (13.1 weeks vs. 30.6 weeks, p = 0.035).</p>" ]

[ "<title>Discussion</title>", "<p id=\"Par24\">This trial was conducted to assess the safety, efficacy, and tolerability of EC for the treatment of patients with metastatic or ABC. CBR and PFS were 79.5% and 16.4 weeks respectively, comparing favorably to historic data of single agent eribulin for ABC (PFS 14.8 weeks) [##UREF##5##14##]. Our data demonstrate that EC has activity in extensively pretreated patients, as 68.2% of patients who enrolled had been treated with three to seven prior lines of chemotherapy. Of note, responses were observed in patients who had received prior anthracycline and taxane therapies.</p>", "<p id=\"Par25\">Patients with both HR + and TNBC were enrolled. In particular, metastatic TNBC is an aggressive breast cancer subtype associated with poor clinical outcomes highlighting the importance of identifying novel treatment approaches. Previous phase III studies demonstrated statistically significant improvements in OS in patients with metastatic TNBC treated with eribulin versus treatment of physician choice in both subgroup and pooled analyses. Specifically, in a pooled analysis of the EMBRACE study and Study 301, eribulin significantly improved OS compared with TPC in patients with TNBC (HR: 0.74, p = 0.006) [##REF##25381136##16##, ##REF##27398025##23##]. In our study, 12 patients with metastatic TNBC received EC, with a CBR of 66.7% and a PR rate of 25%. The median PFS was 10.8 weeks with EC treatment. Among patients with metastatic TNBC, the heavily pre-treated patients had longer responses to EC than the less heavily-pretreated patients, although numbers are small; future larger studies can evaluate this further.</p>", "<p id=\"Par26\">The adverse events of eribulin in this study is consistent with what has been reported in previous studies [##UREF##5##14##, ##REF##25381136##19##]. In this study, the most frequently reported treatment related AEs were fatigue (68.2%), neutropenia (59.1%), nausea (56.8%), constipation (50%), peripheral neuropathy (47.7%), dyspnea (40.9%), headache (36.4%), and anorexia (36.4%), which reflect the known toxicity profiles of eribulin and cyclophosphamide. The most common grade 3/4 AE was neutropenia (47.7%; 6.8% febrile neutropenia). The incidence of neuropathy was 47.7%, but no patient experienced grade 3/4 neuropathy. Similarly, prior clinical trials also report high incidence of neutropenia and neuropathy with the use of eribulin. In the EMBRACE study, 52% of participants experienced neutropenia (grade 3: 8%, grade 4: 1%) and 35% of participants experienced peripheral neuropathy (grade 3: 8%, grade 4: 0.4%) [##UREF##5##14##]. In real-world studies of patients treated with eribulin grade 3/4 neutropenia occurred in 12% of patients and grade 3/4 neuropathy occurred in 2.6% of patients [##REF##30411979##24##]. Overall, given that the patient population was heavily pre-treated in our study, it was reassuring that the toxicity profile of this combination chemotherapy regimen was similar to those previously reported in single agent studies.</p>", "<p id=\"Par27\">In this study, we administered eribulin using a 21-day cycle at a dose of 1.4 mg/m² on days 1 and 8 combined with cyclophosphamide at a fıxed dose of 600 mg/m² on day 1 of each cycle. Previous studies demonstrated that eribulin was more tolerable when administered on a 21-day schedule compared to a 28-day schedule [##UREF##7##25##]. Alternative schedules of eribulin administration have been investigated to provide better tolerance in patients who experienced myelosuppression. A modified biweekly regimen which provides additional time for bone marrow recovery may potentially improve safety compared with the 21-day dosing regimen [##REF##29435730##26##]. In a prospective phase 2 trial, biweekly eribulin (1.4 mg/m² on days 1 and 15 of a 28-day cycle) was tolerable and had comparable antitumor activity in patients who were intolerant of the standard eribulin schedule [##REF##29435730##26##]. Dose reductions due to neutropenia required patients to decrease to 500 mg/m² cyclophosphamide (n = 17) and to 1.1 mg/m² in eribulin (n = 12), consistent with expected hematologic toxicity in this heavily pre-treated population. Only five patients discontinued treatment due to AEs.</p>", "<p id=\"Par28\">Previous studies have suggested that the presence of CTCs in patients with MBC is associated with a worse prognosis [##REF##15317891##27##, ##REF##32770247##28##] and can predict treatment response and progression [##REF##15735118##29##]. Based on these prior work, we performed an exploratory CTC analysis to address whether CTC levels correlate with response to EC. Consistent with previous studies [##REF##15317891##27##, ##REF##32770247##28##], the CTC positivity rate was 53.8% (14 of 26 patients) at baseline. Median PFS was significantly shorter in patients who were CTC-positive at baseline or during treatment compared to those who were CTC-negative.</p>", "<p id=\"Par29\">This study has several notable strengths. First, few studies have evaluated combination chemotherapy in patients with MBC who have received multiple lines of prior chemotherapy [##UREF##5##14##, ##REF##32223649##30##]. Inclusion of this patient population in our study suggests that the EC regimen may be more generalizable to real-world treatment scenarios. Second, this trial included patients with multiple breast cancer subtypes: most patients had HR + /HER2- breast cancer, with a smaller number of patients with TNBC, and one patient with HR + /HER2 + disease. The RR to EC was higher in patients with HR + /HER2- disease compared to patients with TNBC, as expected, but our study was not powered to fully detect differences between subtypes. Third, the CTC data provides interesting correlative data, and we found that the median PFS was significantly shorter in patients who were CTC-positive at baseline compared to those who were CTC-negative, providing rationale to continue to study the prognostic and predictive value of CTCs in ABC.</p>", "<p id=\"Par30\">This study also has several limitations. First, the primary endpoint of CBR provides important clinical information about response to EC, but this trial is not designed to provide data about OS. Second, since patients were heavily pre-treated, treatment history was fairly heterogenous, which clearly impacts both response to therapy and toxicity. This was a single-arm study so it is not possible to determine the efficacy of this regimen compared to others, and further studies are needed to clarify the efficacy of this regimen. Lastly, our study was conducted before regulatory approval of pembrolizumab plus chemotherapy for PD-L1 + metastatic TNBC, sacituzumab govitecan for metastatic TNBC and heavily pre-treated HR + /HER2- MBC, and trastuzumab deruxtecan for HER2-low MBC. However, as patients being treated with eribulin today will be even more heavily pre-treated than the patients in this trial, the activity of this combination regimen may have implications for current therapeutic options.</p>", "<p id=\"Par31\">In conclusion, the results of this trial demonstrate that EC has antitumoral activity in heavily pretreated patients with locally ABC or MBC. Importantly, EC demonstrated a manageable tolerability profile. These results support the additional clinical development of EC as a novel treatment combination for the treatment of ABC.</p>" ]

[]

[ "<title>Purpose</title>", "<p id=\"Par1\">We hypothesized that eribulin combined with cyclophosphamide (EC) would be an effective combination with tolerable toxicity for the treatment of advanced breast cancer (ABC).</p>", "<title>Methods</title>", "<p id=\"Par2\">Patients with histologically confirmed metastatic or unresectable ABC with any number of prior lines of therapy were eligible to enroll. In the dose escalation cohort, dose level 0 was defined as eribulin 1.1 mg/m² and cyclophosphamide 600 mg/m²_, and dose level 1 was defined as eribulin 1.4 mg/m² and cyclophosphamide 600 mg/m². Eribulin was given on days 1 and 8 and cyclophosphamide on day 1 of a 21-day cycle. In the dose expansion cohort, enrollment was expanded at dose level 1. The primary objective was clinical benefit rate (CBR), and secondary objectives were response rate (RR), duration of response (DOR), progression-free survival (PFS), and safety.</p>", "<title>Results</title>", "<p id=\"Par3\">No dose-limiting toxicities were identified in the dose escalation cohort (n = 6). In the dose expansion cohort, an additional 38 patients were enrolled for a total of 44 patients, including 31 patients (70.4%) with hormone receptor-positive (HR +)/HER2- disease, 12 patients (27.3%) with triple-negative breast cancer (TNBC), and 1 patient (2.3%) with HR + /HER2 + disease. Patients had a median age of 56 years (range 33–82 years), 1 prior line of hormone therapy (range 0–6), and 2 prior lines of chemotherapy (range 0–7). CBR was 79.5% (35/44; 7 partial response, 28 stable disease) and the median DOR was 16.4 weeks (range 13.8–21.1 weeks). Median PFS was 16.4 weeks (95% CI: 13.8–21.1 weeks). The most common grade 3/4 adverse event was neutropenia (47.7%, n = 21). Fourteen of 26 patients (53.8%) with circulating tumor cell (CTC) data were CTC-positive ( 5 CTC/7.5 mL) at baseline. Median PFS was shorter in patients who were CTC-positive vs. negative (13.1 vs 30.6 weeks, p = 0.011).</p>", "<title>Conclusion</title>", "<p id=\"Par4\">In heavily pretreated patients with ABC, treatment with EC resulted in an encouraging CBR of 79.5% and PFS of 16.4 weeks, which compares favorably to single-agent eribulin. Dose reduction and delays were primarily due to neutropenia. The contribution of cyclophosphamide to eribulin remains unclear but warrants further evaluation. NCT01554371.</p>", "<title>Supplementary Information</title>", "<p>The online version contains supplementary material available at 10.1007/s10549-023-07073-0.</p>", "<title>Keywords</title>" ]

[ "<title>Supplementary Information</title>", "<p>Below is the link to the electronic supplementary material.</p>" ]

[ "<title>Funding</title>", "<p>This work was supported in part by funding to the UCSF Regents from Eisai, Inc.</p>", "<title>Data availability</title>", "<p>Enquiries about data availability should be directed to the authors.</p>", "<title>Declarations</title>", "<title>Conflict of interest</title>", "<p id=\"Par32\">Amy Jo Chien: Research funding to institution: Merck, Puma, Amgen, Seagen. Michelle E. Melisko: Research funding to institution: Astra Zeneca, Novartis, KCRN Research, Puma, Seattle Genetics. For spouse: Speaker bureau/honoraria: AstraZeneca, Genentech, Gilead. Stock Ownership: Merrimack. John Park: Consulting/honoraria: AstraZenica, Genentech, Diichi, Gilead. Equity/stock ownership: Merrimack Pharma. Hope S. Rugo: Research support for clinical trials through the University of California: Pfizer, Merck, Novartis, Lilly, Roche, Daiichi, Seattle Genetics, Macrogenics, Sermonix, Boehringer Ingelheim, AstraZeneca, Astellas and Gilead. Honoraria from: Puma, Samsung, Mylan, Chugai, Blueprint, and NAPO. The remaining authors, declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.</p>" ]

[ "<fig id=\"Fig1\"><label>Fig. 1</label><caption><p>Kaplan–Meier plot of progression-free survival (PFS). <bold>a</bold> Shown is the Kaplan–Meier plot for PFS for all patients in the study. The median PFS was 16.4 weeks (95% CI: 13.8–21.1 weeks) in all patients. <bold>b</bold> Shown is the Kaplan–Meier plot for PFS for patients with HR + /HER2- disease (red) and TNBC (blue). The median PFS was 18.1 weeks in patients with HR + /HER2- disease and 10.8 weeks in patients with TNBC</p></caption></fig>" ]

[ "<table-wrap id=\"Tab1\"><label>Table 1</label><caption><p>Patient characteristics at baseline</p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th align=\"left\">Characteristics</th><th align=\"left\">EC (n = 44)</th><th align=\"left\">%</th></tr></thead><tbody><tr><td align=\"left\" colspan=\"3\">Age, years</td></tr><tr><td align=\"left\"> Median (range, yrs)</td><td align=\"left\">56 (33–82)</td><td align=\"left\"/></tr><tr><td align=\"left\" colspan=\"3\">Sex</td></tr><tr><td align=\"left\"> Female</td><td align=\"left\">44</td><td align=\"left\">100</td></tr><tr><td align=\"left\"> Male</td><td align=\"left\">0</td><td align=\"left\">0</td></tr><tr><td align=\"left\" colspan=\"3\">Subtype</td></tr><tr><td align=\"left\"> HR + /HER2-</td><td align=\"left\">31</td><td align=\"left\">70.4</td></tr><tr><td align=\"left\"> TNBC</td><td align=\"left\">12</td><td align=\"left\">27.3</td></tr><tr><td align=\"left\"> HR + /HER2 + </td><td align=\"left\">1</td><td align=\"left\">2.3</td></tr><tr><td align=\"left\" colspan=\"3\">Prior lines of chemotherapy</td></tr><tr><td align=\"left\"> 0</td><td align=\"left\">1</td><td align=\"left\">2.3</td></tr><tr><td align=\"left\"> 1</td><td align=\"left\">15</td><td align=\"left\">34.1</td></tr><tr><td align=\"left\"> 2</td><td align=\"left\">10</td><td align=\"left\">22.7</td></tr><tr><td align=\"left\">  3</td><td align=\"left\">18</td><td align=\"left\">40.9</td></tr><tr><td align=\"left\" colspan=\"3\">Prior lines of endocrine therapy</td></tr><tr><td align=\"left\"> 0–1</td><td align=\"left\">24</td><td align=\"left\">54.5</td></tr><tr><td align=\"left\"> 2</td><td align=\"left\">8</td><td align=\"left\">18.2</td></tr><tr><td align=\"left\">  3</td><td align=\"left\">12</td><td align=\"left\">27.3</td></tr><tr><td align=\"left\" colspan=\"3\">Metastatic disease</td></tr><tr><td align=\"left\"> Bone only disease</td><td align=\"left\">1</td><td align=\"left\">2.3</td></tr><tr><td align=\"left\"> Visceral disease</td><td align=\"left\">8</td><td align=\"left\">18.2</td></tr><tr><td align=\"left\"> Bone and visceral disease</td><td align=\"left\">35</td><td align=\"left\">79.5</td></tr><tr><td align=\"left\" colspan=\"3\">Metastatic sites</td></tr><tr><td align=\"left\"> Bone</td><td align=\"left\">34</td><td align=\"left\">77.3</td></tr><tr><td align=\"left\"> Lymph node</td><td align=\"left\">29</td><td align=\"left\">65.9</td></tr><tr><td align=\"left\"> Liver</td><td align=\"left\">29</td><td align=\"left\">65.9</td></tr><tr><td align=\"left\"> Lung</td><td align=\"left\">20</td><td align=\"left\">45.5</td></tr><tr><td align=\"left\"> Soft tissue involvement</td><td align=\"left\">8</td><td align=\"left\">18.2</td></tr><tr><td align=\"left\"> CNS metastasis</td><td align=\"left\">6</td><td align=\"left\">13.6</td></tr><tr><td align=\"left\"> Peritoneum</td><td align=\"left\">6</td><td align=\"left\">13.6</td></tr></tbody></table></table-wrap>", "<table-wrap id=\"Tab2\"><label>Table 2</label><caption><p>Clinical response to EC therapy</p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th align=\"left\">Response endpoints</th><th align=\"left\">EC (N = 44)</th></tr></thead><tbody><tr><td align=\"left\">CBR</td><td align=\"left\">35 (79.5)</td></tr><tr><td align=\"left\">CR, n (%)</td><td align=\"left\">0 (0)</td></tr><tr><td align=\"left\">PR, n (%)</td><td align=\"left\">7 (15.9)</td></tr><tr><td align=\"left\">SD, n (%)</td><td align=\"left\">28 (63.6)</td></tr><tr><td align=\"left\">PD, n (%)</td><td align=\"left\">6 (13.6)</td></tr><tr><td align=\"left\">n/a, n (%)</td><td align=\"left\">3 (6.8)</td></tr><tr><td align=\"left\">CBR (CR + PR + SD 6 months), n (%)</td><td align=\"left\">9 (20.5)</td></tr><tr><td align=\"left\">PFS, weeks (median) 95% CI</td><td align=\"left\">16.4 (13.8–21.1)</td></tr><tr><td align=\"left\">PFS, weeks for HR + vs TNBC patients</td><td align=\"left\">18.1 vs 10.8 (p = 0.067)</td></tr></tbody></table></table-wrap>", "<table-wrap id=\"Tab3\"><label>Table 3</label><caption><p>Individual patient characteristics whose PFS with EC 24 weeks</p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th align=\"left\">Patient no</th><th align=\"left\">Ph</th><th align=\"left\">Subtype</th><th align=\"left\">Prior lines of ET</th><th align=\"left\">Prior lines of CT</th><th align=\"left\">Metastatic sites</th><th align=\"left\">PFS<break/>(weeks)</th></tr></thead><tbody><tr><td align=\"left\">101</td><td align=\"left\">Ib</td><td align=\"left\">HR + /HER2-</td><td align=\"left\">3</td><td align=\"left\">3</td><td align=\"left\">Bone only</td><td char=\".\" align=\"char\">27.3</td></tr><tr><td align=\"left\">103</td><td align=\"left\">Ib</td><td align=\"left\">HR + /HER2-</td><td align=\"left\">0</td><td align=\"left\">1</td><td align=\"left\">Bone and visceral</td><td char=\".\" align=\"char\">40.7</td></tr><tr><td align=\"left\">107</td><td align=\"left\">Ib</td><td align=\"left\">HR + /HER2-</td><td align=\"left\">2</td><td align=\"left\">1</td><td align=\"left\">Bone and visceral</td><td char=\".\" align=\"char\">34.4</td></tr><tr><td align=\"left\">208</td><td align=\"left\">II</td><td align=\"left\">TNBC</td><td align=\"left\">0</td><td align=\"left\">3</td><td align=\"left\">Bone and visceral</td><td char=\".\" align=\"char\">30.6</td></tr><tr><td align=\"left\">213</td><td align=\"left\">II</td><td align=\"left\">HR + /HER2-</td><td align=\"left\">3</td><td align=\"left\">2</td><td align=\"left\">Bone and visceral</td><td char=\".\" align=\"char\">33.0</td></tr><tr><td align=\"left\">219</td><td align=\"left\">II</td><td align=\"left\">HR + /HER2-</td><td align=\"left\">1</td><td align=\"left\">3</td><td align=\"left\">Bone and visceral</td><td char=\".\" align=\"char\">53.3</td></tr><tr><td align=\"left\">228</td><td align=\"left\">II</td><td align=\"left\">HR + /HER2-</td><td align=\"left\">1</td><td align=\"left\">2</td><td align=\"left\">Bone and visceral</td><td char=\".\" align=\"char\">45.0</td></tr><tr><td align=\"left\">235</td><td align=\"left\">II</td><td align=\"left\">TNBC</td><td align=\"left\">0</td><td align=\"left\">2</td><td align=\"left\">Bone and visceral</td><td char=\".\" align=\"char\">28.6</td></tr><tr><td align=\"left\">240</td><td align=\"left\">II</td><td align=\"left\">HR + /HER2-</td><td align=\"left\">2</td><td align=\"left\">2</td><td align=\"left\">Bone and visceral</td><td char=\".\" align=\"char\">29.4</td></tr></tbody></table></table-wrap>", "<table-wrap id=\"Tab4\"><label>Table 4</label><caption><p>Adverse events</p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th align=\"left\">Adverse event</th><th align=\"left\">Any grade, n (%)</th><th align=\"left\">Grade 1–2, n (%)</th><th align=\"left\">Grade 3–4, n (%)</th></tr></thead><tbody><tr><td align=\"left\">Fatigue</td><td char=\"(\" align=\"char\">30 (68.2)</td><td char=\"(\" align=\"char\">28 (63.6)</td><td char=\"(\" align=\"char\">2 (4.5)</td></tr><tr><td align=\"left\">Neutrophil count decrease</td><td char=\"(\" align=\"char\">26 (59.1)</td><td char=\"(\" align=\"char\">5 (11.4)</td><td char=\"(\" align=\"char\">21 (47.7)</td></tr><tr><td align=\"left\">Nausea</td><td char=\"(\" align=\"char\">25 (56.8)</td><td char=\"(\" align=\"char\">25 (56.8)</td><td char=\"(\" align=\"char\">0 (0)</td></tr><tr><td align=\"left\">Constipation</td><td char=\"(\" align=\"char\">22 (50)</td><td char=\"(\" align=\"char\">22 (50)</td><td char=\"(\" align=\"char\">0 (0)</td></tr><tr><td align=\"left\">Peripheral neuropathy</td><td char=\"(\" align=\"char\">21 (47.7)</td><td char=\"(\" align=\"char\">21 (47.7)</td><td char=\"(\" align=\"char\">0 (0)</td></tr><tr><td align=\"left\">Dyspnea</td><td char=\"(\" align=\"char\">18 (40.9)</td><td char=\"(\" align=\"char\">16 (36.4)</td><td char=\"(\" align=\"char\">2 (4.5)</td></tr><tr><td align=\"left\">Headache</td><td char=\"(\" align=\"char\">16 (36.4)</td><td char=\"(\" align=\"char\">16 (36.4)</td><td char=\"(\" align=\"char\">0 (0)</td></tr><tr><td align=\"left\">Anorexia</td><td char=\"(\" align=\"char\">16 (36.4)</td><td char=\"(\" align=\"char\">16 (36.4)</td><td char=\"(\" align=\"char\">0 (0)</td></tr><tr><td align=\"left\">Anemia</td><td char=\"(\" align=\"char\">15 (34.1)</td><td char=\"(\" align=\"char\">14 (31.8)</td><td char=\"(\" align=\"char\">1 (2.3)</td></tr><tr><td align=\"left\">Alopecia</td><td char=\"(\" align=\"char\">14 (31.8)</td><td char=\"(\" align=\"char\">14 (31.8)</td><td char=\"(\" align=\"char\">0 (0)</td></tr><tr><td align=\"left\">Arthralgia</td><td char=\"(\" align=\"char\">14 (31.8)</td><td char=\"(\" align=\"char\">14 (31.8)</td><td char=\"(\" align=\"char\">0 (0)</td></tr><tr><td align=\"left\">Febrile neutropenia</td><td char=\"(\" align=\"char\">3 (6.8)</td><td char=\"(\" align=\"char\">1 (2.3)</td><td char=\"(\" align=\"char\">2 (4.5)</td></tr></tbody></table></table-wrap>" ]

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[ "<table-wrap-foot><p><italic>CNS</italic> central nervous system, <italic>ECOG</italic> Eastern Cooperative Oncology Group</p></table-wrap-foot>", "<table-wrap-foot><p><italic>CBR</italic> clinical benefit rate, <italic>CI</italic> confidence interval, <italic>CR</italic> complete response, <italic>n</italic> number, <italic>PD</italic> progressive disease, <italic>PR</italic> partial response, <italic>PFS</italic>, progression-free survival, <italic>SD</italic> stable disease</p></table-wrap-foot>", "<table-wrap-foot><p><italic>CT</italic> chemotherapy, <italic>ET</italic> endocrine therapy, <italic>PFS</italic> progression-free survival</p></table-wrap-foot>", "<table-wrap-foot><p><italic>n</italic> number</p></table-wrap-foot>", "<fn-group><fn><p><bold>Publisher's Note</bold></p><p>Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.</p></fn><fn><p>Ozge Gumusay and Laura A. Huppert are co-first authors.</p></fn></fn-group>" ]

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[ "<title>Introduction</title>", "<p id=\"Par2\">Many studies support the notion that when parents are consistent in their responses to their child’s misbehavior, children show less externalizing behavior (Barry et al., ##UREF##3##2009##; Gardner, ##REF##2745902##1989##; Gryczkowski et al., ##UREF##8##2010##; Halgunseth et al., ##REF##23544924##2013##; Lengua &amp; Kovacs, ##UREF##10##2005##). However, these studies have focused on global self-reports, which mostly confound consistency of parental reactions across multiple episodes of misbehavior (e.g., punishing misbehavior in one episode, and condoning it in another) with consistency within a single episode of misbehavior (e.g., threatening with punishment, but leaving it in the end). Additionally, observational studies have mostly focused on within-episode consistency (Del Vecchio &amp; O’Leary, ##REF##16597215##2006##; Gardner, ##REF##2745902##1989##). Therefore, it is not known whether within- and across-episode consistency play similar roles in the early development of externalizing problems. In this study, we use a daily diary approach to examine parental consistency both within and across episodes, offering a unique possibility to differentiate these two types of consistency. We examine associations between these aspects of consistency, and how they are associated with child externalizing behavior, both concurrently and longitudinally.</p>", "<title>Differentiating Within- from Across Disruptive Behavior Episode Consistency</title>", "<p id=\"Par3\">Empirically, studies using questionnaire measures of inconsistent discipline have indeed found it to be associated with more externalizing behavior in children (Barry et al., ##UREF##3##2009##; Gryczkowski et al., ##UREF##8##2010##; Halgunseth et al., ##REF##23544924##2013##; Lengua &amp; Kovacs, ##UREF##10##2005##). However, research to date has not separated within-episode consistency from across-episode consistency. For instance, the frequently used Alabama Parenting Questionnaire (Frick et al., ##UREF##7##1999##) includes items in the inconsistency scale asking parents whether they threaten to punish, but then do not do so in the end, or whether they let their child out of punishment early, which are examples of inconsistent responding <italic>within</italic> a single episode of misbehavior. Additionally, items are included asking parents whether their punishment depends on their mood, which concerns consistency <italic>across</italic> episodes of misbehavior. It is important to note that these types of inconsistency do not necessarily co-occur. Parents may be relatively inconsistent in their responses within a single episode, perhaps not feeling competent enough to follow through with an initial course of action and giving up along the way (Deković et al., ##REF##20101464##2010##), yet they may be consistent in this response style across multiple episodes of misbehavior. Alternatively, parents may be consistent within an episode of misbehavior by providing a negative consequence and sticking to it, but respond in different ways to new episodes, for instance switching to ignoring the misbehavior or trying to redirect attention by making a joke. They may switch approaches because they are in a different mood state themselves (Rueger et al., ##UREF##13##2011##), or because they felt their approach was not effective in a previous instance.</p>", "<p id=\"Par4\">Several theoretical frameworks would predict that both types of consistency would be associated with increased externalizing problems. Attachment theory postulates that children form secure attachments to their caregivers if these are consistently responsive to their needs. With consistency, children learn that they can trust their caregiver to provide them with a secure base and a safe haven (Ainsworth et al., ##UREF##1##2015##). With inconsistency in caregiving, in contrast, children learn that their environment is unpredictable and insecure, increasing the risk of attachment problems and problem behavior (Madigan et al., ##REF##26619212##2016##). Supporting this notion, unpredictable behavior from parents has been shown to impact the stress response in both very young children (Noroña-Zhou et al., ##REF##32115696##2020##), as well as older children (Manczak et al., ##REF##28625195##2018##). A dysregulated stress system may in turn result in problems with self-regulation, resulting in heightened disruptive behavior (Wesarg et al., ##UREF##17##2020##). Additionally, unpredictability likely hampers children’s ability to develop a sense of self-efficacy, because it prevents them from developing a sense of control over situations (Bandura, ##UREF##2##1978##; Lippold et al., ##REF##28736495##2016##). Low self-efficacy, in turn, makes it more difficult to regulate anger and frustration, resulting in increased levels of disruptive behavior.</p>", "<p id=\"Par5\">Social learning theories emphasize operant conditioning principles which predict that within-episode inconsistency would be associated with increased problematic behavior because failing to follow-through with negative consequences reinforces the child for showing the misbehavior (Patterson, ##UREF##12##1982##). Additionally, consistent negative consequences across different disruptive behavior episodes, or at least a lack of reward, would quickly result in extinction of problematic behavior, whereas inconsistent discipline – with intermittent patterns of positive or negative reinforcement – makes it difficult for children to learn that their behavior is not acceptable. Both social learning and attachment frameworks would thus predict that within- as well as across-episode consistency would play a role in the maintenance of disruptive behavior.</p>", "<p id=\"Par6\">Although most questionnaire studies have confounded within- and across-episode consistency, some studies have specifically investigated within-episode consistency. Using observations of parent-child interactions, parents with children who were high on externalizing behavior were less likely to follow-through on an initial demand than parents of children who were low on externalizing behavior (Gardner, ##REF##2745902##1989##). Additionally, mothers of aggressive toddlers have been observed to be more likely to react with both overreactivity and laxness to instances of child aggression than mothers of non-aggressive toddlers (Del Vecchio &amp; O’Leary, ##REF##16597215##2006##). However, with observations it is more difficult to assess how this within-episode consistency relates to across episode consistency. As observations are already so time-consuming, observing (enough) episodes of misbehavior to examine across episode consistency would be especially difficult.</p>", "<title>Using Daily Diaries to Assess Parental Consistency</title>", "<p id=\"Par7\">Daily diary assessments allow for overcoming the abovementioned limitations of single-time surveys or observations. Single-time self-reports of parental consistency may be biased, as correctly judging how consistent one is in their reactions might be even more difficult than broadly gauging whether one frequently reacts in a certain way (Lippold et al., ##REF##28736495##2016##). A more valid way to assess parental consistency may therefore be to repeatedly ask parents about their actual behavior, and analyze the consistency across their responses. A small scale study examined parents’ daily fluctuations in their overreactive and lax discipline, with lax discipline indicating inconsistency by not following through. Although overreactivity and laxness were positively correlated at the between-person level, indicating that mothers who were generally more overreactive were also more lax, there was a negative association at the within-person level, indicating that when mothers were overreactive in a certain instance, they were less likely to be lax at that time, and vice versa (Passini et al., ##REF##23458698##2013##). These results indicate that mothers can be inconsistent across episodes with regards to their within-episode consistency. However, associations with child externalizing behavior were not examined here. Another diary study did examine associations with externalizing behavior and found that for mothers with 5- 8-year old children maternal consistency across one week was associated with less child externalizing behavior (Villarreal et al., ##UREF##16##2021##). Consistency was operationalized as the within-person fluctuations in destructive conflict characteristics, which included both maternal punitive behavior as well as child and mother negativity, making it unclear whether this was an association with inconsistency in maternal behavior specifically. Yet another study examined daily reports of parents harsh and warm reactions to child misbehavior and found that consistency in the level of warmth was associated with less child ADHD symptoms, whereas consistency in the level of harshness was not (Li &amp; Lansford, ##REF##29608072##2018##). As these studies operationalized inconsistency as fluctuations in mean levels of parenting behaviors, we do not know whether parents exhibited different types of responses in the same instance of misbehavior. For instance, some parents may be more likely to only punish the child by taking away a privilege, whereas other parents are more likely to yell at the child, take away a privilege and also comfort the child, with the latter type of response perhaps especially confusing to the child. A previous observational study has indeed found that some parents were more likely to use both positive and negative discipline strategies within the same episode than others, but did not examine associations between this inconsistency and externalizing behavior (van Zeijl et al., ##UREF##15##2007##).</p>", "<title>The Present Study</title>", "<p id=\"Par8\">In the present study, we use daily diary data to differentiate consistent responding within a single episode of misbehavior from consistent responding across multiple episodes of misbehavior and examine how they are each associated with the severity of disruptive behavior in children. We assess within-episode consistency as the mean number of different reactions to a specific episode of misbehavior, distinguishing positive attention, positive consequence, negative consequences, negative attention, and ignoring. Across-episode consistency is assessed as the overall dispersion of mothers’ reactions across all possible categories, taking into account the total number of episodes of misbehavior across a week. Parental reactions that were concentrated in fewer reaction categories were indicative of more consistency.</p>", "<p id=\"Par9\">We examine associations between the two types of consistency in two independent samples: a community sample (<italic>N</italic> = 134) of mothers of 1.5 to 3.5 year old children who completed a daily diary for 7 days, and an at-risk sample with heightened disruptive behavior (<italic>N</italic> = 149) of 3 to 8 year old children who filled out a daily diary for 14 days. Including these two samples has several benefits. First, we are able to examine whether associations between the two types of consistency conceptually replicate across multiple samples. Second, we can investigate whether both types of consistency are equally associated with disruptive behavior during a developmental stage when disruptive behavior starts to emerge and is relatively more normative, as during a developmental stage when disruptive behavior for most children has started to decline (Tremblay, ##REF##20146751##2010##) as a result of increases in children’s verbal-skills and overall self-regulation (Kuhn et al., ##REF##27101154##2016##). This allows us to investigate whether the two types of consistency play a similar role in the early emergence as in the maintenance of more persistent problem behavior across development. Contemporary accounts of social learning theories would predict that across-episode consistency may be less relevant for problem behavior that persists into preschool age, as repeated coercive cycling in parent-child dyads is thought to result in increasingly rigid, mutually negative interactions over time (Granic &amp; Patterson, ##REF##16478303##2006##).</p>", "<p id=\"Par10\">We also examine the added value of computing these types of consistency from daily diary data over a single questionnaire assessment. To this end, we assessed whether it is a better predictor of children’s externalizing behavior one year later than a measure of consistency derived from a general questionnaire as administered in a single – less time-consuming baseline assessment – asking parents to estimate how often in the past month they showed the reactions that we also included in the diary study. Although this ‘general consistency’ measure confounds within- and across-episode consistency, it may still be a better measure of consistency than some of the current measures that are used. Rather than asking parents to report on how consistent they are, we merely asked parents to indicate how often they reacted a certain way, and compute consistency by calculating the dispersion of parents’ responses across the different reaction categories. This approach makes it less likely that this association is for instance explained by parents scoring themselves as inconsistent due to a more negative self-view (Smit et al., ##UREF##14##2021##). This likely plays a role in more traditional questionnaire measures, as most parents will realize that threatening with punishment and then not following through is not an effective parenting strategy. Our approach will allow us to examine whether taking multiple days of measurements to assess consistency is really necessary, or whether we have enough information when we just ask parents how often they react a certain way overall. Additionally, associations between the general consistency and within- and across-episode consistency can be examined to provide an indication of the validity of these measures.</p>" ]

[ "<title>Method</title>", "<title>Sample</title>", "<p id=\"Par11\">We included two samples, to allow for conceptual replication: Sample 1 is a community sample of 134 mothers of 1.5–3.5 year old children (<italic>M</italic> = 30 months, 44.3% girls), who reported on their child’s temper tantrums (frequency and severity) and their responses to these tantrums – in general across the past month and daily for 7 days. Mothers were predominantly, but not exclusively, highly educated (79% higher vocational or university education), and were not selected for experiencing any particular difficulties with their child. Seven percent indicated that they raised their child without a partner. No information regarding ethnicity was collected for this sample.</p>", "<p id=\"Par12\">Between February 2016 and June 2017, undergraduate students recruited mothers with children between one and five years old for a research practical. They recruited mothers through online parenting fora and Facebook, and face-to-face outside in Amsterdam. Mothers who participated were also asked to forward the invitation for the study to other mothers. Mothers were informed about the study and gave informed consent in the online study environment. They filled out the general questionnaire regarding: children’s temper tantrums and their own reactions, their personality and sense of parenting competence (<italic>N</italic> = 884). Mothers who indicated that their child was between 1.5 and 3.5 years old were asked if they would like to participate in an additional daily diary study, and <italic>N</italic> = 382 indicated that they would like to receive more information. They were contacted by telephone, with <italic>N</italic> = 220 eventually participating. For this study, we only selected participants if they had participated in at least 4 days of the study (<italic>N</italic> = 185), and who had reported reactions for at least two tantrums, resulting in a final sample of <italic>N</italic> = 134. Participants completed an average of 6.76 days (<italic>SD</italic> = 0.62, range = 4–7 days). For this sample, mothers additionally reported on their child’s externalizing behavior one year later (<italic>n</italic> = 86). Participants who dropped out of the study did not differ significantly from those who participated one year later with regards to age of the mother (<italic>T</italic>(131) = 1.66, <italic>p</italic> = 0.062) or child (<italic>T</italic>(132) = -1.65, <italic>p</italic> = 0.050), the child’s sex (<italic>χ</italic>2(1) = 0.29, <italic>p</italic> = 0.589) or mothers’ educational level (<italic>χ</italic>2(4) = 0.74, <italic>p</italic> = 0.947). Additionally, there were no significant differences in children’s tantrum severity (<italic>T</italic>(132) = 0.90, <italic>p</italic> = 185) and mothers’ within-, across-, or general consistency at T1 (<italic>T</italic>(132) = -0.10,<italic> p</italic> = 0.922; <italic>T</italic>(132) = -0.09, <italic>p</italic> = 0.930; <italic>T</italic>(117) = -0.21, <italic>p</italic> = 0.833, respectively).</p>", "<p id=\"Par13\">Mothers who filled out the general questionnaire had a chance of winning a gift certificate of 50 euros. Mothers who participated at least four days of the diary study received a small gift (a small book for their child – 2 euros) by mail. The study was approved by the ethical review board of the Department of Child Development and Education at the University of Amsterdam (#2015-CDE-6367).</p>", "<p id=\"Par14\">Sample 2 consists of 149 parents (94% mothers) of 3–8 year old children (<italic>M</italic> = 5.88; 46% girls) oversampled for disruptive behavior – 17% had received parenting support for disruptive child behavior prior to the study; seven percent still received support during the study. Parents reported on how they generally responded to their child’s disruptive behavior as well as their daily responses for 14 days. Parents were predominantly, but not exclusively, highly educated (78% higher vocational or university education). Ten percent indicated that they raised their child without a partner. Culturally, 93% identified as Dutch, of which 19% identified as bicultural (mainly other European cultures or Moroccan). Others self-identified as Moroccan, other European, Asian, Surinamese, or Turkish. This roughly represents the Dutch population where around 25% of families has at least parts of their roots outside the Netherlands, most often in Turkey, Morocco and Surinam (Centraal Bureau voor de Statistiek [CBS], ##UREF##4##2020##).</p>", "<p id=\"Par15\">Parents were recruited between March 2020 and June 2021, through social media, primary schools across the Netherlands, and databases from the University of Amsterdam of parents who consented to be contacted for research projects. Children with disruptive behavior problems were oversampled by advertising the study as targeting parents of children with mild to moderate levels of disruptive behavior. Parents who signed up were contacted by phone to explain the study procedures. Parents who agreed to participate signed informed consent, completed a baseline assessment (i.e., demographics and trait measures) with a link to daily online daily questionnaire (<italic>N</italic> = 156). For this study, we only selected participants if they had participated for at least 8 days of the study, and who had reported reactions to at least two disruptive behaviors, resulting in a final sample of <italic>N</italic> = 149. Participants completed an average of 13.22 days (<italic>SD</italic> = 1.34, range = 8–14 days).</p>", "<p id=\"Par16\">Parents received €50 for completing the study. Study procedures were approved by the Ethical Review Board of the department of Child Development and Education of the University of Amsterdam (2019-CDE-11055).</p>", "<title>Measures</title>", "<title>Parental Consistency</title>", "<p id=\"Par17\">In Sample 1, parents reported on how they responded to their child’s tantrums both in general over the past month – before they started the diary study, and for each tantrum that took place during the diary study (for a maximum of seven tantrums a day). Parents rated their responses from a list of 11 behaviors. We made a functional classification based on social learning principles (Patterson, ##UREF##12##1982##), differentiating punishment and reward from lack of punishment or reward, and positive and negative attention: negative consequence (2 items: ‘I sent my child to their room/corner/time-out’, ‘I punished my child’), positive consequence (‘I negotiated with my child’, ‘I gave in to my child’), withholding attention (‘I didn’t, I let my child cool off’, ‘I ignored my child’), negative attention (‘I became angry with my child’, ‘I grabbed my child’, ‘I spoke sternly to my child’), positive attention (‘I distracted my child’, ‘I comforted my child’). In the questionnaire about responses to tantrums in general, participants indicated how often they tended to respond that way (1 = <italic>never</italic>; 2 = <italic>almost never</italic>; 3 = &lt; <italic>half the time</italic>; 4 = <italic>about half the time</italic>; 5 = &gt; <italic>half the time</italic>; 6 = <italic>almost always</italic>; 7 = <italic>always</italic>), and we computed a mean score per category. In the daily diaries parents indicated whether or not they responded that way in that particular instance (0 = <italic>no</italic>; 1 = <italic>yes</italic>), allowing for multiple responses. Parents received a score of 1 in a category when answered yes to at least one of the responses in that category.</p>", "<p id=\"Par18\">For our measure of across-episode consistency from the daily diary data and the general consistency measure from the baseline questionnaire, we calculated the Index of Qualitative Variation (IQV), which is a measure of variation for nominal variables – where a standard deviation cannot be computed due to qualitative rather than quantitative differences between categories, using the following formula (Frankfort-Nachmias &amp; Leon-Guerrero, ##UREF##6##2018##):</p>", "<p id=\"Par19\">For each response category we first computed what proportions of the total number of responses they were for each individual. The squared proportions of each of the categories are summed and then subtracted from 1 and multiplied by K, which is the number of categories (5 in our study). This is then divided by the number of categories minus 1. The resulting value can range from 0 to 1.00, with higher scores indicating greater inconsistency. Therefore, we subtracted this value from 1, so that higher scores indicated greater consistency.</p>", "<p id=\"Par20\">From the diary data, we additionally computed a measure of within-episode consistency. For each tantrum, we summed the total number of responses in the different categories (potential range 1–5), and then computed a mean score across all tantrums that were reported during the study. The observed range was 1–3 with higher values indicating less consistency. For ease of interpretation we recoded this variable so that higher values indicated greater consistency by subtracting the values from 3. The final variable thus ranged from 0–2. The intraclass correlation coefficient (ICC) for within-episode consistency was 0.20. This value is similar to ICCs that have previously been reported for parenting variables in diary studies, such as psychological control and autonomy support (Mabbe et al., ##UREF##11##2018##), with somewhat higher levels of around 0.33 also reported for psychological control (Aunola et al., ##REF##23750527##2013##).</p>", "<p id=\"Par21\">In Sample 2, parents were asked how they responded to their child’s disruptive behavior in general at the start of the study, and how they responded to their child’s most challenging disruptive behavior that particular day (if any) in the diary study. Parents rated their responses from a list of 13 behaviors, which we grouped into the same five categories as for Sample 1: negative consequence (4 items: ‘I sent my child to their room for at least an hour’, ‘I gave my child a short time-out, away from others’, ‘I took something nice away from my child (e.g., toys or screen time)’, I gave my child extra chores (e.g., set the table)’), positive consequence (2 items: ‘I gave my child his/her way’, ‘I gave in to my child’), withholding attention (2 items: ‘I did nothing’, ‘I talked about it with my child afterwards’), negative attention (3 items: ‘I yelled/swore’, ‘I said things I didn’t mean’, ‘I threatened with punishment, but did not punish’), positive attention (2 items: ‘I begged my child to stop’, ‘I used humor to distract my child’). In the general questionnaire, participants indicated how often they tended to respond that way on a 5-point Likert type scale (1 = <italic>less than once a week,</italic> 2 = <italic>once a week</italic>; 3 = <italic>few times a week</italic>; 4 = <italic>once a day</italic>; 5 = <italic>several times a day</italic>), and we computed a mean score per category. In the daily diaries study parents indicated whether or not they responded that way in that particular instance (0 = <italic>no</italic>, 1 = <italic>yes</italic>), allowing for multiple responses. Parents received a score of 1 in a category when answered yes to at least one of the responses in that category. When parents reported that their child had not shown any disruptive behavior that day, the response category was coded as missing.</p>", "<p id=\"Par22\">Like in Sample 1, we computed the IQV as a measure of across-episode consistency from the daily diary data and a measure of general consistency from the baseline questionnaire, and the mean number of selected categories of responses per episode as our measure of within-episode consistency. For within-episode consistency, the ICC was 0.18.</p>", "<title>Child Externalizing Behavior</title>", "<p id=\"Par23\">In Sample 1, we calculated a measure of severity of the child’s tantrum behavior from the daily diary reports, by summing for each tantrum the total number of aggressive (hitting, kicking, biting, throwing an object, pushing/pulling, spitting, grabbing) and self-injurious behaviors (banging head, holding breath, freezing). A previous study on this sample found that a profile with elevated levels on these behaviors was predictive of both internalizing and externalizing problems above and beyond tantrum frequency and duration (Van den Akker et al., ##REF##35316228##2022##). The ICC for tantrum severity was 0.24.</p>", "<p id=\"Par24\">One year later (T2), parents in Sample 1 filled out 24 items of the Externalizing Problem Behavior Scale (the attention problem and aggressive behavior problem subscales, e.g., “My child does not seem to feel guilty after misbehavior”) of the Dutch version of the Child Behavior Checklist (1, 5–5) (Achenbach &amp; Rescorla, ##UREF##0##2000##). Parents were instructed to indicate for the past 2 months how characteristic the item was of their child's behavior, with each item rated as 0 (<italic>not true</italic>), 1 (<italic>sometimes/somewhat true</italic>), or 2 (<italic>often/very true</italic>). Cronbach's alpha for the present sample was 0.89.</p>", "<p id=\"Par25\">For Sample 2, rather than indicating how many disruptive behaviors children had displayed, parents rated children’s overall level of disruptive child behavior at T1 each day (i.e., “how disruptive was your child’s behavior today?”) on a 1 − 10 scale. A mean score across the 14 days was computed. The ICC was 0.32.</p>", "<title>Analysis Plan</title>", "<p id=\"Par26\">Hypotheses and analyses were registered on the Open science Framework (<ext-link ext-link-type=\"uri\" xlink:href=\"https://osf.io/tecr4/?view_only=0c6f1e3d6b2f46e49c5599c4c168be3c\">https://osf.io/tecr4/?view_only=0c6f1e3d6b2f46e49c5599c4c168be3c</ext-link>).</p>", "<p id=\"Par27\">We first winsorized outliers (outside 1.5* IQR) to the nearest value if there was a gap in data between that range and the outlier. In Sample 1, for the within-episode consistency measures as derived from the daily diaries we identified two outliers, and for the across-episode consistency derived from the questionnaire asking about tantrums in general, we identified one outlier. For the severity of daily disruptive behavior we identified six outliers, and for externalizing behavior we identified two outliers. In Sample 2, for the within-episode consistency measure as derived from the daily diaries we identified one outlier, and for the across-episode consistency we identified four outlier. For the severity of daily disruptive behavior we identified four outliers. To answer our first research question- whether our measures of within- and across-episode consistency measure different but related aspects of consistency – we computed correlations. Next, we performed regression analyses to predict the severity of daily disruptive behavior from the within- and across-episode consistency measures to see whether they were uniquely associated. In a next step, we examined whether associations were significant above and beyond mean levels of the daily parental reactions. These analyses control for child sex and age and parental educational level and are performed on both Samples 1 and 2. As 11 parents in Sample 2 received parenting support for their child’s behavior, we also controlled for received support in Sample 2. Finally, we performed regression analysis in SPSS (version 28) to examine – in Sample 1 – whether within- and across- episode consistency as derived from the daily diary reports longitudinally predict child externalizing behavior problems one year later (T2), over and above a measure of consistency derived from estimates of parental behavior across the past month, controlling for the severity of temper tantrum behavior as reported in the diary study at T1.</p>" ]

[ "<title>Results</title>", "<p id=\"Par28\">On average, children in Sample 1 had an average 5.90 tantrums during the 7-day period (<italic>SD</italic> = 4.21, range 2–20), and for children in Sample 2 the mean level of disruptive behavior was rated 3.32 on the 10 point scale across the 14 days (<italic>SD</italic> = 1.21, range 1.14–6.54). Descriptives and intercorrelations for Samples 1 and 2 are provided in Table ##TAB##0##1##. In both samples, within- and across episode consistently were significantly associated. Associations were strong, but not so strong as to indicate that they would actually be measuring the same thing. In Sample 1, only across-episode consistency was negatively associated with disruptive behavior severity; in Sample 2, both within- and across-episode consistency were negatively associated with disruptive behavior severity.</p>", "<title>Within- and Across-episode Consistency and Severity of Child Disruptive Behavior</title>", "<p id=\"Par29\">To examine whether within- and across-episode consistency were uniquely associated with the severity of daily disruptive behavior, we performed regression analyses. In Sample 1, the first step, controlling for age and sex of the child and educational level of the parent was not significant (<italic>F</italic>(3,130) = 0.47, <italic>p</italic> = 0.707, <italic>R</italic>^{<italic>2</italic>} = 0.01). Adding across-episode and within-episode consistency resulted in a significant improvement of the model (Δ<italic>F</italic>(2,128) = 6.95, <italic>p</italic> = 0.001, Δ<italic>R</italic>^{<italic>2</italic>} = 0.10): when parents were more consistent across disruptive behavior episodes, children displayed less severe disruptive behavior, whereas within-episode consistency was not significantly associated with severity of daily disruptive behavior (Table ##TAB##1##2##).</p>", "<p id=\"Par30\">Results of Sample 2 conceptually replicated the findings of Sample 1. The first step, controlling for sex of the child and educational level of the parent was not significant (<italic>F</italic>(3,144) = 0.32, <italic>p</italic> = 0.808, <italic>R</italic>^{<italic>2</italic>} = 0.01). Adding across-episode and within-episode consistency resulted in a significant improvement of the model (Δ<italic>F</italic>(2,142) = 9.66, <italic>p</italic> &lt; 0.001, Δ<italic>R</italic>^{<italic>2</italic>} = 0.12): only across-episode consistency, not within-episode consistency, was significantly associated with severity of daily disruptive behavior (Table ##TAB##2##3##).</p>", "<p id=\"Par31\">In a next set of regression analyses, we examined whether within- and across-episode consistency predicted the severity of daily disruptive behavior, above and beyond mean levels of the different response categories. In Sample 1, adding the mean levels of the proportions of the five parental responses across the seven days did not result in a significant improvement over the model including only age and sex of the child and educational level of the parent (Δ<italic>F</italic>(5,125) = 0.47, <italic>p</italic> = 0.801, Δ<italic>R</italic>^{<italic>2</italic>} = 0.02), indicating that how much parents displayed a certain type of reaction was not predictive of the child’s disruptive behavior. Adding within- and across-episode consistency to the model did result in a significant improvement (Δ<italic>F</italic>(2,123) = 8.63, <italic>p</italic> &lt; 0.001, Δ<italic>R</italic>^{<italic>2</italic>} = 0.12). Across-episode consistency was predictive of daily disruptive behavior severity, whereas within-episode consistency was not. For model coefficients, see Table ##TAB##1##2##.</p>", "<p id=\"Par32\">Different from Sample 1, in Sample 2, adding the mean levels of the proportions of the five parental responses did result in a significant improvement over the model including only sex and educational level of the parent (Δ<italic>F</italic>(5,139) = 6.21, <italic>p</italic> &lt; 0.001, Δ<italic>R</italic>^{<italic>2</italic>} = 0.18). Providing negative consequences and giving negative attention to disruptive behavior, were each associated with more severe daily disruptive behavior. Here, adding within- and across-episode consistency to the model did not result in a significant improvement (Δ<italic>F</italic>(2,137) = 0.55, <italic>p</italic> = 0.581, Δ<italic>R</italic>^{<italic>2</italic>} = 0.01), indicating that the association between consistency and child disruptive behavior was explained by the individual negative responses. For model coefficients, see Table ##TAB##2##3##.</p>", "<title>Prediction of Externalizing Problems One Year Later</title>", "<p id=\"Par33\">In Sample 1, we examined whether the consistency measures predicted externalizing behavior one year later, controlling for the severity of daily disruptive behavior at T1 and for parent-reported consistency as derived from a one-time questionnaire about general responses to tantrums. The first step was significant (<italic>F</italic>(4,73) = 5.01, <italic>p</italic> = 0.001, Δ<italic>R</italic>^{<italic>2</italic>} = 0.22): more severe daily disruptive behavior was predictive of more externalizing behavior one year later. Above and beyond this effect, less consistency as derived from parents’ reports of how frequently they generally displayed certain responses to their child’s tantrums (i.e. general consistency as computed from the baseline measure), was predictive of more externalizing problems. Importantly however, adding within- and across-episode consistency to the model did not result in a significant improvement (Δ<italic>F</italic>(2,71) = 0.001, <italic>p</italic> = 0.999, Δ<italic>R</italic>^{<italic>2</italic>} = 0.00). These results indicate that consistency in parental responses as derived from their reports of how often in the last month they displayed certain reactions, was longitudinally predictive of externalizing problems, whereas within- and across-episode consistency as derived from the daily diary measures were not. For model coefficients, see Table ##TAB##3##4##.</p>" ]

[ "<title>Discussion</title>", "<p id=\"Par34\">Aim of this study was to investigate how within- and across-episode parental consistency in responding to misbehavior are associated to externalizing problem behavior in children, using a daily diary approach. Within- and across-episode consistency were moderately strongly correlated with each other, but only across-episode consistency was associated with the severity of daily disruptive behavior. In Sample 1, this association was significant above and beyond the content of the parental reactions, whereas in Sample 2, the association was explained by the fact that parents who were more consistent across episodes were less likely to provide negative consequences or negative attention for the disruptive behavior. When we compared the longitudinal predictive value of the measures of consistency derived from the daily diaries to a measure of consistency derived from how often parents indicate they usually react, we found that the measures from the daily diary did not predict externalizing behavior problems one year later, whereas parental consistency as derived from the general questionnaire did.</p>", "<title>Within- and Across-episode Consistency</title>", "<p id=\"Par35\">In both samples, we found that the two types of consistency were significantly associated with each other, but the associations were not so strong that they indicated that they reflected the same underlying construct. This indicates that some parents were relatively higher on across-episode consistency whereas others were relatively higher on within-episode consistency. These results support the idea that it is relevant to separate the two types of consistency. The correlations between the two types of consistency were quite similar across the two different samples, as were the associations between the two types of consistency as computed from the daily diary data and the ‘trait’ measure of consistency that was computed based on how parents indicated that they generally responded to disruptive behavior in the baseline measure. These associations provide some validation of these measures.</p>", "<p id=\"Par36\">Interestingly, despite moderately strong correlations between within- and across episode consistency, when associations between the two types of consistency and the severity of daily disruptive behavior were examined, across-episode consistency was significantly associated with the severity of daily disruptive behavior in both samples, whereas within-episode consistency was not. That within-episode consistency was not associated with disruptive behavior severity is not in line with observational findings that lower within-episode consistency differentiated mother-child dyads with conduct-problems from those without (Gardner, ##REF##2745902##1989##), and mothers of aggressive toddlers from those without (Del Vecchio &amp; O’Leary, ##REF##16597215##2006##). These findings may indicate that with regards to within-episode consistency, the rewarding nature of the final response – when a parent eventually gives in or does not follow through on their initial demand – is more important in explaining this effect of within-episode consistency rather than the mere variation of types of responses as was assessed by our measure. Parents who reward the child for misbehavior are likely to first provide negative attention for instance, scolding the child, and only give in after a sequence of different types of reactions (Gardner, ##REF##2745902##1989##). Alternatively it may mean that across-episode consistency is actually more strongly associated with disruptive behavior. As previous studies examining these rewarding interaction sequences have not controlled for across-episode consistency, more research is necessary to examine whether this association also disappears when across-episode consistency is taken into account.</p>", "<p id=\"Par37\">We add to previous findings that inconsistency in responses across episodes of misbehavior may be specifically associated with more severe disruptive behavior, regardless of the variation in types of responses within single episodes. In Sample 1, parents who were less consistent not only varied more within- or across-episode in how they responded to children’s tantrum, but also more frequently used each of the responses, both positive (e.g., positive consequences such as ‘giving in’) and negative (e.g., negative consequences such as ‘punishing’). Importantly, it was the variation between responses rather than the frequency of the individual responses that was associated with child disruptive behavior. This might indicate that parental consistency in responding is more important for lowering child disruptive behavior than how parents respond specifically. Alternatively, it might mean that when children show more disruptive behavior, parents are more likely to try out different ways of responding in an attempt to deal with it. Other studies have found that behavioral or emotional variation is associated with more maladjustment in young children as observed at a more micro time-scale, across real-time interaction. For instance, variability in affective displays has been related to more externalizing problems in mother-toddler dyads (Lunkenheimer et al., ##REF##23786697##2011##), as has behavioral variability (Lunkenheimer et al., ##REF##33180517##2020##).</p>", "<p id=\"Par38\">Our findings support the idea that, in a non-clinical sample, predictable parental responses are most important in reducing disruptive behavior. Unpredictable behavior from parents has been shown to impact the stress response in infants, with a blunted cortisol response to a painful stressors for infants of mothers who’s behavior was less predictable (Noroña-Zhou et al., ##REF##32115696##2020##), and variability in the affective quality of mother-child interactions and even in the timing of leisure activities, has been associated with an increased production of proinflammatory cytokines, an index of stress-reactivity, for youth (Manczak et al., ##REF##28625195##2018##). More research is necessary to understand whether these processes play a role in explaining the association between across-episode consistency in parenting behavior and child disruptive behavior.</p>", "<p id=\"Par39\">In Sample 2, a sample with older children who were at-risk for problem behavior, across-episode consistency was no longer associated with disruptive behavior above and beyond the individual reactions, whereas the frequency of providing negative consequences and negative attention were associated with more severe disruptive behavior. It seems that, whereas in Sample 1 it did not matter so much what parents did to reduce child disruptive behavior, as long as they did it consistently across episodes, in this sample negative responding was specifically associated with disruptive child behavior. Perhaps for families with older children with elevated levels of disruptive behavior as in Sample 2, parent and child have more strongly established patterns of negative responding to each other (Granic &amp; Patterson, ##REF##16478303##2006##). In support of this idea, in studies of school aged children, affective variability has been associated with less rather than more behavioral problems (Granic et al., ##REF##17549621##2007##; Hollenstein et al., ##REF##15648527##2004##). Settling into a rigid, negative interaction style is a process that takes place in the interaction between parent and child over several years. Heightened variability in other areas may still have negative effects in older children and adolescents. For instance, higher variability in experienced stressors has been associated with worse emotional adjustment in adolescents (Zheng et al., ##REF##36273075##2022##), and higher variability in daily activities is associated with lower psychological well-being in young adults (Lee et al., ##UREF##9##2018##).</p>", "<title>Daily Diary Measures</title>", "<p id=\"Par40\">In this study, daily disruptive behavior was associated with externalizing problems one year later, and within- and across-episode consistency were associated with our measure of general consistency. Thus, it appears that the daily diary measures were tapping some of the micro-level processes giving rise to increases in problems at a developmental timescale (Granic &amp; Patterson, ##REF##16478303##2006##). However, it also appears that these associations between parenting and child behavior did not cross-over from one level to the other, as the general parental consistency measure was predictive of externalizing problems one year later, whereas the daily measures of consistency (within- and across-episode consistency) were not. Additionally, general consistency was in turn not associated with daily disruptive behavior severity, whereas the daily measure of across-episode consistency was. It thus seems that the parenting and child behavior measures that were measured on a more similar timescale were more likely to be associated with each other. A previous study had similar findings in this regard, with daily measures of parenting variability associated with global parenting measures, but only global measures associated with a measure of the child’s ADHD symptoms (Li &amp; Lansford, ##REF##29608072##2018##). Although there, ADHD symptoms became significantly associated with variability in parental warmth after controlling for parental ADHD symptoms and several types of stress, and daily symptom expression was not assessed. More research is necessary to understand how inconsistency in daily parent-child interactions may eventually increase externalizing problems over months and years.</p>", "<p id=\"Par41\">Although daily diary measures are especially helpful in differentiating within- from across-episode consistency, the measure of how often parents indicated to react to their children’s disruptive behavior a certain way over the past month – the general consistency measure – was more predictive longitudinally of externalizing behavior than the daily diary measures. Although this measure again confounds within- and across-episode consistency, it may still be a better measure of consistency than some of the other measures of general consistency. As we asked directly about very specific reactions, our measure is likely a more valid measure of actual consistency in responding (Morsbach &amp; Prinz, ##REF##16636897##2006##). At the same time, we would like to note that the validity of this measure deserves further scrutiny.</p>", "<title>Strengths and Limitations</title>", "<p id=\"Par42\">This study has several strengths. First, we included two samples of daily diary data that allowed us to differentiate within- from across-episode consistency and examine how our results would replicate across samples. Second, the analyses were registered on the OSF before conducting them. In addition to these strengths, some limitations are also worth mentioning. First, we did not differentiate different types of disruptive behavior episodes. Perhaps some episodes were more similar to each other than others, with a child yelling after not getting what it wanted in separate instances more similar than hitting a sibling in frustration about losing a game. Parents might react differently to different types of misbehavior, but consistently so within the types of misbehavior. Relatedly, in the functionally based categorization of behaviors in this study, certain responses were collapsed into categories as they were highly similar in their function – with these categorizations preregistered. Categories of positive attention, and ‘getting what you want’ were differentiated as they are likely different enough to be inconsistent, as are receiving negative attention or being punished for instance. An even higher level of abstraction could also be chosen, where anything ‘positive’ is contrasted with anything ‘negative’. At present, it is not known how different responses must be to contribute to inconsistency. Relatedly, the categorization of the parental reactions was based on a social learning theory perspective. However, inconsistencies in other aspects of the response might also be relevant. For instance, for several of the reactions that parents could choose from, it would be possible to be quite calm or quite frustrated while doing so, and these differences in affective quality and intensity might also contribute to inconsistency. More research is necessary to investigate whether inconsistency computed from other aspects of parental responses shows similar associations as the inconsistency measures derived from the categorization we made here. Second, both samples consisted of families with mostly highly-educated parents, raising questions about how generalizable these findings are to populations with different educational backgrounds. Additionally, whereas for Sample 2 it was clear that it was representative of ethnicities in the Netherlands, for Sample 1 information about ethnic diversity of the sample was not collected, making it impossible to draw any conclusions about this. Third, although similar measures were available for both samples included in this study, these studies were not designed to be the same, and as a result varied in multiple design aspects, making it impossible to draw any conclusions about why the results may have differed between them. Fourth, there are other aspects of consistency that we have not included in this study. For instance, consistency between different caregivers’ reactions might also play a role adjustment problems (Dwairy, ##UREF##5##2010##).</p>", "<title>Conclusion</title>", "<p id=\"Par43\">Results of this study show that it is meaningful to separate parental consistency within- from consistency across-episodes of misbehavior as they are correlated, but not strongly so. Furthermore, the aspects of consistency may be differentially important for the severity of child disruptive behavior as it is displayed in daily life, and there is some indication that across-episode consistency might be more important than actual responses, at least in a general population sample of toddlers. However, the actual responses were more important in our sample of early elementary school aged children from an at-risk population. Findings thus suggest that different risk factors (across-episode consistency or negative responding specifically) for disruptive behavior might apply to different subpopulations.</p>" ]

[ "<title>Conclusion</title>", "<p id=\"Par43\">Results of this study show that it is meaningful to separate parental consistency within- from consistency across-episodes of misbehavior as they are correlated, but not strongly so. Furthermore, the aspects of consistency may be differentially important for the severity of child disruptive behavior as it is displayed in daily life, and there is some indication that across-episode consistency might be more important than actual responses, at least in a general population sample of toddlers. However, the actual responses were more important in our sample of early elementary school aged children from an at-risk population. Findings thus suggest that different risk factors (across-episode consistency or negative responding specifically) for disruptive behavior might apply to different subpopulations.</p>" ]

[ "<p id=\"Par1\">Consistent discipline is thought to reduce early child externalizing behavior. It is unclear, however, whether consistency is important mainly within episodes of misbehavior (e.g., threatening with discipline but then giving in) or across episodes of misbehavior (e.g., disciplining each instance of misbehavior). Using a daily diary approach, we examine whether these two types of consistency are associated with disruptive child behavior, concurrently and prospectively. We included two samples (Sample 1: <italic>N</italic> = 134, <italic>M</italic>_agechild = 30 months, 44% girls; Sample 2: <italic>N</italic> = 149, <italic>M</italic>_agechild = 5.88 years; 46% girls, at-risk sample) with daily reports of child disruptive behavior and parental responses (Sample 1 = 7 days; Sample 2 = 14 days). Sample 1 parents additionally reported on their reactions over the past month and their child’s externalizing behavior one year later. Within-episode consistency was assessed by the average number of parental reactions per episode; across-episode consistency by the Index of Qualitative Variation; and general consistency by parents’ report of how they had responded to child disruptive behavior in the past month. In both samples correlations between within- and across-episode consistency were significant, but not so strong that they were not differentiated. Again in both samples, regression analyses provided evidence for unique predictive value of across-episode, not within-episode, consistency for daily disruptive behavior. Parental general consistency was longitudinally associated with fewer externalizing problems, whereas within- and across-episode consistency were not. It appears meaningful to differentiate within- from across-episode consistency to better understand the relevance of different aspects of consistency.</p>", "<title>Keywords</title>" ]

[]

[ "<title>Appendix I: Questions as included in the daily diaries</title>", "<title>Parental Reactions</title>", "<title>Sample 1</title>", "<p id=\"Par46\">For each tantrum during that day in daily diary, question:</p>", "<title>Sample 2</title>", "<p id=\"Par48\">For the most difficult to manage disruptive behavior reported that day, question:</p>", "<p id=\"Par50\">For answer options, see Table ##TAB##4##5##. Answer categories for each option in both samples were: check box, coded unchecked = 0, checked = 1. Multiple checks allowed.\n</p>", "<title>Child Externalizing Behavior</title>", "<title>Sample 1</title>", "<p id=\"Par51\">For each tantrum that occurred that day, question:</p>", "<p id=\"Par53\">For answer options, see Table ##TAB##5##6##. Answer categories for each option in both samples were: check box, coded unchecked = 0, checked = 1. Multiple checks allowed.\n</p>", "<title>Sample 2</title>", "<p id=\"Par54\">Question: ‘<italic>How disruptive was your child’s behavior today?’.</italic></p>", "<p id=\"Par55\">Answer categories were: slider 1 − 10.</p>", "<title>Author Contribution</title>", "<p>All authors contributed to the study conception and design. Material preparation and data collection were performed by Alithe Van den akker, Patty Leijten, and Peter Hoffenaar, data analysis was performed by Alithe Van den akker. The first draft of the manuscript was written by Alithe Van den akker and all authors commented on previous versions of the manuscript. All authors read and approved the final manuscript.</p>", "<title>Funding</title>", "<p>Study 2 was funded by the Dutch Organization for Health Research and Development (ZonMw) under grant number 636320007 to Patty Leijten. The funder had no role in the design of this protocol, the collection of data, the data analysis, or the interpretation or publication of the study results.</p>", "<title>Data Availability</title>", "<p>Data are available from the first author upon request.</p>", "<title>Compliance with Ethical Standards</title>", "<title>Conflict of Interest</title>", "<p id=\"Par44\">The authors declare that they have no conflict of interests.</p>", "<title>Ethical Approval</title>", "<p id=\"Par45\">All procedures performed in studies involving human participants were in accordance with the ethical standards of the institutional research committee and with the 1964 Helsinki Declaration and its later amendments or comparable ethical standards.</p>" ]

[]

[ "<table-wrap id=\"Tab1\"><label>Table 1</label><caption><p>Descriptives and Intercorrelations for the Study Variables</p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th align=\"left\">Measures</th><th align=\"left\">1.</th><th align=\"left\">2.</th><th align=\"left\">3.</th><th align=\"left\">4.</th><th align=\"left\">5.</th><th align=\"left\">6.</th><th align=\"left\">7</th><th align=\"left\">8.</th><th align=\"left\">9.</th><th align=\"left\">10.</th><th align=\"left\"><italic>M(SD)</italic></th></tr></thead><tbody><tr><td align=\"left\">1.Within-episode consistency</td><td char=\".\" align=\"char\">–</td><td char=\".\" align=\"char\">0.53**</td><td char=\".\" align=\"char\">–</td><td char=\".\" align=\"char\">-0.28**</td><td char=\".\" align=\"char\">–</td><td char=\".\" align=\"char\">-0.34**</td><td char=\".\" align=\"char\">-0.31**</td><td char=\".\" align=\"char\">-0.27**</td><td char=\".\" align=\"char\">-0.50**</td><td char=\".\" align=\"char\">-0.56**</td><td char=\".\" align=\"char\">1.81(0.35)</td></tr><tr><td align=\"left\">2.Across-episode consistency</td><td char=\".\" align=\"char\">0.60**</td><td char=\".\" align=\"char\">–</td><td char=\".\" align=\"char\">–</td><td char=\".\" align=\"char\">-0.31**</td><td char=\".\" align=\"char\">–</td><td char=\".\" align=\"char\">-0.40**</td><td char=\".\" align=\"char\">-0.42**</td><td char=\".\" align=\"char\">0.20*</td><td char=\".\" align=\"char\">-0.44**</td><td char=\".\" align=\"char\">-0.38**</td><td char=\".\" align=\"char\">0.25(0.18)</td></tr><tr><td align=\"left\">3.General consistency</td><td char=\".\" align=\"char\">0.40**</td><td char=\".\" align=\"char\">0.49**</td><td char=\".\" align=\"char\">–</td><td char=\".\" align=\"char\">–</td><td char=\".\" align=\"char\">–</td><td char=\".\" align=\"char\">–</td><td char=\".\" align=\"char\">–</td><td char=\".\" align=\"char\">–</td><td char=\".\" align=\"char\">–</td><td char=\".\" align=\"char\">–</td><td char=\".\" align=\"char\">–</td></tr><tr><td align=\"left\">4.Daily disruptive behavior severity</td><td char=\".\" align=\"char\">-0.11</td><td char=\".\" align=\"char\">-0.29**</td><td char=\".\" align=\"char\">-0.16</td><td char=\".\" align=\"char\">–</td><td char=\".\" align=\"char\">–</td><td char=\".\" align=\"char\">0.30**</td><td char=\".\" align=\"char\">0.16</td><td char=\".\" align=\"char\">0.03</td><td char=\".\" align=\"char\">0.30**</td><td char=\".\" align=\"char\">0.19**</td><td char=\".\" align=\"char\">3.32(1.21)</td></tr><tr><td align=\"left\">5.Externalizing problems T2</td><td char=\".\" align=\"char\">-0.08</td><td char=\".\" align=\"char\">-0.22*</td><td char=\".\" align=\"char\">-0.32**</td><td char=\".\" align=\"char\">0.34**</td><td char=\".\" align=\"char\">–</td><td char=\".\" align=\"char\">–</td><td char=\".\" align=\"char\">–</td><td char=\".\" align=\"char\">–</td><td char=\".\" align=\"char\">–</td><td char=\".\" align=\"char\">–</td><td char=\".\" align=\"char\">–</td></tr><tr><td align=\"left\">6.Negative consequence</td><td char=\".\" align=\"char\">-0.23**</td><td char=\".\" align=\"char\">-0.24**</td><td char=\".\" align=\"char\">-0.31**</td><td char=\".\" align=\"char\">-0.03</td><td char=\".\" align=\"char\">0.10</td><td char=\".\" align=\"char\">–</td><td char=\".\" align=\"char\">0.25**</td><td char=\".\" align=\"char\">-0.06</td><td char=\".\" align=\"char\">0.11</td><td char=\".\" align=\"char\">-0.04</td><td char=\".\" align=\"char\">0.27(0.25)</td></tr><tr><td align=\"left\">7.Positive consequence</td><td char=\".\" align=\"char\">-0.34**</td><td char=\".\" align=\"char\">-0.34**</td><td char=\".\" align=\"char\">-0.21*</td><td char=\".\" align=\"char\">-0.03</td><td char=\".\" align=\"char\">-0.03</td><td char=\".\" align=\"char\">0.05</td><td char=\".\" align=\"char\">–</td><td char=\".\" align=\"char\">-0.05</td><td char=\".\" align=\"char\">0.15</td><td char=\".\" align=\"char\">0.12</td><td char=\".\" align=\"char\">0.08(0.11)</td></tr><tr><td align=\"left\">8.Withhold attention</td><td char=\".\" align=\"char\">-0.20*</td><td char=\".\" align=\"char\">-0.30**</td><td char=\".\" align=\"char\">-0.16</td><td char=\".\" align=\"char\">0.12</td><td char=\".\" align=\"char\">0.04</td><td char=\".\" align=\"char\">-0.09</td><td char=\".\" align=\"char\">-0.24**</td><td char=\".\" align=\"char\">–</td><td char=\".\" align=\"char\">-0.05</td><td char=\".\" align=\"char\">0.12</td><td char=\".\" align=\"char\">0.51(0.24)</td></tr><tr><td align=\"left\">9.Negative attention</td><td char=\".\" align=\"char\">-0.43**</td><td char=\".\" align=\"char\">-0.29**</td><td char=\".\" align=\"char\">-0.33**</td><td char=\".\" align=\"char\">0.03</td><td char=\".\" align=\"char\">0.20</td><td char=\".\" align=\"char\">0.22*</td><td char=\".\" align=\"char\">-0.00</td><td char=\".\" align=\"char\">0.01</td><td char=\".\" align=\"char\">–</td><td char=\".\" align=\"char\">0.17*</td><td char=\".\" align=\"char\">0.18(0.17)</td></tr><tr><td align=\"left\">10.Positive attention</td><td char=\".\" align=\"char\">-0.22*</td><td char=\".\" align=\"char\">0.09</td><td char=\".\" align=\"char\">0.30**</td><td char=\".\" align=\"char\">-0.09</td><td char=\".\" align=\"char\">-0.08</td><td char=\".\" align=\"char\">-0.17</td><td char=\".\" align=\"char\">0.00</td><td char=\".\" align=\"char\">-0.17</td><td char=\".\" align=\"char\">-0.18*</td><td char=\".\" align=\"char\">–</td><td char=\".\" align=\"char\">0.20(0.22)</td></tr><tr><td align=\"left\" rowspan=\"2\"><italic>M(SD)</italic></td><td char=\".\" align=\"char\">1.52</td><td char=\".\" align=\"char\">0.40</td><td char=\".\" align=\"char\">0.07</td><td char=\".\" align=\"char\">4.15</td><td char=\".\" align=\"char\">0.60</td><td char=\".\" align=\"char\">0.07</td><td char=\".\" align=\"char\">0.12</td><td char=\".\" align=\"char\">0.30</td><td char=\".\" align=\"char\">0.21</td><td char=\".\" align=\"char\">0.42</td><td char=\".\" align=\"char\">–</td></tr><tr><td char=\".\" align=\"char\">(0.47)</td><td char=\".\" align=\"char\">(0.29)</td><td char=\".\" align=\"char\">(0.05)</td><td char=\".\" align=\"char\">(4.04)</td><td char=\".\" align=\"char\">(0.31)</td><td char=\".\" align=\"char\">(0.16)</td><td char=\".\" align=\"char\">(0.21)</td><td char=\".\" align=\"char\">(0.28)</td><td char=\".\" align=\"char\">(0.26)</td><td char=\".\" align=\"char\">(0.30)</td><td char=\".\" align=\"char\"/></tr></tbody></table></table-wrap>", "<table-wrap id=\"Tab2\"><label>Table 2</label><caption><p>Results of Regression Analyses Predicting Severity of Daily Disruptive Behavior in Sample 1</p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th align=\"left\"/><th align=\"left\"/><th align=\"left\"/><th align=\"left\"/><th align=\"left\" colspan=\"4\">Severity of daily disruptive behavior</th></tr><tr><th align=\"left\">variables</th><th align=\"left\"><italic>b(SE)</italic></th><th align=\"left\">β</th><th align=\"left\"><italic>p</italic></th><th align=\"left\">variables</th><th align=\"left\"><italic>b(SE)</italic></th><th align=\"left\">β</th><th align=\"left\"><italic>p</italic></th></tr><tr><th align=\"left\" colspan=\"4\">Step 1</th><th align=\"left\" colspan=\"4\">Step 1</th></tr></thead><tbody><tr><td align=\"left\">  age child</td><td char=\".\" align=\"char\">-0.04(0.04)</td><td char=\".\" align=\"char\">-0.08</td><td char=\".\" align=\"char\">0.370</td><td align=\"left\">  Age child</td><td char=\".\" align=\"char\">-0.04(0.04)</td><td char=\".\" align=\"char\">-0.08</td><td char=\".\" align=\"char\">0.370</td></tr><tr><td align=\"left\">  Sex child</td><td char=\".\" align=\"char\">-0.41(0.72)</td><td char=\".\" align=\"char\">-0.05</td><td char=\".\" align=\"char\">0.572</td><td align=\"left\">  Sex child</td><td char=\".\" align=\"char\">-0.41(0.72)</td><td char=\".\" align=\"char\">-0.05</td><td char=\".\" align=\"char\">0.572</td></tr><tr><td align=\"left\">  Education Level</td><td char=\".\" align=\"char\">-0.23(0.38)</td><td char=\".\" align=\"char\">-0.05</td><td char=\".\" align=\"char\">0.553</td><td align=\"left\">  Education Level</td><td char=\".\" align=\"char\">-0.23(0.38)</td><td char=\".\" align=\"char\">-0.05</td><td char=\".\" align=\"char\">0.553</td></tr><tr><td align=\"left\" colspan=\"4\">Step 2</td><td align=\"left\" colspan=\"4\">Step 2</td></tr><tr><td align=\"left\">  Age child</td><td char=\".\" align=\"char\">-0.04(0.04)</td><td char=\".\" align=\"char\">-0.12</td><td char=\".\" align=\"char\">0.294</td><td align=\"left\">  Age child</td><td char=\".\" align=\"char\">-0.04(0.04)</td><td char=\".\" align=\"char\">-0.08</td><td char=\".\" align=\"char\">0.385</td></tr><tr><td align=\"left\">  Sex child</td><td char=\".\" align=\"char\">-0.65(0.69)</td><td char=\".\" align=\"char\">-0.07</td><td char=\".\" align=\"char\">0.353</td><td align=\"left\">  Sex child</td><td char=\".\" align=\"char\">-0.30(0.73)</td><td char=\".\" align=\"char\">-0.04</td><td char=\".\" align=\"char\">0.685</td></tr><tr><td align=\"left\">  Education Level</td><td char=\".\" align=\"char\">-0.24(0.37)</td><td char=\".\" align=\"char\">-0.05</td><td char=\".\" align=\"char\">0.525</td><td align=\"left\">  Education Level</td><td char=\".\" align=\"char\">-0.14(0.40)</td><td char=\".\" align=\"char\">-0.03</td><td char=\".\" align=\"char\">0.725</td></tr><tr><td align=\"left\">  Within-episode consistency</td><td char=\".\" align=\"char\">0.70(0.94)</td><td char=\".\" align=\"char\">0.06</td><td char=\".\" align=\"char\">0.455</td><td align=\"left\">  negative consequence</td><td char=\".\" align=\"char\">-0.77(2.40)</td><td char=\".\" align=\"char\">-0.03</td><td char=\".\" align=\"char\">0.748</td></tr><tr><td align=\"left\">  Across-episode consistency</td><td char=\".\" align=\"char\">-4.92(1.47)</td><td char=\".\" align=\"char\">-0.36</td><td char=\".\" align=\"char\">0.001</td><td align=\"left\">  positive consequence</td><td char=\".\" align=\"char\">0.30(1.79)</td><td char=\".\" align=\"char\">0.02</td><td char=\".\" align=\"char\">0.866</td></tr><tr><td align=\"left\" rowspan=\"14\" colspan=\"4\"/><td align=\"left\">  withholding attention</td><td char=\".\" align=\"char\">1.39(1.33)</td><td char=\".\" align=\"char\">0.10</td><td char=\".\" align=\"char\">0.297</td></tr><tr><td align=\"left\">  negative attention</td><td char=\".\" align=\"char\">0.56(1.47)</td><td char=\".\" align=\"char\">0.04</td><td char=\".\" align=\"char\">0.705</td></tr><tr><td align=\"left\">  positive attention</td><td char=\".\" align=\"char\">-0.96(1.27)</td><td char=\".\" align=\"char\">-0.07</td><td char=\".\" align=\"char\">0.454</td></tr><tr><td align=\"left\" colspan=\"4\">Step 3</td></tr><tr><td align=\"left\">  Age child</td><td char=\".\" align=\"char\">-0.02(0.04)</td><td char=\".\" align=\"char\">-0.04</td><td char=\".\" align=\"char\">0.644</td></tr><tr><td align=\"left\">  Sex child</td><td char=\".\" align=\"char\">-0.80(0.71)</td><td char=\".\" align=\"char\">-0.10</td><td char=\".\" align=\"char\">0.259</td></tr><tr><td align=\"left\">  Education Level</td><td char=\".\" align=\"char\">-0.33(0.38)</td><td char=\".\" align=\"char\">-0.08</td><td char=\".\" align=\"char\">0.382</td></tr><tr><td align=\"left\">  negative consequence</td><td char=\".\" align=\"char\">-4.00(2.42)</td><td char=\".\" align=\"char\">-0.16</td><td char=\".\" align=\"char\">0.101</td></tr><tr><td align=\"left\">  positive consequence</td><td char=\".\" align=\"char\">-4.05(2.06)</td><td char=\".\" align=\"char\">-0.21</td><td char=\".\" align=\"char\">0.051</td></tr><tr><td align=\"left\">  withholding attention</td><td char=\".\" align=\"char\">-1.74(1.51)</td><td char=\".\" align=\"char\">-0.12</td><td char=\".\" align=\"char\">0.252</td></tr><tr><td align=\"left\">  negative attention</td><td char=\".\" align=\"char\">-1.78(1.65)</td><td char=\".\" align=\"char\">-0.11</td><td char=\".\" align=\"char\">0.282</td></tr><tr><td align=\"left\">  positive attention</td><td char=\".\" align=\"char\">-1.50(1.38)</td><td char=\".\" align=\"char\">-0.11</td><td char=\".\" align=\"char\">0.282</td></tr><tr><td align=\"left\">  Within-episode consistency</td><td char=\".\" align=\"char\">-0.55(1.17)</td><td char=\".\" align=\"char\">-0.06</td><td char=\".\" align=\"char\">0.638</td></tr><tr><td align=\"left\">  Across-episode consistency</td><td char=\".\" align=\"char\">-6.05(1.67)</td><td char=\".\" align=\"char\">-0.44</td><td char=\".\" align=\"char\">&lt; 0.001</td></tr></tbody></table></table-wrap>", "<table-wrap id=\"Tab3\"><label>Table 3</label><caption><p>Results of Regression Analyses Predicting Severity of Daily Disruptive Behavior in Sample 2</p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th align=\"left\"/><th align=\"left\"/><th align=\"left\"/><th align=\"left\"/><th align=\"left\" colspan=\"4\">Severity of daily disruptive behavior</th></tr><tr><th align=\"left\">variables</th><th align=\"left\"><italic>b(SE)</italic></th><th align=\"left\">β</th><th align=\"left\"><italic>p</italic></th><th align=\"left\">variables</th><th align=\"left\"><italic>b(SE)</italic></th><th align=\"left\">β</th><th align=\"left\"><italic>p</italic></th></tr><tr><th align=\"left\" colspan=\"4\">Step 1</th><th align=\"left\" colspan=\"4\">Step 1</th></tr></thead><tbody><tr><td align=\"left\">  Age child</td><td char=\".\" align=\"char\">-0.00(0.01)</td><td char=\".\" align=\"char\">-0.03</td><td char=\".\" align=\"char\">0.736</td><td align=\"left\">  Age child</td><td char=\".\" align=\"char\">-0.00(0.01)</td><td char=\".\" align=\"char\">-0.01</td><td char=\".\" align=\"char\">0.736</td></tr><tr><td align=\"left\">  Sex child</td><td char=\".\" align=\"char\">-0.12(0.20)</td><td char=\".\" align=\"char\">-0.05</td><td char=\".\" align=\"char\">0.541</td><td align=\"left\">  Sex child</td><td char=\".\" align=\"char\">-0.12(0.20)</td><td char=\".\" align=\"char\">-0.05</td><td char=\".\" align=\"char\">0.541</td></tr><tr><td align=\"left\">  Education Level</td><td char=\".\" align=\"char\">0.06(0.10)</td><td char=\".\" align=\"char\">0.05</td><td char=\".\" align=\"char\">0.562</td><td align=\"left\">  Education Level</td><td char=\".\" align=\"char\">0.06(0.10)</td><td char=\".\" align=\"char\">0.05</td><td char=\".\" align=\"char\">0.562</td></tr><tr><td align=\"left\">  Received support</td><td char=\".\" align=\"char\">0.71(0.38)</td><td char=\".\" align=\"char\">0.16</td><td char=\".\" align=\"char\">0.062</td><td align=\"left\">  Received support</td><td char=\".\" align=\"char\">0.71(0.38)</td><td char=\".\" align=\"char\">0.16</td><td char=\".\" align=\"char\">0.062</td></tr><tr><td align=\"left\" colspan=\"4\">Step 2</td><td align=\"left\" colspan=\"4\">Step 2</td></tr><tr><td align=\"left\">  Age child</td><td char=\".\" align=\"char\">-0.00(0.01)</td><td char=\".\" align=\"char\">-0.01</td><td char=\".\" align=\"char\">0.988</td><td align=\"left\">  Age child</td><td char=\".\" align=\"char\">0.00(0.01)</td><td char=\".\" align=\"char\">0.02</td><td char=\".\" align=\"char\">0.776</td></tr><tr><td align=\"left\">  Sex child</td><td char=\".\" align=\"char\">-0.09(0.19)</td><td char=\".\" align=\"char\">-0.04</td><td char=\".\" align=\"char\">0.639</td><td align=\"left\">  Sex child</td><td char=\".\" align=\"char\">-0.06(0.19)</td><td char=\".\" align=\"char\">-0.02</td><td char=\".\" align=\"char\">0.770</td></tr><tr><td align=\"left\">  Education Level</td><td char=\".\" align=\"char\">0.03(0.10)</td><td char=\".\" align=\"char\">0.02</td><td char=\".\" align=\"char\">0.785</td><td align=\"left\">  Education Level</td><td char=\".\" align=\"char\">0.05(0.10)</td><td char=\".\" align=\"char\">0.04</td><td char=\".\" align=\"char\">0.612</td></tr><tr><td align=\"left\">  Received support</td><td char=\".\" align=\"char\">0.83(0.36)</td><td char=\".\" align=\"char\">0.18</td><td char=\".\" align=\"char\">0.021</td><td align=\"left\">  Received support</td><td char=\".\" align=\"char\">0.98(0.36)</td><td char=\".\" align=\"char\">0.21</td><td char=\".\" align=\"char\">0.007</td></tr><tr><td align=\"left\">  Within-episode consistency</td><td char=\".\" align=\"char\">-0.59(0.32)</td><td char=\".\" align=\"char\">-0.17</td><td char=\".\" align=\"char\">0.063</td><td align=\"left\">  negative consequence</td><td char=\".\" align=\"char\">1.45(0.39)</td><td char=\".\" align=\"char\">0.30</td><td char=\".\" align=\"char\">&lt; 0.001</td></tr><tr><td align=\"left\">  Across-episode consistency</td><td char=\".\" align=\"char\">-1.60(0.61)</td><td char=\".\" align=\"char\">-0.24</td><td char=\".\" align=\"char\">0.010</td><td align=\"left\">  positive consequence</td><td char=\".\" align=\"char\">0.07(0.90)</td><td char=\".\" align=\"char\">0.01</td><td char=\".\" align=\"char\">0.938</td></tr><tr><td align=\"left\" rowspan=\"15\" colspan=\"4\"/><td align=\"left\">  withholding attention</td><td char=\".\" align=\"char\">0.04(0.39)</td><td char=\".\" align=\"char\">0.01</td><td char=\".\" align=\"char\">0.901</td></tr><tr><td align=\"left\">  negative attention</td><td char=\".\" align=\"char\">1.77(0.54)</td><td char=\".\" align=\"char\">0.26</td><td char=\".\" align=\"char\">0.001</td></tr><tr><td align=\"left\">  positive attention</td><td char=\".\" align=\"char\">0.83(0.44)</td><td char=\".\" align=\"char\">0.15</td><td char=\".\" align=\"char\">0.060</td></tr><tr><td align=\"left\" colspan=\"4\">Step 3</td></tr><tr><td align=\"left\">  Age child</td><td char=\".\" align=\"char\">0.00(0.01)</td><td char=\".\" align=\"char\">0.02</td><td char=\".\" align=\"char\">0.785</td></tr><tr><td align=\"left\">  sex</td><td char=\".\" align=\"char\">-0.05(0.19)</td><td char=\".\" align=\"char\">-0.02</td><td char=\".\" align=\"char\">0.778</td></tr><tr><td align=\"left\">  Education Level</td><td char=\".\" align=\"char\">0.04(0.10)</td><td char=\".\" align=\"char\">0.03</td><td char=\".\" align=\"char\">0.686</td></tr><tr><td align=\"left\">  Received support</td><td char=\".\" align=\"char\">0.96(0.36)</td><td char=\".\" align=\"char\">0.21</td><td char=\".\" align=\"char\">0.008</td></tr><tr><td align=\"left\">  negative consequence</td><td char=\".\" align=\"char\">1.42(0.46)</td><td char=\".\" align=\"char\">0.29</td><td char=\".\" align=\"char\">0.002</td></tr><tr><td align=\"left\">  positive consequence</td><td char=\".\" align=\"char\">-0.06(0.97)</td><td char=\".\" align=\"char\">-0.01</td><td char=\".\" align=\"char\">0.955</td></tr><tr><td align=\"left\">  withholding attention</td><td char=\".\" align=\"char\">0.22(0.44)</td><td char=\".\" align=\"char\">0.04</td><td char=\".\" align=\"char\">0.624</td></tr><tr><td align=\"left\">  negative attention</td><td char=\".\" align=\"char\">1.77(0.67)</td><td char=\".\" align=\"char\">0.26</td><td char=\".\" align=\"char\">0.009</td></tr><tr><td align=\"left\">  positive attention</td><td char=\".\" align=\"char\">0.86(0.59)</td><td char=\".\" align=\"char\">0.16</td><td char=\".\" align=\"char\">0.142</td></tr><tr><td align=\"left\">  Within-episode consistency</td><td char=\".\" align=\"char\">0.26(0.46)</td><td char=\".\" align=\"char\">0.08</td><td char=\".\" align=\"char\">0.578</td></tr><tr><td align=\"left\">  Across-episode consistency</td><td char=\".\" align=\"char\">-0.60(0.74)</td><td char=\".\" align=\"char\">-0.09</td><td char=\".\" align=\"char\">0.421</td></tr></tbody></table></table-wrap>", "<table-wrap id=\"Tab4\"><label>Table 4</label><caption><p>Results of Regression Analysis Predicting Externalizing Behavior one year Later</p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th align=\"left\">variables</th><th align=\"left\"><italic>b(SE)</italic></th><th align=\"left\">β</th><th align=\"left\"><italic>p</italic></th></tr><tr><th align=\"left\" colspan=\"4\">Step 1</th></tr></thead><tbody><tr><td align=\"left\">  Age Child</td><td char=\".\" align=\"char\">-0.01(0.00)</td><td char=\".\" align=\"char\">-0.15</td><td char=\".\" align=\"char\">0.199</td></tr><tr><td align=\"left\">  Sex Child</td><td char=\".\" align=\"char\">0.00(0.07)</td><td char=\".\" align=\"char\">0.01</td><td char=\".\" align=\"char\">0.955</td></tr><tr><td align=\"left\">  Education Level</td><td char=\".\" align=\"char\">-0.01(0.03)</td><td char=\".\" align=\"char\">-0.04</td><td char=\".\" align=\"char\">0.676</td></tr><tr><td align=\"left\">  Daily disruptive behavior severity T1</td><td char=\".\" align=\"char\">0.02(0.01)</td><td char=\".\" align=\"char\">0.31</td><td char=\".\" align=\"char\">0.005</td></tr><tr><td align=\"left\">  General consistency</td><td char=\".\" align=\"char\">-1.90(0.68)</td><td char=\".\" align=\"char\">-0.31</td><td char=\".\" align=\"char\">0.006</td></tr><tr><td align=\"left\" colspan=\"4\">Step 2</td></tr><tr><td align=\"left\">  Age Child</td><td char=\".\" align=\"char\">-0.01(0.00)</td><td char=\".\" align=\"char\">-0.13</td><td char=\".\" align=\"char\">0.260</td></tr><tr><td align=\"left\">  Sex Child</td><td char=\".\" align=\"char\">-0.01(0.07)</td><td char=\".\" align=\"char\">-0.01</td><td char=\".\" align=\"char\">0.927</td></tr><tr><td align=\"left\">  Education Level</td><td char=\".\" align=\"char\">-0.01(0.04)</td><td char=\".\" align=\"char\">-0.04</td><td char=\".\" align=\"char\">0.711</td></tr><tr><td align=\"left\">  Daily disruptive behavior severity T1</td><td char=\".\" align=\"char\">0.02(0.01)</td><td char=\".\" align=\"char\">0.30</td><td char=\".\" align=\"char\">0.010</td></tr><tr><td align=\"left\">  General consistency</td><td char=\".\" align=\"char\">-1.76(0.82)</td><td char=\".\" align=\"char\">-0.29</td><td char=\".\" align=\"char\">0.036</td></tr><tr><td align=\"left\">  Within-episode consistency</td><td char=\".\" align=\"char\">0.05(0.09)</td><td char=\".\" align=\"char\">0.08</td><td char=\".\" align=\"char\">0.553</td></tr><tr><td align=\"left\">  Across-episode consistency</td><td char=\".\" align=\"char\">-0.12(0.17)</td><td char=\".\" align=\"char\">-0.11</td><td char=\".\" align=\"char\">0.464</td></tr></tbody></table></table-wrap>", "<table-wrap id=\"Tab5\"><label>Table 5</label><caption><p>Answer categories for each parental reaction for the two studies</p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th align=\"left\"><bold>Study 1</bold></th><th align=\"left\"><bold>Study 2</bold></th></tr></thead><tbody><tr><td align=\"left\" colspan=\"2\"><bold>Negative consequence</bold></td></tr><tr><td align=\"left\">I sent my child to their room/corner/time-out</td><td align=\"left\">I sent my child to their room for at least an hour</td></tr><tr><td align=\"left\">I punished my child</td><td align=\"left\">I gave my child a short time-out, away from others</td></tr><tr><td align=\"left\"/><td align=\"left\">I took something nice away from my child (e.g., toys or screen time)</td></tr><tr><td align=\"left\"/><td align=\"left\">I gave my child extra chores (e.g., set the table)</td></tr><tr><td align=\"left\" colspan=\"2\"><bold>Positive consequence</bold></td></tr><tr><td align=\"left\">I negotiated with my child</td><td align=\"left\">I gave my child his/her way</td></tr><tr><td align=\"left\">I gave in to my child</td><td align=\"left\">I gave in to my child</td></tr><tr><td align=\"left\" colspan=\"2\"><bold>Withholding attention</bold></td></tr><tr><td align=\"left\">I didn’t, I let my child cool off</td><td align=\"left\">I did nothing</td></tr><tr><td align=\"left\">I ignored my child</td><td align=\"left\">I talked about it with my child afterwards</td></tr><tr><td align=\"left\" colspan=\"2\"><bold>Negative attention</bold></td></tr><tr><td align=\"left\">I became angry with my child</td><td align=\"left\">I yelled/swore</td></tr><tr><td align=\"left\">I grabbed my child</td><td align=\"left\">I said things I didn’t mean</td></tr><tr><td align=\"left\">I spoke sternly to my child</td><td align=\"left\">I threatened with punishment, but did not punish</td></tr><tr><td align=\"left\" colspan=\"2\"><bold>Positive attention</bold></td></tr><tr><td align=\"left\">I comforted my child</td><td align=\"left\">I begged my child to stop</td></tr><tr><td align=\"left\">I distracted my child</td><td align=\"left\">I used humor to distract my child</td></tr></tbody></table></table-wrap>", "<table-wrap id=\"Tab6\"><label>Table 6</label><caption><p>Answer categories for the aggressive tantrum behaviors</p></caption><table frame=\"hsides\" rules=\"groups\"><tbody><tr><td align=\"left\" colspan=\"2\">Hitting</td></tr><tr><td align=\"left\" colspan=\"2\">Kicking</td></tr><tr><td align=\"left\" colspan=\"2\">Biting</td></tr><tr><td align=\"left\" colspan=\"2\">Throwing an object</td></tr><tr><td align=\"left\" colspan=\"2\">Pushing/pulling</td></tr><tr><td align=\"left\" colspan=\"2\">Spitting</td></tr><tr><td align=\"left\" colspan=\"2\">Grabbing</td></tr><tr><td align=\"left\" colspan=\"2\">Banging head</td></tr><tr><td align=\"left\" colspan=\"2\">Holding breath</td></tr><tr><td align=\"left\" colspan=\"2\">freezing</td></tr></tbody></table></table-wrap>" ]

[ "<disp-formula id=\"Equa\"><alternatives><tex-math id=\"M1\">\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym} \n\t\t\t\t\\usepackage{amsfonts} \n\t\t\t\t\\usepackage{amssymb} \n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\mathrm{IQV}=\\frac{\\left(\\mathrm{K}\\left(1-\\left(\\sum \\mathrm{Prop}2\\right)\\right)\\right)}{\\left(\\mathrm{K}-1\\right)}$$\\end{document}</tex-math><mml:math id=\"M2\" display=\"block\"><mml:mrow><mml:mi mathvariant=\"normal\">IQV</mml:mi><mml:mo>=</mml:mo><mml:mfrac><mml:mfenced close=\")\" open=\"(\"><mml:mi mathvariant=\"normal\">K</mml:mi><mml:mfenced close=\")\" open=\"(\"><mml:mn>1</mml:mn><mml:mo>-</mml:mo><mml:mfenced close=\")\" open=\"(\"><mml:mo>∑</mml:mo><mml:mi mathvariant=\"normal\">Prop</mml:mi><mml:mn>2</mml:mn></mml:mfenced></mml:mfenced></mml:mfenced><mml:mfenced close=\")\" open=\"(\"><mml:mi mathvariant=\"normal\">K</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mfenced></mml:mfrac></mml:mrow></mml:math></alternatives></disp-formula>" ]

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[ "<disp-quote><p id=\"Par47\"><italic>‘How did you react to tantrum x?’ (x = 1–7 max)</italic></p></disp-quote>", "<disp-quote><p id=\"Par49\"><italic>‘How did you react to this disruptive behavior?’</italic></p></disp-quote>", "<disp-quote><p id=\"Par52\">‘<italic>Which of the following behaviors did your child show during tantrum x?’ (x</italic>=<italic>1-7)</italic></p></disp-quote>" ]

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[ "<table-wrap-foot><p>Estimates for Sample 1 are provided below the diagonal, and for Sample 2 above the diagonal</p><p>*<italic>p</italic> &lt; 0.05, **<italic>p</italic> &lt; 0.01</p></table-wrap-foot>", "<fn-group><fn><p><bold>Publisher’s Note</bold></p><p>Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.</p></fn></fn-group>" ]

[ "<graphic xlink:href=\"10802_2023_1073_Article_Equa.gif\" position=\"anchor\"/>" ]

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{ "acronym": [], "definition": [] }

[ "<title>Introduction</title>", "<p id=\"Par5\">Multiparametric MRI (mpMRI) of the prostate is nowadays considered the key tool for the diagnosis of clinically significant prostate cancer (csPCa) [##UREF##0##1##].</p>", "<p id=\"Par6\">The quality of MRI has progressively increased in the last years and standardised multiparametric sequences acquisition and reporting have been established, with the development of the Prostate Imaging – Reporting and Data System (PI-RADS) scoring system, first published in 2012 by the European Society of Urogenital Radiology (ESUR) [##REF##22322308##2##].</p>", "<p id=\"Par7\">The updated PI-RADS versions 2 and 2.1 are nowadays broadly used in clinical practice, overcoming some ambiguities and limitations related to the overall scoring system of the former version [##REF##26427566##3##, ##REF##30898406##4##].</p>", "<p id=\"Par8\">Nevertheless, PI-RADS has not being designed for staging purposes; on this regard, the ESUR developed another score system based on capsular alterations detectable at MRI and related to the likelihood of an extracapsular extension (ECE) of the lesion [##REF##25504428##5##].</p>", "<p id=\"Par9\">During treatment planning, the identification of ECE as a strategy for local staging is crucial in order to achieve an adequate balance between cancer control and preservation of potency and continence, ultimately obtaining the best surgical, oncological and functional results.</p>", "<p id=\"Par10\">The main objective of this study was to identify clinical, pathological and radiological parameters associated to ECE in a single institution cohort of patients undergoing mpMRI prior to radical prostatectomy (RP), as by whole-mount prostate sections for definitive histological assessment. Furthermore, we evaluated the usefulness and the inter-observer variability of the overall LIKERT (subjective operator assessment for the likelihood of a csPCa, from 1 to 5), ECE-LIKERT (subjective operator assessment for the likelihood of ECE, from 1 to 5), PI-RADS v2 and ESUR-ECE, in order to evaluate their performance in predicting the ECE risk in the same cohort of patients.</p>" ]

[ "<title>Patients and methods</title>", "<title>Study design and population</title>", "<p id=\"Par11\">This is a retrospective analysis of patients who had undergone mpMRI before RP at Fundació Puigvert – Barcelona (ES), between April 2013 (date of the implementation of mpMRI driven pathway) and December 2017. The mpMRI was requested at urology consultant discretion, either before or after the diagnostic biopsy, as no specific recommendations for requesting an mpMRI were still put in place during the period in observation nor in the international guidelines neither in our internal protocol.</p>", "<p id=\"Par12\">Overall, 126 patients met the selection criteria (mpMRI undertaken within 6 months before surgery, either before or after the diagnostic biopsy; availability of full set of data in observation) and their medical records were collected and reviewed. The patients’ data were managed according to our institutional review board protocol, in full compliance with both the principles of the latest version of the Declaration of Helsinki and of the Spanish adaptation of the General Data Protection Regulation (organic law 3/2018, December the 5th 2018).</p>", "<title>mpMRI protocol</title>", "<p id=\"Par13\">A 3-Tesla mpMRI examination with a pelvic phased-array surface coil was performed for all patients. The mpMRI protocol-included T2-weighted (T2W) sequences in three planes, Diffusion Weighted Imaging (DWI) sequences with high <italic>b</italic>-values (&gt; 1200 s/mm²) and apparent diffusion coefficient (ADC) map, and Dynamic Contrast-Enhanced (DCE) sequences with a bolus of gadolinium contrast medium injection. Imaging acquisition protocol was rigorously compliant with the PIRADS v1 guidelines, in force during the period in observation [##REF##22322308##2##]. It is Important to note that these guidelines also included recommendations for waiting a period of 4–6 weeks between a biopsy and the eventual MRI to minimise the effect of eventual haemorrhage, and in case of substantial persisting artifacts to repeat the MRI within further 4 weeks or so. These recommendations were duly followed, and interestingly remained substantially unvaried along the updated v2 and v2.1 versions [##REF##26427566##3##, ##REF##30898406##4##]. All images were retrospectively and independently reassessed by two expert radiologists (L.G. and J.H.) with at least 4 years of experience in prostate mpMRI, both blinded to clinical and histological data; PI-RADS v2 was used to score the MRI explorations, as the updated v2.1 was published posteriorly to the radiological revision of the imaging tests. The maximal index lesion size (ILS), length of capsular involvement (LCI) by tumour, number of lesions and location were recorded whenever visible from the dominant sequence involved. The radiologists also used an overall-LIKERT and ECE-LIKERT scores, as by subjective impression of the likelihood of significant malignancy and extracapsular extension (1 = very unlikely; 5 = very likely), according to the recommendation of the PREDICT (Prostate Diagnostic Imaging Consensus Meeting) panel [##REF##28111879##6##]. Furthermore, the likelihood of ECE was evaluated using the ESUR MRI scoring guidelines of extra-prostatic disease [##REF##25504428##5##].</p>", "<title>Reference standard</title>", "<p id=\"Par14\">Whole-mount histological sections from the RP specimens were used as the reference standard. The specimens were fixed in 10% buffered formalin, and sectioned into horizontal sections of 3–4 mm. All tissues were paraffin-embedded, and 3–4 microns sections were obtained and stained with hematoxylin–eosin; then, the sections of the tissue were assessed by a single expert uropathologist (F.A.), blinded to mpMRI data. The uropathologist recorded cancer location, size, volume, and Gleason grade group according to the International Society of Urological Pathology (ISUP) consensus conference of 2014 [##REF##26492179##7##].</p>", "<title>Outcomes and statistical analysis</title>", "<p id=\"Par15\">The primary outcome consisted in identifying the parameters associated to the ECE at the RP specimen. Descriptive data were expressed as median and interquartile range (IQR, 25–75 quartile). Analysis between groups was performed using Student’s <italic>t</italic> test (Mann–Whitney <italic>U</italic> test in variables without normal distribution) for continuous variables, and Chi-square (Fisher’s exact test with observed frequencies &lt; 5) for categorical variables.</p>", "<p id=\"Par16\">Quantitative (continuous) variables were transformed to binary (categorical) variables before inclusion in logistic models using the best predictive cut-off point obtained with Receiver Operating Characteristics (ROC) curve analysis.</p>", "<p id=\"Par17\">Univariate and multivariate logistic regression models were performed including ECE as a dependent variable. Preoperative clinical, pathological and mpMRI variables with <italic>p</italic> value &lt; 0.2 at the univariate analysis were included as independent variables using the backward stepwise logistic regression analysis. Predictors from the final model were used to calculate the likelihood of ECE according to the following equation: Exp(<italic>β</italic>)/[1 + Exp(<italic>β</italic>)], where <italic>β</italic> = [− 3.00 + <italic>X</italic>*(predictor <italic>A</italic>) + <italic>Y</italic>*(predictor <italic>B</italic>)] for two predictors. The quality of the final model was assessed using the Hosmer–Lemeshow goodness-of-fit-test.</p>", "<p id=\"Par18\">Inter-observer agreement between the two radiologists regarding the MRI features/scores was assessed using the intraclass correlation coefficient (ICC) with the 95% confidence interval, applying a two-way ICC with a random rater assumption. The agreement (match) between a PCa lesion detected at mpMRI and an equivalent lesion at whole-mount histological sections from the RP specimens was assessed using Cohen Kappa coefficient, whose results were categorised in standardised ranges  &lt; 0.4 poor agreement; 0.4–0.6 moderate agreement; 0.61–0.8 substantial agreement; 0.81–1 as excellent. A <italic>p</italic>-value &lt; 0.05 was considered statistically significant for all cases. Statistical analysis was performed using R studio (V2.5) package.</p>" ]

[ "<title>Results</title>", "<p id=\"Par19\">The clinical, bioptic and MRI features are summarised in Table ##TAB##0##1##. The median age at prostate biopsy, PSA and PSA-density were 66.6 years, 7.2 ng/ml and 0.2 ng/ml², respectively.</p>", "<p id=\"Par20\">Overall, 41 patients (32.5%) had T2 or T3 clinical stage. Intraprostatic perineural invasion (IPNI) and ISUP group grade &gt; 3 were observed in 26 (20.6%) and 29 (23%) patients, respectively.</p>", "<p id=\"Par21\">The median index lesion size (ILS) and length of capsular involvement (LCI) were 12 and 9 mm (mm), respectively. MRI readings based on PI-RADS v2, ESUR and LIKERT scores are available in the Supplementary material.</p>", "<p id=\"Par22\">Pathology data and ECE analysis are summarised in Table ##TAB##1##2## and Supplementary material, respectively. The median ILS of RP specimen was 16 mm. Accordingly, MRI underestimated the ILS by a 25% in comparison to the true specimen size.</p>", "<p id=\"Par23\">Overall, ECE was found in 35 (27.8%) patients; definitive ISUP grade &gt; 3 was observed in 40 (31.7%) patients.</p>", "<p id=\"Par24\">The following variables were significantly associated to ECE: length of biopsy core (LBC; median: 7 vs 4 mm, <italic>p</italic> &lt; 0.001), intraprostatic perineural invasion (48.6% vs 9.9%, <italic>p</italic> &lt; 0.001), ILS (median: 20 vs 10 mm, <italic>p</italic> &lt; 0.001), LCI (median: 17 vs 7 mm, <italic>p</italic> &lt; 0.001), PSA density (0.24 vs 0.14, <italic>p</italic> = 0.001), and biopsy ISUP grade &gt; 3 (40% vs 16.4%, <italic>p</italic> = 0.003).</p>", "<p id=\"Par25\">At ROC Curve analysis, the best cut-off points for LBC, LCI and ILS were identified as 5.5, 9.5 and 11 mm, respectively.</p>", "<p id=\"Par26\">Overall, both LCI and IPNI showed statistical significance (<italic>p</italic> &lt; 0.001) at multivariate logistic regression model (Table ##TAB##2##3##). The probability to detect ECE with the generated model was 81.4% by including the two variables (Supplementary material). The model was calibrated with an overall p-value of 0.985 by using the Hosmer–Lemeshow test (<italic>R</italic>² = 66%), and the predictive accuracy was evaluated through ROC curve analysis, with an area under the curve (AUC) of 0.83 [95% CI (0.76–0.90)], <italic>p</italic> &lt; 0.001 (Fig. ##FIG##0##1##).</p>", "<p id=\"Par27\">Univariate and multivariate logistic regression analyses were also conducted to identify predictors in the subgroup of patients with seminal vesicle invasion (SVI), but they were not included in the final model because of the low number of events (<italic>n</italic> = 8). Data analysis and comparison of ROC curve between final model and imaging scores models (PI-RADS v2, LIKERT, ESUR) are available in the Supplementary material.</p>", "<p id=\"Par28\">Concordance between the two radiologists was evaluated for ILS, PI-RADS v2, LIKERT and ESUR score, with an overall ICC &gt; 0.6 (<italic>p</italic> &lt; 0.001) in all parameters examined (Supplementary material). Correlation for ILS was 0.66 and 0.62 for L.G. and J.H., respectively (<italic>p</italic> &lt; 0.001).</p>" ]

[ "<title>Discussion</title>", "<p id=\"Par29\">The prognosis of PCa is highly related to tumour stage, and ECE is the most common adverse feature found in histopathology after RP [##REF##23083655##8##]. Inadequate selection of patients with ECE to nerve-sparing RP increases the risk of positive surgical margins with subsequent need for either adjuvant or salvage radiotherapy. In order to improve preoperative risk assessment, numerous nomograms have been developed in the last years, mostly based on the association of multiple clinical risk factors (PSA, biopsy ISUP grade, clinical stage, etc.) [##REF##15076291##9##–##REF##16469587##12##].</p>", "<p id=\"Par30\">The Partin tables were among the first tools developed for the prediction of pathological stage, and have been widely used for several years for surgical planning [##REF##9145716##10##]. Other predictive tools were subsequently developed using different variables (e.g. percentage of positive biopsy cores), but none of them reached a comparable popularity as for the Partin tables [##REF##15076291##9##, ##REF##16469587##12##]. Nevertheless, they could not distinguish unilateral from bilateral ECE [##REF##11744442##13##].</p>", "<p id=\"Par31\">With the advent of mpMRI, further nomograms have been developed in an attempt to improve the accuracy to predict the ECE. Feng et al. first reported that mpMRI could improve the performance of Partin tables and MSKCC nomogram regarding ECE prediction [##REF##26194289##14##].</p>", "<p id=\"Par32\">Giganti et al. developed a nomogram exploiting clinical and MRI parameters with strong accuracy for ECE prediction [##UREF##1##15##], subsequently confirmed by an external validation conducted by Alves et al. [##REF##32307562##16##]. One of the most important features of their model was the excellent concordance between MRI-tumour volume and the ADC map of tumour lesion.</p>", "<p id=\"Par33\">More recently, Gandaglia et al. developed a model to predict ECE, SVI and stage upgrading in patients diagnosed with MRI-targeted and concomitant systematic biopsies [##REF##31547938##17##], achieving an AUC of 73% (ECE), 81% (SVI) and 73% (upgrading) at internal validation.</p>", "<p id=\"Par34\">Nevertheless, a meta-analysis of de Rooij et al. reported high specificity but low sensitivity for mpMRI accuracy in local staging; overall, staging based on MRI alone lacks sensitivity in detecting ECE, especially in case of focal, minimal or microscopic extension because of limitation in spatial resolution [##REF##26215604##18##, ##REF##26260000##19##]. Moreover, the degree of underestimation increases with smaller radiologic tumour size and lower PI-RADS scores [##REF##33026934##20##].</p>", "<p id=\"Par35\">The MRI and histology biopsy features have been variably reported in literature in the recent past as predictors of ECE [##REF##33272865##21##].</p>", "<p id=\"Par36\">Baco et al. found that MRI-tumour LCI well correlated with ECE, with an AUC that outperformed the Partin tables; they also found higher accuracy for microscopic-ECE detection with a 20 mm threshold [##REF##25150643##22##].</p>", "<p id=\"Par37\">Similarly, Kongnyuy et al. identified MRI-LCI as a promising predictor of ECE, positive pathological lymph nodes and biochemical recurrence. The 12.5 mm cut-off showed the highest sensitivity (77%) and specificity (59%) in predicting ECE. The AUC was comparable to that of the Partin tables, outperforming them with LCI and PSA combination [##UREF##2##23##].</p>", "<p id=\"Par38\">Moreover, in a recent meta-analysis of Li et al., the LCI showed high diagnostic performance in predicting ECE, with a pooled sensitivity and specificity of 0.79 and 0.77, respectively. When subgroup analysis was performed comparing different threshold values, lower LCI cut-off values yielded slightly better sensitivity and comparable specificity, without substantial differences between sub-groups [##REF##34881183##24##].</p>", "<p id=\"Par39\">In addition to LCI, IPNI is another acknowledged parameter often associated with ECE, possibly because in the 85% of cases the ECE goes through neurovascular bundles by dissection of the intraprostatic perineural spaces for tissue planes of least resistance [##REF##17495178##25##]</p>", "<p id=\"Par40\">Perineural invasion is defined as the tumour invasion into the perineural sheath, and during the years, it has been associated with tumour progression and prognosis of several malignancies, including prostate cancer [##REF##34994908##26##]. In a recent study on upper urinary tract urothelial carcinomas, Lin et al. found that PNI-positive patients had unfavourable pathological features, including high pathological stage, high tumour grade and lymphovascular invasion, leading to worse progression-free survival (PFS) [##REF##34994908##26##].</p>", "<p id=\"Par41\">Algaba et al. found that IPNI is correlated to cancer volume and higher percentage of extraprostatic cancer [##REF##16084008##27##]. Same finding was reported more recently by Leyh-Bannurah et al., being IPNI the only histological parameter found significantly associated to ECE in their nomogram [##REF##32248363##28##] Nevertheless, IPNI has not yet been reported among the most powerful histological features even in the latest version of the EAU guidelines, so that our finding might prompt its inclusion among the reporting recommendations for the prostatic biopsy.</p>", "<p id=\"Par42\">In our study, we identified both the LCI and the IPNI as predictors of ECE: the LCI cut-off that best correlated to ECE was 9.5 mm, which was a similar finding reported also by Li et al. [##REF##34881183##24##].</p>", "<p id=\"Par43\">Interestingly, in our cohort, the ILS at MRI did not show the same degree of association to the ECE as by the LCI; furthermore, MRI underestimated pathological tumour size by 25%.</p>", "<p id=\"Par44\">Overall, these data may have several implications: (1) Prostate mpMRI alone, including singular features and scoring systems, do not adequately predict ECE, except LCI—especially in combination to a histology biopsy feature, as IPNI in our series. (2) A change of PI-RADS score 5 definition should be prompted in the future PI-RADS updated version: score 5 is attributed to ILS ≥ 15 mm, but this threshold was chosen by the PI-RADS steering committee on the basis of old studies of the’90, when csPCa and ECE were found to be correlated to a tumour volume ≥ 0.5 cm³ (15 mm in major axis) at whole-mount RP specimen [##REF##7506797##29##, ##REF##8709307##30##]. If differences of PI-RADS scores 4 to 5 are to be based on the risk of ECE, LCI should be the preferred MRI feature and with a lower cut-off, as the 15 mm cut-off at MRI might underestimate for a quarter the actual tumour size at specimen. (3) We found excellent or substantial inter-readers agreement for relevant MRI variables, thus strengthening the importance of high-quality imaging acquisition and readers’ skills for their adequate assessment. This latter matter has been popularised with the introduction of a dedicate score (PIQUAL) about the quality of the MRI sequences and the ability to make decision on the basis of it [##REF##32646850##31##].</p>", "<p id=\"Par45\">Main limitations of the study includes (1) the retrospective design of our analysis, even though significant efforts have been done in reviewing MRI images by two radiologists blinded to final histology; (2) the number of cases is limited, especially because MRI implementation in the clinical practice has substantially increased in more recent years; (3) the MRI images were reassessed according to the PI-RADS v2, as the version 2.1 became available on a later stage to that phase of our study. Nevertheless, it is very unlikely that the minor changes in score reporting of the latest version would have had an impact on the outcome of our study, as shown in a recent publication comparing the v2.0 vs v2.1 diagnostic performance with no difference in concordance rates between targeted biopsy and radical prostatectomy (doi: 10.2214/AJR.23.29964).</p>" ]

[ "<title>Conclusions</title>", "<p id=\"Par46\">The mpMRI confirms limitations in predicting the ECE at the RP specimen, even when involving tailored scoring systems. On the other side, MRI-LCI is the most robust factor associated to ECE; in our series, we found a strong predictive accuracy when combined with the IPNI presence. This outcome may prompt a change in the definition of PI-RADS score 5, by reducing IL\nsize cut-off to 10–12mm, or by replacing IL size reporting with LCI (cut-off 9–10mm).</p>" ]

[ "<title>Objectives</title>", "<p id=\"Par1\">To identify the predictive factors of prostate cancer extracapsular extension (ECE) in an institutional cohort of patients who underwent multiparametric MRI of the prostate prior to radical prostatectomy (RP).</p>", "<title>Patients and methods</title>", "<p id=\"Par2\">Overall, 126 patients met the selection criteria, and their medical records were retrospectively collected and analysed; 2 experienced radiologists reviewed the imaging studies. Logistic regression analysis was conducted to identify the variables associated to ECE at whole-mount histology of RP specimens; according to the statistically significant variables associated, a predictive model was developed and calibrated with the Hosmer–Lomeshow test.</p>", "<title>Results</title>", "<p id=\"Par3\">The predictive ability to detect ECE with the generated model was 81.4% by including the length of capsular involvement (LCI) and intraprostatic perineural invasion (IPNI). The predictive accuracy of the model at the ROC curve analysis showed an area under the curve (AUC) of 0.83 [95% CI (0.76–0.90)], <italic>p</italic> &lt; 0.001. Concordance between radiologists was substantial in all parameters examined (<italic>p</italic> &lt; 0.001). Limitations include the retrospective design, limited number of cases, and MRI images reassessment according to PI-RADS v2.0.</p>", "<title>Conclusion</title>", "<p id=\"Par4\">The LCI is the most robust MRI factor associated to ECE; in our series, we found a strong predictive accuracy when combined in a model with the IPNI presence. This outcome may prompt a change in the definition of PI-RADS score 5.</p>", "<title>Supplementary Information</title>", "<p>The online version contains supplementary material available at 10.1007/s00345-023-04720-5.</p>", "<title>Keywords</title>", "<p>Open access funding provided by Università degli Studi di Sassari within the CRUI-CARE Agreement.</p>" ]

[ "<title>Supplementary Information</title>", "<p>Below is the link to the electronic supplementary material.</p>" ]

[ "<title>Acknowledgements</title>", "<p>We are thankful to Cristina Esquinas Lopez, Professor of Statistics at at Vall d’Hebron Research Institute, Universitat Autonoma de Barcelona (Spain) for undertaking the statistical analysis on the manuscript. <underline>[email protected]</underline></p>", "<title>Funding</title>", "<p>Open access funding provided by Università degli Studi di Sassari within the CRUI-CARE Agreement. None.</p>", "<title>Declarations</title>", "<title>Conflict of interest</title>", "<p id=\"Par47\">The authors have nothing to disclose.</p>" ]

[ "<fig id=\"Fig1\"><label>Fig. 1</label><caption><p>ROC curve analysis for predictive model</p></caption></fig>" ]

[ "<table-wrap id=\"Tab1\"><label>Table 1</label><caption><p>Patients’ clinical data</p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th align=\"left\">Variables</th><th align=\"left\">Median (IQR)</th><th align=\"left\"><italic>N</italic> (%)</th></tr></thead><tbody><tr><td align=\"left\">Age at biopsy (yr)</td><td char=\"(\" align=\"char\">66.6 (61.5–68.9)</td><td char=\"(\" align=\"char\"/></tr><tr><td align=\"left\" colspan=\"3\">Clinical stage</td></tr><tr><td align=\"left\"> T1</td><td char=\"(\" align=\"char\"/><td char=\"(\" align=\"char\">85 (67.5)</td></tr><tr><td align=\"left\"> T2 + T3</td><td char=\"(\" align=\"char\"/><td char=\"(\" align=\"char\">41 (32.5)</td></tr><tr><td align=\"left\">Baseline PSA at biopsy (ng/ml)</td><td char=\"(\" align=\"char\">7.2 (5.4–10.4)</td><td char=\"(\" align=\"char\"/></tr><tr><td align=\"left\">Max length core (mm)</td><td char=\"(\" align=\"char\">5 (3–8)</td><td char=\"(\" align=\"char\"/></tr><tr><td align=\"left\" colspan=\"3\">Intraprostatic perineural invasion</td></tr><tr><td align=\"left\"> Yes</td><td char=\"(\" align=\"char\"/><td char=\"(\" align=\"char\">26 (20.6)</td></tr><tr><td align=\"left\"> No</td><td char=\"(\" align=\"char\"/><td char=\"(\" align=\"char\">100 (79.4)</td></tr><tr><td align=\"left\">Prostatic volume Observer #1 (cc)</td><td char=\"(\" align=\"char\">45.5 (32–65.2)</td><td char=\"(\" align=\"char\"/></tr><tr><td align=\"left\">Prostatic volume Observer #2 (cc)</td><td char=\"(\" align=\"char\">50 (35–74)</td><td char=\"(\" align=\"char\"/></tr><tr><td align=\"left\">PSA-density at biopsy (ng/cc)</td><td char=\"(\" align=\"char\">0.17 (0.11–0.26)</td><td char=\"(\" align=\"char\"/></tr><tr><td align=\"left\">Number of targeted biopsies (n. 49, 38.89%)</td><td char=\"(\" align=\"char\">3 (3–4)</td><td char=\"(\" align=\"char\"/></tr><tr><td align=\"left\">Number of positive targeted biopsy (n. 49, 38.89%)</td><td char=\"(\" align=\"char\">2 (0–3)</td><td char=\"(\" align=\"char\"/></tr><tr><td align=\"left\" colspan=\"3\">ISUP group at biopsy</td></tr><tr><td align=\"left\"> 1</td><td char=\"(\" align=\"char\"/><td char=\"(\" align=\"char\">41 (32.5)</td></tr><tr><td align=\"left\"> 2</td><td char=\"(\" align=\"char\"/><td char=\"(\" align=\"char\">47 (37.3)</td></tr><tr><td align=\"left\"> 3</td><td char=\"(\" align=\"char\"/><td char=\"(\" align=\"char\">9 (7.1)</td></tr><tr><td align=\"left\"> 4</td><td char=\"(\" align=\"char\"/><td char=\"(\" align=\"char\">18 (14.3)</td></tr><tr><td align=\"left\"> 5</td><td char=\"(\" align=\"char\"/><td char=\"(\" align=\"char\">11 (8.8)</td></tr><tr><td align=\"left\" colspan=\"3\">Type of MRI</td></tr><tr><td align=\"left\"> T2w + DWI + DCE</td><td char=\"(\" align=\"char\"/><td char=\"(\" align=\"char\">123 (97.6)</td></tr><tr><td align=\"left\"> T2w + DWI</td><td char=\"(\" align=\"char\"/><td char=\"(\" align=\"char\">3 (2.4)</td></tr><tr><td align=\"left\"><p>Index Lesion size (mm), Observer #1</p><p>Median (IQR)</p></td><td char=\"(\" align=\"char\">12 (8–19)</td><td char=\"(\" align=\"char\"/></tr><tr><td align=\"left\"><p>Index Lesion size (mm), Observer #2</p><p>Median (IQR)</p></td><td char=\"(\" align=\"char\">12 (8–18)</td><td char=\"(\" align=\"char\"/></tr><tr><td align=\"left\"><p>Length of capsular involvement (mm)</p><p>Median (IQR)</p></td><td char=\"(\" align=\"char\">9 (3.7–15.2)</td><td char=\"(\" align=\"char\"/></tr></tbody></table></table-wrap>", "<table-wrap id=\"Tab2\"><label>Table 2</label><caption><p>Patients’ pathology data (<italic>N</italic> = 126 patients)</p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th align=\"left\">Variables</th><th align=\"left\">Median (IQR)</th><th align=\"left\"><italic>N</italic> (%)</th></tr></thead><tbody><tr><td align=\"left\">Index Lesion size (mm)</td><td char=\"–\" align=\"char\">16 (10–21)</td><td char=\"(\" align=\"char\"/></tr><tr><td align=\"left\" colspan=\"3\">Histology</td></tr><tr><td align=\"left\"> pT0</td><td char=\".\" align=\"char\"/><td char=\"(\" align=\"char\">2 (1.6)</td></tr><tr><td align=\"left\"> pT2a</td><td char=\".\" align=\"char\"/><td char=\"(\" align=\"char\">25 (19.9)</td></tr><tr><td align=\"left\"> pT2b</td><td char=\".\" align=\"char\"/><td char=\"(\" align=\"char\">8 (6.3)</td></tr><tr><td align=\"left\"> pT2c</td><td char=\".\" align=\"char\"/><td char=\"(\" align=\"char\">56 (44.4)</td></tr><tr><td align=\"left\"> pT3a</td><td char=\".\" align=\"char\"/><td char=\"(\" align=\"char\">27 (21.4)</td></tr><tr><td align=\"left\"> pT3b</td><td char=\".\" align=\"char\"/><td char=\"(\" align=\"char\">8 (6.4)</td></tr><tr><td align=\"left\" colspan=\"3\">ISUP grade group</td></tr><tr><td align=\"left\"> 0</td><td char=\".\" align=\"char\"/><td char=\"(\" align=\"char\">2 (1.6)</td></tr><tr><td align=\"left\"> 1</td><td char=\".\" align=\"char\"/><td char=\"(\" align=\"char\">15 (11.9)</td></tr><tr><td align=\"left\"> 2</td><td char=\".\" align=\"char\"/><td char=\"(\" align=\"char\">51 (40.5)</td></tr><tr><td align=\"left\"> 3</td><td char=\".\" align=\"char\"/><td char=\"(\" align=\"char\">18 (14.4)</td></tr><tr><td align=\"left\"> 4</td><td char=\".\" align=\"char\"/><td char=\"(\" align=\"char\">23 (18.2)</td></tr><tr><td align=\"left\"> 5</td><td char=\".\" align=\"char\"/><td char=\"(\" align=\"char\">17 (13.4)</td></tr></tbody></table></table-wrap>", "<table-wrap id=\"Tab3\"><label>Table 3</label><caption><p>Logistic Regression for ECE</p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th align=\"left\" rowspan=\"2\"/><th align=\"left\" colspan=\"3\">Univariate</th><th align=\"left\" colspan=\"3\">Multivariate</th><th align=\"left\" colspan=\"3\">Multivariate*</th></tr><tr><th align=\"left\">OR</th><th align=\"left\">95% CI</th><th align=\"left\"><italic>p</italic> value</th><th align=\"left\">OR</th><th align=\"left\">95% CI</th><th align=\"left\"><italic>p</italic> value</th><th align=\"left\">OR</th><th align=\"left\">95% CI</th><th align=\"left\"><italic>p</italic> value</th></tr></thead><tbody><tr><td align=\"left\">Clinical stage</td><td align=\"left\"/><td align=\"left\"/><td align=\"left\"/><td align=\"left\"/><td align=\"left\"/><td align=\"left\"/><td align=\"left\" colspan=\"3\"/></tr><tr><td align=\"left\"> T1</td><td align=\"left\">Ref</td><td align=\"left\">–</td><td align=\"left\" rowspan=\"2\"><bold><italic>0.002</italic></bold></td><td align=\"left\">Ref</td><td align=\"left\">–</td><td align=\"left\" rowspan=\"2\"><italic>0.971</italic></td><td align=\"left\" rowspan=\"8\" colspan=\"3\"/></tr><tr><td align=\"left\"> T2 + T3</td><td align=\"left\">3.72</td><td align=\"left\">1.65–8.58</td><td align=\"left\">1.02</td><td align=\"left\">0.31–3.22</td></tr><tr><td align=\"left\">PSA density (ng/ml²)</td><td align=\"left\"/><td align=\"left\"/><td align=\"left\"/><td align=\"left\"/><td align=\"left\"/><td align=\"left\"/></tr><tr><td align=\"left\">  ≤ 0.15</td><td align=\"left\">Ref</td><td align=\"left\">–</td><td align=\"left\" rowspan=\"2\"><bold><italic>0.003</italic></bold></td><td align=\"left\">Ref</td><td align=\"left\">–</td><td align=\"left\" rowspan=\"2\"><italic>0.406</italic></td></tr><tr><td align=\"left\">  &gt; 0.15</td><td align=\"left\">4.16</td><td align=\"left\">1.71–11.29</td><td align=\"left\">1.75</td><td align=\"left\">0.48–6.97</td></tr><tr><td align=\"left\">Max length core (mm)</td><td align=\"left\"/><td align=\"left\"/><td align=\"left\"/><td align=\"left\"/><td align=\"left\"/><td align=\"left\"/></tr><tr><td align=\"left\">  ≤ 5.5</td><td align=\"left\">Ref</td><td align=\"left\">–</td><td align=\"left\" rowspan=\"2\"><bold> &lt; </bold><bold><italic>0.001</italic></bold></td><td align=\"left\">Ref</td><td align=\"left\">–</td><td align=\"left\" rowspan=\"2\"><italic>0.096</italic></td></tr><tr><td align=\"left\">  &gt; 5.5</td><td align=\"left\">5.93</td><td align=\"left\">2.57–14.55</td><td align=\"left\">2.85</td><td align=\"left\">0.85–10.36</td></tr><tr><td align=\"left\">IPNI</td><td align=\"left\"/><td align=\"left\"/><td align=\"left\"/><td align=\"left\"/><td align=\"left\"/><td align=\"left\"/><td align=\"left\"/><td align=\"left\"/><td align=\"left\"/></tr><tr><td align=\"left\"> No</td><td align=\"left\">Ref</td><td align=\"left\">-</td><td align=\"left\" rowspan=\"2\"><bold> &lt; </bold><bold><italic>0.001</italic></bold></td><td align=\"left\">Ref</td><td align=\"left\">–</td><td align=\"left\" rowspan=\"2\"><bold><italic>0.001</italic></bold></td><td align=\"left\">Ref</td><td align=\"left\">–</td><td align=\"left\" rowspan=\"2\"><bold> &lt; </bold><bold><italic>0.001</italic></bold></td></tr><tr><td align=\"left\"> Yes</td><td align=\"left\">8.61</td><td align=\"left\">3.31–22.37</td><td align=\"left\">10.24</td><td align=\"left\">2.57–40.85</td><td align=\"left\">8.17</td><td align=\"left\">2.78–26.32</td></tr><tr><td align=\"left\">ISUP grade at biopsy</td><td align=\"left\"/><td align=\"left\"/><td align=\"left\"/><td align=\"left\"/><td align=\"left\"/><td align=\"left\"/><td align=\"left\" colspan=\"3\"/></tr><tr><td align=\"left\">  ≤ 3</td><td align=\"left\">Ref</td><td align=\"left\">–</td><td align=\"left\" rowspan=\"2\"><bold> &lt; </bold><bold><italic>0.001</italic></bold></td><td align=\"left\">Ref</td><td align=\"left\">–</td><td align=\"left\" rowspan=\"2\"><italic>0.078</italic></td><td align=\"left\" rowspan=\"2\" colspan=\"3\"/></tr><tr><td align=\"left\">  &gt; 3</td><td align=\"left\">4.39</td><td align=\"left\">1.93–10.66</td><td align=\"left\">2.81</td><td align=\"left\">0.91–9.37</td></tr><tr><td align=\"left\">LCI (mm)</td><td align=\"left\"/><td align=\"left\"/><td align=\"left\"/><td align=\"left\"/><td align=\"left\"/><td align=\"left\"/><td align=\"left\"/><td align=\"left\"/><td align=\"left\"/></tr><tr><td align=\"left\">  ≤ 9.5</td><td align=\"left\">Ref</td><td align=\"left\">–</td><td align=\"left\" rowspan=\"2\"><bold> &lt; </bold><bold><italic>0.001</italic></bold></td><td align=\"left\">Ref</td><td align=\"left\">–</td><td align=\"left\" rowspan=\"2\"><bold><italic>0.002</italic></bold></td><td align=\"left\">Ref</td><td align=\"left\">–</td><td align=\"left\" rowspan=\"2\"><bold> &lt; </bold><bold><italic>0.001</italic></bold></td></tr><tr><td align=\"left\">  &gt; 9.5</td><td align=\"left\">11.61</td><td align=\"left\">4.41–36.77</td><td align=\"left\">14.34</td><td align=\"left\">2.87–95.10</td><td align=\"left\">11.09</td><td align=\"left\">3.91–38.18</td></tr><tr><td align=\"left\">ILS at MRI (mm)</td><td align=\"left\"/><td align=\"left\"/><td align=\"left\"/><td align=\"left\"/><td align=\"left\"/><td align=\"left\"/><td align=\"left\" colspan=\"3\"/></tr><tr><td align=\"left\">  ≤ 11</td><td align=\"left\">Ref</td><td align=\"left\">–</td><td align=\"left\" rowspan=\"2\"><bold> &lt; </bold><bold><italic>0.001</italic></bold></td><td align=\"left\">Ref</td><td align=\"left\">–</td><td align=\"left\" rowspan=\"2\"><italic>0.673</italic></td><td align=\"left\" rowspan=\"2\" colspan=\"3\"/></tr><tr><td align=\"left\">  &gt; 11</td><td align=\"left\">8.86</td><td align=\"left\">3.38–27.96</td><td align=\"left\">0.69</td><td align=\"left\">0.12–3.81</td></tr></tbody></table></table-wrap>" ]

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[ "<supplementary-material content-type=\"local-data\" id=\"MOESM1\"></supplementary-material>" ]

[ "<table-wrap-foot><p><italic>yr</italic> years, <italic>ng</italic> nanograms, <italic>ml</italic> millilitres, <italic>mm</italic> millimetres, <italic>cc</italic> cubic centimetres</p></table-wrap-foot>", "<table-wrap-foot><p><italic>mm</italic> millimetres</p></table-wrap-foot>", "<table-wrap-foot><p><bold>Multivariate</bold>: Analysis conducted considering all parameters; <bold>Multivariate*</bold>: Analysis conducted considering all parameters with stepwise method for statistical significance (<italic>p</italic> &lt; 0.05)</p><p><italic>ECE</italic> extracapsular extension, <italic>OR</italic> odds-ratio, <italic>IPNI</italic> intraprostatic perineural invasion, <italic>LCI</italic> length of capsular involvement, <italic>ILS</italic> index lesion size</p><p>Bold Italic are reported the statistically significant <italic>p</italic>-values</p></table-wrap-foot>", "<fn-group><fn><p><bold>Publisher's Note</bold></p><p>Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.</p></fn></fn-group>" ]

[ "<graphic xlink:href=\"345_2023_4720_Fig1_HTML\" id=\"MO1\"/>" ]

[ "<media xlink:href=\"345_2023_4720_MOESM1_ESM.docx\"><caption><p>Supplementary file1 (DOCX 129 KB)</p></caption></media>" ]

{ "acronym": [], "definition": [] }

[ "<title>Introduction</title>", "<p id=\"Par16\">Invasive lobular carcinoma (ILC) is the second most common histological subtype of breast cancer after invasive ductal carcinoma (IDC), representing 10–15% of all cases [##REF##25848941##1##]. In the metastatic setting, ILC differs in its pattern of metastatic sites, often involving the bone, and gastrointestinal tract [##REF##28757653##2##–##REF##37020372##4##]. Additionally, several studies demonstrate worse overall survival (OS) in metastatic ILC compared to IDC, even when evaluating patients with similar receptor subtypes [##REF##37020372##4##–##REF##18704988##6##].</p>", "<p id=\"Par17\">While investigators have evaluated surgical outcomes by histologic subtype in early-stage disease, there are scant data evaluating the use of primary-site surgery in the metastatic setting in those with ILC versus IDC. The current recommended therapy for metastatic breast cancer is systemic therapy, with local therapy reserved for palliation of symptoms [##REF##17891444##7##]. While retrospective studies and institutional series have found associations between primary tumor resection and longer survival in those with metastatic breast cancer [##REF##17522944##8##], most randomized control trials have not demonstrated such a survival advantage [##REF##34995128##9##–##REF##26363985##11##]. A previous study found that ILC patients with bone-only metastases had longer OS than those with visceral metastases when given a combination of chemotherapy and surgery, but it is unclear whether this reflects the more indolent course of osseous metastases, or an ILC specific effect of treatment [##REF##37020372##4##, ##REF##17687611##12##].</p>", "<p id=\"Par18\">As such, whether histologic subtype should factor into patient selection for primary-site surgery is unknown. Prior analyses have suggested that if surgery of the primary tumor is associated with improved survival, this may be more likely in those with hormone receptor (HR) positive disease, or bone-only metastases [##REF##29777404##10##, ##REF##20101736##13##]. Given that ILC is largely HR-positive, HER2-negative, and has a propensity for bone metastases, we wondered if primary-site surgery use differed in patients with metastatic ILC compared to IDC.</p>", "<p id=\"Par19\">We used the National Cancer Database (NCDB) to evaluate differences in practice patterns and management of patients with metastatic ILC compared to metastatic IDC. Specifically, we investigated the following questions: whether rates of primary-site surgery differ by histologic subtype and whether selection factors associated with undergoing primary-site surgery differ by histologic subtype. As secondary endpoints, we evaluated the use of chemotherapy and radiotherapy relative to surgery by histologic subtype, and the association between primary-site surgery and OS in ILC and IDC cohorts in both unmatched and propensity score matched multivariable models.</p>" ]

[ "<title>Methods</title>", "<title>Data source and study cohort</title>", "<p id=\"Par20\">The NCDB is a national comprehensive clinical surveillance resource representing over 70% of all newly diagnosed cancer cases in the United States and includes patient demographics, clinical information, and survival outcomes [##REF##19636004##14##, ##REF##28241198##15##]. Participants User Files from 2010 to 2016 were used. Due to the de-identified nature of the public-access user files, the study was exempted from institutional review board approval.</p>", "<p id=\"Par21\">Since most ILC tumors are HR-positive and HER2-negative, we limited analysis to tumors with this receptor subtype. Tumors that were estrogen receptor (ER) and/or progesterone receptor (PR) positive were considered HR-positive. Histology codes were used to identify cohorts, with the ILC cohort comprising those with codes for ILC or mixed ILC/IDC (histology codes 8520, 8522, and 8524 if invasive behavior), and the IDC cohort comprising codes for IDC or invasive mammary carcinoma not otherwise specified (histology codes 8500, 8501, 8502, 8503, and 8523 if invasive behavior). We excluded patients with stage I-III disease, histologic subtypes other than IDC or ILC, individuals who died within 6 months of their diagnosis, and those missing critical clinical information including disease stage, HR-status, HER2-status, or treatment type.</p>", "<title>Clinical measures</title>", "<p id=\"Par22\">Charlson-Deyo Co-Morbidity Index (CDCI) was recorded as a measure of severity of co-morbid conditions. Age at diagnosis was subdivided into under 50 years and over 50 years to estimate pre- and post-menopausal status, respectively. Metastatic disease sites were categorized as bone-only, visceral-only, bone and visceral, or unknown [##REF##37020372##4##]. We utilized data on treatment facility type, insurance status, and median income in univariate and multivariate analyses.</p>", "<title>Statistical methods</title>", "<p id=\"Par23\">We compared clinicopathologic and demographic features between the ILC and IDC cohorts using chi-square tests for categorical variables and t-tests for continuous variables. We investigated factors associated with receiving surgery for the primary tumor, radiotherapy, chemotherapy, and timing of chemotherapy relative to surgery by histologic subtype. For univariate analyses, we used Kaplan-Meier plots and log-rank tests to assess associations between receipt of surgery and OS by histologic subtype, and by timing of chemotherapy and receipt of radiotherapy. We also evaluated treatment facility type, insurance type, and median income quartiles by surgery and histologic subtype.</p>", "<p id=\"Par24\">For multivariate analysis, we used Cox proportional hazards models to account for confounders with OS. The multivariable model included age, CDCI (0/1+), metastatic site (bone-only versus all other), and receptor subtype (ER-positive, PR-positive, HER2-negative versus ER-positive, PR-negative, HER2-negative).</p>", "<p id=\"Par25\">Finally, we performed propensity score matching including age, tumor grade, receptor subtype, site of metastatic disease, CDCI (0/1+), and treating facility variables to account for likelihood of having primary-site surgery to determine the association between primary-site surgery and OS. Within each histology category among those who had survival data available patients who had surgery (ILC n = 1,444; IDC n = 4,924) were matched to patients who did not have surgery (ILC n = 3,553; IDC n = 10,894) using the greedy nearest neighbor matching algorithm. Matching was restricted to observations that had propensity scores in the extended common support region (ILC 0.05–0.71; IDC 0.06–0.66), which extends the common support region by 0.25 times a pooled estimate of the common standard deviation of the logit of the propensity score. The PSMATCH procedure in SAS version 9.4 was used to perform matching. To account for the matched nature of the sample, log-rank tests and Cox models were stratified on the matched pairs.</p>", "<p id=\"Par26\">Hypothesis tests were two-sided, and the significance threshold was set to 0.05. Statistical analyses were performed using Stata 16 and SAS version 9.4.</p>" ]

[ "<title>Results</title>", "<title>Study cohort</title>", "<p id=\"Par27\">There were 100,147 patients with stage IV breast cancer, of whom 25,294 had HR-positive, HER2-negative invasive lobular or ductal histology, and met study criteria (Fig. ##FIG##0##1##). Of these patients, 19,171 (75.8%) had IDC and 6,123 (24.2%) had ILC. Within the ILC cohort, 4,484 (73.2%) had pure ILC, with the remaining having mixed ILC/IDC. Median follow-up time of the ILC cohort was 27.2 months (IQ range 14.7–41.5 months), which was similar to the median follow-up time of 26.8 months (IQ range 14.6–42.6 months) for the IDC cohort (Table ##TAB##0##1##). Patients with ILC were slightly older than those with IDC (mean age 64 years versus 61 years, <italic>p</italic> &lt; 0.001) and differed significantly by race (<italic>p</italic> &lt; 0.001), with a higher proportion of White patients (79.4% versus 74.2%, <italic>p</italic> &lt; 0.0001) (Table ##TAB##0##1##). Additionally, ILC and IDC patients differed by tumor grade (<italic>p</italic> &lt; 0.001), with ILC patients having a higher proportion of grade 1 tumors (22.1% versus 9.48%, p &lt; 0.0001) and more bone-only metastases than those with IDC (60.8% versus 45.2%, <italic>p</italic> &lt; 0.001).</p>", "<p id=\"Par28\">\n\n</p>", "<p id=\"Par53\">\n\n</p>", "<p id=\"Par30\">While overall most patients were treated at community cancer centers, slightly more ILC patients than IDC patients were treated at academic centers (ILC 36.0% versus IDC 33.5%, <italic>p</italic> = 0.001) (Table ##TAB##0##1##). ILC patients were significantly more likely to have public insurance (ILC 54.7% versus IDC 51.6%) and less likely to be uninsured (ILC 3.82% versus IDC 5.15%, <italic>p</italic> &lt; 0.0001). Patients with ILC had higher median income, with 39.7% in the highest quartile compared to 35.7% of IDC patients in the highest median income quartile (<italic>p</italic> &lt; 0.0001).</p>", "<title>Primary-site surgery by histologic subtype</title>", "<p id=\"Par31\">In the overall study population, 7,158 (28.3%) underwent primary-site surgery. Although the absolute difference is small, primary-site surgery was performed less often in those with metastatic ILC than those with metastatic IDC (n = 1,644 [26.8%] versus n = 5,514 [28.8%] respectively, <italic>p</italic> = 0.004) (Table ##TAB##1##2##). This difference was more pronounced when restricting the analysis to patients with pure ILC, where only 23.8% underwent primary-site surgery. Additionally, the slightly lower rate of primary-site surgery among patients with ILC compared to IDC was observed among those with bone-only or unknown site of metastases, but not among those with visceral metastases (Fig. ##FIG##1##2##). Among those who had primary-site surgery in both histologic cohorts, the site of metastatic disease was significantly more likely to be bone-only compared to other sites (ILC 63.5%; IDC 48.5%, <italic>p</italic> &lt; 0.001) (Table ##TAB##2##3##).</p>", "<p id=\"Par54\">\n\n</p>", "<p id=\"Par32\">\n\n</p>", "<p id=\"Par33\">For both cohorts, patients with private insurance were significantly more likely to receive primary-site surgery (ILC 46.4%; IDC 49.4%, <italic>p</italic> &lt; 0.0001) compared to patients with public insurance (Table ##TAB##2##3##). Both ILC and IDC patients who received surgery were equally likely to have been treated at an academic treating facility (ILC n = 415 [26.3%]; IDC n = 1,318 [26.2%]) compared to a community setting (ILC n = 1,164 [73.7%]; IDC n = 3,714 [73.8%]).</p>", "<p id=\"Par55\">\n\n</p>", "<p id=\"Par34\">Among those who underwent primary-site surgery (n = 7,158), the differences by histology reflected those seen in the overall study population. Those with ILC were older (ILC mean age 61.7 years versus IDC 58.1 years, <italic>p</italic> &lt; 0.001), had more T3 tumors (ILC 30.8% versus IDC 15.5%, <italic>p</italic> &lt; 0.001), had more N3 nodal status (ILC 35.3% versus IDC 20.5%, <italic>p</italic> &lt; 0.001), and had more grade 2 disease (ILC 61.4% versus IDC 45.9%, <italic>p</italic> &lt; 0.001) (Table ##TAB##0##1##). There was no significant difference in mastectomy rate between the two cohorts (Table ##TAB##1##2##). Positive surgical margins were significantly more common in those with ILC who underwent lumpectomy compared to IDC (15.7% versus 11.2%, <italic>p =</italic> 0.025) (Table ##TAB##0##1##).</p>", "<p id=\"Par35\">The factors associated with undergoing primary-site surgery were similar in the ILC and IDC cohorts. In both groups, primary-site surgery was less common in older patients, and more common in those with larger tumors (except T4) and higher N category (Table ##TAB##3##4##). The odds of undergoing primary-site surgery were highest for ILC patients with pathologic stage T3 disease versus T1 (OR 2.65, 95% CI 1.17–3.51, <italic>p</italic> = 0.002; Table ##TAB##3##4##) whereas the odds of surgery for patients with IDC were highest in pathologic stage T2 disease versus T1 (OR 1.71, 95% CI 1.30–2.25, p &lt; 0.001). In both groups, those with T4 disease had significantly lower odds of primary-site surgery compared to T1 (Table ##TAB##3##4##). Over time, the odds of undergoing primary-site surgery decreased. Specifically, the odds of surgery decreased by 16% per each additional year of diagnosis (OR 0.84 and <italic>p</italic> &lt; 0.001 for both groups; 95% CI 0.82–0.87 in ILC group; 95% CI 0.83–0.85 in IDC group, Table ##TAB##3##4##).</p>", "<p id=\"Par56\">\n\n</p>", "<title>Radiotherapy by histologic subtype</title>", "<p id=\"Par36\">The use of radiotherapy overall was lower for patients with ILC than IDC (29.1% versus 37.9%, <italic>p &lt;</italic> 0.001) (Table ##TAB##1##2##). In IDC patients who had surgery, 51.5% also had radiation, while in ILC patients who had surgery, 42.5% also had radiation (Table ##TAB##2##3##). Among those who received radiotherapy, there was no difference in the rate of radiation to local versus distant sites by histologic subtype (<italic>p</italic> = 0.55).</p>", "<title>Use and timing of chemotherapy</title>", "<p id=\"Par37\">More IDC patients received chemotherapy (41.3% ILC versus 47.4% IDC, <italic>p</italic> &lt; 0.001), while those with ILC were more likely to receive endocrine therapy (83.5% ILC versus 78.6% IDC, <italic>p</italic> &lt; 0.001). For those who had primary-site surgery, the sequence of chemotherapy and surgery differed by histologic subtype; while 40.5% of patients with IDC had chemotherapy prior to surgery, only 29.0% of patients with ILC had chemotherapy prior to surgery (<italic>p</italic> &lt; 0.001).</p>", "<title>Survival analyses in unmatched cohorts</title>", "<p id=\"Par38\">Overall, patients with ILC had slightly but significantly shorter OS than those with IDC (median 38 months ILC versus 40 months IDC, <italic>p</italic> = 0.006). In both cohorts, primary-site surgery was associated with significantly improved OS (Table ##TAB##0##1##). In the ILC cohort, undergoing primary-site surgery was associated with 35% lower risk of death compared to those who did not undergo surgery (HR 0.65, 95% CI 0.57–0.68, <italic>p</italic> &lt; 0.001) (Fig. ##FIG##2##3##). This association persisted when controlling for age, CDCI (0/1+), metastatic site, and receptor subtype (HR 0.64, 95% CI 0.58–0.70, <italic>p</italic> &lt; 0.001). The timing of surgery (before or after systemic chemotherapy) was not significantly associated with OS among those with ILC in unadjusted analysis nor after controlling for age, CDCI (0/1+), metastatic site, and receptor subtype.</p>", "<p id=\"Par39\">\n\n</p>", "<p id=\"Par40\">Similarly, patients in the IDC cohort who underwent primary-site surgery had an associated 40% lower risk of death compared to those without surgery (HR 0.60, 95% CI 0.57–0.63, <italic>p</italic> &lt; 0.001) (Fig. ##FIG##2##3##). This remained true after controlling for age, CDCI (0/1+), metastatic site, and receptor subtype (HR 0.61, 95% CI 0.58–0.64, <italic>p</italic> &lt; 0.001). Unlike for those with ILC, timing of surgery was significantly associated with OS. Patients with IDC who had chemotherapy before surgery had 24% less risk of death compared to those who had surgery prior to chemotherapy (HR 0.76, 95% CI 0.70–0.84, <italic>p</italic> &lt; 0.001). This association persisted after controlling for age, CDCI (0/1+), site of metastasis, and receptor subtype (HR 0.83, 95% CI 0.76–0.92, <italic>p</italic> &lt; 0.001).</p>", "<p id=\"Par41\">The type of surgery (mastectomy versus lumpectomy) was not significantly associated with different OS in either group. However, we found a significant statistical interaction between having surgery for the primary tumor and the site of radiation. Primary-site surgery was associated with a greater reduction in risk of death among those who had local radiation compared to those who had distant radiation (HR 0.35, 95% CI 0.3–0.4 versus HR 0.67, 95% CI 0.61–0.74 respectively, test of interaction <italic>p</italic> &lt; 0.001). This interaction between primary-site surgery and site of radiation was similar in both the ILC and IDC cohorts separately.</p>", "<title>Survival analyses in propensity score matched cohorts</title>", "<p id=\"Par42\">The propensity score model included age, tumor grade, receptor subtype, metastatic site, CDCI (0/1+), and treating facility variables, with 3,089 ILC patients (991 with surgery, 2,098 without surgery) and 11,216 IDC patients (3,429 with surgery, 7,787 without surgery) in each cohort with available data for matching. In both the ILC and IDC matched samples there was still a significant association between having surgery for the primary tumor and improved OS (ILC HR 0.71, 95% CI 0.59–0.85, <italic>p</italic> &lt; 0.001; IDC HR 0.67, 95% CI 0.61–0.74, <italic>p</italic> &lt; 0.001).</p>" ]

[ "<title>Discussion</title>", "<p id=\"Par43\">Recent randomized trial data suggest that the role of primary-site surgery in the management of patients with metastatic breast cancer is limited to local control in select cases, with no evidence of impact on OS [##REF##34995128##9##–##REF##26363985##11##]. However, optimal selection criteria for primary-site surgery are unknown, with decisions being made on an individualized basis in clinical practice [##REF##20101736##13##]. Given the known differences in surgical management in the early stage setting and disease patterns in the metastatic setting between ILC and IDC, we explored whether the use of primary-site surgery differs in HR-positive HER2-negative metastatic lobular versus ductal breast cancer.</p>", "<p id=\"Par44\">In this cohort of 25,294 patients from the NCDB, we found that a high proportion of patients overall underwent primary-site surgery in the setting of metastatic breast cancer (28.3%). It is important to note that these data represent patients diagnosed between 2010 and 2016, prior to the publication of randomized trials demonstrating the lack of OS benefit from primary-site surgery [##REF##34995128##9##–##REF##26363985##11##]. Indeed, we found that the odds of undergoing primary-site surgery significantly decreased over time. While primary-site surgery was more common in those with bone-only metastases, and those with ILC were more likely to have such a disease pattern, the overall usage of primary-site surgery in the ILC cohort was slightly but significantly lower than in the IDC cohort. Although a slightly smaller proportion of patients with ILC had primary-site surgery, the majority of factors associated with receiving surgery did not differ between the lobular and ductal groups; in both groups, primary-site surgery was more common among younger patients, those with T2 or T3 tumors, more nodal disease, and private insurance.</p>", "<p id=\"Par45\">Interestingly, while patients with ILC had larger tumors than those with IDC, there was no difference in the rate of mastectomy by histologic subtype. This differs from the early-stage setting, where lobular histology is associated with higher mastectomy rates. Similar to the early-stage setting, however, those with metastatic ILC who had primary-site surgery experienced significantly higher positive margin rates after lumpectomy than those with metastatic IDC. This suggests that the local control benefit of primary-site surgery might be attenuated in those with ILC, who may require more extensive surgery to achieve negative margins. We did find an association between local radiotherapy and improved OS in this cohort; whether this association reflects a relationship between improved local control and survival outcomes versus improved outcomes in those selected to have radiation is unknown. Of note, patients with ILC were significantly less likely to receive radiation than those with IDC, which is consistent with other studies [##REF##35949419##16##, ##REF##33415073##17##].</p>", "<p id=\"Par46\">Interestingly, we found significantly lower odds of primary-site surgery in patients with T4 tumors in both lobular and ductal groups. Since palliation is the most accepted purpose of primary-site surgery in the stage IV setting, we would have expected higher rates of surgery in those with T4 tumors. Alternatively, these tumors may have been deemed unresectable; one of the challenges of analyzing this retrospective dataset is the inability to discern the reasons for performing primary-site surgery.</p>", "<p id=\"Par47\">This limitation likely impacts the strong association between primary-site surgery and improved OS that we found in both groups. For example, in both the ILC and IDC cohorts, patients who had private insurance were more likely to have surgery compared to patients who had public insurance. The improved outcomes associated with primary-site surgery may reflect improved access to care as opposed to a biologic effect of surgery. This is consistent with prior data; retrospective series tend to show a survival advantage associated with primary-site surgery [##REF##37020372##4##, ##REF##17687611##12##, ##REF##20101736##13##], whereas recent randomized trial data do not [##REF##34995128##9##, ##REF##26363985##11##]. Such discrepancies suggest selection bias in which patients undergo primary-site surgery. We are likely unable to account for the many factors that influence why surgery would be used in some patients versus others despite using propensity score matched models.</p>", "<p id=\"Par48\">Of more interest, perhaps, is the finding that the use of pre-operative systemic therapy was associated with improved OS in the IDC cohort, but not in the ILC cohort. We suspect that pre-operative systemic therapy in the IDC cohort may have helped to select patients who would have more durable response to therapy, and therefore have improved OS. In contrast, response to therapy in those with ILC may be more difficult to ascertain, or less likely to be associated with outcomes.</p>", "<p id=\"Par49\">For systemic therapy, those with ILC were significantly more likely to receive endocrine therapy than those with IDC, despite all studied cases being HR-positive. Likewise, those with IDC were more likely to receive chemotherapy. This treatment pattern has been observed in previous literature and may point to the notion that early-stage ILC has reduced sensitivity to chemotherapies, or perceived as such, and is therefore utilized less frequently [##REF##35347549##18##, ##REF##27482285##19##]. However, more recent studies show that in the metastatic setting, response to eribulin and CDK4/6 inhibitors may be similar between ILC and IDC [##REF##30578311##20##, ##REF##31859246##21##]. These findings highlight the need to identify lobular specific therapies for those with metastatic disease.</p>", "<p id=\"Par50\">As a secondary endpoint, we looked at OS by histology. Similar to our findings, worse OS in those with metastatic ILC has been shown in other studies as well [##REF##28757653##2##, ##REF##15084238##3##, ##REF##18458044##22##]. While the underlying reason for this difference is unclear, it suggests that ILC is indeed biologically different than IDC, given differential outcomes despite restricting the study population to those with HR-positive, HER2-negative tumor types, and ILC tumors being of lower grade than IDC tumors. One potential explanation could be that those with metastatic ILC may have an overall higher burden of disease than is typically detected on standard imaging modalities [##REF##33520704##23##].</p>", "<p id=\"Par51\">To our knowledge, this is the largest reported study evaluating primary-site surgery by histologic subtype in metastatic breast cancer. However, this study is subject to several limitations, including selection bias, lack of detailed systemic therapy information, radiation field data, and the absence of local recurrence events as an endpoint. However, the findings reflect real-world management patterns which appear to differ by histologic subtype.</p>", "<p id=\"Par52\">While ILC has long been regarded as a less aggressive tumor type, our findings from this large NCDB study are consistent with others showing worse outcomes in ILC than IDC. The differences between the IDC and ILC groups in this study were relatively small, however, it is interesting to note that histology appears to be influencing management. The use of primary-site surgery was slightly lower, and the use of both radiotherapy and chemotherapy were much lower in those with metastatic ILC compared to metastatic IDC. It is unclear what is driving the lower usage of chemotherapy and radiotherapy in ILC cases; this may reflect an underlying bias that lobular tumors are more indolent and slow growing. Coupled with shorter OS in the ILC cohort, these findings reinforce the need for further study to determine histologic subtype-specific management options. In regard to surgical management, the significantly larger tumor size and higher positive margin rates in the ILC cohort suggest that if primary-site surgery is to be utilized, one should consider a larger excision and likely incorporate radiotherapy to maximize potential benefit of locoregional intervention. Further work is needed to improve management outcomes for those with metastatic ILC.</p>" ]

[]

[ "<title>Purpose</title>", "<p id=\"Par1\">Primary site surgery for metastatic breast cancer improves local control but does not impact overall survival. Whether histologic subtype influences patient selection for surgery is unknown. Given differences in surgical management between early-stage lobular versus ductal disease, we evaluated the impact of histology on primary site surgery in patients with metastatic breast cancer.</p>", "<title>Methods</title>", "<p id=\"Par2\">The National Cancer Database (NCDB, 2010–2016) was queried for patients with stage IV HR-positive, HER2-negative invasive lobular carcinoma (ILC) and invasive ductal carcinoma (IDC). We compared clinicopathologic features, primary site surgery rates, and outcomes by histologic subtype. Multivariable Cox proportional hazard models with and without propensity score matching were used for overall survival (OS) analyses.</p>", "<title>Results</title>", "<p id=\"Par3\">In 25,294 patients, primary site surgery was slightly but significantly less common in the 6,123 patients with ILC compared to the 19,171 patients with IDC (26.9% versus 28.8%, <italic>p</italic> = 0.004). Those with ILC were less likely to receive chemotherapy (41.3% versus 47.4%, <italic>p</italic> &lt; 0.0001) or radiotherapy (29.1% versus 37.9%, <italic>p &lt;</italic> 0.0001), and had shorter OS. While mastectomy rates were similar, those with ILC who underwent lumpectomy had significantly higher positive margin rates (ILC 15.7% versus IDC 11.2%, <italic>p =</italic> 0.025). In both groups, the odds of undergoing surgery decreased over time, and were higher in younger patients with T2/T3 tumors and higher nodal burden.</p>", "<title>Conclusion</title>", "<p id=\"Par4\">Lobular histology is associated with less primary site surgery, higher positive margin rates, less radiotherapy and chemotherapy, and shorter OS compared to those with HR-positive HER2-negative IDC. These findings support the need for ILC-specific data and treatment approaches in the setting of metastatic disease.</p>", "<title>Keywords</title>" ]

[]

[ "<title>Author contributions</title>", "<p>Supervision, study concept and design, and interpretation of data were provided by Rita A Mukhtar. The first draft of the manuscript was written by Harriet Rothschild and all authors commented on previous versions of the manuscript. Amy Shui performed statistical analysis and data interpretation. All authors read and approved the final manuscript.</p>", "<title>Funding</title>", "<p>Author HTR was supported by the National Center for Advancing Translational Sciences, National Institutes of Health (NIH) (Grant number TL1 TR 001871). The contents of this manuscript are solely the responsibility of the authors and do not necessarily represent the official views of the NIH. Author AMS is part of the Biostatistics Core that is generously supported by the University of California, San Francisco Department of Surgery. Author RAM is supported by National Cancer Institute Award K08CA256047. For the remaining authors no sources of funding were declared.</p>", "<title>Data Availability</title>", "<p>The datasets analyzed during the current study are available in the National Cancer Database, <ext-link ext-link-type=\"uri\" xlink:href=\"https://www.facs.org/quality-programs/cancer-programs/national-cancer-database/\">https://www.facs.org/quality-programs/cancer-programs/national-cancer-database/</ext-link>.</p>", "<title>Declarations</title>", "<title>Ethics approval</title>", "<p id=\"Par73\">Due to the de-identified nature of the public-access user files in the National Cancer Database, the study was deemed exempt from institutional review board approval.</p>", "<title>Competing interests</title>", "<p id=\"Par310\">The authors have no relevant financial or non-financial interests to disclose.</p>" ]

[ "<fig id=\"Fig1\"><label>Fig. 1</label><caption><p>CONSORT flow diagram for study population of stage IV, hormone receptor positive, HER2-negative breast cancer patients from the National Cancer Database (PUF 2010–2016). <italic>ER</italic>, estrogen receptor; <italic>PR</italic>, progesterone receptor; <italic>HER2</italic>, human epidermal growth factor receptor 2; <italic>IDC</italic>, invasive ductal carcinoma; <italic>ILC</italic>, invasive lobular carcinoma; <italic>PUF</italic>, Participant User Files</p></caption></fig>", "<fig id=\"Fig2\"><label>Fig. 2</label><caption><p>Proportion of patients undergoing primary-site surgery by histologic subtype grouped by site of metastatic disease. For patients with bone metastasis only or unknown metastatic site, use of primary-site surgery was significantly less common in those with ILC compared to IDC</p></caption></fig>", "<fig id=\"Fig3\"><label>Fig. 3</label><caption><p>Kaplan-Meier survival curves based on primary-site surgery and histology for stage IV hormone receptor positive, HER2-negative patients in the National Cancer Database (PUF 2010–2016). Estimated overall survival in patients with invasive ductal carcinoma (IDC) and invasive lobular carcinoma (ILC) with or without surgery (unmatched, unadjusted cohorts)</p></caption></fig>" ]

[ "<table-wrap id=\"Tab1\"><label>Table 1</label><caption><p>Comparison of patient characteristics between invasive ductal carcinoma (IDC) and invasive lobular carcinoma (ILC) cohorts in unmatched population</p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th align=\"left\">Patient Characteristics</th><th align=\"left\">IDC, n (%)<break/>Total n = 19,171</th><th align=\"left\">ILC, n (%)<break/>Total n = 6,123</th><th align=\"left\"><italic>P-</italic>value</th></tr></thead><tbody><tr><td align=\"left\"><bold>Mean Age</bold> [S.D.]</td><td align=\"left\">61.3 [± 13.9]</td><td align=\"left\">63.6 [± 12.6]</td><td char=\".\" align=\"char\">&lt; 0.0001</td></tr><tr><td align=\"left\"> Age &lt; 50</td><td align=\"left\">3,861 (20.1%)</td><td align=\"left\">836 (13.7%)</td><td char=\".\" align=\"char\">&lt; 0.0001</td></tr><tr><td align=\"left\"> Age ≥ 50</td><td align=\"left\">15,310 (79.8%)</td><td align=\"left\">5,287 (86.3%)</td><td align=\"left\"/></tr><tr><td align=\"left\"><bold>Median Follow up time</bold>, months [IQ range]</td><td align=\"left\">26.8 [14.6–42.6]</td><td align=\"left\">27.2 [17.7–41.5]</td><td align=\"left\"/></tr><tr><td align=\"left\">\n<bold>Race</bold>\n</td><td align=\"left\"/><td align=\"left\"/><td align=\"left\"/></tr><tr><td align=\"left\"> White</td><td align=\"left\">14,116 (74.2%)</td><td align=\"left\">4,825 (79.4%)</td><td char=\".\" align=\"char\">&lt; 0.0001</td></tr><tr><td align=\"left\"> African American</td><td align=\"left\">3,010 (15.8%)</td><td align=\"left\">753 (12.4%)</td><td align=\"left\"/></tr><tr><td align=\"left\"> East Asian</td><td align=\"left\">130 (0.68%)</td><td align=\"left\">27 (0.44%)</td><td align=\"left\"/></tr><tr><td align=\"left\"> Other</td><td align=\"left\">1,763 (9.27%)</td><td align=\"left\">470 (7.74%)</td><td align=\"left\"/></tr><tr><td align=\"left\">\n<bold>Hispanic</bold>\n</td><td align=\"left\"/><td align=\"left\"/><td char=\".\" align=\"char\">0.19</td></tr><tr><td align=\"left\"> Hispanic</td><td align=\"left\">1,081 (5.82%)</td><td align=\"left\">319 (5.36%)</td><td align=\"left\"/></tr><tr><td align=\"left\"> Non-Hispanic</td><td align=\"left\">17,508 (94.2%)</td><td align=\"left\">5,629 (94.6%)</td><td align=\"left\"/></tr><tr><td align=\"left\">\n<bold>Treatment Facility Type</bold>\n</td><td align=\"left\"/><td align=\"left\"/><td char=\".\" align=\"char\">0.001</td></tr><tr><td align=\"left\"> Academic</td><td align=\"left\">6,003 (33.5%)</td><td align=\"left\">2146 (36%)</td><td align=\"left\"/></tr><tr><td align=\"left\"> Community^a</td><td align=\"left\">11,927 (66.5%)</td><td align=\"left\">3810 (64%)</td><td align=\"left\"/></tr><tr><td align=\"left\">\n<bold>Primary Payer</bold>\n</td><td align=\"left\"/><td align=\"left\"/><td char=\".\" align=\"char\">&lt; 0.0001</td></tr><tr><td align=\"left\"> No insurance</td><td align=\"left\">987 (5.15%)</td><td align=\"left\">234 (3.82%)</td><td align=\"left\"/></tr><tr><td align=\"left\"> Private insurance</td><td align=\"left\">7,990 (41.7%)</td><td align=\"left\">2,475 (40.4%)</td><td align=\"left\"/></tr><tr><td align=\"left\"> Public insurance</td><td align=\"left\">9,890 (51.6%)</td><td align=\"left\">3,348 (54.7%)</td><td align=\"left\"/></tr><tr><td align=\"left\"> Unknown insurance</td><td align=\"left\">304 (1.59%)</td><td align=\"left\">66 (1.08%)</td><td align=\"left\"/></tr><tr><td align=\"left\">\n<bold>Median Income Quartiles</bold>\n^{<bold>b</bold>}\n</td><td align=\"left\"/><td align=\"left\"/><td char=\".\" align=\"char\">&lt; 0.0001</td></tr><tr><td align=\"left\"> &lt;$40,227</td><td align=\"left\">3,724 (19.7%)</td><td align=\"left\">1,030 (17.1%)</td><td align=\"left\"/></tr><tr><td align=\"left\"> $40,227–50,353</td><td align=\"left\">4,053 (21.4%)</td><td align=\"left\">1,212 (20.1%)</td><td align=\"left\"/></tr><tr><td align=\"left\"> $50,354–63,332</td><td align=\"left\">4,389 (23.2%)</td><td align=\"left\">1,398 (23.2%)</td><td align=\"left\"/></tr><tr><td align=\"left\"> ≥$63,333</td><td align=\"left\">6,739 (35.6%)</td><td align=\"left\">2,394 (39.7%)</td><td align=\"left\"/></tr><tr><td align=\"left\">\n<bold>Receptor Status</bold>\n</td><td align=\"left\"/><td align=\"left\"/><td char=\".\" align=\"char\">0.017</td></tr><tr><td align=\"left\"> ER positive / PR positive</td><td align=\"left\">16,170 (84.3%)</td><td align=\"left\">5,086 (83.1%)</td><td align=\"left\"/></tr><tr><td align=\"left\"> ER positive / PR negative</td><td align=\"left\">3,001 (15.7%)</td><td align=\"left\">1,037 (16.9%)</td><td align=\"left\"/></tr><tr><td align=\"left\">\n<bold>Grade</bold>\n</td><td align=\"left\"/><td align=\"left\"/><td char=\".\" align=\"char\">&lt; 0.0001</td></tr><tr><td align=\"left\"> 1</td><td align=\"left\">1,580 (9.48%)</td><td align=\"left\">1,020 (22.1%)</td><td align=\"left\"/></tr><tr><td align=\"left\"> 2</td><td align=\"left\">8,822 (52.9%)</td><td align=\"left\">2,861 (62%)</td><td align=\"left\"/></tr><tr><td align=\"left\"> 3</td><td align=\"left\">6,272 (37.6%)</td><td align=\"left\">732 (15.9%)</td><td align=\"left\"/></tr><tr><td align=\"left\">\n<bold>N Stage</bold>\n</td><td align=\"left\"/><td align=\"left\"/><td char=\".\" align=\"char\">&lt; 0.001</td></tr><tr><td align=\"left\"> 1</td><td align=\"left\">1,932 (47.6%)</td><td align=\"left\">480 (36.5%)</td><td align=\"left\"/></tr><tr><td align=\"left\"> 2</td><td align=\"left\">1,295 (31.9%)</td><td align=\"left\">372 (28.3%)</td><td align=\"left\"/></tr><tr><td align=\"left\"> 3</td><td align=\"left\">831 (20.5%)</td><td align=\"left\">465 (35.3%)</td><td align=\"left\"/></tr><tr><td align=\"left\">\n<bold>T Stage</bold>\n</td><td align=\"left\"/><td align=\"left\"/><td char=\".\" align=\"char\">&lt; 0.001</td></tr><tr><td align=\"left\"> 1</td><td align=\"left\">1,217 (23.4%)</td><td align=\"left\">328 (21.0%)</td><td align=\"left\"/></tr><tr><td align=\"left\"> 2</td><td align=\"left\">2,056 (39.6%)</td><td align=\"left\">570 (36.5%)</td><td align=\"left\"/></tr><tr><td align=\"left\"> 3</td><td align=\"left\">803 (15.5%)</td><td align=\"left\">481 (30.8%)</td><td align=\"left\"/></tr><tr><td align=\"left\"> 4</td><td align=\"left\">1,118 (21.5%)</td><td align=\"left\">182 (11.7%)</td><td align=\"left\"/></tr><tr><td align=\"left\">\n<bold>Charlson-Deyo Score</bold>\n</td><td align=\"left\"/><td align=\"left\"/><td char=\".\" align=\"char\">0.007</td></tr><tr><td align=\"left\"> 0</td><td align=\"left\">15,894 (82.9%)</td><td align=\"left\">4,984 (81.4%)</td><td align=\"left\"/></tr><tr><td align=\"left\"> ≥1</td><td align=\"left\">3,277 (17.1%)</td><td align=\"left\">1,139 (18.6%)</td><td align=\"left\"/></tr><tr><td align=\"left\">\n<bold>Positive Surgical Margins</bold>\n</td><td align=\"left\"/><td align=\"left\"/><td align=\"left\"/></tr><tr><td align=\"left\"> Lumpectomy</td><td align=\"left\">182/1621 (11.2%)</td><td align=\"left\">70/445 (15.7%)</td><td char=\".\" align=\"char\">0.025</td></tr><tr><td align=\"left\"> Mastectomy</td><td align=\"left\">262/3893 (6.7%)</td><td align=\"left\">99/1199 (8.2%)</td><td char=\".\" align=\"char\">0.14</td></tr><tr><td align=\"left\">\n<bold>Metastasis site</bold>\n</td><td align=\"left\"/><td align=\"left\"/><td char=\".\" align=\"char\">&lt; 0.0001</td></tr><tr><td align=\"left\"> Bone-only</td><td align=\"left\">8,510 (45.2%)</td><td align=\"left\">3,641 (60.8%)</td><td align=\"left\"/></tr><tr><td align=\"left\"> All other</td><td align=\"left\">10,334 (54.8%)</td><td align=\"left\">2,346 (39.2%)</td><td align=\"left\"/></tr><tr><td align=\"left\">\n<bold>Overall Survival in months (Median (95% CI))</bold>\n</td><td align=\"left\"/><td align=\"left\"/><td align=\"left\"/></tr><tr><td align=\"left\"> All patients</td><td align=\"left\">39.7 (39.0-40.6)</td><td align=\"left\">38.4 (47.2–39.7)</td><td char=\".\" align=\"char\">0.006</td></tr><tr><td align=\"left\"> With surgery</td><td align=\"left\">50.9 (49.1–52.9)</td><td align=\"left\">47.4 (44.9–50.6)</td><td char=\".\" align=\"char\">&lt; 0.001</td></tr><tr><td align=\"left\"> Without surgery</td><td align=\"left\">35.3 (34.4–36.0)</td><td align=\"left\">34.7 (33.5–35.9)</td><td char=\".\" align=\"char\">&lt; 0.001</td></tr></tbody></table></table-wrap>", "<table-wrap id=\"Tab2\"><label>Table 2</label><caption><p>Comparison of treatment patterns by histology. <italic>IDC</italic>, invasive ductal carcinoma; <italic>ILC</italic>, invasive lobular carcinoma; <italic>CNS</italic>, central nervous system</p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th align=\"left\">Treatment</th><th align=\"left\">IDC, n (%)<break/>Total n = 19,171</th><th align=\"left\">ILC, n (%)<break/>Total n = 6,123</th><th align=\"left\"><italic>p-</italic>value</th></tr></thead><tbody><tr><td align=\"left\">\n<bold>Any surgery</bold>\n</td><td align=\"left\">5,514 (28.8%)</td><td align=\"left\">1,644 (26.8%)</td><td char=\".\" align=\"char\">0.004</td></tr><tr><td align=\"left\">\n<bold>Surgery type</bold>\n</td><td align=\"left\"/><td align=\"left\"/><td char=\".\" align=\"char\">0.19</td></tr><tr><td align=\"left\"> Lumpectomy</td><td align=\"left\">1,621 (27.5%)</td><td align=\"left\">445 (27.1%)</td><td align=\"left\"/></tr><tr><td align=\"left\"> Mastectomy</td><td align=\"left\">3,788 (64.2%)</td><td align=\"left\">1,166 (70.9%)</td><td align=\"left\"/></tr><tr><td align=\"left\"> Radical mastectomy</td><td align=\"left\">105 (8.38%)</td><td align=\"left\">33 (2.01%)</td><td align=\"left\"/></tr><tr><td align=\"left\">\n<bold>Chemotherapy</bold>\n</td><td align=\"left\"/><td align=\"left\"/><td char=\".\" align=\"char\">&lt; 0.001</td></tr><tr><td align=\"left\"> No chemotherapy</td><td align=\"left\">9,592 (50%)</td><td align=\"left\">3,406 (55.6%)</td><td align=\"left\"/></tr><tr><td align=\"left\"> Yes chemotherapy</td><td align=\"left\">9,084 (47.4%)</td><td align=\"left\">2,529 (41.3%)</td><td align=\"left\"/></tr><tr><td align=\"left\"> Unknown chemotherapy</td><td align=\"left\">495 (2.58%)</td><td align=\"left\">188 (3.1%)</td><td align=\"left\"/></tr><tr><td align=\"left\">\n<bold>Chemotherapy timing</bold>\n</td><td align=\"left\"/><td align=\"left\"/><td char=\".\" align=\"char\">&lt; 0.001</td></tr><tr><td align=\"left\"> Preoperative chemotherapy</td><td align=\"left\">2,231 (40.5%)</td><td align=\"left\">477 (29%)</td><td align=\"left\"/></tr><tr><td align=\"left\"> Postoperative chemotherapy</td><td align=\"left\">3,283 (59.5%)</td><td align=\"left\">1,167 (71%)</td><td align=\"left\"/></tr><tr><td align=\"left\">\n<bold>Endocrine therapy</bold>\n</td><td align=\"left\"/><td align=\"left\"/><td char=\".\" align=\"char\">&lt; 0.001</td></tr><tr><td align=\"left\"> No endocrine therapy</td><td align=\"left\">3,520 (18.4%)</td><td align=\"left\">869 (14.2%)</td><td align=\"left\"/></tr><tr><td align=\"left\"> Yes endocrine therapy</td><td align=\"left\">15,075 (78.6%)</td><td align=\"left\">5,112 (83.5%)</td><td align=\"left\"/></tr><tr><td align=\"left\"> Unknown endocrine therapy</td><td align=\"left\">576 (3%)</td><td align=\"left\">142 (2.32%)</td><td align=\"left\"/></tr><tr><td align=\"left\">\n<bold>Any radiation therapy</bold>\n</td><td align=\"left\">7,260 (37.9%)</td><td align=\"left\">1,781 (29.1%)</td><td char=\".\" align=\"char\">&lt; 0.001</td></tr><tr><td align=\"left\">\n<bold>Radiation location</bold>\n</td><td align=\"left\"/><td align=\"left\"/><td char=\".\" align=\"char\">0.55</td></tr><tr><td align=\"left\"> Local radiation</td><td align=\"left\">2,365 (35.9%)</td><td align=\"left\">564 (35.1%)</td><td align=\"left\"/></tr><tr><td align=\"left\"> Distant radiation</td><td align=\"left\">4,220 (64.1%)</td><td align=\"left\">1,042 (64.9%)</td><td align=\"left\"/></tr><tr><td align=\"left\">\n<bold>Detailed radiation location</bold>\n</td><td align=\"left\"/><td align=\"left\"/><td align=\"left\"/></tr><tr><td align=\"left\"> CNS/head</td><td align=\"left\">600 (8.26%)</td><td align=\"left\">168 (9.43%)</td><td char=\".\" align=\"char\">0.02</td></tr><tr><td align=\"left\"> Viscera</td><td align=\"left\">86 (1.18%)</td><td align=\"left\">28 (1.57%)</td><td align=\"left\"/></tr><tr><td align=\"left\"> Breast</td><td align=\"left\">2,365 (3.25%)</td><td align=\"left\">564 (31.7%)</td><td align=\"left\"/></tr><tr><td align=\"left\"> Bone</td><td align=\"left\">3,534 (48.7%)</td><td align=\"left\">846 (47.5%)</td><td align=\"left\"/></tr><tr><td align=\"left\"> Other</td><td align=\"left\">383 (5.28%)</td><td align=\"left\">79 (4.43%)</td><td align=\"left\"/></tr><tr><td align=\"left\"> Unknown</td><td align=\"left\">292 (4.02%)</td><td align=\"left\">96 (5.4%)</td><td align=\"left\"/></tr></tbody></table></table-wrap>", "<table-wrap id=\"Tab3\"><label>Table 3</label><caption><p>Sociodemographic and treatment patterns broken down by histology and surgery status. Total number of patients, (n= ), unless otherwise stated. <italic>IDC</italic>, invasive ductal carcinoma; <italic>ILC</italic>, invasive lobular carcinoma; <italic>CNS</italic>, central nervous system</p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th align=\"left\" rowspan=\"2\"/><th align=\"left\" colspan=\"4\">Invasive Ductal Carcinoma</th><th align=\"left\"/><th align=\"left\" colspan=\"4\">Invasive Lobular Carcinoma</th></tr><tr><th align=\"left\">All IDC (n = 19,171)</th><th align=\"left\">IDC w/surgery<break/>(n = 5,514)</th><th align=\"left\">IDC w/out surgery (n = 13,657)</th><th align=\"left\"><italic>P-</italic>value</th><th align=\"left\"/><th align=\"left\">All ILC (n = 6,123)</th><th align=\"left\">ILC w/surgery<break/>(n = 1,644)</th><th align=\"left\">ILC w/out surgery (n = 4,479)</th><th align=\"left\"><italic>P-</italic>value</th></tr></thead><tbody><tr><td align=\"left\">\n<bold>Treatment Facility</bold>\n^{<bold>a</bold>}\n</td><td align=\"left\"/><td align=\"left\"/><td align=\"left\"/><td char=\".\" align=\"char\">&lt; 0.0001</td><td align=\"left\"/><td align=\"left\"/><td align=\"left\"/><td align=\"left\"/><td char=\".\" align=\"char\">&lt; 0.0001</td></tr><tr><td align=\"left\"> Academic</td><td char=\".\" align=\"char\">6,003 (33.5%)</td><td char=\".\" align=\"char\">1,318 (26.2%)</td><td align=\"left\">4,685 (36.3%)</td><td align=\"left\"/><td align=\"left\"/><td align=\"left\">2,146 (36%)</td><td char=\".\" align=\"char\">415 (26.3%)</td><td align=\"left\">1,731 (39.6%)</td><td align=\"left\"/></tr><tr><td align=\"left\"> Community</td><td char=\".\" align=\"char\">11,927 (66.5%)</td><td char=\".\" align=\"char\">3,714 (73.8%)</td><td align=\"left\">8,213 (63.7%)</td><td align=\"left\"/><td align=\"left\"/><td align=\"left\">3,810 (64%)</td><td char=\".\" align=\"char\">1,164 (73.7%)</td><td align=\"left\">2,646 (60.4%)</td><td align=\"left\"/></tr><tr><td align=\"left\">\n<bold>Primary Payer</bold>\n^{<bold>b</bold>}\n</td><td align=\"left\"/><td align=\"left\"/><td align=\"left\"/><td align=\"left\"/><td align=\"left\"/><td align=\"left\"/><td align=\"left\"/><td align=\"left\"/><td align=\"left\"/></tr><tr><td align=\"left\"> No insurance</td><td char=\".\" align=\"char\">987 (5.15%)</td><td char=\".\" align=\"char\">207 (3.75%)</td><td align=\"left\">780 (5.71%)</td><td char=\".\" align=\"char\">&lt; 0.0001</td><td align=\"left\"/><td align=\"left\">234 (3.82%)</td><td char=\".\" align=\"char\">41 (2.49%)</td><td align=\"left\">193 (4.31%)</td><td char=\".\" align=\"char\">&lt; 0.0001</td></tr><tr><td align=\"left\"> Private insurance</td><td char=\".\" align=\"char\">7,990 (41.7%)</td><td char=\".\" align=\"char\">2,723 (49.4%)</td><td align=\"left\">5,267 (38.6%)</td><td align=\"left\"/><td align=\"left\"/><td align=\"left\">2,475 (40.4%)</td><td char=\".\" align=\"char\">763 (46.4%)</td><td align=\"left\">1,712 (38.2%)</td><td align=\"left\"/></tr><tr><td align=\"left\"> Public insurance</td><td char=\".\" align=\"char\">9,890 (51.6%)</td><td char=\".\" align=\"char\">2,509 (45.5%)</td><td align=\"left\">7,381 (54.1%)</td><td align=\"left\"/><td align=\"left\"/><td align=\"left\">3,348 (54.7%)</td><td char=\".\" align=\"char\">824 (50.1%)</td><td align=\"left\">2,524 (56.4%)</td><td align=\"left\"/></tr><tr><td align=\"left\"> Unknown</td><td char=\".\" align=\"char\">304 (1.59%)</td><td char=\".\" align=\"char\">75 (1.36%)</td><td align=\"left\">229 (1.68%)</td><td align=\"left\"/><td align=\"left\"/><td align=\"left\">66 (1.08%)</td><td char=\".\" align=\"char\">16 (1.12%)</td><td align=\"left\">50 (1.12%)</td><td align=\"left\"/></tr><tr><td align=\"left\">\n<bold>Median Income Quartile</bold>\n^{<bold>c</bold>}\n</td><td align=\"left\"/><td align=\"left\"/><td align=\"left\"/><td char=\".\" align=\"char\">&lt; 0.0001</td><td align=\"left\"/><td align=\"left\"/><td align=\"left\"/><td align=\"left\"/><td char=\".\" align=\"char\">&lt; 0.0001</td></tr><tr><td align=\"left\"> &lt;$40,227</td><td char=\".\" align=\"char\">3,724 (19.7%)</td><td char=\".\" align=\"char\">1,084 (20.0%)</td><td align=\"left\">2,640 (19.6%)</td><td align=\"left\"/><td align=\"left\"/><td align=\"left\">1,030 (17.1%)</td><td char=\".\" align=\"char\">260 (16.1%)</td><td align=\"left\">770 (17.4%)</td><td align=\"left\"/></tr><tr><td align=\"left\"> $40,227–50,353</td><td char=\".\" align=\"char\">4,053 (21.4%)</td><td char=\".\" align=\"char\">1,175 (21.6%)</td><td align=\"left\">2,878 (21.4%)</td><td align=\"left\"/><td align=\"left\"/><td align=\"left\">1,212 (20.1%)</td><td char=\".\" align=\"char\">351 (21.7%)</td><td align=\"left\">861 (19.5%)</td><td align=\"left\"/></tr><tr><td align=\"left\"> $50,354–63,332</td><td char=\".\" align=\"char\">4,389 (23.2%)</td><td char=\".\" align=\"char\">1,261 (23.2%)</td><td align=\"left\">3,128 (23.2%)</td><td align=\"left\"/><td align=\"left\"/><td align=\"left\">1,398 (23.2%)</td><td char=\".\" align=\"char\">365 (22.6%)</td><td align=\"left\">1,033 (23.4%)</td><td align=\"left\"/></tr><tr><td align=\"left\"> ≥$63,333</td><td char=\".\" align=\"char\">6,739 (35.7%)</td><td char=\".\" align=\"char\">1,911 (35.2%)</td><td align=\"left\">4,828 (35.8%)</td><td align=\"left\"/><td align=\"left\"/><td align=\"left\">2,394 (39.7%)</td><td char=\".\" align=\"char\">641 (39.6%)</td><td align=\"left\">1,753 (39.7%)</td><td align=\"left\"/></tr><tr><td align=\"left\">\n<bold>Metastatic Site</bold>\n</td><td align=\"left\"/><td align=\"left\"/><td align=\"left\"/><td align=\"left\"/><td align=\"left\"/><td align=\"left\"/><td align=\"left\"/><td align=\"left\"/><td align=\"left\"/></tr><tr><td align=\"left\"> Bone metastasis only</td><td char=\".\" align=\"char\">8510 (44.4%)</td><td char=\".\" align=\"char\">2,677 (48.5%)</td><td align=\"left\">5833 (42.7%)</td><td char=\".\" align=\"char\">&lt; 0.001</td><td align=\"left\"/><td align=\"left\">3641 (59.5%)</td><td char=\".\" align=\"char\">1044 (63.5%)</td><td align=\"left\">2597 (58%)</td><td char=\".\" align=\"char\">&lt; 0.001</td></tr><tr><td align=\"left\"> Visceral metastasis only</td><td char=\".\" align=\"char\">3226 (16.8%)</td><td char=\".\" align=\"char\">1,059 (19.2%)</td><td align=\"left\">2167 (15.9%)</td><td char=\".\" align=\"char\">&lt; 0.001</td><td align=\"left\"/><td align=\"left\">515 (8.4%)</td><td char=\".\" align=\"char\">150 (9.12%)</td><td align=\"left\">365 (8.15%)</td><td char=\".\" align=\"char\">&lt; 0.001</td></tr><tr><td align=\"left\"> Bone and visceral metastases</td><td char=\".\" align=\"char\">5424 (28.3%)</td><td char=\".\" align=\"char\">873 (15.8%)</td><td align=\"left\">4551 (33.3%)</td><td char=\".\" align=\"char\">&lt; 0.001</td><td align=\"left\"/><td align=\"left\">1167 (19.0%)</td><td char=\".\" align=\"char\">163 (9.91%)</td><td align=\"left\">1004 (22.4%)</td><td char=\".\" align=\"char\">&lt; 0.001</td></tr><tr><td align=\"left\"> Unknown metastatic site</td><td char=\".\" align=\"char\">2011 (10.5%)</td><td char=\".\" align=\"char\">905 (16.4%)</td><td align=\"left\">1106 (8.1%)</td><td char=\".\" align=\"char\">&lt; 0.001</td><td align=\"left\"/><td align=\"left\">800 (13.1%)</td><td char=\".\" align=\"char\">287 (17.5%)</td><td align=\"left\">513 (11.5%)</td><td char=\".\" align=\"char\">&lt; 0.001</td></tr><tr><td align=\"left\">\n<bold>Radiation frequency</bold>\n</td><td char=\".\" align=\"char\">7260 (37.9%)</td><td char=\".\" align=\"char\">2,842 (51.5%)</td><td align=\"left\">4418 (32.3%)</td><td char=\".\" align=\"char\">&lt; 0.001</td><td align=\"left\"/><td align=\"left\">1781 (29.1%)</td><td char=\".\" align=\"char\">698 (42.5%)</td><td align=\"left\">1083 (24.2%)</td><td char=\".\" align=\"char\">&lt; 0.001</td></tr><tr><td align=\"left\">\n<bold>Radiation binary</bold>\n</td><td align=\"left\"/><td align=\"left\"/><td align=\"left\"/><td char=\".\" align=\"char\">&lt; 0.001</td><td align=\"left\"/><td align=\"left\"/><td align=\"left\"/><td align=\"left\"/><td char=\".\" align=\"char\">&lt; 0.001</td></tr><tr><td align=\"left\"> Local</td><td char=\".\" align=\"char\">2,365 (35.9%)</td><td char=\".\" align=\"char\">1835 (69.2%)</td><td align=\"left\">530 (13.5%)</td><td align=\"left\"/><td align=\"left\"/><td align=\"left\">564 (35.1%)</td><td char=\".\" align=\"char\">458 (69.5%)</td><td align=\"left\">106 (11.2%)</td><td align=\"left\"/></tr><tr><td align=\"left\"> Distant</td><td char=\".\" align=\"char\">4,220 (64.1%)</td><td char=\".\" align=\"char\">817 (30.8%)</td><td align=\"left\">3,403 (86.5%)</td><td align=\"left\"/><td align=\"left\"/><td align=\"left\">1042 (64.8%)</td><td char=\".\" align=\"char\">201 (30.5%)</td><td align=\"left\">841 (88.8%)</td><td align=\"left\"/></tr><tr><td align=\"left\">\n<bold>Radiation detailed</bold>\n</td><td align=\"left\"/><td align=\"left\"/><td align=\"left\"/><td char=\".\" align=\"char\">&lt; 0.001</td><td align=\"left\"/><td align=\"left\"/><td align=\"left\"/><td align=\"left\"/><td char=\".\" align=\"char\">&lt; 0.001</td></tr><tr><td align=\"left\"> CNS/head</td><td char=\".\" align=\"char\">600 (8.26%)</td><td char=\".\" align=\"char\">87 (3.06%)</td><td align=\"left\">513 (11.6%)</td><td align=\"left\"/><td align=\"left\"/><td align=\"left\">168 (9.43%)</td><td char=\".\" align=\"char\">18 (2.58%)</td><td align=\"left\">150 (13.9%)</td><td align=\"left\"/></tr><tr><td align=\"left\"> Viscera</td><td char=\".\" align=\"char\">86 (1.18%)</td><td char=\".\" align=\"char\">39 (1.37%)</td><td align=\"left\">47 (1.06%)</td><td align=\"left\"/><td align=\"left\"/><td align=\"left\">28 (1.57%)</td><td char=\".\" align=\"char\">8 (1.15%)</td><td align=\"left\">20 (1.84%)</td><td align=\"left\"/></tr><tr><td align=\"left\"> Breast</td><td char=\".\" align=\"char\">2365 (32.6%)</td><td char=\".\" align=\"char\">1,835 (64.6%)</td><td align=\"left\">530 (12%)</td><td align=\"left\"/><td align=\"left\"/><td align=\"left\">564 (31.7%)</td><td char=\".\" align=\"char\">458 (65.6%)</td><td align=\"left\">106 (9.79%)</td><td align=\"left\"/></tr><tr><td align=\"left\"> Bone</td><td char=\".\" align=\"char\">3534 (48.7%)</td><td char=\".\" align=\"char\">691 (24.3%)</td><td align=\"left\">2843 (64.4%)</td><td align=\"left\"/><td align=\"left\"/><td align=\"left\">846 (47.5%)</td><td char=\".\" align=\"char\">175 (25.1%)</td><td align=\"left\">671 (62%)</td><td align=\"left\"/></tr><tr><td align=\"left\"> Other</td><td char=\".\" align=\"char\">383 (5.28%)</td><td char=\".\" align=\"char\">100 (3.52%)</td><td align=\"left\">283 (6.41%)</td><td align=\"left\"/><td align=\"left\"/><td align=\"left\">79 (4.44%)</td><td char=\".\" align=\"char\">18 (2.58%)</td><td align=\"left\">61 (5.63%)</td><td align=\"left\"/></tr><tr><td align=\"left\"> Unknown</td><td char=\".\" align=\"char\">292 (4.02%)</td><td char=\".\" align=\"char\">90 (3.17%)</td><td align=\"left\">202 (4.57%)</td><td align=\"left\"/><td align=\"left\"/><td align=\"left\">96 (5.39%)</td><td char=\".\" align=\"char\">21 (3.01%)</td><td align=\"left\">75 (6.93%)</td><td align=\"left\"/></tr></tbody></table></table-wrap>", "<table-wrap id=\"Tab4\"><label>Table 4</label><caption><p>Factors associated with receiving primary-site surgery in those with metastatic invasive ductal carcinoma and those with metastatic invasive lobular carcinoma. <italic>OR odds ratio; CI confidence interval</italic></p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th align=\"left\"/><th align=\"left\">Unadjusted OR</th><th align=\"left\">95% CI</th><th align=\"left\" colspan=\"2\"><italic>p</italic>-value</th></tr></thead><tbody><tr><td align=\"left\">\n<bold>Invasive Ductal Carcinoma</bold>\n</td><td align=\"left\"/><td align=\"left\" colspan=\"2\"/><td align=\"left\"/></tr><tr><td align=\"left\">Age^a</td><td align=\"left\">0.83</td><td align=\"left\" colspan=\"2\">(0.81–0.85)</td><td align=\"left\">&lt; 0.001</td></tr><tr><td align=\"left\">Stage 0 vs. T1</td><td align=\"left\">0.50</td><td align=\"left\" colspan=\"2\">(0.31–0.81)</td><td align=\"left\">0.0045</td></tr><tr><td align=\"left\">Stage T2 vs. T1</td><td align=\"left\">1.71</td><td align=\"left\" colspan=\"2\">(1.30–2.25)</td><td align=\"left\">0.0001</td></tr><tr><td align=\"left\">Stage T3 vs. T1</td><td align=\"left\">1.68</td><td align=\"left\" colspan=\"2\">(1.17–2.41)</td><td align=\"left\">0.0050</td></tr><tr><td align=\"left\">Stage T4 vs. T1</td><td align=\"left\">0.63</td><td align=\"left\" colspan=\"2\">(0.49–0.82)</td><td align=\"left\">0.0006</td></tr><tr><td align=\"left\">Node N1 vs. N0</td><td align=\"left\">0.73</td><td align=\"left\" colspan=\"2\">(0.60–0.88)</td><td align=\"left\">0.0008</td></tr><tr><td align=\"left\">Node N2 vs. N0</td><td align=\"left\">4.21</td><td align=\"left\" colspan=\"2\">(3.18–5.58)</td><td align=\"left\">&lt; 0.001</td></tr><tr><td align=\"left\">Node N3 vs. N0</td><td align=\"left\">2.91</td><td align=\"left\" colspan=\"2\">(2.17–3.91)</td><td align=\"left\">&lt; 0.001</td></tr><tr><td align=\"left\">Year of diagnosis</td><td align=\"left\">0.84</td><td align=\"left\" colspan=\"2\">(0.83–0.85)</td><td align=\"left\">&lt; 0.001</td></tr><tr><td align=\"left\">\n<bold>Invasive Lobular Carcinoma</bold>\n</td><td align=\"left\"/><td align=\"left\" colspan=\"2\"/><td align=\"left\"/></tr><tr><td align=\"left\">Age^a</td><td align=\"left\">0.85</td><td align=\"left\" colspan=\"2\">(0.81–0.89)</td><td align=\"left\">&lt; 0.001</td></tr><tr><td align=\"left\">Stage 0 vs. T1</td><td align=\"left\">0.08</td><td align=\"left\" colspan=\"2\">(0.04–0.20)</td><td align=\"left\">&lt; 0.001</td></tr><tr><td align=\"left\">Stage T2 vs. T1</td><td align=\"left\">2.03</td><td align=\"left\" colspan=\"2\">(1.17–3.51)</td><td align=\"left\">0.0113</td></tr><tr><td align=\"left\">Stage T3 vs. T1</td><td align=\"left\">2.65</td><td align=\"left\" colspan=\"2\">(1.43–4.90)</td><td align=\"left\">0.0020</td></tr><tr><td align=\"left\">Stage T4 vs. T1</td><td align=\"left\">0.45</td><td align=\"left\" colspan=\"2\">(0.27–0.78)</td><td align=\"left\">0.0042</td></tr><tr><td align=\"left\">Node N1 vs. N0</td><td align=\"left\">0.79</td><td align=\"left\" colspan=\"2\">(0.57–1.09)</td><td align=\"left\">0.1529</td></tr><tr><td align=\"left\">Node N2 vs. N0</td><td align=\"left\">19.65</td><td align=\"left\" colspan=\"2\">(9.28–41.61)</td><td align=\"left\">&lt; 0.001</td></tr><tr><td align=\"left\">Node N3 vs. N0</td><td align=\"left\">12.12</td><td align=\"left\" colspan=\"2\">(6.88–21.35)</td><td align=\"left\">&lt; 0.001</td></tr><tr><td align=\"left\">Year of diagnosis</td><td align=\"left\">0.84</td><td align=\"left\" colspan=\"2\">(0.82–0.87)</td><td align=\"left\">&lt; 0.001</td></tr></tbody></table></table-wrap>" ]

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[ "<table-wrap-foot><p>^aCommunity treatment facility includes Community Cancer Programs, Comprehensive Community Cancer Programs, and Integrated Network Cancer Programs</p><p>^bMedian Income Quartiles from 2012–2016</p></table-wrap-foot>", "<table-wrap-foot><p>^a Treatment facility data available for 17,930 IDC and 5,956 ILC patients</p><p>^b Primary payer data available for 19,171 IDC and 6,123 ILC patients</p><p>^c Median income quartile data available for 18,905 IDC and 6,034 ILC patients</p></table-wrap-foot>", "<table-wrap-foot><p>^a In this analysis, patient age at diagnosis is scaled to every 10 years</p></table-wrap-foot>", "<fn-group><fn><p><bold>Publisher’s Note</bold></p><p>Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.</p></fn></fn-group>" ]

[ "<graphic xlink:href=\"10549_2023_7125_Fig1_HTML\" id=\"d32e462\"/>", "<graphic xlink:href=\"10549_2023_7125_Fig2_HTML\" id=\"d32e1318\"/>", "<graphic xlink:href=\"10549_2023_7125_Fig3_HTML\" id=\"d32e2276\"/>" ]

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{ "acronym": [ "CDCI", "ER", "HER2", "HR", "IDC", "ILC", "NCDB", "OS", "PR" ], "definition": [ "Charlson-Deyo Co-Morbidity Index", "Estrogen receptor", "Human epidermal growth factor receptor 2", "Hormone receptor", "Invasive ductal carcinoma", "Invasive lobular carcinoma", "National Cancer Database", "Overall survival", "Progesterone receptor" ] }

[ "<title>Introduction</title>", "<p id=\"Par5\">Terpenoids are a large group of natural compounds constructed from isoprene units (C5). Many different terpenoids are used in the food, beverage, cosmetic, and pharmaceutical industries (Tetali ##REF##30467631##2019##). The current industrial supply of terpenoids relies mainly on chemical synthesis or extraction from plants. The chemical synthesis of terpenoids harboring complex structures has been actively researched (Phil ##REF##29635919##2018##). However, the multi-stereo center in the complex terpenoids requires diastereoselective synthesis, resulting in multi-step total synthesis with high costs and low yield (Santosh et al. ##UREF##2##2020##; Le et al. ##REF##36296479##2022##). Therefore, the chemical synthesis of terpenoids harboring complex structures is expensive and requires fine-controlled and complex processes. Plant extraction has been also used to supply terpenoids; however, it is limited by the presence of various compounds harboring similar physicochemical properties in plants and the difficulty of separating these compounds (Rodger et al. ##REF##21855882##2011##). In addition, natural plant supplies are limited owing to their finite amount (Nur and Henrik ##REF##29187859##2017##). Therefore, the engineering of microorganisms to produce terpenoids has emerged as a promising approach.</p>", "<p id=\"Par6\">Research on terpenoid production using microorganisms has rapidly increased in the twenty-first century. <italic>Saccharomyces cerevisiae</italic> and <italic>Escherichia coli</italic> have been used to produce plant terpenoids by heterologous expression of their genes because they have advantages for production, such as the presence of a native terpenoid synthetic pathway and facile genetic manipulation (Navale et al. ##REF##33394155##2021##; Sun et al. ##REF##33616900##2020##). <italic>Saccharomyces cerevisiae</italic> offers several advantages for terpenoid production, including a high sugar catabolic rate, a relatively fast growth rate, and its GRAS status (Claudia et al. ##REF##28623722##2017##). In addition, <italic>saccharomyces cerevisiae</italic> has an inherent terpenoid production pathway, the mevalonate (MVA) pathway, in which isopentenyl diphosphate (IPP) or dimethylallyl diphosphate (DMAPP), essential intermediates of sterols and ubiquinones, are synthesized from acetyl-CoA (Guo et al. ##REF##32786348##2020##). Metabolic reactions in the MVA pathway proceed in the cytosol, which, in turn, are used for terpenoid production. Recently, terpenoid production using organelles has attracted attention owing to their ability to produce acetyl-CoA.</p>", "<p id=\"Par7\">Peroxisomes are considered potential acetyl-CoA pools because β-oxidation of fatty acids proceeds in them (Hammer and Avalos ##REF##28853733##2017##). In fact, a terpenoid, β-amyrin, was produced at high level (57.8 mg/g dry cell weight (DCW)) by introducing two enzymes of the MVA pathway into peroxisomes and four downstream enzymes into the cytosol or peroxisomes (Du et al. ##REF##34955018##2022##). The concentration of acetyl-CoA in the mitochondria is as much as 20–30 times higher than that in the cytosol because of the high enzymatic activity of the mitochondrial pyruvate dehydrogenase complex (Weinert et al. ##UREF##3##2014##). Thus, mitochondria are considered suitable organelles for the introduction of the MVA pathway for terpenoid production (Duran et al. ##REF##32592388##2020##). For example, patchoulol, a terpenoid, was produced at a high level (19.24 mg/L) by introducing all the enzymes of the MVA pathway into the mitochondria and a fusion protein of the downstream enzymes into the cytosol (Tao et al. ##REF##35254611##2022##). In another example, amorphadiene was produced at 400 mg/L by introducing several enzymes of the MVA pathway into the mitochondria and two downstream enzymes into the cytosol (Yuan and Ching ##REF##27471067##2016##).</p>", "<p id=\"Par8\">Although introducing the MVA pathway into the mitochondria is attractive for producing terpenoids in yeast, there may be a limit because the mitochondria comprise a small percentage of the entire cell (Uchida et al. ##REF##21360734##2011##). We hypothesized that terpenoid production will increase by introducing the MVA pathway into the mitochondria when mitochondrial volume increases. In the present study, we constructed yeast strains to analyze squalene production or β-carotene production and investigated the relationship between mitochondria-mediated terpenoid (squalene or β-carotene) production and mitochondrial volume.</p>" ]

[ "<title>Materials and methods</title>", "<title>Genes for yeast strain construction</title>", "<p id=\"Par9\">The genes used for the construction of the yeast strains were prepared as described in the Supplementary Information. The plasmids, primers, and DNA fragments used are listed in Tables ##SUPPL##0##S1##, ##SUPPL##0##S2##, and ##SUPPL##0##S3##, respectively. <italic>Escherichia coli</italic> NovaBlue (Merck Millipore, Darmstadt, Germany) was used for plasmid preparation using the Orthodox method.</p>", "<title>Yeast strains</title>", "<p id=\"Par10\"><italic>Saccharomyces cerevisiae</italic> BY4741 was used as the host strain. The strains used in this study are listed in Table ##TAB##0##1##. The methods used for strain construction are described in the Supplementary Information.</p>", "<title>Mitochondrial volume analysis</title>", "<p id=\"Par11\">Mutants with defective mitochondrial morphology were selected from the Yeast Knockout Collection (Open Biosystems, Huntsville, AL, USA) database. The total mitochondrial volume per cell was quantified using a previously described method (Viana et al. ##REF##25640425##2015##). The mutant strains were transformed with pGK426-MLS-GFP and cultured in 5 mL of synthetic complete media without uracil (2% glucose, 1.46 g/L Yeast Synthetic Drop-out Medium Supplements; Merck, Darmstadt, Germany) overnight at 30 °C with shaking at 200 rpm. The strains were transferred to YPD media (2% glucose, 2% peptone, and 1% yeast extract) at 0.1–0.4 OD₆₀₀ and cultured further for 4 h under the same conditions. When the OD₆₀₀ reached 0.5–1.0, the cultures were diluted 8–16-fold with YPD, transferred to a 96-well glass plate coated with concanavalin-A, and incubated at 30 °C for 20 min. The supernatant was removed and fresh YPD was added to each well. MLS-GFP signals in mutants were detected using an Olympus FV1000 confocal microscope (Olympus, Tokyo, Japan). Z-stack images of MLS-GFP were constructed and mitochondrial volumes were calculated using ImageJ and MitoGraph software. The mitochondrial volume in one cell was calculated as the average of at least 10 cells per strain.</p>", "<title>Analyses for IPP/DMAPP, squalene, and β-carotene contents</title>", "<p id=\"Par12\">For squalene production, strains, pre-incubated in 5 mL of YPD at 30 °C overnight, were inoculated into 30 mL of YPD at 0.01 OD₆₀₀ and incubated at 30 °C for 24 h in 250-mL baffled flask with shaking at 200 rpm. For β-carotene production, YPD supplemented with 200 μg/mL hygromycin (Nacalai Tesque, Japan) was used, and sampling was performed 48 h after inoculation. DCW was determined using the equation, DCW/OD₆₀₀ = 0.561 derived beforehand.</p>", "<p id=\"Par13\">Intracellular IPP/DMAPP were extracted as previously described (Luo et al. ##REF##32758537##2020##). Briefly, 2 mL of each culture was centrifuged at 5,000 × <italic>g</italic> for 2 min, and the supernatant was removed. Cells were lysed in an acetonitrile/methanol/water (40:40:20) solution. The solution was chilled at –20 °C for 20 min, followed by centrifugation at 16,000 × <italic>g</italic> for 10 min. The supernatant was collected, dried, and resuspended in 50 μL of water. IPP/DMAPP were analyzed using liquid chromatography-tandem mass spectrometry (LC–MS/MS) (Agilent Nexera 1260 series high-performance liquid chromatography system and 6460 Triple Quad LC/MS; Agilent Technologies, Santa Clara, CA, USA). Separation, detection, and quantification of IPP /DMAPP were performed as previously described (Kato et al. ##REF##22280965##2012##); however, IPP and DMAPP could not be separated from each other. Thus, in this study, the contents are presented as the sum of IPP and DMAPP (referred to as IPP/DMAPP).</p>", "<p id=\"Par14\">Intracellular squalene was extracted as previously described (Zhu et al. ##REF##34710614##2021##). Briefly, 600 μL of each culture was centrifuged at 5,000 × <italic>g</italic> for 2 min and the supernatant was removed. 600 μL of ethyl acetate and approximately 200 μL of grass beads (diameter of 0.5 mm) were added to the cells, which were then lysed by shaking at 1,500 rpm for 10 min using a grinder (shake Master NEO BMS, Japan) with a pre-cooled sample holder at 4 °C. The cell lysate was centrifuged at 21,880 × <italic>g</italic> for 10 min and the upper organic solvent was collected. A GCMS-QP2010 (Shimadzu, Japan) equipped with a DB-5MS column (15 m × 0.25 mm, 0.25 μm film thickness; Agilent Technologies) was used to analyze squalene. The oven temperature was maintained at 100 °C for 2 min, gradually increased to 250 °C at a rate of 20 °C/min, maintained for 2 min, then increased to 325 °C at a rate of 50 °C/min and maintained for 2 min.</p>", "<p id=\"Par15\">Intracellular β-carotene was extracted as previously described (Ma et al. ##REF##35102143##2022##). Briefly, 100 μL of each culture was centrifuged at 5,000 × <italic>g</italic> for 2 min and the supernatant was removed. A total of 900 μL of dimethyl sulfoxide was added to the cells, and β-carotene was extracted after incubation at 50 °C for 1 h. After the incubation, 450 μL of methanol was added and the mixtures were centrifuged at 21,880 × <italic>g</italic> for 5 min. The supernatants were collected. A high-performance liquid chromatography (HPLC) (Shimadzu, Japan) equipped with a BDS Hypersil C18 column (4.6 × 150 mm², 5 μm particle size; Thermo Fisher Scientific, Waltham, MA, USA) and a SPD-20A UV–VIS detector (Shimadzu) was used to analyze β-carotene. The oven temperature was maintained at 25 °C. Methanol and acetonitrile were used as mobile phases at a flow rate of 0.28 mL/min and 0.52 mL/min. β-carotene was detected by absorbance at 450 nm.</p>", "<p id=\"Par16\">The representative chromatograms of IPP/DMAPP, squalene, and β-carotene were shown in Figs, ##SUPPL##0##S1##, ##SUPPL##0##S2##, and ##SUPPL##0##S3##.</p>", "<title>Measurement of intracellular oxidation levels</title>", "<p id=\"Par17\">The reactive oxygen species (ROS) level was analyzed using ROS Assay Kit -Photo-oxidation Resistant DCFH-DA- (Dojindo, kumamoto, Japan). Briefly, strains, pre-incubated in 5 mL of YPD at 30 °C overnight, were inoculated into 5 mL of fresh YPD at 1.0 OD₆₀₀ and incubated at 30 °C for 4 h in test tubes with shaking at 200 rpm. Then, 1 mL of each culture was centrifuged at 5,000 × <italic>g</italic> for 2 min, and the supernatant was removed. The cells were treated according to the manufacturer’s protocol and the fluorescence was measured with λ_ex = 500 nm and λ_em = 540 nm using a plate-reader (INFINITE M NANO + ; Tercan, Männedorf, Switzerland).</p>", "<title>Spot assay using dithiothreitol (DTT)</title>", "<p id=\"Par18\">The effect of DTT addition to media on growth was analyzed as previously described (Bode et al. ##REF##23198688##2013##). Briefly, strains, pre-incubated in 5 mL of YPD at 30 °C overnight, were inoculated at 1.0 OD₆₀₀ into 5 mL of fresh YPD and incubated at 30 °C for 4 h in a test tube with shaking at 200 rpm. OD₆₀₀ was adjusted to 1, and tenfold serial dilutions were spotted at 3μL onto YPD, YPD + DTT 0.5 mM, or YPD + DTT 5 mM plates. The plates were incubated at 30 °C for 24 h.</p>" ]

[ "<title>Results</title>", "<title>The strain expressing the MVA pathway in mitochondria</title>", "<p id=\"Par19\">Proteins can be delivered to the mitochondria by fusion with a mitochondrial localization signal (MLS); however, their enzymatic activities are sometimes interrupted in the mitochondria if the MLS remains fused to them (Duran et al. ##REF##32592388##2020##). To introduce the entire MVA pathway into the mitochondria and activate it, we constructed a BY4741 background strain, SSY1, expressing all the enzymes of the MVA pathway fused with the MLS of CoxIV (Fig. ##FIG##0##1##a), because it is cleaved away in mitochondria. The MLS of CoxIV consists of 26 amino acids; the first 25 amino acids and last glutamine are required for translocation and cleavage, respectively. To ascertain the action of MLS, we constructed a strain, expressing a GFP fused with MLS, and examined the localization of GFP as compared with MitoTracker. As shown in Fig. ##FIG##0##1##b, the signal of the MLS-fused GFP coincided with that of MitoTracker, confirming that the MLS of CoxIV worked effectively in translocation to the mitochondria. Thus, we hypothesized that all enzymes of the MVA pathway fused with MLS were also translocated to the mitochondria. Next, we examined whether the MVA pathway introduced into the mitochondria functioned correctly by measuring the amount of IPP/DMAPP in SSY1 compared to that in BY4741. The IPP/DMAPP content in SSY1 was 164 nmol/g DCW, which was 15-fold higher than that in BY4741, 11 nmol/g DCW (Fig. ##FIG##0##1##c). This result indicates that the MVA pathway introduced into the mitochondria functioned and was beneficial for terpenoid production, as previously reported (Lv et al. ##UREF##0##2016##).</p>", "<title>The total mitochondrial volumes in mutants with defective mitochondrial morphology</title>", "<p id=\"Par20\">Mitochondrial volume has been reported to be decreased in a deletion mutant of <italic>MGM1</italic>, a mitochondrial morphology-related gene (Bernhardt et al. ##REF##25601284##2015##). We hypothesized that mutants with defective mitochondrial morphology could be used to examine the effect of total mitochondrial volume on terpenoid production in yeast expressing the MVA pathway in their mitochondria. We selected 13 genes, <italic>FZO1</italic>, <italic>MGM1</italic>, <italic>UGO1</italic>, <italic>DNM1</italic>, <italic>FIS1</italic>, <italic>MDV1</italic>, <italic>CAF4</italic>, <italic>MMM1</italic>, <italic>MDM10</italic>, <italic>MDM12</italic>, <italic>MDM31</italic>, <italic>MDM32</italic>, and <italic>MDM33</italic> (Ohsumi and Shimoda ##UREF##1##2007##). GFP fused with the MLS of CoxIV was expressed in BY4741 and the deletion mutants, and mitochondrial volumes were calculated from the fluorescent signals of GFP using the confocal microscopic Z-stack method (Fig. ##FIG##1##2##). As expected, we found differences in mitochondrial volume among them. The only strain with a higher volume than that in BY4741 was Δ<italic>mdm32</italic> (2.2 m³/cell), which was 1.8-fold higher than that in BY4741 (1.2 μm³/cell). The strains with volumes lower than those in BY4741 were Δ<italic>mdm10</italic>, Δ<italic>mdm12</italic>, Δ<italic>fzo1</italic>, Δ<italic>mmm1,</italic> Δ<italic>mdm31,</italic> Δ<italic>mdm33,</italic> Δ<italic>mgm1</italic>, and Δ<italic>ugo1</italic>. The strain with the lowest volume was Δ<italic>ugo1</italic> (0.4 μm³/cell), which was threefold lower than that in BY4741.</p>", "<title>The effect of mitochondrial volume on mitochondria-mediated terpenoid production</title>", "<p id=\"Par21\">We hypothesized that an increase in mitochondrial volume may augment the terpenoid production in yeast expressing the MVA pathway in their mitochondria. To examine the effect of mitochondrial volume on terpenoid production, we used mutants with defective mitochondrial morphology, Δ<italic>mdm32</italic>, Δ<italic>fzo1</italic>, Δ<italic>mgm1</italic>, and Δ<italic>ugo1</italic>. Four mutants were constructed from SSY1, and the relationship between mitochondrial volume and IPP/DMAPP or squalene content was examined. In SSY2 (Δ<italic>mdm32</italic>) harboring a large mitochondrial volume, both the contents of IPP/DMAPP (25 nmol/g DCW) and squalene (707 nmol/ g DCW) increased 1.3- and 2.8-fold higher than that in SSY1 (19 nmol/g DCW for IPP/DMAPP, 256 nmol/ g DCW for squalene) (Fig. ##FIG##2##3##a and b). In contrast, in SSY3 (Δ<italic>fzo1</italic>), SSY4 (Δ<italic>mgm1</italic>), and SSY5 (Δ<italic>ugo1</italic>) harboring a low of mitochondrial volume, the contents of IPP/DMAPP and squalene decreased compared with that in SSY1, with the exception of squalene in SSY3. The contents of IPP/DMAPP in SSY3, 4, and 5 were 4.65, 1.29, and 1.32 nmol/g DCW, which were 4.2-, 15.1-, and 14.7-fold less than those in SSY1. The concentrations of squalene in SSY3, 4, and 5 were 233, 14, and 25 nmol/ g DCW, which were 1.1-, 18.2-, and 10.1-fold less than those in SSY1. Furthermore, we found that the contents of both IPP/DMAPP and squalene were clearly correlated with the mitochondrial volume (Fig. ##FIG##2##3##c and d).</p>", "<p id=\"Par22\">In addition, we investigated the effect of the mitochondrial volume on β-carotene production to evaluate the effectiveness of our approach on other terpenoid production. In SSY7 (Δ<italic>mdm32</italic>) harboring a large mitochondrial volume, the contents of IPP/DMAPP (15 nmol/g DCW) and β-carotene (1609 nmol/ g DCW) were 1.2- and 1.4-fold higher than those in SSY6 (12.4 nmol/g DCW for IPP/DMAPP, 1132 nmol/ g DCW for β-carotene) (Fig. ##FIG##3##4##a and b). In contrast, the contents of IPP/DMAPP in SSY8 (Δ<italic>fzo1</italic>), 9 (Δ<italic>mgm1</italic>), and 10 (Δ<italic>ugo1</italic>) were 3.0-, 2.7-, and 12.4-fold lower than those in SSY6 at 4.1, 4.6, and 1.0 nmol/g DCW, respectively. The concentrations of β-carotene in SSY8, 9, and 10 were 1030, 491, and 678 nmol/ g DCW, 1.1-, 2.3-, and 1.7-fold lower than those in SSY6, respectively. Furthermore, the contents of both IPP/DMAPP and β-carotene were correlated with the mitochondrial volume (Fig. ##FIG##3##4##c and d). These data indicate that the increase in mitochondrial volume augmented terpenoid production in yeast expressing the MVA pathway in the mitochondria.</p>" ]

[ "<title>Discussion</title>", "<p id=\"Par23\">In this study, we demonstrated the beneficial effect of the MVA pathway introduced into the mitochondria for terpenoid production by measuring IPP/DMAPP, an essential precursor of terpenoids. Furthermore, using mutants with defective mitochondrial morphology, we found that the contents of IPP/DMAPP and terpenoids (squalene or β-carotene) were positively correlated with the mitochondrial volume and showed differences in the mitochondrial volume. These results not only support the advantages of introducing the MVA pathway into mitochondria for terpenoid production but also suggest that increasing the mitochondrial volume might enhance the production.</p>", "<p id=\"Par24\">Both IPP/DMAPP and terpenoids contents were positively correlated with the mitochondrial volume. However, notably, there was a difference in the degree of increase between IPP/DMAPP and terpenoids in SSY2 (Δ<italic>mdm32</italic>) and SSY7 (Δ<italic>mdm32</italic>) (1.3-fold for IPP/DMAPP and 2.8-fold for squalene in squalene production, 1.2-fold for IPP/DMAPP and 1.4-fold for in β-carotene production). IDI1 was not introduced into the mitochondria of any of the strains used in this study. Thus, DMAPP was not synthesized in the mitochondria. The growth defect compared to the wild-type strain is believed to be caused by the accumulation of the pyrophosphate compounds, mevalonate-5-PP, IPP, and DMAPP in the mitochondria (Zhu et al. ##REF##34710614##2021##), which was also observed in this study (Fig. ##FIG##0##1## C). Pyrophosphate compounds can react with adenosine monophosphate-amino acids through aminoacyl tRNA synthetases and convert them to toxic ATP analogs, which inhibit mitochondrial adenine nucleotide translocase and F1-ATPase (Mönkkönen et al. ##REF##16402039##2006##, Mookerjee-Basu et al. ##REF##20483757##2010##). However, we first speculated that the defective mitochondrial morphology or the introduction of MVA pathway enzymes into the mitochondria might cause mitochondrial dysfunction and induce oxidative stress. Therefore, we measured the ROS levels and growth of BY4741, <italic>mdm32</italic> deletion strain, SSY1, and SSY2 (Δ<italic>mdm32</italic>) and found that the ROS levels in SSY2 (Δ<italic>mdm32</italic>) tended to be slightly higher compared to those in BY4741 (Fig. ##SUPPL##0##S4##). Moreover, the addition of DTT as a reducing agent to media improved the growth of the strain, which was exposed to oxidative stress (Bode et al. ##REF##23198688##2013##). Thus, a DTT assay was performed using the SSY2 (Δ<italic>mdm32</italic>) strain, which revealed that DDT addition to YPD media did not improve the growth of SSY2 (Δ<italic>mdm32</italic>) (Fig. ##SUPPL##0##S5##). Therefore, the slight increase in ROS levels in SSY2 (Δ<italic>mdm32</italic>) was not the cause of the growth defect of SSY2 (Δ<italic>mdm32</italic>), supporting the theory that growth defects are attributed to the accumulation of the pyrophosphate compounds mentioned in the previous study (Zhu et al. ##REF##34710614##2021##).</p>", "<p id=\"Par25\">Thus, it is plausible that IPP/DMAPP mainly contains IPP accumulated in the mitochondria. After production in the mitochondrial matrix, IPP is exported to the cytosol, where squalene production proceeds (Zhu et al. ##REF##34710614##2021##). During export, IPP must pass through the inner and outer mitochondrial membranes. Several transporters selectively pass through small molecules in the inner membranes (Duran et al ##REF##32592388##2020##). Porin is present in the outer membrane and allows small molecules (&lt; 5 kDa) to pass freely between the cytosol and intermembrane space. Thus, passage through the inner membrane is thought to be critical for IPP export from the mitochondria to the cytosol. Mdm32 is a protein present in the mitochondrial inner membrane, and loss of Mdm32 leads to defects in mitochondrial morphology (Okamoto and Shaw ##REF##16285870##2005##). Although the mechanism underlying the IPP export remains to be elucidated, structural changes in the inner mitochondrial membrane may enhance IPP export or diffusion.</p>", "<p id=\"Par26\">Regarding the Δ<italic>fzo1</italic> strains (SSY3 (Δ<italic>fzo1</italic>) and SSY8 (Δ<italic>fzo1</italic>)), although both IPP and terpenoid decreased in the Δ<italic>fzo1</italic> strains compared to the normal strains (SSY1 and SSY6), we also observed a difference in the degree of decrease between IPP and squalene content: 76% for IPP and 9% for squalene (Fig. ##FIG##2##3##a and c) and that between IPP and β-carotene content: 68% for IPP and 9% for β-carotene (Fig. ##FIG##3##4##a and c). These differences seem to arise from the balance between IPP synthesis in the mitochondria and IPP export. Due to the low mitochondrial volume, a sufficient amount of enzymes in the MVA pathway could not be transported to the mitochondria; thus, the rate of IPP synthesis decreased. The IPP produced via the mitochondria in the Δ<italic>fzo1</italic> strains seem to be exported smoothly to the cytosol, because of its low synthetic activity, and is subsequently metabolized to squalene or β-carotene. Thus, although the terpenoid level in the Δ<italic>fzo1</italic> strains was slightly lower than that in the normal strains, the difference in terpenoid levels between the normal strains and the Δ<italic>fzo1</italic> strains was not as drastic as that in IPP. In contrast, the IPP/DMAPP levels in the Δ<italic>fzo1</italic> strains decreased drastically compared with those in the normal strains (Figs. ##FIG##2##3##a and ##FIG##3##4## a). The Fzo1 protein resides in the outer mitochondrial membrane, and the loss of <italic>Fzo1</italic> may not affect IPP export or diffusion due to the presence of porins in the outer membrane, suggesting that there may be no difference in the export ability of mitochondria. There may be a limit to export, and the IPP in the normal strains probably accumulates in the mitochondrial matrix.</p>", "<p id=\"Par27\">In the present study, SSY1 to 10 showed growth defects compared to the wild-type strain, BY4741. The defects were severe in the Δ<italic>mdm32</italic> strains (SSY2 and SSY7). The deletion of <italic>FZO1</italic>, <italic>MGM1</italic>, <italic>UGO1</italic>, or <italic>MDM32</italic> leads growth defects by dysfunction of mitochondria (Okamoto and Shaw ##REF##16285870##2005##). In the normal and Δ<italic>mdm32</italic> strains, the IPP/DMAPP levels were high (Figs. ##FIG##2##3##a and ##FIG##3##4##a), indicating that pyrophosphate compounds may accumulate in the mitochondria. Therefore, the severe growth defects in the Δ<italic>mdm32</italic> strains is probably due to both mitochondrial dysfunction and pyrophosphate compound accumulation in the mitochondria. As the normal strains had no mitochondrial morphological defects. Thus, the growth defect in the normal strains may only be due to the accumulation of pyrophosphate compounds in the mitochondria. In contrast, the growth defects in Δ<italic>fzo1</italic> strains, Δ<italic>mgm1</italic> strains (SSY4 and SSY9), and Δ<italic>ugo1</italic> strains (SSY5 and SSY10) may be due to mitochondrial dysfunction because their IPP levels, that is, their activities in the MVA pathway introduced in the mitochondria, were considerably low. Further approaches to expand the volume of mitochondria without growth defects by mitochondrial dysfunction and enhance the export of IPP, inhibiting pyrophosphate compound accumulation, would be required to increase terpenoid production using our proposed mitochondria-based strategy.</p>" ]

[]

[ "<title>Abstract</title>", "<p id=\"Par1\">Terpenoids are widely used in the food, beverage, cosmetics, and pharmaceutical industries. Microorganisms have been extensively studied for terpenoid production. In yeast, the introduction of the mevalonate (MVA) pathway in organelles in addition to the augmentation of its own MVA pathway have been challenging. Introduction of the MVA pathway into mitochondria is considered a promising approach for terpenoid production because acetyl-CoA, the starting molecule of the MVA pathway, is abundant in mitochondria. However, mitochondria comprise only a small percentage of the entire cell. Therefore, we hypothesized that increasing the total mitochondrial volume per cell would increase terpenoid production. First, we ascertained that the amounts of isopentenyl diphosphate (IPP) and dimethylallyl diphosphate (DMAPP), the final molecules of the MVA pathway, were 15-fold higher of the strain expressing the MVA pathway in mitochondria than in the wild-type yeast strain. Second, we found that different deletion mutants induced different mitochondrial volumes by measuring the mitochondrial volume in various deletion mutants affecting mitochondrial morphology; for example,Δ<italic>mdm32</italic> increased mitochondrial volume, and Δ<italic>fzo1</italic> decreased it. Finally, the effects of mitochondrial volume on amounts of IPP/DMAPP and terpenoids (squalene or β-carotene) were investigated using mutants harboring large or small mitochondria expressing the MVA pathway in mitochondria. Amounts of IPP/DMAPP and terpenoids (squalene or β-carotene) increased when the mitochondrial volume expanded. Introducing the MVA pathway into mitochondria for terpenoid production in yeast may become more attractive by enlarging the mitochondrial volume.</p>", "<title>Key points</title>", "<p id=\"Par2\">\n<italic>• IPP/DMAPP content increased in the strain expressing the MVA pathway in mitochondria</italic>\n</p>", "<p id=\"Par3\">\n<italic>• IPP/DMAPP and terpenoid contents are positively correlated with mitochondrial volume</italic>\n</p>", "<p id=\"Par4\">\n<italic>• Enlarging the mitochondria may improve mitochondria-mediated terpenoid production</italic>\n</p>", "<title>Supplementary Information</title>", "<p>The online version contains supplementary material available at 10.1007/s00253-023-12922-5.</p>", "<title>Keywords</title>", "<p>Open Access funding provided by Kobe University.</p>" ]

[ "<title>Supplementary Information</title>", "<p>Below is the link to the electronic supplementary material.</p>" ]

[ "<title>Acknowledgements</title>", "<p>The authors are grateful for the support from NEDO Projects No. P16009 (Development of production techniques for highly functional biomaterials using plant and other organism smart cells).</p>", "<title>Author contribution</title>", "<p>SY designed and performed the experiments, interpreted the results, and helped write the manuscript. TB was involved in planning and performing the experiments. TK performed an interpretation of the results and helped write the manuscript. AK reviewed the research plan and manuscript. TH was the supervisor in research planning, experiments, interpretation of the results, and writing of the manuscript. All authors read and approved the final manuscript.</p>", "<title>Funding</title>", "<p>Open Access funding provided by Kobe University.</p>", "<title>Data availability</title>", "<p>The data generated and/or analyzed during the current study are available from the corresponding author upon reasonable request.</p>", "<title>Declarations</title>", "<title>Ethical approval</title>", "<p id=\"Par28\">This article does not contain any studies with human participants or animals performed by any of the authors.</p>", "<title>Competing interests</title>", "<p id=\"Par29\">The authors declare no competing interests.</p>" ]

[ "<fig id=\"Fig1\"><label>Fig. 1</label><caption><p>Introduction of the MVA pathway into mitochondria. <bold>a</bold> Introduction of the MVA pathway into mitochondria. Native and engineered pathways are shown in black and blue, respectively. <bold>b</bold> The localization of the MLS-fused GFP. The left and right panels show the fluorescent images of the MLS-fused GFP and MitoTracker. IPP production in BY4741 and SSY1. <bold>c</bold> The amount of IPP/DMAPP (columns) and OD₆₀₀ (dots) were measured 24 h after inoculation. Data are presented as the mean ± standard deviation (SD). P-values were determined using two-tailed Student’s <italic>t</italic>-tests (* <italic>P</italic> &lt; 0.05). Biological replication was achieved by three individual cultures</p></caption></fig>", "<fig id=\"Fig2\"><label>Fig. 2</label><caption><p>Mitochondrial volume in the mutants with defective mitochondrial morphology. All strains expressed MLS-fused GFP. Mitochondrial volume was calculated from the fluorescent signals of GFP using the confocal microscopic Z-stack method. Data are presented as the mean ± SD. <italic>P</italic>-values were determined using two-tailed Student’s <italic>t</italic>-tests (* <italic>P</italic> &lt; 0.05) compared with BY4741 cells. Biological replication was achieved using at least ten individual cells</p></caption></fig>", "<fig id=\"Fig3\"><label>Fig. 3</label><caption><p>Effect of mitochondrial volume on squalene production. <bold>a</bold> Effect of mitochondrial volume on IPP/DMAPP level. The amounts of IPP/DMAPP (columns) and OD₆₀₀ (dots) were measured 24 h after inoculation. Biological replication was achieved using three individual cultures. <italic>P</italic>-values were determined using two-tailed Student’s <italic>t</italic>-tests (* <italic>P</italic> &lt; 0.05) compared with SSY1. <bold>b</bold> Squalene production; the amount of squalene (columns) and OD₆₀₀ (dots) were measured 24 h after inoculation. Biological replication and P-values were the same in (<bold>a</bold>). <bold>c</bold> The relationship between the amount of IPP/DMAPP and mitochondrial volume. <bold>d</bold> The relationship between squalene production and mitochondrial volume</p></caption></fig>", "<fig id=\"Fig4\"><label>Fig. 4</label><caption><p>Effect of mitochondrial volume on β-carotene production. <bold>a</bold> Effect of mitochondrial volume on IPP/DMAPP level. The amounts of IPP/DMAPP (columns) and OD₆₀₀ (dots) were measured 48 h after inoculation. Biological replication was achieved using three individual cultures. P-values were determined using two-tailed Student’s <italic>t</italic>-tests (* P &lt; 0.05) compared with SSY6. <bold>b</bold> β-carotene production; the amount of β-carotene (columns) and OD₆₀₀ (dots) were measured 48 h after inoculation. Biological replication and P-values were the same in (<bold>a</bold>). <bold>c</bold> The relationship between the amount of IPP/DMAPP and mitochondrial volume. <bold>d</bold> The relationship between β-carotene production and mitochondrial volume</p></caption></fig>" ]

[ "<table-wrap id=\"Tab1\"><label>Table 1</label><caption><p>Strains used or constructed in this study</p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th align=\"left\">Strain</th><th align=\"left\">Genotype</th></tr></thead><tbody><tr><td align=\"left\">BY4741</td><td align=\"left\">MATa, <italic>his3</italic>Δ<italic>1</italic>, <italic>leu2</italic>Δ<italic>0</italic>, <italic>met15</italic>Δ<italic>0</italic>, <italic>ura3</italic>Δ<italic>0</italic></td></tr><tr><td align=\"left\">SSY1</td><td align=\"left\">BY4741, ARS208::T_TDH3-<italic>ERG10</italic>-MLS-P_TDH3-P_ADH1-MLS-<italic>ERG13</italic>-T_ADH1; ARS308::T_TDH3-<italic>tHMG1</italic>-MLS-P_TDH3-P_ADH1-MLS-<italic>ERG12</italic>-T_ADH1; ARS416::T_TDH3-<italic>ERG19</italic>-MLS-P_TDH3-P_ADH1-MLS-<italic>ERG8</italic>-T_ADH1</td></tr><tr><td align=\"left\">SSY2</td><td align=\"left\">SSY1, Δ<italic>mdm32</italic></td></tr><tr><td align=\"left\">SSY3</td><td align=\"left\">SSY1, Δ<italic>fzo1</italic></td></tr><tr><td align=\"left\">SSY4</td><td align=\"left\">SSY1, Δ<italic>mgm1</italic></td></tr><tr><td align=\"left\">SSY5</td><td align=\"left\">SSY1, Δ<italic>ugo1</italic></td></tr><tr><td align=\"left\">SSY6</td><td align=\"left\">SSY1, pCrtYBI-BTS1</td></tr><tr><td align=\"left\">SSY7</td><td align=\"left\">SSY1, Δ<italic>mdm32</italic>, pCrtYBI-BTS1</td></tr><tr><td align=\"left\">SSY8</td><td align=\"left\">SSY1, Δ<italic>fzo1</italic>, pCrtYBI-BTS1</td></tr><tr><td align=\"left\">SSY9</td><td align=\"left\">SSY1, Δ<italic>mgm</italic>, pCrtYBI-BTS1</td></tr><tr><td align=\"left\">SSY10</td><td align=\"left\">SSY1, Δ<italic>ugo1</italic>, pCrtYBI-BTS1</td></tr></tbody></table></table-wrap>" ]

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[ "<media xlink:href=\"253_2023_12922_MOESM1_ESM.pdf\"><caption><p>Supplementary file1 (PDF 1399 KB)</p></caption></media>" ]

{ "acronym": [], "definition": [] }

[ "<title>Introduction</title>", "<p id=\"Par6\">The asymmetric reduction of C = C double bonds is widely applied in the pharmaceutical, fine chemical, and agrochemical industries, as up to two stereogenic centers can be created in those processes. Unfortunately, the chemo-catalyzed reduction of olefins using hydrogen gas and transition metal catalysts with expensive chiral ligands suffers from poor environmental sustainability, narrow substrate range, or low enantioselectivity (Hollmann et al. ##UREF##2##2021##). The biocatalyzed reduction provides an alternative green synthetic route with a broad substrate range and excellent enantioselectivity. Ene-reductases (ERs), NAD(P)H-dependent flavoproteins from the old yellow enzyme (OYEs) family, are powerful biocatalysts for the asymmetric reduction of activated alkenes, generating the products up to two new stereogenic centers with excellent enantioselectivities (Hall and Bommarius ##REF##21692484##2011##; Toogood and Scrutton ##REF##31157123##2018##).</p>", "<p id=\"Par7\">Since the first OYE was isolated from yeast, many OYEs have been identified from bacteria, yeasts, fungi, plants, and algae (Parmeggiani et al. ##REF##34586700##2022##). The OYEs could be used in the reduction of C = C double bonds bearing an electron-withdrawing group, such as nitro, aldehyde, ketone, carboxylic acid, ester, imide, or nitrile moiety. The enzymes of the OYE family were proposed to be divided into five subclasses based on the phylogenetic and biochemical analysis, called class I (classical OYEs), class II, class III (thermophilic-like OYEs), class IV, and class IV (fungal OYEs). Most of the known OYEs belong to the classical OYE subgroup, which contains OYE1-like OYEs from fungi, OYE3 from <italic>Saccharomyces</italic> sp<italic>.</italic>, PETNR-like OYEs from bacteria, and OPR-like OYEs from plants (Peters et al. ##REF##30758121##2019##). These enzymes have been used in the synthesis of various industrial interested building blocks, such as (<italic>R</italic>)-citronellal (Zheng et al. ##UREF##11##2018##), amino acid derivatives (Winkler et al. ##REF##22498437##2012##), and (<italic>R</italic>)-3-Hydroxy-2-methylpropanoate (Stueckler et al. ##UREF##9##2010##).</p>", "<p id=\"Par8\">Thermophilic enzymes have great potential for industrial use as they have high resistance to temperature and organic solvents. Since the identification of the first thermophilic-like OYE YqjM from <italic>Bacillus subtilis</italic> (Kitzing et al. ##REF##15890652##2005##), another 10 thermophilic-like OYEs belonging to class III have been isolated and characterized, including XenA from <italic>Pseudomonas putida</italic> (Griese et al. ##REF##16822524##2006##), <italic>Ts</italic>OYE from <italic>Thermus scotoductus</italic> SA-01 (Opperman et al. ##REF##18263719##2008##), TOYE from <italic>Thermoanaerobacter pseudethanolicus</italic> (Adalbjornsson et al. ##REF##19943268##2010##), <italic>Gk</italic>OYE from <italic>Geobacillus kaustophilus</italic> (Schittmayer et al. ##UREF##6##2011##), <italic>Chr</italic>-OYE3 from <italic>Chryseobacterium</italic> sp. CA49 (Xu et al. ##UREF##10##2014##), <italic>Geo</italic>ER from <italic>Geobacillus</italic> sp. 30 (Tsuji et al. ##REF##24927695##2014##), OYERo2 from <italic>Rhodococcus opacus</italic> 1CP (Riedel et al. ##REF##26483784##2015##), <italic>F</italic>OYE-1 from <italic>Ferrovum</italic> sp. JA12 (Scholtissek et al. ##REF##27542380##2017##), <italic>Ca</italic>OYE from <italic>Chloroflexus aggregans</italic> (Robescu et al. ##REF##33925162##2021##), and <italic>Pf</italic>ER2 from <italic>Pseudomonas fluorescens</italic> (Shi et al. ##REF##34489033##2021##). Moreover, four thermal-tolerant OYEs were also identified from other subclasses, including <italic>Bc</italic>OYE (class IV) from <italic>Bacillus coagulans</italic> (Zhou et al. ##UREF##12##2014##), YqiG (class IV) from <italic>B. subtilis</italic> (Sheng et al. ##UREF##7##2016##), <italic>Ct</italic>OYE (class I) from <italic>Chroococcidiopsis thermalis</italic>, and <italic>Gs</italic>OYE (class I) from <italic>Galdieria sulphuraria</italic> (Robescu et al. ##REF##31930452##2020##). All the thermophilic-like OYEs showed activity optima around 40–55 °C exhibited high thermostability and high resistance toward organic solvents, so they have huge potential for industrial application.</p>", "<p id=\"Par9\">Several thermophilic-like OYE structures have been revealed, including YqjM (PDB: 1Z41) (Kitzing et al. ##REF##15890652##2005##), XenA (PDB: 2H8X) (Griese et al. ##REF##16822524##2006##), CrS (PDB: 3HF3) (Opperman et al. ##REF##20138824##2010##), TOYE (PDB: 3KRU) (Adalbjornsson et al. ##REF##19943268##2010##), <italic>Gk</italic>OYE (PDB: 3GR7) (Schittmayer et al. ##UREF##6##2011##), <italic>Rm</italic>ER (PDB: 5OCS) (Opperman ##REF##28833623##2017##), and <italic>Ca</italic>OYE (PDB: 7O0T) (Robescu et al. ##REF##33925162##2021##), which showed that OYE homologs are highly conserved. The general catalytic mechanism of OYE-mediated reduction is well understood. The electron-withdrawing group moiety forms an H-bond interaction with two donor residues, usually histidine and asparagine, and then the addition of hydride from the reduced flavin to the Cβ of the activated alkene, followed by proton transfer to the Cα of the substrate from a tyrosine or cysteine residue.</p>", "<p id=\"Par10\">Although many OYEs have been identified over the last years, the demand for new OYEs keeps growing to overcome application limitations, such as low catalytic activity and poor stability in harsh reaction conditions. Fungi appear to be valuable sources of OYEs, but most of the fungal OYEs are unexplored. <italic>Aspergillus</italic> sp<italic>.</italic> produces many secondary metabolites, and it has been proposed that ene-reductases participate in the synthetic pathways. For example, the <italic>Aspergillus fumigatus</italic> old yellow enzyme EasA reduces chanoclavine-I aldehyde to dihydrochanoclavine aldehyde in the ergot alkaloid pathways (Cheng et al. ##REF##20102147##2010##). More recently, four OYEs were identified from <italic>Aspergillus niger</italic> and <italic>Botryotinia fuckeliana</italic> (Robescu et al. ##UREF##5##2022a##, ##REF##35328465##2022b##). Herein, we mined the OYEs in the filamentous fungus <italic>Aspergillus flavus</italic> to expand the ene-reductase toolbox. A new thermophilic-like OYE (<italic>Af</italic>OYE1) was identified and carefully characterized, which exhibited broad substrate scope and excellent enantioselectivity.</p>" ]

[ "<title>Materials and methods</title>", "<title>Reagents</title>", "<p id=\"Par11\">Alkene substrates and racemic reduction products were purchased from MilliporeSigma (MA, USA), Aladdin (Shanghai, China), or TCI (Shanghai, China). Glucose dehydrogenase from <italic>Bacillus</italic> was purchased from Aladdin (G139687, Shanghai, China), and NADH disodium salt hydrate was purchased from TCI (Shanghai, China). All protein commercial crystallization kits were purchased from Hampton Research (CA, USA), MiTeGen LLC (NY, USA), Qiagen (Hilden, Germany), or Molecular Dimensions (OH, USA). All other reagents and solvents were commercially available and were used without further purification.</p>", "<title>General analysis methods</title>", "<p id=\"Par12\">¹H NMR spectra were acquired on a Bruker-400 spectrometer in CDCl₃, and all signals were reported in parts per million (ppm) downfield relative to tetramethylsilane. Optical rotations were measured with a Perkin Elmer 341 polarimeter. Gas chromatographic analyses were performed on a Thermo Scientific TRACE 1300 gas chromatograph equipped with a CHIRASIL-DEX CB column (Agilent Technologies, USA), and using a flame ionization detector, nitrogen was used as the carrier gas at 5 mL/min, the split ratio was 1:20 (v/v), and the column temperature was programmed as being kept at 50 °C for 1 min and then upgraded to 150 °C at the rate of 3 °C/min. The protein purities were analyzed using SDS-PAGE using 4–12% polyacrylamide gradient gel (SurePAGE, Genscript, Nanjing, China) running in 50 mM MOPS, 50 mM Tris base, 0.1% SDS, and 1 mM EDTA, at pH 7.7, and stained with Coomassie Blue. Protein concentrations were measured using the Bradford assay (Coomassie Brilliant Blue G-250) and bovine serum albumin as the protein standard.</p>", "<title>Mining OYE from Aspergillus flavus and plasmid construction</title>", "<p id=\"Par13\">The thermophilic-like old yellow enzyme YqjM was submitted to a BLASTp program against the <italic>A. flavus</italic> strain NRRL3357 proteins in UniProtKB/Swiss-Prot database, which returned an NADPH dehydrogenase afvA (UniProt: B8N8Q9, GenBank: XP_041144250.1) with 41.7% identities and an NADP-dependent oxidoreductase lnbE (UniProt: B8NWW6) with 34% identities. Here, the NADPH dehydrogenase afvA was selected, and its gene sequence with an N-terminal His6-tag followed by a tobacco etch virus (TEV) protease cleavage site was codon-optimized by OPTIMIZER (<ext-link ext-link-type=\"uri\" xlink:href=\"http://genomes.urv.es/OPTIMIZER/\">http://genomes.urv.es/OPTIMIZER/</ext-link>, See Supplementary Information for nucleotide sequence) (Puigbò et al. ##REF##17439967##2007##) for the expression in <italic>E. coli</italic> and synthesized by GENEWIZ (Nanjing, China). The synthesized gene was inserted into the pET28b vector between the <italic>Nco</italic>I and <italic>Xho</italic>I enzyme sites, creating the plasmid pET28b-<italic>afvA</italic>, and the sequence was confirmed via sequencing (Shenggong, Shanghai, China).</p>", "<title>Protein production and purification</title>", "<p id=\"Par14\">The plasmid pET28b-<italic>afvA</italic> was transferred into <italic>E. coli</italic> BL21 (DE3) for protein production. LB medium containing kanamycin (50 μg/mL) was inoculated with <italic>E. coli</italic> BL21 (pET28b-<italic>afvA</italic>), and the culture was incubated at 37 °C and 220 rpm. Then, isopropyl-β-D-thio-galactoside (IPTG) was added into the culture at the final concentration of 1 mM when the OD₆₀₀ reached about 0.5. The culture was further cultured at 16 °C for 18 h for protein expression, and then the cells were harvested by centrifugation (5000 g, 4 °C) for 6 min.</p>", "<p id=\"Par15\">The cell pellets were re-suspended in HEPES buffer (25 mM, pH 7.5, 10 M NaOH solution was used to adjust pH) containing 300 mM NaCl, 20 mM imidazole, and 1 mM tris(2-carboxyethyl)phosphine (TCEP), and then the cells were lysed by low-temperature ultra-high‐pressure homogenizer (JN-mini, JNBio, China) at 1100 bar and 4 °C. The lysis was centrifuged at 18,000 g and 4 °C for 1 h to remove the cell debris. The clear lysate was filtered through a surfactant-free cellulose acetate membrane with a 0.45 μm pore size and loaded on the His Trap™ HP column, which was pre-equilibrated with HEPES buffer (25 mM, pH 7.5) containing 300 mM NaCl, 20 mM imidazole, and 1 mM TCEP. Proteins were eluted by HEPES buffer (25 mM, pH 7.5) containing 300 mM NaCl and 1 mM TCEP with a linear imidazole concentration gradient from 20 to 500 mM. The peak fractions were collected and followed by His6-TEV protease cleavage and dialysis against HEPES buffer (25 mM, pH 7.5) containing 300 mM NaCl and 1 mM TCEP overnight. The dialysate was loaded onto a His Trap™ HP column again, which was pre-equilibrated with HEPES buffer (25 mM, pH 7.5) containing 300 mM NaCl and 1 mM TCEP. Flow through and wash fractions that contain the desired purified protein were collected and checked by SDS-PAGE. The fractions containing desired purified protein were pooled and concentrated by Amicon® Ultra-15 Centrifugal Filter Unit (10 K, Merck Millipore) and used as the catalyst.</p>", "<p id=\"Par16\">To prepare protein samples for crystallization, the purified protein was further purified using gel filtration (HiLoad 16/60 Superdex pg 200, GE Healthcare) with HEPES buffer (25 mM, pH 7.5) containing 100 mM NaCl and 1 mM TCEP. The peak fractions were concentrated and used for crystallization.</p>", "<title>General procedures of AfOYE1-catalyzed reductions</title>", "<p id=\"Par17\">The purified <italic>Af</italic>OYE1 was used in all the measurements. All the reaction mixtures contained 5 mL potassium phosphate buffer (100 mM, pH 7.0), 0.1 μM purified <italic>Af</italic>OYE1, 5 U glucose dehydrogenase from <italic>Bacillus</italic>, 50 mM glucose, 20 mM NADH, and 8 mM substrate, and then the reactions were conducted at 45 °C for 10 min in triplicates. Then, the mixtures were extracted with ethyl acetate (1 mL), and then the yields were determined by GC with an external standard calibration curve. One unit of enzyme activity was defined as the amount of enzyme producing 1 μmol product per minute under the assay condition.</p>", "<p id=\"Par18\">The irreversible thermal inactivation (<italic>T</italic>_1/2) of <italic>Af</italic>OYE1 was analyzed by heating the purified <italic>Af</italic>OYE1 in potassium phosphate buffer (pH 7.0) at different temperatures (30–70 °C) for 5 min, chilled on ice. The residual activities on 2-cyclopentenone were then measured under standard assay conditions.</p>", "<p id=\"Par19\">To determine the optimal reaction conditions, 8 mM 2-cyclopentenone was used as a substrate, and the reactions were initiated by the addition of purified <italic>Af</italic>OYE1. The reaction mixtures in 5 mL potassium phosphate buffer (100 mM, pH 7.0), were conducted at 25–60 °C for 10 min, or in 5 mL potassium phosphate buffer (100 mM, pH 5.5–8.0) were conducted at 45 °C for 10 min. Then, the mixtures were extracted with ethyl acetate (1 mL), and the yield of cyclopentanone was determined by GC with an external standard calibration curve.</p>", "<p id=\"Par20\">Preparative biotransformation was carried out in 50 mL potassium phosphate buffer (100 mM, pH 7.0), 0.5 μM purified <italic>Af</italic>OYE1, 20 U glucose dehydrogenase from <italic>Bacillus</italic>, 200 mM glucose, 20 mM NADH, and 50 mM substrate, and then the reactions were conducted at 45 °C for 10 h. Then, the mixtures were extracted with ethyl acetate (3 × 50 mL), the organic phase was combined and dried by anhydrous NaSO₄, the solvents were removed under certain reduced pressure, and then the final product was purified with column chromatography and subjected to NMR and optical rotation analysis.</p>", "<p id=\"Par21\">The kinetic parameters <italic>K</italic>_{<italic>m</italic>} and <italic>k</italic>_cat of <italic>Af</italic>OYE1 for 2-cyclopentenone were determined by measuring the production of cyclopentanone. The reaction mixture containing varying concentrations of 2-cyclopentenone in 5 mL potassium phosphate buffer (100 mM, pH 7.0) was incubated under the optimal reaction conditions (pH 7.0, and 45 °C) at 220 rpm for 2.0 min. The mixtures were extracted with ethyl acetate (1 mL), and then the yield of cyclopentanone was determined by GC. The parameters were calculated using the Prism program (GraphPad, San Diego, CA, USA).</p>", "<title>Determination of the flavin cofactor</title>", "<p id=\"Par22\">The flavin content of <italic>Af</italic>OYE1 was determined spectrophotometrically from absorption scans (350–750 nm) using a SpectraMax® M2e (Molecular Devices, USA) microplate reader. Free flavin was released from <italic>Af</italic>OYE1 by the protein denaturation through incubating in a boiling water bath for 10 min; then, the samples were centrifuged at 15,000 g for 10 min. The supernatant, <italic>Af</italic>OYE1 solution, and standard flavin mononucleotide (FMN) solution were analyzed by microplate readers to determine flavin content.</p>", "<title>Crystallization</title>", "<p id=\"Par23\">Initial crystallization screening of <italic>Af</italic>OYE1 (20 mg/mL) was carried out at 18 °C with high-throughput sparse matrix crystallization trails using a Gryphon crystallization robot (Art Robbins Instruments). MRC 96-well two-drop standard plates were adopted with the sitting-drop vapor diffusion method by mixing 0.5 μL protein and 0.5 μL reservoir solution in the initial screening. A 24-well plate was used with the hanging-drop vapor diffusion method by mixing 1 μL protein and 1 μL reservoir solution in manual optimization steps. Well-diffracted crystals were achieved in the condition containing 0.1 M Bis–Tris propane, pH 6.5, 15% (w/v) PEG 3350, and 0.2 M sodium nitrate, after seeding with smaller crystals initially obtained in the same condition. Crystals were cryoprotected in mother liquor supplemented with 25% (v/v) ethylene glycerol and snap-frozen in liquid nitrogen for data collection.</p>", "<title>Data collection and structural determination</title>", "<p id=\"Par24\">The X-ray diffraction data were collected at 100 K Crystallography Beamline 18U1 (BL18U1) using a Pilatus 3 S 6 M detector at Shanghai Synchrotron Radiation Facility (SSRF, Shanghai, China) and at the wavelength of 0.979 Å. Datasets were processed using XDS (Kabsch ##UREF##3##2010##). The structure was solved by molecular replacement using a modified monomer model of ene-reductase (PDB code: 5OCS, sharing 42% identity with <italic>Af</italic>OYE1) (Opperman ##REF##28833623##2017##) with all amino acids mutating to alanine in coot as a search model. An <italic>Af</italic>OYE1 dimer was positioned in the asymmetric unit using PHENIX Phaser-MR (McCoy et al. ##REF##19461840##2007##). All refinement and model-building procedures were carried out by PHENIX.refine (Adams et al. ##UREF##0##2010##) and COOT (Emsley and Cowtan ##UREF##1##2004##). FMN was automatically imported from the Coot dictionary. The final parameters obtained for the best models reached a Rfactor/Rfree of 0.144/0.167 at 1.55 Å (Table ##SUPPL##0##S1##). All the structure figures were prepared by PyMOL.</p>", "<title>Phylogenetic analysis</title>", "<p id=\"Par25\">The sequence alignment of 83 known OYEs (Table ##SUPPL##0##S2##) and <italic>Af</italic>OYE1 was done by ClustalW, and then phylogenetic analysis was conducted using the Maximum Likelihood statistical method based on the Jones-Taylar-Thornton (JTT) model in MEGAX. The initial tree for the heuristic search was obtained by applying NJ/BioNJ algorithms to a matrix of pairwise distance that was estimated by a <italic>p</italic>-distance and then selecting the topology with superior log likelihood value (− 28,266.79). The visualizing phylogenetic tree was produced by an online tree viewer iTOL Version 6.5.8, and the assignment of the subclasses was based on a previous evolutionary study.</p>", "<title>Construction of point mutations</title>", "<p id=\"Par26\">The mutants were constructed by site-directed mutagenesis using primers listed in Table ##SUPPL##0##S3## and confirmed by sequencing (Shenggong, Shanghai, China). The correct plasmid was transformed into <italic>E. coli</italic> BL21 for protein production.</p>" ]

[ "<title>Results</title>", "<title>Mining OYEs from A. flavus and sequence analysis</title>", "<p id=\"Par27\">To identify new OYEs from <italic>A. flavus</italic>, the thermophilic-like OYE YqjM from <italic>Bacillus subtilis</italic> was used as a reference to search the <italic>A. flavus</italic> proteins in UniProtKB/Swiss-Prot database. The returned results showed that <italic>A. flavus</italic> contains two OYEs, including a putative NADPH dehydrogenase afvA (UniProt: B8N8Q9) and an NADP-dependent oxidoreductase lnbE (UniProt: B8NWW6). The afvA and lnbE have 41.7% and 34% identities with YqjM, respectively. Here, the afvA was selected since it has higher identities with the known YqjM, and then it was overexpressed in <italic>E. coli</italic> for further analysis. Functional analysis showed that afvA has the ability to reduce 2-cyclopentenone into 2-cyclopentanone. The <italic>Af</italic>OYE1 was previously named afvA, which was annotated as an NADPH dehydrogenase. afvA was proposed to catalyze the hydroxylation of siderin and 7-demethyl siderin to the corresponding hydroxylsiderin and hydroxydemethylsiderin (Uka et al. ##REF##33337039##2020##). However, we found that afvA could not hydroxylate the siderin analogs, such as 6-hydroxy-4-methylcoumarin, 6-methylcoumarin, and 7-methylcoumarin, which indicated that afvA might not be able to catalyze the hydroxylation. Here, we annotated afvA as <italic>Af</italic>OYE1 based on the functional analysis.</p>", "<p id=\"Par28\">The phylogenetic analysis of <italic>Af</italic>OYE1 and 83 known OYEs was conducted, and the results showed that <italic>Af</italic>OYE1 belongs to the group of Class III OYEs (previous thermophilic-like OYEs group) (Fig. ##FIG##0##1##) (Peters et al. ##REF##30758121##2019##). The sequence alignments showed that <italic>Af</italic>OYE1 conserved the two fingerprint motifs of the thermophilic-like OYE homologs and the catalytic residues of OYEs, including His211, His214, and Tyr216 (Fig. ##SUPPL##0##S1##) (Kitzing et al. ##REF##15890652##2005##). <italic>Af</italic>OYE1 has the highest similarities to the OYEs from <italic>Botryotinia fuckeliana</italic> (<italic>Bf</italic>OYE4) and <italic>Aspergillus niger</italic> (<italic>An</italic>OYE8), with 53.6% and 51.7% identities, respectively (Robescu et al. ##UREF##5##2022a##). The phylogenetic analysis and sequence properties also indicated that <italic>Af</italic>OYE1 is an ene-reductase other than a hydroxylase.</p>", "<title>Expression and characterization of the recombinant</title>", "<p id=\"Par29\">The heterologous expression of <italic>Af</italic>OYE1 with an N-terminal His6-tag was conducted in the host <italic>E. coli</italic> BL21 (DE3) at 16 °C. SDS-PAGE analysis showed that <italic>Af</italic>OYE1 was produced with a molecular mass of around 43 kDa (<bold>Fig. S2</bold>), which was in accordance with the predicted molecular mass derived from the amino acid sequence.</p>", "<p id=\"Par30\">The UV–VIS absorption spectra of the purified <italic>Af</italic>OYE1, released flavin, and standard FMN suggested that <italic>Af</italic>OYE1 contained an FMN molecule as its prosthetic group and the FMN was non-covalently bound to the protein (Fig. ##FIG##1##2##). The supernatant after boiling of <italic>Af</italic>OYE1 and centrifugation turned into bright yellow, and both the released flavin and standard FMN had identical spectra, which had maxima absorbance at about 370 and 450 nm. The flavin contained <italic>Af</italic>OYE1 exhibited maxima absorbance at 370 and 460 nm.</p>", "<title>Evaluation of the thermostability of AfOYE1 and optimal reaction conditions</title>", "<p id=\"Par31\">The <italic>Af</italic>OYE1 thermal stability was evaluated by detecting the midpoint for thermal inactivation (<italic>T</italic>_1/2). The irreversible thermal deactivation curve showed that the <italic>T</italic>_1/2 of <italic>Af</italic>OYE1 was about 60 °C (Fig. ##FIG##2##3##A). Moreover, the catalytic activity of <italic>Af</italic>OYE1 decreased slowly after 5 min heat treatment at the temperature below 55 °C, and then a drastic decline was observed at the temperature above 55 °C.</p>", "<p id=\"Par32\">To further characterize <italic>Af</italic>OYE1, the effects of temperature and pH on its catalytic activity with 2-cyclopentenone were investigated carefully. <italic>Af</italic>OYE1 exhibited high catalytic activities at temperatures 40–50 °C and observed maximum activity (0.85 U/mg) at 45 °C (Fig. ##FIG##2##3##B). The data showed that <italic>Af</italic>OYE1 still conserved 70% of the maximal activity at 60 °C (0.6 U/mg) and over 60% of the maximal activity when the temperature was down to 20 °C (0.52 U/mg). Moreover, <italic>Af</italic>OYE1 had high catalytic activity in a wide range of pH (pH 5.5–8.0) (Fig. ##FIG##2##3##C). The highest catalytic activity (1.06 U/mg) of <italic>Af</italic>OYE1 was observed at pH 7.0, and it displayed over 85% of the maximal activity at pH 5.5 (0.91 U/mg) and pH 8.0 (0.94 U/mg).</p>", "<title>Evaluation of the catalytic activity of AfOYE1 toward activated alkenes and kinetic analysis</title>", "<p id=\"Par33\">The <italic>Af</italic>OYE1-catalyzed bioreduction system was further applied to different kinds of activated α, β-unsaturated alkenes. The results showed that cyclic enones (<bold>1</bold>, <bold>2</bold>, <bold>3</bold>, and <bold>4</bold>), acrylamide (<bold>5</bold>), nitroalkenes (<bold>6</bold> and <bold>7</bold>), and α, β-unsaturated aldehydes (<bold>8</bold>) could be reduced by <italic>Af</italic>OYE1 (Fig. ##FIG##3##4##). However, cyclic imide (<bold>9</bold>) and α, β-unsaturated esters (<bold>12</bold> and <bold>13</bold>) could not be accepted as substrates for <italic>Af</italic>OYE1. <italic>Af</italic>OYE1 displayed the highest activity toward 2-cyclohexenone with TTN of 2050. For the <italic>Af</italic>OYE1-catalyzed bioreduction of cyclic enones, it had higher activity toward cyclohexenones than cyclopentenones (<bold>3</bold> vs. <bold>1</bold>, <bold>4</bold> vs. <bold>2</bold>). Similar to other OYEs in the thermophilic-like group, <italic>Af</italic>OYE1 was shown to have a restricted substrate spectrum (Amato and Stewart ##REF##25940546##2015##).</p>", "<p id=\"Par34\">The results showed that the substituent on the cyclic moiety had an important effect on the catalytic activity. <italic>Af</italic>OYE1 could catalyze the reduction of α-methyl substituted cyclopentenone (<bold>2</bold>) and cyclohexenone (<bold>4</bold>). However, it could not convert the β-methyl substituted cyclopentenone (<bold>10</bold>) and cyclohexenone (<bold>11</bold>) into the corresponding product. Moreover, the <italic>Af</italic>OYE1-catalyzed reduction had excellent enantioselectivities toward the prochiral alkenes. The bioreduction of 2-methylcyclopentenone (<bold>2</bold>) and 2-methylcyclohexenone (<bold>4</bold>) yielded the corresponding (<italic>S</italic>)-2-methylcyclopentanone ([α]²⁵_D =  + 16.5° (c 0.4, CHCl₃) ((<italic>S</italic>), [α]²⁵_D =  + 114.9° (c 0.52, CHCl₃) (Shimoda et al. ##UREF##8##2004##)) and (<italic>S</italic>)-2-methylcyclohexanone with &gt; 99% ee.</p>", "<p id=\"Par35\">The kinetic parameters <italic>K</italic>_{<italic>m</italic>} and <italic>k</italic>_cat of <italic>Af</italic>OYE1 were investigated with varying concentrations of 2-cyclopentenone (Fig. ##SUPPL##0##S3##). The <italic>K</italic>_{<italic>m</italic>} value determined by non-linear regression was 0.061 ± 0.005 mM⁻¹, which is significantly smaller than those of most of the known OYEs (Robescu et al. ##REF##35328465##2022b##, ##REF##31930452##2020##; Xu et al. ##UREF##10##2014##). The apparent <italic>k</italic>_cat of the <italic>Af</italic>OYE1 on 2-cyclopentenone was 34.12 ± 0.74 min⁻¹, resulting in catalytic efficiencies of 9.32 mM⁻¹ s⁻¹.</p>", "<title>Crystal structure of AfOYE1</title>", "<p id=\"Par36\">The structure of <italic>Af</italic>OYE1 (PDB: 8J59) was determined to have a maximum resolution of 1.55 Å in the P1211 space group with an elongated dimer per asymmetric unit (Table ##SUPPL##0##S2##). The overall structure of <italic>Af</italic>OYE1 showed an expected eight-stranded (α, β)-barrel fold of triosephosphate isomerase (TIM), which is highly conserved in the thermophilic-like OYEs (Fig. ##FIG##4##5##A). On the top of the β-barrel core of each monomer encircles one FMN cofactor, which points toward the C-terminal side and is buried within the active site cavity. Similar to other thermophilic-like OYEs, the PDBePISA analysis (<ext-link ext-link-type=\"uri\" xlink:href=\"https://www.ebi.ac.uk/msd-srv/prot_int/cgi-bin/piserver\">https://www.ebi.ac.uk/msd-srv/prot_int/cgi-bin/piserver</ext-link>) of the <italic>Af</italic>OYE1 structure showed that the inverse dimeric architecture with an interacting surface of 492 Ų is contributed by α1-helix, the peculiar long N-terminal loops, and the C-terminal helix of each protomer, which contains 7 hydrogen bonds and 6 salt bridges among the 41 interfacing residues (Fig. ##FIG##4##5##B and Table ##SUPPL##0##S4##) (Krissinel and Henrick ##REF##17681537##2007##). Meanwhile, <italic>Af</italic>OYE1 has a slightly negative surface but a highly positive inner surface at the entrance of the active site, which endows its ability to bind the negatively charged FMN cofactor (Fig. ##SUPPL##0##S4##).</p>", "<p id=\"Par37\">The DoGSiteScorer analysis (<ext-link ext-link-type=\"uri\" xlink:href=\"https://proteins.plus\">https://proteins.plus</ext-link>) of the structure showed that the catalytic cavity of <italic>Af</italic>OYE1 is quite deep, and the FMN cofactor is heavily buried at the bottom of the catalytic hole with a depth of 24.23 Å (Table ##SUPPL##0##S4##). As shown by the difference electron density map <italic>F</italic>_{<italic>o</italic>}-<italic>F</italic>_{<italic>c</italic>}, the <italic>si</italic>-face of the FMN faced the solvent, while the <italic>re</italic>-side was in contact with the protein backbone tightly via the extensive hydrogen bonding and hydrophobic interactions with the side chain and main chain elements (Fig. ##FIG##4##5##C and Fig. ##SUPPL##0##S5##) (Robescu et al. ##REF##35328465##2022b##). Meanwhile, the histidine pair (His211/His214) together with the highly conserved proton donor Tyr216 interacts with a buffer-sourced chloride anion and is positioned on the <italic>si</italic>-face of the isoalloxazine ring of FMN. It indicated that His/His together with Tyr 216 participated in the binding and correct orientation of substrate (Fig. ##FIG##4##5##C) (Robescu et al. ##REF##35328465##2022b##, ##REF##31930452##2020##, ##REF##33925162##2021##).</p>", "<p id=\"Par38\">Until now, two OYE structures from filamentous fungi, including OYE4 from <italic>B. fuckeliana</italic> (<italic>Bf</italic>OYE4, PDB: 7BLF) (Robescu et al. ##UREF##5##2022a##) and OYE from <italic>A. niger</italic> (<italic>An</italic>OYE8, PDB: 7QFX) (Robescu et al. ##REF##35328465##2022b##), have been reported. The superposition structures of <italic>Af</italic>OYE1 with <italic>Bf</italic>OYE4 and <italic>An</italic>OYE8, with an r.m.s.d. of 0.694 Å and 0.588 Å, respectively, showed that they have similar monomer folding, except the α6-helix for the residues (from Phe271 to Glu280), the loop region (from Ser316 to Gly325), and the cap subdomain region (from Ala132 to Thr182) (Fig. ##SUPPL##0##S6##). The protruding α6-helix and the long N-terminal loops of <italic>Af</italic>OYE1 work dimerization function in multi-asymmetry units (Fig. ##SUPPL##0##S7##). However, the protruding α6-helix shows random loops in other OYEs (Fig. ##SUPPL##0##S6##). The loop region from Ser316 to Gly325 runs on the enzyme surface and reaches the active site entrance, which contributes to the size and feature of the catalytic cavity (Fig. ##SUPPL##0##S8##A). The residues Ile318 and Ile320 in the loop have a distance to the FMN flavin group of 5.1 Å and 4.1 Å, respectively. Meanwhile, the residue Ile320 points its side chain toward the top of the catalytic cavity, which is stabilized by a hydrogen bond with Arg371 at the C-terminal α-helix (from Arg371 to Gln375) (Fig. ##SUPPL##0##S8##B). The cap region of <italic>Af</italic>OYE1 with large unstructured turns is located on the top of the entrance of the active site (from Ala132 to Thr182), while the <italic>Bf</italic>OYE4 shows α-helices and the <italic>Gs</italic>OYE (PDB: 6S0G) shows β-hairpins (Robescu et al. ##REF##31930452##2020##) (Fig. ##SUPPL##0##S8##C), which may involve interacting with NADH/NADPH and adjusting the selectivity toward FMN cofactor (Adalbjornsson et al. ##REF##19943268##2010##; Knaus et al. ##REF##26727612##2016##; Pompeu et al. ##UREF##4##2012##; Pudney et al. ##REF##17939663##2007##). The C-terminal α helix of each monomer (from Thr395 to Phe401) points to the active sites of the adjacent monomer, and the “finger” residue Trp399 interacts with the respective FMN flavin group at a distance of 3.8 Å (Fig. ##SUPPL##0##S8##B). This may play an important role in enzyme assembling, TIM barrel catalytic domain surface stabling, and two protomers dimerization (Fig. ##SUPPL##0##S8##B).</p>", "<title>Analysis of the catalytic residues of AfOYE1</title>", "<p id=\"Par39\">Both the sequence alignment and structure analysis indicated that <italic>Af</italic>OYE1 highly conserved the catalytic residues, including His211, His214, and Tyr216. Here, the function of those residues was investigated. Both His211 and His214 were mutated into leucine, creating the variants H211L and H214L. The catalytic activity analysis showed that these two variants still conserved about 30% of catalytic activity of the wild type <italic>Af</italic>OYE1 (Fig. ##FIG##5##6##). Previously, the histidine pair has been considered as the binding motifs of OYE homologs, such as His164 and His167 in YqjM (Shi et al. ##REF##32830294##2020##), His172 and His175 in Crs (Opperman et al. ##REF##20138824##2010##), and His182 and Asn185 in “thermophilic-like” OYE homologs (Riedel et al. ##REF##26483784##2015##). The variants H211L and H214L had only 30% catalytic activity of the wild type, which indicated that the histidine residues in the catalytic center were important for substrate binding. Furthermore, the tyrosine residue in the catalytic center of OYE was proposed as a proton donor of OYE, which contributed to the transfer of proton to substrate at the α-position (Kitzing et al. ##REF##15890652##2005##). Therefore, Tyr216 was mutated to phenylalanine (Y216F) to investigate whether Tyr216 serves as the proton donor in <italic>Af</italic>OYE1. The results showed that the variant Y216F conserved 30% catalytic activity of the wild type (Fig. ##FIG##5##6##), which indicated that Tyr216 did act as a proton donor in <italic>Af</italic>OYE1. However, alternative proton donors might exist, because the variant Y216F was still able to reduce 2-cyclopentenone.</p>" ]

[ "<title>Discussion</title>", "<p id=\"Par40\">The <italic>Af</italic>OYE1 was previously named afvA, which was annotated as an NADPH dehydrogenase. The proposed function of afvA was to hydroxylate siderin and 7-demethyl siderin in the biosynthesis of aflavarin (Cary et al. ##REF##26209694##2015##). Here, the recombinant <italic>Af</italic>OYE1 was shown to have the ability to reduce the activated alkenes. However, it could not catalyze the hydroxylation of the siderin analogs, such as 6-hydroxy-4-methylcoumarin, 6-methylcoumarin, and 7-methylcoumarin. These data indicated that <italic>Af</italic>OYE1 might not play as a hydroxylase in the biosynthesis of aflavarin. Similar to other OYEs in the thermophilic-like group, the recombinant <italic>Af</italic>OYE1 was shown to have a restricted substrate spectrum (Amato and Stewart ##REF##25940546##2015##). It has the catalytic activity toward cyclic enones, acrylamide, nitroalkenes, and α, β-unsaturated aldehydes, but the cyclic imide and α, β-unsaturated esters could not be accepted as substrate.</p>", "<p id=\"Par41\">Until now, over twenty OYE homologs from Class III OYEs have been identified, including <italic>Bf</italic>OYE4 from filamentous fungus <italic>B. fuckeliana</italic> and <italic>An</italic>OYE8 from filamentous fungus <italic>A. niger</italic>. Although <italic>Bf</italic>OYE4 and <italic>An</italic>OYE8 were phylogenetically clustered in the group of Class III OYEs, the thermostability of these two filamentous fungi source OYEs is similar to those of non-thermostable OYEs (Robescu et al. ##REF##35328465##2022b##). This newly identified <italic>Af</italic>OYE1 from filamentous fungus <italic>A. flavus</italic> had high catalytic activity at 40–50 °C and with the optima temperature at 45 °C. Moreover, the thermal stability analysis showed that the midpoint for thermal inactivation (<italic>T</italic>_1/2) of <italic>Af</italic>OYE1 was 60 °C. All the data indicated that <italic>Af</italic>OYE1 is a thermostable OYE. Therefore, the first thermostable OYE is identified from <italic>A. flavus</italic>, which provides a new optional thermostable OYE in asymmetric reduction of activated alkenes.</p>", "<p id=\"Par42\"><italic>Af</italic>OYE1 catalyzed the bioreduction of 2-methylcyclopentenone and 2-methylcyclohexenone with excellent (<italic>S</italic>)-enantioselectivity, producing the corresponding (<italic>S</italic>)-2-methylcyclopentanone and (<italic>S</italic>)-2-methylcyclohexanone with &gt; 99% ee. Until now, most of the known OYEs are (<italic>R</italic>)-stereoselectivity, and only 4 OYEs, including KYE1 from <italic>Kluyveromyces lactis</italic> (Yanto et al. ##REF##21510626##2011##), YersER from <italic>Yersinia bercovieri</italic> (Yanto et al. ##REF##21510626##2011##), <italic>Ct</italic>OYE from <italic>Chroococcidiopsis thermalis</italic> (Yanto et al. ##REF##21510626##2011##), and <italic>Gs</italic>OYE from <italic>Galdieria sulphuraria</italic> (Robescu et al. ##REF##31930452##2020##), were (<italic>S</italic>)-stereoselectivity in the reduction of 2-methylcyclohexenone to 2-methylcyclohexanone. However, <italic>Af</italic>OYE1 only shared 23–33% of identities with those four OYEs. The identification of a new thermostable <italic>Af</italic>OYE1 would provide a robust enzyme for (<italic>S</italic>)-enantioselective bioreductions.</p>", "<p id=\"Par43\">The novel structure of the loop of Ser316 to Gly325 narrows the entrance of the catalytic pocket. This 10aa-long loop of <italic>Af</italic>OYE1 spans the entrance of the catalytic pocket and bumps close to the entrance of the catalytic pocket, resulting in a narrow entrance of the catalytic center (Fig. ##SUPPL##0##S9##A). However, the corresponding loop in <italic>Ca</italic>OYE (Ser265 to Try280), <italic>An</italic>OYE8 (Ser308 to Phe323), <italic>Bf</italic>OYE4 (Ser324 to Tyr339), YqjM (His 255 to Pro262), <italic>Ts</italic>OYE (Ser260 to Val277), and <italic>Gk</italic>OYE (Ser250 to Gln265) was away from the entrance of the catalytic pocket (Fig. ##SUPPL##0##S9##A), resulting in an open catalytic pocket (Robescu et al. ##REF##35328465##2022b##, ##REF##33925162##2021##). Meanwhile, similar to the residue Arg325 of <italic>Ca</italic>OYE and Phe368 of <italic>Rm</italic>ER, the “finger” residue Trp399 of <italic>Af</italic>OYE1, which is located in the C-terminal α helix of each monomer and points to the FMN flavin group, and together with the cap region (residues Ala132 to Thr182) may interact with NADH/NADPH and adjust the selectivity of FMN cofactor (Opperman ##REF##28833623##2017##).</p>", "<p id=\"Par44\">The functional analysis of the conserved catalytic residues His211, His214, and Tyr216 of <italic>Af</italic>OYE1 showed that all three residues were important to its catalytic activity. However, the mutagenesis of those three residues did not eliminate their catalytic activity. The histidine pair has been considered as the binding motifs of OYE homologs, such as His164 and His167 in YqjM (Shi et al. ##REF##32830294##2020##), His172 and His175 in Crs (Opperman et al. ##REF##20138824##2010##), and His182 and Asn185 in “thermophilic-like” OYE homologs (Riedel et al. ##REF##26483784##2015##). The <italic>Af</italic>OYE1 variants H211L and H214L conserved 30% catalytic activity of the wild type, which showed that those two histidine residues played an important role in the substrate binding. Meanwhile, the variants H211L and H214L still exhibited activity, which indicated that other residues in the catalytic center of <italic>Af</italic>OYE1 might enable the enzyme to bind the substrate when the histidine was replaced. The replacement of Tyr216 by phenylalanine resulted in a dramatic decrease in catalytic activity, indicating Tyr216 acts as a proton donor in the <italic>Af</italic>OYE1 (Kitzing et al. ##REF##15890652##2005##). Thus, a common catalytic mechanism can be anticipated for <italic>Af</italic>OYE1 and other OYEs. On the other hand, the variant Y216F could still reduce 2-cyclopentenone into 2-cyclopentanone, which suggested that the proton transfer process could be compensated. Previously, it has also been shown that the replacement of Tyr177 in CrS resulted in the variant only harboring 20% of the catalytic efficiency (Opperman et al. ##REF##20138824##2010##).</p>", "<p id=\"Par45\">In summary, a new thermophilic-like OYE <italic>Af</italic>OYE1 was identified from <italic>A. flavus</italic>. It could reduce a spectrum of activated alkenes, had an optimal temperature of 45 °C, and exhibited high reduction activity in a wide pH range (pH 5.5–8.0). Unlike most of the known OYEs, the <italic>Af</italic>OYE1-catalyzed asymmetric reduction had (<italic>S</italic>)-selective with excellent enantioselectivity, which expands the toolbox of thermostable (<italic>S</italic>)-selective OYEs. The crystal structure of <italic>Af</italic>OYE1 revealed that it had a special catalytic cavity, which may be relevant to its enantioselectivity. Our results indicate that fungi are good sources for the identification of new OYEs. We believe the newly identified <italic>Af</italic>OYE1 will give an alternative for asymmetric alkene hydrogenation in industrial processes.</p>" ]

[]

[ "<title>Abstract</title>", "<p id=\"Par1\">Old yellow enzymes (OYEs) have been proven as powerful biocatalysts for the asymmetric reduction of activated alkenes. Fungi appear to be valuable sources of OYEs, but most of the fungal OYEs are unexplored. To expand the OYEs toolbox, a new thermophilic-like OYE (<italic>Af</italic>OYE1) was identified from <italic>Aspergillus flavus</italic> strain NRRL3357. The thermal stability analysis showed that the <italic>T</italic>_1/2 of <italic>Af</italic>OYE1 was 60 °C, and it had the optimal temperature at 45 °C. Moreover, <italic>Af</italic>OYE1 exhibited high reduction activity in a wide pH range (pH 5.5–8.0). <italic>Af</italic>OYE1 could accept cyclic enones, acrylamide, nitroalkenes, and α, β-unsaturated aldehydes as substrates and had excellent enantioselectivity toward prochiral alkenes (&gt; 99% ee). Interestingly, an unexpected (<italic>S</italic>)-stereoselectivity bioreduction toward 2-methylcyclohexenone was observed. The further crystal structure of <italic>Af</italic>OYE1 revealed that the “cap” region from Ala132 to Thr182, the loop of Ser316 to Gly325, α short helix of Arg371 to Gln375, and the C-terminal “finger” structure endow the catalytic cavity of <italic>Af</italic>OYE1 quite deep and narrow, and flavin mononucleotide (FMN) heavily buried at the bottom of the active site tunnel. Furthermore, the catalytic mechanism of <italic>Af</italic>OYE1 was also investigated, and the results confirmed that the residues His211, His214, and Tyr216 compose its catalytic triad. This newly identified thermophilic-like OYE would thus be valuable for asymmetric alkene hydrogenation in industrial processes.</p>", "<title>Key points</title>", "<p id=\"Par2\">\n<list list-type=\"bullet\"><list-item><p id=\"Par3\"><italic>A new thermophilic-like OYE AfOYE1 was identified from Aspergillus flavus, and the T</italic>_1/2\n<italic>of AfOYE1 was 60 °C</italic></p></list-item><list-item><p id=\"Par4\"><italic>AfOYE1 catalyzed the reduction of 2-methylcyclohexenone with (S)-stereoselectivity</italic></p></list-item><list-item><p id=\"Par5\"><italic>The crystal structure of AfOYE1 was revealedv</italic></p></list-item></list></p>", "<title>Supplementary Information</title>", "<p>The online version contains supplementary material available at 10.1007/s00253-023-12963-w.</p>", "<title>Keywords</title>" ]

[ "<title>Supplementary Information</title>", "<p>Below is the link to the electronic supplementary material.</p>" ]

[ "<title>Acknowledgements</title>", "<p>We thank SSRF 18U1 beamline scientists for their help with data collection and the NMR center in the College of Science at Henan Agricultural University for taking NMR spectra. We acknowledge English editorial assistance by Maria Ajmal (Henan Agricultural University, China).</p>", "<title>Author contribution</title>", "<p>NL, YW, and YM carried out the experimental work and drafted the manuscript; YL, SZ, SW, and PM analyzed the data and revised the manuscript; NL, YH, and HL designed the experiments and revised the manuscript. All authors have read and approved the final manuscript.</p>", "<title>Funding</title>", "<p>This study was funded by the National Natural Science Foundation of China (Nos. 32171472 and 31900876), the Key Scientific Research Projects for Higher Education of Henan Province (No. 22A180013), the Key Scientific and Technological Project of Henan Province (No. 232102311151), and High-level Talent Scientific Research Startup Fund Program of Henan University of Technology (2019BS020).</p>", "<title>Data availability</title>", "<p>All data generated or analyzed during this study are included in this published article (and its supplementary information files).</p>", "<title>Declarations</title>", "<title>Ethics approval</title>", "<p id=\"Par46\">This article does not contain any studies with human participants or animals performed by any of the authors.</p>", "<title>Conflict of interest</title>", "<p id=\"Par47\">The authors declare no competing interests.</p>" ]

[ "<fig id=\"Fig1\"><label>Fig. 1</label><caption><p>Phylogenetic analysis of <italic>Af</italic>OYE1 and previously described OYEs. The phylogenetic analysis was conducted using the Maximum Likelihood statistical method based on the Jones-Taylar-Thornton (JTT) model in MEGAX. The corresponding alignment was done by ClustalW. The accession numbers of the previously described OYEs are listed in Table ##SUPPL##0##S2##</p></caption></fig>", "<fig id=\"Fig2\"><label>Fig. 2</label><caption><p>UV–visible absorption spectra of purified <italic>Af</italic>OYE1 (solid line), released flavin after thermal denaturation (100 °C for 10 min, dotted line), and standard FMN (dash line)</p></caption></fig>", "<fig id=\"Fig3\"><label>Fig. 3</label><caption><p>Thermostability <bold>(A)</bold>, temperature <bold>(B)</bold>, and pH <bold>(C)</bold> optima of <italic>Af</italic>OYE1. The standard deviations of triplicates are represented by error bars</p></caption></fig>", "<fig id=\"Fig4\"><label>Fig. 4</label><caption><p><italic>Af</italic>OYE1-catalyzed reduction of activated α, β-unsaturated alkenes. Reaction conditions: 5 mL PBS (100 mM, pH 7.0), 0.1 μM <italic>Af</italic>OYE1, 5 U glucose dehydrogenase, 50 mM glucose, 20 mM NADH, and 8 mM substrate, at 45 °C for 10 min. TTN (total turnover number): the total moles of product yield divided by the moles of <italic>Af</italic>OYE1</p></caption></fig>", "<fig id=\"Fig5\"><label>Fig. 5</label><caption><p>Crystal structure of <italic>Af</italic>OYE1. <bold>A</bold> Overall cartoon structure of <italic>Af</italic>OYE1. The two protomers of <italic>Af</italic>OYE1 are colored green and cyan, with FMN (yellow) and Cl (magenta) buried in each protomer’s active pocket. <bold>B</bold> Surface representation showing the two protein chains forming the dimeric architecture contributed deeply by the interaction of α1-helix, N-terminal loops, and C-terminal helix of each protomer. <bold>C</bold> Details of <italic>Af</italic>OYE1 catalytic cavity. The omit |<italic>F</italic>_{<italic>o</italic>}|-|<italic>F</italic>_{<italic>c</italic>}| map is shown in gray mesh contoured at 2σ level indicating locations of FMN and ethylene glycol (EDO). FMN and EDO bound in the active site are shown with C atoms in cyan and Cl in the green sphere together with H₂O in the red dot. Relevant residues of the catalytic cavity are shown with green C atoms, Red O, and Blue N. Hydrogen bonds are shown as yellow dashed lines</p></caption></fig>", "<fig id=\"Fig6\"><label>Fig. 6</label><caption><p>The catalytic activity of the variants and wild type <italic>Af</italic>OYE1 toward 2-cyclopentenone</p></caption></fig>" ]

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[ "<supplementary-material content-type=\"local-data\" id=\"MOESM1\"></supplementary-material>" ]

[ "<fn-group><fn><p><bold>Publisher's Note</bold></p><p>Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.</p></fn><fn><p>Na Li and Yuan Wang contributed equally to the work.</p></fn></fn-group>" ]

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[ "<title>Introduction</title>", "<p id=\"Par5\">The model yeast species <italic>Saccharomyces cerevisiae</italic> is used in many fundamental and applied research applications, such as production of many industrially relevant compounds (Krivoruchko and Nielsen ##REF##25544013##2015##) and as biosensors to various substances, for example, heavy metals or estrogens (Martin-Yken ##REF##32413968##2020##). The compounds produced in yeast include D-lactic acid (DLA) (Baek et al. ##REF##26596574##2016##), which is used for production of stereocomplex type poly-lactic acid, a promising biodegradable polymer (de Albuquerque et al. ##REF##34273343##2021##). Genetically engineered yeast strains produce up to 112 g/L (1.24 M) of DLA in neutralizing conditions or over 53.2 g/L (0.59 M) of this substance without neutralizing agents (Ishida et al. ##REF##16569615##2006##, ##UREF##3##2011##; Baek et al. ##REF##26596574##2016##; Yamada et al. ##REF##28475210##2017##; Mitsui et al. ##REF##32960291##2020##). Efficient sensing systems for DLA would also be relevant in medicine, as D-lactic acidosis is a rare but serious neurologic condition specific to individuals with short bowel syndrome (Kowlgi and Chhabra ##UREF##4##2015##; Petersen ##REF##16306301##2005##).</p>", "<p id=\"Par6\">However, a systematic evaluation of yeast cell response is scarce not only for DLA but even for its incomparably more common enantiomer L-lactic acid (LLA). The response to LLA has been studied in several yeast strains on the levels of viability or growth rate reduction, as well as transcription. Specifically, it was found that lactic acid (LA) in industrially relevant concentrations of 90 or even 280 mM LA (presumably LLA) had a rather limited effect on the metabolism of <italic>S. cerevisiae</italic> in thermostat cultures but did affect the energy status of the cell by provoking a reduction in the ATP content (Thomsson and Larsson ##REF##16317544##2006##). Another study estimated the minimal inhibitory concentration of LA (presumably LLA) as 2.5% w/v (278 mM), while 0.2% began to stress the cells (Narendranath et al. ##REF##11420658##2001##). There are also two studies on the transcriptional effect of LLA. One of them dealt with LLA in comparison with acetic and hydrochloric acids with DNA microarrays in shake flask cultures. The authors found that these organic acids triggered relatively similar gene expression perturbations and affected cell wall and metal metabolism; the latter was intermediated by the Atp1p transcription factor (Kawahata et al. ##REF##16911514##2006##). Another study, using chemostat cultures, also found iron metabolism remodeling; it was very pronounced at pH 5 and 500 mM LLA and much less severe at pH 3 and 900 mM LLA (Abbott et al. ##REF##18676708##2008##). To the best of our knowledge, there is no published data on the effect of DLA on yeast transcriptome or proteome.</p>", "<p id=\"Par7\">In this study, we evaluated the transcriptional response of <italic>S. cerevisiae</italic> to varying concentrations of DLA (from 0.05 mM to 45 mM) and 45 mM of LLA in order to check if this model species possessed any promoters that would quantitatively respond to DLA but not LLA and could thus be promising for designing a yeast-based stereo-specific biosensor to lactic acid. Such transcriptome-based approach has already been successfully applied to find a 1-butanol sensing promoter in <italic>S. cerevisiae</italic> (Shi et al. ##REF##28712783##2017##). In addition, we aimed at enriching the data on transcriptional response to DLA in comparison to LLA. We found that the concentrations of DLA of 0.05 or 0.5 mM did not trigger any changes in gene expression compared to the control samples. The genes activated in response to 5 mM DLA were enriched in those controlling cell wall organization, while the genes upregulated upon 45 mM DLA treatment included several genes functioning in lactate metabolism and iron uptake. Finally, the genes responding to LLA contained many genes known to respond to this and other weak acids.</p>" ]

[ "<title>Materials and methods</title>", "<title>Yeast cultivation</title>", "<p id=\"Par8\">The yeast strain used for this work was BY4742 (<italic>MATα his3Δ1 leu2Δ0 lys2Δ0 ura3Δ0</italic>; Baker Brachmann et al. ##REF##9483801##1998##). For the experimental exposures, overnight suspension cultures were inoculated from an independent BY4742 colony on solid YPD medium (1% yeast extract, 2% peptone, 2% D-glucose, and 2% agar) and grown in synthetic medium containing 0.67% (w/v) yeast nitrogen base, 2% (w/v) glucose, 150 mM NaCl, 20 mg/L L-histidine HCl, 100 mg/L L-leucine, 30 mg/L L-lysine HCl, and 20 mg/L uracil at 28 °C with orbital shaking (120 rpm). The addition of NaCl had the purpose of mimicking the conditions in the blood plasma for further use in sensor applications. The next day, approximately 1.5 optical density units of each culture were collected by centrifugation (1700 × g for 5 min at room temperature) and resuspended in 3 ml of the same media (control) or media with lactic acid. OD600 was recorded at the beginning of the exposure and in 3–4 h. After this time, the experimental cultures were collected by centrifugation and frozen at -80 °C for RNA extraction. The relative growth rate was calculated as the difference between logarithmic (base 2) final OD600 and initial OD600 divided over incubation time in hours. This experiment was performed in total seven times with independent suspension cultures with 0.05, 0.5, 5, or 45 mM DLA, or 45 mM LLA and a control medium; three of these replicates were used for the RNA sequencing. Moreover, similar exposures were performed with eight independent suspension cultures with 45 mM DLA, 45 mM sodium D-lactate (DLS), and a control medium; five of these replicates were used for qPCR testing.</p>", "<title>RNA extraction, sequencing, and quantitative PCR (qPCR)</title>", "<p id=\"Par9\">RNA extraction, library construction, and sequencing were performed by the CeGaT company (Tübingen, Germany). RNA isolation was performed with the RNeasy kit (Qiagen, Hilden, Germany) according to manual (RNeasy Mini Handbook) with slight modifications. Cells were homogenized by mechanical disruption. After the addition of RLT buffer (Qiagen, Hilden, Germany) and glass beads, the samples were vortexed three times for 3 min. After each vortexing step, the samples were cooled on ice. Then, the lysate was centrifuged for 2 min at maximum speed, and the supernatant was transferred to new microcentrifuge tubes, combined with the same volume of 70% ethanol and mixed by pipetting. Sequencing libraries were prepared with the TruSeq Stranded mRNA kit (Illumina Inc., CA, USA) and sequenced at 2 × 100 bp with a NovaSeq 6000 (Illumina Inc., USA). Demultiplexing of the sequencing reads was performed with Illumina bcl2fastq v2.20, and adapters were trimmed with Skewer v 0.2.2 (Jiang et al. ##REF##24925680##2014##). For each sample, between 2.8 and 7.4 Gb were sequenced.</p>", "<p id=\"Par10\">RNA extraction for qPCR-based gene expression analysis was performed with the RNASwift method according to the original protocol (Nwokeoji et al. ##REF##27495141##2016##) using GeneJET spin columns (Thermo Scientific, Waltham, MA, USA) and buffers provided with the RNeasy kit (Qiagen, Hilden, Germany) at the last step. RNA purification was performed according to the recommendation of the buffer manufacturer. Then, RNA was treated with RapidOut DNA removal kit (Thermo Scientific, Waltham, MA, USA) to remove residual genomic DNA. Then, RNA concentration was measured with the Nano-300 (Allsheng, Hangzhou, China) micro-spectrophotometer, and approximately 60–70 ng of DNA-free RNA was used for reverse transcription, which was performed with RevertAid reverse transcriptase and the corresponding buffer (Thermo Scientific, Waltham, MA, USA), RiboLock RNase inhibitor (Thermo Scientific, Waltham, MA, USA), dNTPs and Oligo(dT)18 primers (Thermo Scientific, Waltham, MA, USA) according to the recommendation of the enzyme manufacturer. Then, 1 μL of the resulting cDNA of the resulting solution was used for 10-μL qPCR. The amplification was performed using a StepOne Plus instrument (Thermo Scientific, Massachusetts, USA) with the 5X qPCRmix-HS SYBR Hi-Rox (Evrogen, Moscow, Russia) and primers (5 pmol each) specific for the following genes: <italic>ACT1</italic> and <italic>CDC19</italic> used as reference genes; <italic>AQR1</italic>; <italic>DLD3</italic>; <italic>FIT2</italic>; and <italic>YPS3</italic>. Primer sequences (Supplemental Table ##SUPPL##0##S1##) for <italic>ACT1</italic> and <italic>CDC19</italic> were taken from the work by Cankorur-Cetinkaya et al. (##REF##22675547##2012##); the other primer pairs were designed with NCBI Primer Blast (Ye et al. ##UREF##10##2012##). Amplification efficiency was tested for each primer pair with serial dilutions of a control cDNA sample and lied in the range of 88–100% (Supplemental Table ##SUPPL##0##S1##).</p>", "<title>Data analysis and availability</title>", "<p id=\"Par11\">Quality control of raw data was performed with FastQC v0.11.9 and summarized with MultiQC v1.13 (Ewels et al. ##REF##27312411##2016##). The R64-1–1 release of <italic>S. cerevisiae</italic> strain S288C genome (Engel et al. ##UREF##2##2014##) was downloaded from Ensembl (Cunningham et al. ##REF##34791404##2022##) release 108 and used as a reference. The reads were aligned to the genome with hisat2 (Kim et al. ##REF##31375807##2019##) v2.2.1, sorted with samtools (Li et al. ##REF##19505943##2009##) v1.9 and quantified with featureCounts (Liao et al. ##REF##24227677##2014##) from subread v2.0.4.</p>", "<p id=\"Par12\">The resulting count table was further processed with the DESeq2 (Love et al. ##REF##25516281##2014##) v1.34.0 for R (R Core Team ##UREF##7##2022##) v4.1.2 to compare expression levels. The figures were prepared using the ggplot2 (Wickham ##UREF##8##2016##) v3.4.2, enhancedVolcano (Blighe et al. ##UREF##0##2023##) v1.12.0 and DEGReport (Pantano et al. ##UREF##6##2023##) v1.30.3 packages for R. The data from the Abbott et al. (##REF##18676708##2008##) manuscript, which were used for comparison, were downloaded from the NCBI GEO database (accession number GSE10066) with the script generated by the GEO2R service (Edgar et al. ##REF##11752295##2002##), which utilizes the GEOquery (Davis and Meltzer ##REF##17496320##2007##) 2.62.2, limma (Ritchie et al. ##REF##25605792##2015##) 3.50.3, and DESeq2 packages for R. Gene ontology (GO) term and publication enrichment analyses were performed with YeastMine (Balakrishnan et al. ##REF##22434830##2012##; Cherry et al. ##REF##22110037##2012##) using the database of 1 Apr 2023.</p>", "<p id=\"Par13\">All the code used is available at GitHub (Drozdova ##UREF##1##2023##). The raw and processed RNA sequencing data are also available from the NCBI GEO repository under the accession number GSE231937.</p>" ]

[ "<title>Results</title>", "<title>Overview of transcriptional response to lactic acid enantiomers</title>", "<p id=\"Par14\">The performed analysis revealed differentially expressed genes (hereafter DEGs; absolute log2 fold change &gt; 1 and adjusted <italic>p</italic>-value &lt; 0.05) only in the case of the two highest DLA concentrations (5 mM and 45 mM), as well as in the case of 45 mM LLA (Fig. ##FIG##0##1##a–c). The concentrations of 0.5 mM DLA and below did not produce any significant transcriptional response (Fig. ##FIG##0##1##d, e). Overall, the presence/absence of DEGs correlated with the growth inhibition: whenever growth was inhibited, we recorded differential expression (Supplemental Fig. ##SUPPL##0##S1##; Supplemental Table ##SUPPL##1##S2##). Finally, there were 10 genes that were differentially expressed between the maximal concentration of DLA and the same concentration of LLA (see below in the section “Transcriptional response to LLA and search for DLA-specific genes”).</p>", "<p id=\"Par15\">Furthermore, we functionally characterized DEG lists with gene ontology terms using YeastMine (Table ##TAB##0##1##). We found that the genes upregulated in response to 45 mM were enriched with those participating in lactate biosynthesis and metabolism (these genes will be characterized in detail below) and siderophore transport. The genes upregulated in response to a lower concentration of DLA (5 mM) were connected to the cell wall, while those downregulated in these conditions contained three genes regulating leucine biosynthesis. In the case of 45 mM LLA, we only found one very general enriched GO term for downregulated genes, generation of precursor metabolites, and energy.\n</p>", "<title>Transcriptional response to DLA</title>", "<p id=\"Par16\">In the case of the highest concentration of DLA, we found significant enrichment of two GO terms connected to lactate (Table ##TAB##0##1##). The DEGs annotated with the terms “lactate metabolic process” (GO:0006089) and “lactate biosynthetic process” (GO:0019249) largely overlapped and contained <italic>DLD1</italic> (YDL174C), <italic>DLD3</italic> (YEL071W), <italic>SNO4</italic> (YMR322C), and <italic>HSP32</italic> (YPL280W). The former two genes indeed encode D-lactate dehydrogenases, mitochondrial Dld1 and cytoplasmic Dld3 (Pallotta ##REF##22460278##2012##); according to the literature, Dld3 can also oxidize D-2-hydroxyglutarate to α-ketoglutarate (Becker-Kettern et al. ##REF##26774271##2016##). The latter two are specific small chaperones (Gong et al. ##REF##19536198##2009##). Unfortunately, none of these four genes was both DLA-specific and quantitatively responding to DLA (Fig. ##FIG##1##2##a).</p>", "<p id=\"Par17\">In the case of 5 mM DLA, the main groups of upregulated genes were those associated with cell wall biogenesis (Table ##TAB##0##1##; Supplemental Table ##SUPPL##0##S4##). This effect is not probably specific for DLA, as a similar effect was found for different organic acids (Kawahata et al. ##REF##16911514##2006##). These genes also reacted to 45 mM DLA, even though less strongly (Supplemental Fig. ##SUPPL##0##S2##). Overall, the transcriptional responses to 5 mM DLA and 45 mM DLA correlated quite well (Fig. ##FIG##1##2##b, c).</p>", "<title>Transcriptional response to LLA and search for DLA-specific genes</title>", "<p id=\"Par18\">We found fewer DEGs in response to LLA in comparison to DLA; moreover, there were no overrepresented GO terms for upregulated genes and only a rather vague term “generation of precursor metabolites and energy” in the case of downregulated genes. However, the lists of genes differentially expressed in response to LLA were associated with many (over 20) publications enriched in some of these genes (Supplemental Table ##SUPPL##0##S4##). Impressively, four of the six manuscripts enriched in LLA-upregulated genes dealt with the Haa1 transcription factor, which was indeed shown to mediate the adaptation of yeast cells to lactic and other weak acids (Fernandes et al. ##REF##16176797##2005##; Mira et al. ##UREF##5##2010##, ##REF##21586585##2011##; Sugiyama et al. ##REF##24682296##2014##).</p>", "<p id=\"Par19\">Generally, the changes triggered by LLA correlated well to the changes observed in response to the same concentration of DLA (Fig. ##FIG##2##3##a, b). In order to reveal if there were any genes that quantitatively responded to DLA and did not respond to LLA, we performed a clustering analysis of 214 genes that were differentially expressed in at least one condition. Within the obtained six clusters, none had the desired pattern for a DLA sensor, i.e., monotonous increase or decrease in response to DLA and absence or very slight response to LLA (Fig. ##FIG##2##3##c). It is worth mentioning that the first cluster featured genes which were seemingly affected more by the high concentrations of DLA than by LLA, but in fact, the changes were very subtle (Supplemental Fig. ##SUPPL##0##S3##).</p>", "<p id=\"Par20\">In addition, we analyzed the genes differentially expressed between 45 mM DLA and 45 mM LLA (Fig. ##FIG##0##1##f). There were ten such genes, YHL028W (<italic>WSC4</italic>), YLR054C (<italic>OSW2</italic>), YLR121C (<italic>YPS3</italic>), YGR189C (<italic>CRH1</italic>), YGR146C (<italic>ECL1</italic>), YGL255W (<italic>ZRT1</italic>), YHR209W (<italic>CRG1</italic>), YLR205C (<italic>HMX1</italic>), YKR091W (<italic>SRL3</italic>), and YCR005C (<italic>CIT2</italic>). Some of these genes (<italic>WSC4</italic>, <italic>YPS3</italic>, <italic>CHR1</italic>, and <italic>OSW2</italic>) regulate cell wall assembly. Wsc4, Yps3, and Crh1 have functions in maintaining cell wall integrity (Verna et al. ##REF##9391108##1997##; Krysan et al. ##REF##16087741##2005##; Cabib et al. ##REF##17302808##2007##). Osw2 is a protein of unknown function, which is putatively involved in spore wall assembly (Coluccio et al. ##REF##15590821##2004##). Several genes (<italic>ZRT1</italic>, <italic>ECL1</italic>, and <italic>HMX1</italic>) are implicated in metal transport. Zrt1 is a zinc transporter (Zhao and Eide ##REF##8637895##1996##). Ecl1 is a protein of unknown function upregulated by overexpression of the other iron deprivation-responding transcription factor, Aft2 (Rutherford et al. ##REF##12756250##2003##). <italic>HMX1</italic> encodes a heme oxygenase, and expression of this gene is regulated by the iron deprivation-responding transcription factor Aft1 (Protchenko and Philpott ##REF##12840010##2003##). Finally, the link between some of the genes and LA stress was unclear. Crg1 is a small molecule methyltransferase regulating lipid homeostasis in response to a drug cantharidin (Lissina et al. ##REF##22028670##2011##), Cit2 is a citrate synthase (Kim et al. ##REF##3023912##1986##), and Srl3 (or Whi7) participates in cell cycle regulation (Gomar-Alba et al. ##REF##28839131##2017##). Of these genes, five reacted to both 5 mM and 45 mM DLA but not to 45 mM LLA (Fig. ##FIG##2##3##d). These could be candidates for qualitative sensors for DLA but required further exploration.</p>", "<title>Neutralization compensates for the effect of the DLA on growth rate and transcription of selected genes</title>", "<p id=\"Par21\">During all previous analyses, we found several groups of genes that reacted to one (45 mM) or two (5 and 45 mM) concentrations of DLA, and we also found that these treatments slowed down yeast growth (Supplemental Fig. ##SUPPL##0##S1##). In order to check if the slow growth was caused by the lower pH values, we performed the same treatment but with 45 mM sodium D-lactate (DLS) obtained with addition of the same amount of NaOH (the concentration of additional NaCl in the media was adjusted to sum up to 150 mM) and indeed observed compensation of the growth defect (Fig. ##FIG##3##4##a). Thus, the observed slow growth in yeast treated with 45 mM DLA is explained by pH shift and not by the influence of the D-lactate ion.</p>", "<p id=\"Par22\">Moreover, we checked if the expression of several genes that we found to be DLA-responsive changed in response to DLS. For this analysis, we chose the <italic>AQR1</italic>, <italic>DLD3</italic>, and <italic>FIT2</italic> genes, which were activated to 45 mM DLA but did not respond to 5 mM DLA, as well as the <italic>YPS3</italic> gene that responded to both concentrations of DLA. These genes belong to different functional groups. <italic>AQR1</italic> (YNL065W) encodes a membrane protein from the major facilitator superfamily, which provides the cells with resistance to short-chain monocarboxylic acids (Tenreiro et al. ##REF##11922628##2002##). <italic>FI</italic><italic>T2</italic> (YOR382W) is a cell wall mannoprotein involved in the siderophore transport (Protchenko et al. ##REF##11673473##2001##), and we chose it as a representative of a larger group of iron uptake-related proteins (Table ##TAB##0##1##). <italic>DLD3</italic> (YEL071W) codes for a protein with D-lactate dehydrogenase activity in vitro, but there is evidence that in vivo it contributes to D-lactate synthesis (Chelstowska et al. ##UREF##11##1999##; Becker-Kettern et al. ##REF##26774271##2016##). Finally, <italic>YPS3</italic> (YLR121C) encodes an aspartic protease required for cell wall integrity (Olsen et al. ##REF##10191273##1999##), and we chose it as a representative of the cell wall integrity-related genes and also because it had strong changes in expression in response to both DLA concentrations but not to LLA (Fig. ##FIG##2##3##d) and thus could act as qualitative DLA sensor. However, we found that the relative expression levels of all of these genes were very similar in the control samples and in those treated with 45 mM DLS. Taken together, our data suggest that the expression changes in response to DLA were mostly triggered by the change in the pH value.</p>" ]

[ "<title>Discussion</title>", "<p id=\"Par23\">In this study, we explored the transcriptional response of <italic>S. cerevisiae</italic> to LA enantiomers. One of the goals of this work was to find genes that would specifically and quantitatively respond to DLA but not to LLA. We did not find such genes. In general, we found that the response to high concentration of DLA and LLA was quite similar and even more pronounced to DLA than to LLA. It is possible that the reason for this difference is that LLA is metabolized faster. There was no response to the concentrations of DLA of 0.5 mM and below. It is possible that higher DLA concentrations (&gt; 50 mM) or longer exposures (about a day) could have made the effect more pronounced and reveal differentially expressed genes. However, higher concentrations would be outside the range of DLA concentrations typically found in biological fluids. While normal levels of LLA in mammal blood plasma are about 1–2 mM, the levels of DLA are about two orders of magnitude lower, 0.01–0.07 mM (Ewaschuk et al. ##REF##15987839##2005##). In general, the levels of &lt; 0.2 mM are considered normal (Zhang et al. ##REF##12612331##2003##). The most well-known condition leading to D-lactic acidosis in human is short bowel syndrome, during which plasma DLA levels may reach millimolar concentrations (Zhang et al. ##REF##12612331##2003##; Yilmaz et al. ##REF##30089656##2018##). Similar symptoms in ruminants appear upon elevated carbohydrate level in their diet (Lorenz and Gentile ##REF##24980724##2014##) and may lead to DLA plasma levels as high as about 25 mM (Ewaschuk et al. ##REF##15987839##2005##). Thus, an ideal biosensor for DLA should be sensitive at 0.1–5 mM concentration range. Similarly, the need to use longer exposures of sensor yeast would also hinder its usage in biosensor applications. So, we find it unlikely that a quantitative yeast sensor to DLA may be constructed based on the native yeast transcriptional networks, but the possibility of integrating a heterologous cassette remains open. Such a cassette has been described for <italic>Pseudomonas</italic>, but its sensitivity starts from about 20 mM DLA (Singh et al. ##REF##31517075##2019##), which is also far from ideal for monitoring DLA levels in biological fluids.</p>", "<p id=\"Par24\">Importantly, we present the first dataset on yeast transcriptional response to DLA. Generally, we found that if the particular concentration did not inhibit growth, we did not see a transcriptional response either. This result is very similar to the study, in which the authors compared the transcriptome-wide responses to different alcohols in search for 1-butanol sensor and found that the samples treated with ethanol, which did not cause significant growth inhibition, clustered with the control samples, while samples treated with 1-butanol and 1-propanol, which were much more toxic, clustered separately (Shi et al. ##REF##28712783##2017##). Intriguingly, the response to 5 mM and 45 mM was seemingly different if judging by enriched GO terms (5 mM DLA caused overexpression of cell wall-related genes, while 45 mM led to increased expression of lactate metabolism and siderophore transport genes), but the transcriptional profiles in response to 5 mM and 45 mM DLA were highly correlated (Fig. ##FIG##1##2##c), and more than half of the DEGs were shared between the two comparisons (Fig. ##FIG##0##1##g, h).</p>", "<p id=\"Par25\">In general, the groups of cell-wall related genes and genes of iron/siderophore uptake have already been described to respond to weak acid stress (Kawahata et al. ##REF##16911514##2006##; Abbott et al. ##REF##17484738##2007##, ##REF##18676708##2008##), so our findings fully corroborate the previously published data but at the same time suggest that this response might be pH-dependent rather than specific for the lactic acid. However, comparison of four weak acids, benzoate, sorbate, acetate, and propionate, showed that the transcriptional responses were largely specific (Abbott et al. ##REF##17484738##2007##), proving that the response to pH was not the only reason for gene expression changes. We have checked this hypothesis for the <italic>FIT2</italic> gene, which codes for cell wall mannoprotein involved in the siderophore transport (Protchenko et al. ##REF##11673473##2001##) as a representative of the iron/siderophore uptake functional group and <italic>YPS3</italic>, the gene encoding an aspartic protease required for cell wall integrity (Olsen et al. ##REF##10191273##1999##), as a representative of the cell wall genes group. Both genes did not respond to D-lactate treatment if pH was compensated (Fig. ##FIG##3##4##b). Moreover, in this experiment we also measured the expression levels of the <italic>AQR1</italic> gene. It encodes a membrane protein from the major facilitator superfamily, which provides the cells with resistance to short-chain monocarboxylic acids (Tenreiro et al. ##REF##11922628##2002##). We found that <italic>AQR1</italic> also only responded to DLA at low pH. Interestingly, the authors of the original manuscript noted that the expression of this gene was not stimulated by weak acid stress (Tenreiro et al. ##REF##11922628##2002##). We found that it was upregulated in response to 45 mM DLA, while Abbott et al. (##REF##17484738##2007##) found it as a common gene downregulated in response to the four weak organic acids they tested. Aqr1 has not been shown to have a role in lactic acid transport, but it acts as a lactic acid exporter and has been shown to be important for yeast co-cultivation with lactic acid bacteria (Velasco et al. ##REF##15590823##2004##; Kapetanakis et al. ##REF##34887841##2021##).</p>", "<p id=\"Par26\">While the transcriptional response to DLA has not been explored before, there are studies on transcriptional changes in response to LLA (Kawahata et al. ##REF##16911514##2006##; Abbott et al. ##REF##18676708##2008##). We have compared the changes in the genes differentially expressed in response to 45 mM LLA according to our data and each of the two studies and found substantial positive correlation (Supplemental Fig. ##SUPPL##0##S4##), even though all experimental designs were different.</p>", "<p id=\"Par27\">In the case of the work by Kawahata et al. (##REF##16911514##2006##), experimental exposures were performed in shake flask cultures with 0.3% LLA (about 33 mM), and the S288C strain (parental to BY472) was used. Two experimental designs were used. First, acid shock was performed by pre-growing the cultures to the optical density at 660 nm (OD660) of 1.0 and exposing them to LLA for 30 min. The second design, acid adaptation, involved diluting overnight cultures to the OD660 = 0.1 in the media with LLA and growing until OD660 reached 1. The number of overlapping DEGs was quite low in both cases (five genes), but the changes in the transcription of these genes were mostly similar to our results (Supplemental Fig. ##SUPPL##0##S4##a, b).</p>", "<p id=\"Par28\">In the study by Abbott et al. (##REF##18676708##2008##), chemostat cultures of the CEN.PK 113-7D strain (not closely related to S288C and BY4742) were subjected to quite high concentrations of LLA, namely, 500 mM LLA at pH 3 and 900 mM LLA at pH 5. The lists of DEGs shared in our results and these data were larger (over 30 genes in each case), and the correlation of our 45 mM-LLA exposure was much higher with 500 mM LLA than with 900 mM LLA.</p>", "<p id=\"Par29\">We were particularly interested in the D-lactate metabolism genes <italic>DLD1</italic> and <italic>DLD3</italic>. Originally, both of these genes were shown to code for a mitochondrial and cytoplasmic D-lactate dehydrogenases, respectively (Lodi and Ferrero ##REF##8492799##1993##; Chelstowska et al. ##UREF##11##1999##). According to our results (Fig. ##FIG##1##2##a; Supplemental Table ##SUPPL##2##S3##), <italic>DLD1</italic> was upregulated in response to 5 mM DLA (fold change = 1.92 and adjusted <italic>p</italic> = 0.04), as well as in response to 45 mM DLA (fold change = 2.76 and adjusted <italic>p</italic> = 0.0001) and had a similar trend in response to 45 mM LLA, even though the difference did not reach statistical significance (fold change = 1.85 and adjusted <italic>p</italic> = 0.06). Interestingly, Abbott et al. (##REF##18676708##2008##) also found upregulation of <italic>DLD1</italic> to the highest LLA concentration used in their experimental design, 900 mM (fold change = 4 and adjusted <italic>p</italic> = 0.0002). This non-stereo-specific regulation could be interesting to explore further.</p>", "<p id=\"Par30\">While the Dld1 enzyme is the major D-lactate dehydrogenase, Dld3 is a minor D-lactate dehydrogenase and mostly acts as a transhydrogenase coupling D-2-hydroxyglutarate degradation to DLA synthesis (Lodi and Ferrero ##REF##8492799##1993##; Chestowska et al. 1999; Becker-Kettern et al. ##REF##26774271##2016##). In our experiment, <italic>DLD3</italic> was only upregulated in response to 45 mM DLA but not 5 mM DLA or lower concentrations and did not react to DLS, also corroborating the idea of its very minor role as a D-lactate dehygrogenase. It is possible that this protein acts at high DLA concentrations to prevent cell damage by low pH.</p>", "<p id=\"Par31\">In general, our data enrich our understanding of the yeast transcriptome-wide response to LLA and provide the first description of the response to DLA. We found that even though the response to different stereoisomers of lactic acid had quite significant similarities to the response to other weak acids tested previously and largely dependent on pH, there are large differences between DLA and LLA responses, which probably reflect the difference in their role in yeast biology. The role of pH in the DLA response highlights the importance of controlling and optimizing lactic acid production in yeast under neutralizing and non-neutralizing conditions separately, which is also corroborated by a recent work on a recombinant yeast strain with improved lactic acid tolerance and lactic acid yield under non-neutralizing conditions (Yamada et al. ##UREF##9##2021##).</p>" ]

[]

[ "<title>Abstract</title>", "<p id=\"Par1\">The model yeast, <italic>Saccharomyces cerevisiae</italic>, is a popular object for both fundamental and applied research, including the development of biosensors and industrial production of pharmaceutical compounds. However, despite multiple studies exploring <italic>S. cerevisiae</italic> transcriptional response to various substances, this response is unknown for some substances produced in yeast, such as D-lactic acid (DLA). Here, we explore the transcriptional response of the BY4742 strain to a wide range of DLA concentrations (from 0.05 to 45 mM), and compare it to the response to 45 mM L-lactic acid (LLA). We recorded a response to 5 and 45 mM DLA (125 and 113 differentially expressed genes (DEGs), respectively; &gt; 50% shared) and a less pronounced response to 45 mM LLA (63 DEGs; &gt; 30% shared with at least one DLA treatment). Our data did not reveal natural yeast promoters quantitatively sensing DLA but provide the first description of the transcriptome-wide response to DLA and enrich our understanding of the LLA response. Some DLA-activated genes were indeed related to lactate metabolism, as well as iron uptake and cell wall structure. Additional analyses showed that at least some of these genes were activated only by acidic form of DLA but not its salt, revealing the role of pH. The list of LLA-responsive genes was similar to those published previously and also included iron uptake and cell wall genes, as well as genes responding to other weak acids. These data might be instrumental for optimization of lactate production in yeast and yeast co-cultivation with lactic acid bacteria.</p>", "<title>Key points</title>", "<p id=\"Par2\">\n<italic>• We present the first dataset on yeast transcriptional response to DLA.</italic>\n</p>", "<p id=\"Par3\">\n<italic>• Differential gene expression was correlated with yeast growth inhibition.</italic>\n</p>", "<p id=\"Par4\">\n<italic>• The transcriptome response to DLA was richer in comparison to LLA.</italic>\n</p>", "<title>Supplementary Information</title>", "<p>The online version contains supplementary material available at 10.1007/s00253-023-12863-z.</p>", "<title>Keywords</title>" ]

[ "<title>Supplementary Information</title>", "<p>Below is the link to the electronic supplementary material.</p>" ]

[ "<title>Author contribution</title>", "<p>PD, AG, MT, and EB designed the research. PD, AS, AV, and EI performed the experiments. PD, AG, AV, and EZ analyzed the data and prepared text and figures. MT and EB supervised the project. All authors read and approved the final manuscript.</p>", "<title>Funding</title>", "<p>The work was supported by the Russian Science Foundation (interdisciplinary project 20–64-47011 performed in association with the Petrozavodsk State University, Petrozavodsk, Russia; link to information about the project: <ext-link ext-link-type=\"uri\" xlink:href=\"https://rscf.ru/en/project/20-64-47011/\">https://rscf.ru/en/project/20-64-47011/</ext-link>).</p>", "<title>Data availability</title>", "<p>The datasets generated during and analyzed during the current study are available in the GitHub repository, <ext-link ext-link-type=\"uri\" xlink:href=\"https://github.com/drozdovapb/S_cerevisiae_lactate_transcriptome\">https://github.com/drozdovapb/S_cerevisiae_lactate_transcriptome</ext-link> and the NCBI GEO repository under the accession number GSE231937. </p>", "<title>Declarations</title>", "<title>Ethics approval</title>", "<p id=\"Par32\">This article does not contain any studies with human participants or animals performed by any of the authors.</p>", "<title>Competing interests</title>", "<p id=\"Par33\">The authors declare no competing interests.</p>" ]

[ "<fig id=\"Fig1\"><label>Fig. 1</label><caption><p>Overview of gene expression changes in all comparisons. <bold>a</bold>–<bold>f</bold> show volcano plots for major comparisons with differentially expressed genes (absolute log₂ fold change &gt; 1 and adjusted <italic>p</italic>-value &lt; 0.05) are indicated by red dots (each dot corresponds to one gene). <bold>g</bold>, <bold>h</bold> show intersections of the lists of genes upregulated and downregulated, respectively, in response to different treatments, represented as Venn diagrams. Full expression data are available in Supplemental Table ##SUPPL##2##S3## and from the GEO database (GSE231937)</p></caption></fig>", "<fig id=\"Fig2\"><label>Fig. 2</label><caption><p>Overview of transcriptional response to DLA. Shown are <bold>a</bold> logarithmic (base 2) normalized expression counts of the genes DE in response to 45 mM DLA and annotated with the GO terms “lactate metabolic process” or “lactate biosynthesis process” (ORF (open reading frame) symbols and gene names shown in the titles of the panels), as well as the correlation between expression changes in response to 5/45 mM DLA of shared DEGs (<bold>b</bold>) and all genes (<bold>c</bold>). Pearson’s correlation coefficient = 0.99 for (<bold>b</bold>) and 0.72 for (<bold>c</bold>)</p></caption></fig>", "<fig id=\"Fig3\"><label>Fig. 3</label><caption><p>Comparison of transcriptional responses to LLA and DLA shows significant similarities and reveals several candidate qualitative DLA sensor genes. Correlation between expression changes in response to 45 mM DLA or LLA of shared DEGs (<bold>a</bold>) and all genes (<bold>b</bold>). Pearson’s correlation coefficient = 0.98 for <bold>a</bold> and 0.65 for <bold>b</bold>. <bold>c</bold> Clustering of expression profiles does not reveal any groups with quantitative response to DLA. <bold>d</bold> Expression profiles of potential qualitative sensors, genes responding to DLA but not LLA. The vertical axis shows logarithmic (base 2) normalized expression counts of the genes</p></caption></fig>", "<fig id=\"Fig4\"><label>Fig. 4</label><caption><p>Addition of 45 mM sodium D-lactate compensates for the growth defect caused by 45 mM DLA treatment (<bold>a</bold>) and does not trigger changes in the expression of selective DLA-responsive genes (<bold>b</bold>). <bold>a</bold> Relative growth rate (the difference between logarithmic (base 2) final OD600 and initial OD600 divided over incubation time in hours) of cultures incubated for 3 h in the media with 45 mM DLA or DLS. *<italic>p</italic> &lt; 0.05; ns, not significant (paired pairwise Wilcoxon rank sum test). <bold>b</bold> Expression levels of the <italic>AQR1</italic>, <italic>DLD3</italic>, <italic>FIT2</italic>, and <italic>YPS3</italic> genes relative to the geometric mean of the reference genes <italic>ACT1</italic> and <italic>CDC19</italic> measured with quantitative PCR. Raw qPCR data are presented in Supplemental Table ##SUPPL##3##S5##</p></caption></fig>" ]

[ "<table-wrap id=\"Tab1\"><label>Table 1</label><caption><p>Significantly (<italic>p</italic> &lt; 0.05) overrepresented GO terms</p></caption><table frame=\"hsides\" rules=\"groups\"><tbody><tr><td align=\"left\" colspan=\"2\">45 mM DLA vs. control</td></tr><tr><td align=\"left\">Upregulated (82 genes)</td><td align=\"left\">Downregulated (31 genes)</td></tr><tr><td align=\"left\">Lactate metabolic process (GO:0006089)</td><td align=\"left\">No enrichment found</td></tr><tr><td align=\"left\" colspan=\"2\">Siderophore transport (GO:0015891)</td></tr><tr><td align=\"left\" colspan=\"2\">Lactate biosynthetic process (GO:0019249)</td></tr><tr><td align=\"left\" colspan=\"2\">5 mM DLA vs. control</td></tr><tr><td align=\"left\">Upregulated (74 genes)</td><td align=\"left\">Downregulated (51 genes)</td></tr><tr><td align=\"left\">Cell wall organization or biogenesis (GO:0071554)</td><td align=\"left\">Leucine biosynthetic process (GO:0009098)</td></tr><tr><td align=\"left\" colspan=\"2\">External encapsulating structure organization (GO:0045229)</td></tr><tr><td align=\"left\" colspan=\"2\">Cell wall organization (GO:0071555)</td></tr><tr><td align=\"left\" colspan=\"2\">Fungal-type cell wall organization or biogenesis (GO:0071852)</td></tr><tr><td align=\"left\" colspan=\"2\">Fungal-type cell wall organization (GO:0031505)</td></tr><tr><td align=\"left\" colspan=\"2\">Cell wall biogenesis (GO:0042546)</td></tr><tr><td align=\"left\" colspan=\"2\">45 mM LLA vs. control</td></tr><tr><td align=\"left\">Upregulated (17 genes)</td><td align=\"left\">Downregulated (45 genes)</td></tr><tr><td align=\"left\">No enrichment found</td><td align=\"left\">Generation of precursor metabolites and energy (GO:0006091)</td></tr><tr><td align=\"left\" colspan=\"2\">45 mM DLA vs. 45 mM LLA</td></tr><tr><td align=\"left\">Upregulated (8 genes)</td><td align=\"left\">Downregulated (2 genes)</td></tr><tr><td align=\"left\">No enrichment found</td><td align=\"left\">No enrichment found</td></tr></tbody></table></table-wrap>" ]

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[ "<supplementary-material content-type=\"local-data\" id=\"MOESM1\"></supplementary-material>", "<supplementary-material content-type=\"local-data\" id=\"MOESM2\"></supplementary-material>", "<supplementary-material content-type=\"local-data\" id=\"MOESM3\"></supplementary-material>", "<supplementary-material content-type=\"local-data\" id=\"MOESM4\"></supplementary-material>" ]

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[ "<media xlink:href=\"253_2023_12863_MOESM1_ESM.pdf\"><caption><p>Supplementary file1 (PDF 649 KB)</p></caption></media>", "<media xlink:href=\"253_2023_12863_MOESM2_ESM.xlsx\"><caption><p>Supplementary file2 (XLSX 14 KB)</p></caption></media>", "<media xlink:href=\"253_2023_12863_MOESM3_ESM.xlsx\"><caption><p>Supplementary file3 (XLSX 4.05 MB)</p></caption></media>", "<media xlink:href=\"253_2023_12863_MOESM4_ESM.xlsx\"><caption><p>Supplementary file4 (XLSX 25.2 KB)</p></caption></media>" ]

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[ "<title>Introduction</title>", "<p id=\"Par5\">A substantial proportion of cancers are associated with pathogenic variants (PV) in hereditary cancer genes [##REF##35971132##1##, ##REF##27978560##2##], including an estimated 10% of breast, 10% of colon, and 20% of ovarian cancers. Approximately, 1 in 300 to 500 people in the population will carry a PV in either <italic>BRCA1</italic> or <italic>BRCA2</italic> (<italic>BRCA</italic>; the genes most commonly associated with hereditary breast and ovarian cancer (HBOC)) [##REF##11044354##3##–##REF##10359546##6##], and 1 in 370 individuals will carry a pathogenic variant (PV) in one of the Lynch syndrome genes (the genes most commonly associated with hereditary colorectal cancer) [##REF##35177335##7##]. Early identification of a PV in a cancer susceptibility gene provides individuals with the opportunity for enhanced surveillance and risk-reducing interventions which can significantly reduce the morbidity and mortality of these cancers [##REF##20810374##8##–##REF##21858794##13##]. Identification of carrier status also provides the opportunity for cascade testing in relatives [##REF##31857708##14##–##REF##35115620##16##].</p>", "<p id=\"Par6\">Genetic testing for cancer susceptibility genes is currently offered predominantly to individuals who are considered high-risk for either <italic>BRCA</italic> or Lynch syndrome based on clinical and family history criteria. However, only around half of the carriers of PVs in <italic>BRCA</italic> and Lynch syndrome genes will meet clinical and family history criteria for genetic testing [##REF##29361001##5##, ##REF##31406321##17##, ##REF##24493722##18##]. Additionally, current approaches to testing for hereditary cancer genes are associated with inequities in referral and access to testing [##REF##32720237##19##, ##REF##35380723##20##].</p>", "<p id=\"Par7\">An alternative approach is population-based hereditary cancer genetic testing which may provide a more clinically effective strategy for early identification of high-risk individuals [##REF##29361001##5##, ##REF##26483301##21##]. To date, population-based testing has been primarily studied in the context of <italic>BRCA</italic> testing in the Ashkenazi Jewish population [##REF##25435541##22##, ##REF##20008623##23##], and data are limited in the general population, especially among under-represented racial and ethnic groups. The objective of this study was to determine the yield of hereditary cancer gene PVs among unselected diverse women attending breast imaging centers, as a potential strategy for more complete identification of high-risk individuals who could benefit from enhanced surveillance and/or risk reduction interventions.</p>" ]

[ "<title>Materials and methods</title>", "<title>Study population</title>", "<p id=\"Par8\">This retrospective cohort study included unselected female patients who were offered and underwent genetic testing at the time of breast imaging at three imaging centers (Memorial MRI and Diagnostics, Texas) from November 2020 through March 2022. All patients arriving at the imaging centers were given a written flier with an invitation to undergo genetic testing for a panel of hereditary cancer genes. Patients were also offered the option of an online genetic information session with a board-certified genetic counselor prior to testing. The lead clinician investigator (DM) served as the ordering clinician for the testing at all three centers. A limited number of providers using the imaging centers opted out of having their patients participate. Only patients (including those with a previous history of breast cancer) undergoing routine breast imaging, either by mammogram or ultrasound, were included. Patients undergoing imaging for newly diagnosed breast cancer were excluded.</p>", "<p id=\"Par9\">Clinical, demographic, and family cancer history information were ascertained through test requisition forms and family cancer history questionnaires completed by the patients. Patient questionnaires included clinical questions needed for Tyrer–Cuzick breast cancer risk assessment. Race/ethnicity (ancestry) was self-reported. National Comprehensive Cancer Network (NCCN) guidelines for Genetic/Familial High-Risk Assessment: Breast, Ovarian, and Pancreatic Cancer [##UREF##0##24##] and Lynch Syndrome (LS) were reviewed to determine if genetic testing would have been guideline indicated for a particular patient [##UREF##1##25##]. For the patients unaffected with breast cancer who completed the clinical portion of their questionnaire for breast cancer risk assessment, a Tyrer–Cuzick breast cancer risk 5 years and lifetime risk score was calculated [##REF##14627668##26##]. This score was only reported back to patients if their genetic testing (described below) was negative/uninformative for PVs associated with increased breast cancer risk. It was not available if enough information about personal/family history was not provided to compute a Tyrer–Cuzick score or the patient had a personal history of breast cancer.</p>", "<p id=\"Par10\">If the patients’ clinical and/or family history met NCCN guidelines for hereditary cancer testing, they were given the option to request insurance coverage for testing or self-pay. Patients who did not meet guidelines were offered the option to self-pay and financial assistance options were available to eligible patients, based on their income and family size. Genetic information sessions performed by board-certified genetic counselors (Natera, Inc.) were available to patients on a pre- and post- test basis. All patients with a PV in a breast cancer-associated predisposition gene were offered in person risk counseling by the lead clinician investigator (DM).</p>", "<p id=\"Par11\">This study was granted a waiver of consent process under 45 CFR 46.116(d), a waiver of the requirement for documentation of informed consent according to 45 CFR 46.117(c)(2), and a waiver from the HIPAA Authorization Requirement according to 45 CFR 46.164.512(i) (Salus IRB, ID# 21204—01A).</p>", "<title>Hereditary cancer testing</title>", "<p id=\"Par12\">Next-generation sequencing (NGS)-based hereditary cancer risk assessment was carried out utilizing a multiplex gene panel (40 or 53 genes) testing (Empower™, Natera, Inc. in collaboration with Baylor Genetics). The targeted regions of the genes associated with hereditary cancer syndromes are enriched using a capture-based method and sequenced by next-generation sequencing (NGS) using the Illumina platform. The variants detected in exons and within 20 bp of the exon/intron boundary are reported, unless otherwise specified. Read depth analysis is used to detect copy number variation (CNV) for genes. Positive sequencing results from certain genes or regions with highly homologous sequences in the genome are confirmed by gene-specific long-range PCR and Sanger sequencing. Multiplex ligation-dependent probe amplification (MLPA), PCR-based methods, and/or array comparative genomic hybridization (aCGH) are used to confirm copy number changes involving the genes in the test.</p>", "<p id=\"Par13\">All patients underwent testing for at least 25 clinically actionable genes (Table ##TAB##0##1##). Genes were considered clinically actionable based on the presence of established NCCN and/or peer-reviewed consensus management recommendations for enhanced surveillance, or risk-reducing interventions, and family cascade testing if a PV was detected. Clinically actionable genes were categorized as “high risk” or “moderate risk” based on their reported relative risk for cancer, relative risk of &gt; 4, and relative risk of 2–4, respectively (Table ##TAB##0##1##). Testing included both HBOC and Lynch syndrome (LS) genes (NCCN HBOC, NCCN Colorectal Cancer (CRC)). Variants were classified consistent with guidelines from the American College of Medical Genetics and Genomics and the Association for Molecular Pathology as previously described [##REF##14627668##26##–##REF##30964716##28##]. Only likely pathogenic and pathogenic variants (PVs) were considered in the analysis: benign and likely benign variants, and variants of unknown significance were not considered.</p>", "<p id=\"Par14\">For the purposes of the current analysis, variants were not considered clinically actionable (i.e., had no potential impact on patient care) and were not included if they:<list list-type=\"order\"><list-item><p id=\"Par15\">Were in genes only associated with autosomal recessive disease association (e.g., monoallelic <italic>MUTYH</italic> carriers which may conflate estimates of pathogenic variant prevalence) or</p></list-item><list-item><p id=\"Par16\">Were low penetrance variants in clinically actionable genes, such as <italic>CHEK2</italic> c.470 T &gt; C [p.Ile157Thr].</p></list-item></list></p>", "<title>Analysis</title>", "<p id=\"Par17\">The sociodemographic characteristics and personal and family history of cancer of the study population were explored. Patients’ characteristics were stratified based on the presence and type of PV. The prevalence of P/LP variants was calculated. For patients with a P/LP variant, the proportion who did and did not meet NCCN guidelines for genetic testing was evaluated. We also evaluated the proportion of patients with PV who would have qualified for additional screening based on the empiric risk model (i.e., Tyrer–Cuzick score &gt; = 20%).</p>" ]

[ "<title>Results</title>", "<title>Study population</title>", "<p id=\"Par18\">A total of 1,943 women undergoing breast imaging elected to have hereditary cancer genetic testing during the study period (Table ##TAB##1##2##). Median age was 66 years (range 18–89 years). Self-reported race and ethnicity were Asian 5.0% (<italic>N</italic> = 85); Black 20% (<italic>N</italic> = 339); White 38% (<italic>N</italic> = 650); and Hispanic 32% (<italic>N</italic> = 534).</p>", "<p id=\"Par19\">A personal history of cancer was documented for 7.5% (<italic>N</italic> = 146) (Fig. ##FIG##0##1##), a family history of cancer in 42.3% (<italic>N</italic> = 822), and no personal or family history of cancer in 50.2% (<italic>N</italic> = 975). A personal history of breast or ovarian-related cancers was recorded in 4% (<italic>N</italic> = 80), endometrial cancer in 0.8% (<italic>N</italic> = 15), and colorectal cancer in 0.5% (<italic>N</italic> = 10).</p>", "<p id=\"Par20\">Overall, 18.2% (354/1943) of patients met current NCCN guidelines for hereditary breast and ovarian cancer (HBOC) gene testing, 3.7% (71/1943) met NCCN guidelines for Lynch syndrome genetic testing, and 1.0% (19/1943) met both HBOC and Lynch syndrome guidelines for testing (Table ##TAB##2##3##).</p>", "<title>Prevalence of pathogenic variants</title>", "<p id=\"Par21\">Among 1943 patients who received genetic testing, 39 (2%) were identified as carriers of a PV in an autosomal dominant clinically actionable HBOC-related or LS gene (Table ##TAB##2##3##). Of the 39 PVs identified, 84.6% (<italic>N</italic> = 33) were in HBOC-related genes which corresponds to a prevalence of 1.7% (33/1943) in the total cohort. The remaining 15.4% (<italic>N</italic> = 6) PVs were in LS genes which corresponds to a prevalence of 0.3% (6/1943) in the total cohort. The most common PVs were in <italic>CHEK2</italic> (10/39; 25.6%); <italic>PALB2</italic> (8/39; 20.5%); <italic>BRCA2</italic> (6/39; 15.4%); and <italic>PMS2</italic> (5/39; 12.8%) (Fig. ##FIG##1##2##). Of the 34 PVs where race/ethnicity were known, 47% were detected among non-White patients (Table ##TAB##1##2##). The PV prevalence (%) was distributed across the respective ancestral groups as follows: Black 3/339 (0.89%); White 16/650 (2.4%); Asian 3/85 (3.5%), and Hispanic 9/534 (1.7%).</p>", "<p id=\"Par22\">Patients with a PV were over 50 years of age at the time of their testing in 82.1% (32/39) of cases. This is similar to the 1587/1943 (81.7%) of patients aged 51 or older who were tested in the total cohort and consistent with National screening guidelines for women at average risk [##REF##26757170##27##] (Table ##TAB##1##2##).</p>", "<title>Guideline eligibility for genetic testing or enhanced surveillance breast cancer</title>", "<p id=\"Par23\">Only 38.5% (15/39) of PV carriers met either NCCN guidelines for HBOC testing or LS testing prior to genetic testing. Of these, 25.6% (10/39) met HBOC criteria for genetic testing with PVs in HBOC-related genes and 7.7% (3/39) had a PV in an LS gene, and 5.1% (2/39) met criteria for LS testing with PV in HBOC-related genes.</p>", "<p id=\"Par24\">Notably, 5 out of 6 patients with a <italic>BRCA2</italic> PV and 5 out of 8 patients with a <italic>PALB2</italic> PV did not meet criteria for HBOC testing. The frequencies of clinically actionable PVs identified in high- or moderate-risk genes in patients who either met or did not meet NCCN guidelines for inclusion of the genes are shown in Fig. ##FIG##1##2##.</p>", "<p id=\"Par25\">Data allowing calculation of the patient’s Tyrer–Cuzick score were available for 64.1% (25/39) of patients with a PV in an autosomal dominant clinically actionable HBOC (Table ##TAB##2##3##). Of these, only 8% (2/25) had a Tyrer–Cuzick score ≥ 20 which would have triggered health insurance coverage for increased surveillance for breast cancer, absent in the PV finding (Table ##TAB##2##3##).</p>" ]

[ "<title>Discussion</title>", "<p id=\"Par26\">We found that 2% (39/1943) of women undergoing hereditary cancer gene testing as part of their clinical care at the time of breast imaging had PVs in hereditary cancer genes that had implications for cancer surveillance and clinical management. Importantly, the population undergoing testing was racially and ethnically diverse compared to previous studies [##REF##35380723##20##, ##REF##30964716##28##], predominantly over the age of 50 and without a personal history of cancer (92.5%). This was similar or more diverse compared to CDC population data on race and ethnicity for females aged 55 to 74 years who were resident in Texas in 2021 (4.8% Asian, 12.5% Black, 80.7% White, and 28.1% Hispanic) [##UREF##2##29##].</p>", "<p id=\"Par27\">Among women who had a PV in either a HBOC or LS gene, only 38.5% met NCCN guidelines for testing of either of these conditions. Of those who had a PV in an HBOC gene and had data needed for a Tyrer–Cuzick score, only 8% had an estimated lifetime score of ≥ 20%. Therefore, while testing as part of clinical care at the time of breast imaging identified some women who were eligible for hereditary cancer gene testing based on NCCN guidelines, it predominantly identified women with PVs who neither met NCCN guidelines for testing nor the Tyrer–Cuzick threshold for increased surveillance for breast cancer [##REF##15057881##30##]. Consequently, without this opportunity for genetic testing, women with PVs may not have accessed risk-appropriate enhanced surveillance.</p>", "<p id=\"Par28\">Identification of individuals with PVs in HBOC and LS genes provides opportunities for cancer prevention and earlier detection [##REF##21858794##13##, ##REF##32245629##31##]. For people with PVs in HBOC genes, there are recommendations and options for earlier mammography, screening breast MRI, chemoprophylaxis, and risk-reducing surgeries, such as mastectomy and bilateral salpingo-oophorectomy [##REF##20810374##8##, ##REF##18268356##10##, ##REF##28383664##11##, ##REF##21858794##13##]. Knowledge of specific PVs can also impact decisions about surgery and adjuvant therapies [##REF##35585432##32##–##REF##34081848##34##]. The median age of the population in the current cohort was 66 years and previous research has suggested that HBOC gene testing in younger women would be the most cost-effective approach to testing [##REF##33758026##35##]. Nonetheless, a study of the remaining lifetime risk for the subset of women who were older than 65 years in the population-based CARRIERS project [##REF##33471974##36##] indicated that <italic>BRCA1</italic>, <italic>BRCA2</italic>, and <italic>PALB2</italic> PVs were associated with enough breast cancer risk to warrant high-risk screening [##REF##34292776##37##]. For people who have PVs in LS genes, there is also compelling evidence of the benefit of colonoscopy in reducing mortality from CRC [##REF##19720893##9##, ##REF##33915171##38##]. Additionally, in women with PVs in LS genes, the lifetime risk of endometrial cancer is similar to that of CRC, which can largely be prevented by hysterectomy [##REF##16421367##39##]. For women with PVs who chose to share this information with family members, it can provide more accurate risk assessment and cascade testing [##REF##31857708##14##–##REF##35115620##16##]. The yield of PVs is typically higher (~ 10%) [##REF##21858794##13##] when we use the guideline criteria to screen the eligibility for genetic testing. However, this study demonstrated a meaningful yield (~ 2%) of clinically actionable PVs among participants who did not meet any guideline. Amplifying this point, 5 out of 6 <italic>BRCA2</italic> carriers identified by this unselected approach did not meet any guideline and there is ample evidence of reduction in cancer-specific and all-cause mortality by standard of care gene-specific clinical management [##REF##20810374##8##, ##REF##28914396##40##, ##REF##27087880##41##]. Thus, the universal testing approach and increasingly cost-effective genetic testing are likely to have a high impact despite a modest yield.</p>", "<p id=\"Par29\">Perhaps one of the most striking findings in this clinical cohort was the diversity of the population that accessed the testing. Later stages of cancer at the time of diagnosis and lower cancer survival rates are clearly documented in Black people in the USA [##REF##33603954##42##]. There are data indicating that people of color or Hispanic ancestry are less likely to be referred for genetic counseling or testing and are less likely to take up genetic testing in the absence of a personal history of cancer [##REF##35380723##20##, ##REF##33603954##42##, ##REF##31078448##43##]. As demonstrated in our study, increasing access and convenience of testing may help overcome barriers to more equitable testing [##REF##31078448##43##, ##REF##31239069##44##].</p>", "<p id=\"Par30\">Limitations of this study include that women under 40 years of age without documented increased risk are unlikely to have routine mammography and therefore, may not be represented in this cohort. Thus, <italic>BRCA</italic> carriers may be under-represented. Nonetheless, several <italic>BRCA2</italic> carriers, the majority of which did not meet any testing guidelines, were identified representing a critical opportunity for screening and prevention. Another limitation of the current cohort is that the number of patients who declined hereditary cancer testing or who may have had genetic testing prior to this study was unknown. Given the uncertainty about the total number of women who received the invitation to have genetic testing, the potential benefit of our strategy with regard to the yield of actionable PVs in the imaging center population could be over- or underestimated. Finally, though the racial and ethnic composition of the study participants were exceptionally diverse, we do not know if there were significant differences in the uptake among the respective groups [##REF##36370464##45##]. Nonetheless, (as cited above) the population who underwent testing is diverse and generally representative of the ethnic makeup of the state of Texas. Further, we do not have qualitative or quantitative data on how the patients made their decision to participate and receive testing as it was beyond the scope of our study. However, we believe that this will be important for future research in the context of population health implementation. Finally, the presence of a PV or elevated empiric risk (e.g., &gt; 20%) does not guarantee insurance coverage, and access has been problematic across different healthcare systems and among the underinsured. Nonetheless, we believe that this report provides additional evidence supporting access to risk-appropriate care. Without granular insurance data, we note that all of the individuals in the study at the least had access to the imaging centers.</p>", "<p id=\"Par31\">Additionally, while genetic information sessions both before and after testing were available to all patients who underwent testing, formal pre-test genetic counseling was not required. Requiring pre-test genetic counseling may itself present a barrier to testing [##REF##30526229##46##], so there may be a trade-off between the benchmark of full-genetic counseling and the use of abbreviated genetic information sessions to improve access to potentially life-saving information, especially in population health settings. Additional research is needed to establish what constitutes optimal pre- and post-test counseling and informed consent for patients receiving genetic testing in non-genetic/population health settings. However, in the meantime, the results of this and similar studies suggest that testing in a diverse imaging center population can extend the reach of genetic cancer risk assessment, has a clinically meaningful and actionable yield of cancer-associated PVs, and can help address inequity in access to testing and risk-appropriate screening and prevention.</p>" ]

[]

[ "<title>Purpose</title>", "<p id=\"Par1\">Up to 10% of all breast cancers (BC) are attributed to inherited pathogenic variants (PV) in BC susceptibility genes; however, most carriers of PVs remain unidentified. Here, we sought to determine the yield of hereditary cancer gene PVs among diverse women attending breast imaging centers, who could benefit from enhanced surveillance and/or risk reduction interventions.</p>", "<title>Methods</title>", "<p id=\"Par2\">This cross-sectional retrospective cohort study included consecutive women, unselected for personal or family cancer history, who were offered genetic testing for hereditary cancer genes at the time of breast imaging at three centers (November 2020–March 2022).</p>", "<title>Results</title>", "<p id=\"Par3\">Among 1943 patients (median age: 66 years), self-reported race/ethnicity was White (34.5%), Hispanic (27.7%), African American (17.9%), Asian (4.5%), Ashkenazi Jewish (0.6%), Other (3.5%), and missing (13.0%). Thirty-nine patients (2%) were identified as carriers of a PV in an autosomal dominant clinically actionable hereditary breast and ovarian cancer (HBOC)-related or Lynch syndrome gene, most frequently, <italic>BRCA2</italic> (6/39; 15.4%), <italic>PALB2</italic> (8/39; 20.5%), <italic>CHEK2</italic> (10/39; 25.6%), and <italic>PMS2</italic> (5/39; 12.8%). Of the 34 PVs with known race/ethnicity, 47% were detected among non-White patients. Overall, 354/1,943 (18.2%) of patients met NCCN guidelines for HBOC gene testing and only 15/39 (38.5%) patients with an autosomal dominant clinically actionable PV met guidelines.</p>", "<title>Conclusion</title>", "<p id=\"Par4\">This population health approach extended the reach of genetic cancer risk assessment in a diverse population and highlighted the limits of a guideline-based approach. This may help address inequity in access to risk-appropriate screening and cancer prevention.</p>", "<title>Keywords</title>" ]

[]

[ "<title>Acknowledgements</title>", "<p>We would like to acknowledge Sofia Hurtado, BS and Mayra Rodas, BS from Natera, Inc. for data acquisition and Urmi Sengupta, PhD, from Natera, Inc., for assistance with the development of the manuscript.</p>", "<title>Author contributions</title>", "<p>LW, VS, and JW have contributed to the study concept and design. Data acquisition, analysis, and interpretation were performed by LW, DM, MM, KH, MY, and NK. LW, DM, VS, AR, MM, KH, YS, MY, NK, and JW were involved in the administrative process. LW, VS, and JW supervised the study. Original drafting of the manuscript was performed by LW, VS, and MY. Critical revision and editing of the manuscript were performed by LW, DM, VS, AR, MM, KH, YS, MY, and JW. All authors read and approved the final version of the manuscript for submission.</p>", "<title>Funding</title>", "<p>Not applicable.</p>", "<title>Data availability</title>", "<p>This is descriptive summary data on genetic testing outcome. The specific variant level data are contributed to ClinVar by Baylor Genetics, Houston, Texas.</p>", "<title>Declarations</title>", "<title>Competing interests</title>", "<p id=\"Par32\">VS, MM, KH, YS, MY, RG, and WS are employees of Natera, Inc. with stocks or options to own stocks. LW, DM, and NK declare no conflict of interest.</p>", "<title>Ethical approval</title>", "<p id=\"Par33\">This study was granted a waiver of consent process under 45 CFR 46.116(d), a waiver of the requirement for documentation of informed consent according to 45 CFR 46.117(c)(2), and a waiver from the HIPAA Authorization Requirement according to 45 CFR 46.164.512(i) (Salus IRB, ID# 21204—01A).</p>" ]

[ "<fig id=\"Fig1\"><label>Fig. 1</label><caption><p>Personal cancer history in the full cohort (<italic>N</italic> = 1943)</p></caption></fig>", "<fig id=\"Fig2\"><label>Fig. 2</label><caption><p>Number of clinically actionable PVs identified in high- or moderate-risk genes in patients who either met or did not meet NCCN guidelines for testing of the indicated gene</p></caption></fig>" ]

[ "<table-wrap id=\"Tab1\"><label>Table 1</label><caption><p>List of 25 actionable genes reported for all patients</p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th align=\"left\">High-risk Genes</th><th align=\"left\">Moderate-risk Genes</th></tr></thead><tbody><tr><td align=\"left\"><italic>APC</italic></td><td align=\"left\"><italic>ATM</italic></td></tr><tr><td align=\"left\"><italic>BRCA1</italic></td><td align=\"left\"><italic>BARD1</italic></td></tr><tr><td align=\"left\"><italic>BRCA2</italic></td><td align=\"left\"><italic>BRIP1</italic></td></tr><tr><td align=\"left\"><italic>BMPR1A</italic></td><td align=\"left\"><italic>CHEK2</italic></td></tr><tr><td align=\"left\"><italic>CDH1</italic></td><td align=\"left\"><italic>NF1</italic></td></tr><tr><td align=\"left\"><italic>EPCAM</italic></td><td align=\"left\"><italic>RAD51C</italic></td></tr><tr><td align=\"left\"><italic>MEN1</italic></td><td align=\"left\"><italic>RAD51D</italic></td></tr><tr><td align=\"left\"><italic>MLH1</italic></td><td align=\"left\"/></tr><tr><td align=\"left\"><italic>MSH2</italic></td><td align=\"left\"/></tr><tr><td align=\"left\"><italic>MSH6</italic></td><td align=\"left\"/></tr><tr><td align=\"left\"><italic>MUTYH (Biallelic)</italic></td><td align=\"left\"/></tr><tr><td align=\"left\"><italic>PALB2</italic></td><td align=\"left\"/></tr><tr><td align=\"left\"><italic>PMS2</italic></td><td align=\"left\"/></tr><tr><td align=\"left\"><italic>PTEN</italic></td><td align=\"left\"/></tr><tr><td align=\"left\"><italic>SMAD4</italic></td><td align=\"left\"/></tr><tr><td align=\"left\"><italic>STK11</italic></td><td align=\"left\"/></tr><tr><td align=\"left\"><italic>TP53</italic></td><td align=\"left\"/></tr><tr><td align=\"left\"><italic>VHL</italic></td><td align=\"left\"/></tr></tbody></table></table-wrap>", "<table-wrap id=\"Tab2\"><label>Table 2</label><caption><p>Characteristics of individuals who underwent genetic testing (<italic>N</italic> = 1943)</p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th align=\"left\">Characteristics</th><th align=\"left\">Patients with a PV in HBOC gene<break/> N= 33</th><th align=\"left\">Patients with a PV in a LS gene<break/> N= 6</th><th align=\"left\">Patients without a PV<break/> N= 1904</th><th align=\"left\">All<break/> N= 1943</th></tr></thead><tbody><tr><td align=\"left\">Median age in years (Range)</td><td char=\"(\" align=\"char\">61 (36–78)</td><td char=\"(\" align=\"char\">67 (19–73)</td><td char=\"(\" align=\"char\">65 (18–89)</td><td char=\"(\" align=\"char\">66 (18–89</td></tr><tr><td align=\"left\">&lt; 50 years, n (%)</td><td char=\"(\" align=\"char\">5 (15.2)</td><td char=\"(\" align=\"char\">2 (33.3)</td><td char=\"(\" align=\"char\">349 (18.3)</td><td char=\"(\" align=\"char\">356 (18.3)</td></tr><tr><td align=\"left\"> ≥ 50 years, n (%)</td><td char=\"(\" align=\"char\">28 (84.8)</td><td char=\"(\" align=\"char\">4 (66.7)</td><td char=\"(\" align=\"char\">1555 (81.7)</td><td char=\"(\" align=\"char\">1587 (81.7)</td></tr><tr><td align=\"left\">Race and Ethnicity, n (%)</td><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/><td char=\".\" align=\"char\"/></tr><tr><td align=\"left\">Black</td><td char=\"(\" align=\"char\">2 (6.1)</td><td char=\"(\" align=\"char\">1 (16.7)</td><td char=\"(\" align=\"char\">336 (20)</td><td char=\"(\" align=\"char\">339 (20)</td></tr><tr><td align=\"left\">White</td><td char=\"(\" align=\"char\">14 (42.4)</td><td char=\"(\" align=\"char\">2 (33.3)</td><td char=\"(\" align=\"char\">634 (38.0)</td><td char=\"(\" align=\"char\">650 (38)</td></tr><tr><td align=\"left\">Asian</td><td char=\"(\" align=\"char\">3 (9.1)</td><td char=\"(\" align=\"char\">0 (0.0)</td><td char=\"(\" align=\"char\">82 (5.0)</td><td char=\"(\" align=\"char\">85 (5.0)</td></tr><tr><td align=\"left\">Hispanic</td><td char=\"(\" align=\"char\">8 (24.2)</td><td char=\"(\" align=\"char\">1 (16.7)</td><td char=\"(\" align=\"char\">525 (32)</td><td char=\"(\" align=\"char\">534 (32)</td></tr><tr><td align=\"left\">Multiple races selected</td><td char=\"(\" align=\"char\">1 (3.0)</td><td char=\"(\" align=\"char\">0 (0.0)</td><td char=\"(\" align=\"char\">23 (1.4)</td><td char=\"(\" align=\"char\">24 (1.4)</td></tr><tr><td align=\"left\">Other</td><td char=\"(\" align=\"char\">1 (3.0)</td><td char=\"(\" align=\"char\">1 (16.7)</td><td char=\"(\" align=\"char\">56 (3.4)</td><td char=\"(\" align=\"char\">58 (3.4)</td></tr><tr><td align=\"left\">Missing</td><td char=\"(\" align=\"char\">4 (12.1)</td><td char=\"(\" align=\"char\">1 (16.7)</td><td char=\"(\" align=\"char\">248 (13.0)</td><td char=\"(\" align=\"char\">253 (13.0)</td></tr><tr><td align=\"left\">Personal history of cancer (any type)</td><td char=\"(\" align=\"char\">1 (3.0)</td><td char=\"(\" align=\"char\">0 (0.0)</td><td char=\"(\" align=\"char\">145 (7.6)</td><td char=\"(\" align=\"char\">146 (7.5)</td></tr><tr><td align=\"left\">Family history of cancer (in patients with no personal cancer history)</td><td char=\"(\" align=\"char\">25 (75.8)</td><td char=\"(\" align=\"char\">6 (100.0)</td><td char=\"(\" align=\"char\">791 (41.5)</td><td char=\"(\" align=\"char\">822 (42.3)</td></tr><tr><td align=\"left\">No known personal or family history of cancer</td><td char=\"(\" align=\"char\">7 (21.2)</td><td char=\"(\" align=\"char\">0 (0.0)</td><td char=\"(\" align=\"char\">968 (50.8)</td><td char=\"(\" align=\"char\">975 (50.2)</td></tr></tbody></table></table-wrap>", "<table-wrap id=\"Tab3\"><label>Table 3</label><caption><p>Number (percentage) of patients who met NCCN criteria for hereditary cancer testing based on personal and/or family history</p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th align=\"left\">Characteristics</th><th align=\"left\">Patients with a PV in HBOC gene<break/> N= 33 (%)</th><th align=\"left\">Patients with a PV in a LS gene<break/> N= 6 (%)</th><th align=\"left\">Patients without a PV<break/> N= 1904 (%)</th><th align=\"left\">All<break/> N= 1943 (%)</th></tr></thead><tbody><tr><td align=\"left\">HBOC</td><td char=\"(\" align=\"char\">10 (30.3)</td><td char=\"(\" align=\"char\">3 (50.0)</td><td char=\"(\" align=\"char\">341 (17.9)</td><td char=\"(\" align=\"char\">354 (18.2)</td></tr><tr><td align=\"left\">Lynch</td><td char=\"(\" align=\"char\">1 (3.0)</td><td char=\"(\" align=\"char\">0 (0.0)</td><td char=\"(\" align=\"char\">70 (3.7)</td><td char=\"(\" align=\"char\">71 (3.7)</td></tr><tr><td align=\"left\">Both HBOC and Lynch</td><td char=\"(\" align=\"char\">1 (3.0)</td><td char=\"(\" align=\"char\">0 (0.0)</td><td char=\"(\" align=\"char\">18 (0.9)</td><td char=\"(\" align=\"char\">19 (1.0)</td></tr><tr><td align=\"left\"> Pre-test Tyrer-Cuzick* risk score calculated, n (%)</td><td char=\"(\" align=\"char\">25 (75.8)</td><td char=\"(\" align=\"char\">4(66.6)</td><td char=\"(\" align=\"char\">1405 (73.8)</td><td char=\"(\" align=\"char\">1434 (73.8)</td></tr><tr><td align=\"left\"> Pre-test Tyrer-Cuzick* risk &gt;20%</td><td char=\"(\" align=\"char\">2 (8.0) *</td><td char=\"(\" align=\"char\">1 (16.7)</td><td char=\"(\" align=\"char\">66 (6.0)</td><td char=\"(\" align=\"char\">67 (4.7)</td></tr></tbody></table></table-wrap>" ]

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[ "<table-wrap-foot><p>HBOC, Hereditary Breast and Ovarian Cancer-related genes <italic>(BRCA1, BRCA2, PALB2, ATM, CHEK2, BARD1, NF1, BRIP1)</italic></p><p>LS, Lynch syndrome genes <italic>(MSH2, PMS2)</italic></p></table-wrap-foot>", "<table-wrap-foot><p>*Pre-test Tyrer–Cuzick score was calculated for the patients with no prior history of breast cancer</p></table-wrap-foot>", "<fn-group><fn><p><bold>Publisher's Note</bold></p><p>Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.</p></fn></fn-group>" ]

[ "<graphic xlink:href=\"10549_2023_7137_Fig1_HTML\" id=\"MO1\"/>", "<graphic xlink:href=\"10549_2023_7137_Fig2_HTML\" id=\"MO2\"/>" ]

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{ "acronym": [], "definition": [] }

[ "<title>Introduction</title>", "<p id=\"Par5\">Robotic surgery has revolutionized the treatment of several diseases given the improved clinical outcomes, such as less blood loss, shorter length of hospital stay and fewer complications compared with laparoscopy or open approach. The main limitation has been the high cost due to purchase and maintenance of the da Vinci™ robotic surgical system (Intuitive Surgical, Sunnyvale, CA, USA) that represented the only alternative of the global market so far. After the Intuitive’s patent expiry, novel robotic platforms became available, including the Revo-I, the Senhance, the Versius, Avatera, Hinotori, and Hugo RAS. Some of these new systems share features differing from those of the da Vinci, both in terms of console, robotic units, and arm; thus, the introduction of novel platforms poses specific challenges to be addressed by the whole operating team; information and training about indications, setup, and outcomes are mandatory.</p>", "<p id=\"Par6\">The Versius CMR is one of those new systems that has been approved in the UK in 2018. It displays an open surgical console with hand controllers and a head-up display (HUD). Versius is a multi-modular system with independent bedside unit (BSU), one dedicated to visualization with an endoscopic camera. The HUD provides the surgeon with a three-dimensional, high-definition visualization. The system carries the potentials of versatility, due to the open console design—improving communication in the OR—and adaptability with up to 4 BSU located around the operating table with multiple allowed configurations. The hand controllers are ergonomically designed and, unlike the Da Vinci and other new systems, accommodate all functions—including camera movement and energy delivery—without pedal control [##REF##37325289##1##–##REF##35047780##3##].</p>", "<p id=\"Par7\">After strong evidence of feasibility developed in the pre-clinical setting [##REF##35807035##4##], clinical outcomes from the official “Versius Robotic Surgical System Registry” are yet to come; a low rate of conversion to alternative technique, serious adverse events, and 90-day mortality is anticipated [##REF##37036097##5##].</p>", "<p id=\"Par8\">However, even if recognized viable from wide series, the description of each individual Versius procedures with specific challenges is still crucial.</p>", "<p id=\"Par9\">The aim of the article is to report a comprehensive analysis of the Versius system for pelvic surgery, by describing indications, setup, and early outcomes in a multicentric study.</p>" ]

[ "<title>Methods</title>", "<p id=\"Par10\">This is a retrospective observational study involving two Institutions, ASST Santi Paolo and Carlo, Milan, and at the Apuane Hospital, Massa, Italy.</p>", "<p id=\"Par11\">The Versius CMR was installed by September, 2022, at the former institution, and by 2021 in the latter. The console surgeons involved were urologists (BR and DM), gynecologists (GG, MF), and general surgeons (AP). All except two were prior Da Vinci users; the latter (AP an MF) had an extensive laparoscopic background.</p>", "<p id=\"Par12\">All interventions performed in the pelvic area with the Versius were included in the analysis. Data about indications, intra-, and post-operative course were prospectively collected.</p>", "<title>Training</title>", "<p id=\"Par13\">The training involved the whole OR Team (console surgeon, bedside assistant, scrub nurse, circulating nurse). For surgeons, the Versius training package consisted of a 3.5-day program following 10 h of online didactic training; it includes dry box exercises and wet lab sessions simulated in an operating room using cadaveric models.</p>", "<title>Interventions</title>", "<p id=\"Par14\">Urologic pelvic procedures, gynecological procedures, pelvic procedures from general surgeons were considered. Colectomy and hernia repair were excluded. Interventions could be either performed for oncological or non-oncological purposes; reconstructive interventions with the use of devices (i.e., mesh) were included as well.</p>", "<title>Data collection and analysis</title>", "<p id=\"Par15\">The following variables were collected:<list list-type=\"bullet\"><list-item><p id=\"Par16\">Demographics: age, gender;</p></list-item><list-item><p id=\"Par17\">Surgical indications</p></list-item><list-item><p id=\"Par18\">Intra-operative data: OR setup, number of ports, number of robotic arms, console time, estimated blood loss (EBL), conversion to alternative approaches, complications invoking a change in surgical strategy; use of additional devices for repair (i.e., mesh); malfunctioning of the system requiring a re-start</p></list-item><list-item><p id=\"Par19\">Post-operative data: complications classified by Clavien– Dindo, transfusion rate, length of stay (LOS), 30- and 90-day re-admission</p></list-item></list></p>", "<p id=\"Par20\">Data were inserted in a dedicated data base (Excel, Microsoft) and analyzed by an external reviewer (ST) with formal habilitation as a coordinator with the Versius and the Da Vinci systems. A descriptive analysis of all variables was performed. Console Time, EBL, and LOS were provided as mean value and range.</p>" ]

[ "<title>Results</title>", "<p id=\"Par21\">Overall, a total of 171 interventions were performed with the Versius CMR at the ASST Santi Paolo and Carlo (120) and at the Apuane Hospital (51). Forty-two of them involved pelvic procedures. A full list of interventions stratified by center and specialty is reported in Table ##TAB##0##1##. Twenty-two interventions had an oncological indication (localized prostate cancer) whereas the remaining ones had a non-oncological or functional purpose. The mostly performed procedures were radical prostatectomy (22) and annexectomy (9). The OR setup for urological, gynecological and general surgical procedure (rectopexy and sigmoidectomy) is depicted in Fig. ##FIG##0##1##a–d, respectively. No intra-operative complication invoking a change in surgical strategy occurred nor conversion to open or laparoscopic surgery. Two Clavien II and one IIIb complications were evident (pelvic hematoma requiring transfusion, a urinary tract infection treated with antibiotic therapy and a bowel obstruction due to port-site herniation). Table ##TAB##1##2## reports console time, EBL, complications, and LOS stratified by procedure. Malfunctioning/alarms requiring the whole system re-activation occurred in 2 different cases. An adjustment in trocar placement according to patients’ height was required in 2 patients undergoing RALP, in which the trocar was moved caudally. In two cases, a pelvic organ prolapse (POP) was repaired concomitant with other gynecological procedures: an hysterectomy, bilateral salpingo-oophorectomy, and lateral suspension with a titanium-coated polypropylene mesh, according to Dubuisson technique. A single case, a 53-year-old woman, underwent a ventral rectopexy for full-thickness rectal external prolapse.</p>", "<p id=\"Par22\">Positive margin (PM) rate can be evaluated for radical prostatectomies. The outcomes are variable, one series showed 83.3% of PM rate on 18 cases (11 pT3, published data) [##REF##37604773##6##] and in the other one (4 pT2c cases), no PM were found. At a follow-up ranging from 9 to 11 months, a single case of biochemical recurrence was noticed. One re-admission at 30 days (herniation) was recorded, and none at 90 days.</p>" ]

[ "<title>Discussion</title>", "<p id=\"Par23\">The current series represents the first one completely focused on pelvic surgery with the new Versius surgical system.</p>", "<p id=\"Par24\">Sparse reports evaluating single indications with Versius [##REF##37221826##7##–##UREF##0##9##] are available yet; similarly, some series reported on the use of Versius for abdominal surgery, the majority including preliminary procedures—hernia repair, cholecystectomy—to become familiar with the system [##REF##36002682##10##, ##REF##32989548##11##]. In the present series, we intentionally addressed the pelvic area, to evaluate if robotic surgery with a novel platform may address the challenges posed by such interventions, often made up by complex tasks (dissection and reconstruction) in small spaces with limited accessibility and range of movements. A comprehensive analysis of pelvic approach with Versius has been already reported in the pre-clinical setting: Vasdess et al. [##REF##32989548##11##] described the possible setup for prostate surgery on cadaveric models, exploring multiple options of trocar placement with either 3- and 4-arm configurations. To note, authors evaluated also the feasibility of radical cystectomy (even if in vivo cystectomy has not yet performed so far with the Versius).</p>", "<p id=\"Par25\">Our series confirms the feasibility of Versius pelvic surgery in different settings without major complication; actually, we did not face the need for conversion to other approaches. In preliminary series, conversion has been reported in up to 4–6% of Versius cases [##REF##37334028##12##, ##REF##35752748##13##] together with a certain variable degree of adverse events [##REF##35752748##13##–##REF##35861102##15##]. Herein, a single Clavien IIIb complication was reported (bowel herniation through abdominal wall requiring surgical repair); the occurrence is seemingly unrelated to the use of a novel robotic system. As far as surgical margin status is concerned, radical prostatectomy is the only oncological intervention we included. In this setting, PM rate from a single institution series appears of importance (83.3%); the small sample size and a high rate of pT3 could have accounted for the occurrence [##REF##37604773##6##]. Moreover, PM is counterbalanced by low BCR, (a single case showing a raise in post-operative PSA out of 22 RALP), provided long-term PSA is still awaited.</p>", "<p id=\"Par26\">Unlike other series dealing with the Versius, some complex cases have been performed as well in the present article: this is the case of surgery for endometriosis and of a case of post-radiation sigmoidectomy.</p>", "<p id=\"Par27\">Some specifics to pelvic surgery may apply to the Versius system.</p>", "<p id=\"Par28\">Surgical instruments are shorter than those of the Da Vinci (30 cm): the issue should be considered and an estimation of the distance to the target area (and to other areas to be reached, i.e., pelvic nodal dissection) should be pre-planned. Port placement can be, therefore, moved caudally than the conventional Da Vinci configuration, especially considering patient’s height. In two RALP cases, we had to move one port more caudally to make the instrument reach the target area (urethral stump). The BMI of the patient is another issue that needs to be taken into account: in the current series, extreme BMIs have been successfully managed with a prior accurate planning of port and BSU placement. This was the case of gynecological surgery, which encompassed a range of BMI from 16 to 43 (non-published data).</p>", "<p id=\"Par29\">One of the major concerns of multi-modular robotic system is the likelihood of external clashing between arms, a point somehow counterbalancing the versatility invoked by new systems with independent units [##REF##35637355##16##]. Overall, whereas the Da Vinci Xi allows for a linear trocar placement with a standardized docking, such a step with the Versius should be accurately planned to minimize clashing or limitation in instruments motion [##REF##35637355##16##]. Another feature typical of the Versius system is the docking of the robotic arm to the instrument and not to the trocar, as occurs with the Da Vinci or Hugo RAS; if the use of a non-dedicated trocar may represent an advantage, on the other side collisions between the robotic arm and the skin of the patient are recognized by the system and given as an alarm. The issue should be taken into account as well during the setup, especially in pelvic surgery in which a Trendelenburg of the patient is required and angles with the trocar could be relevant.</p>", "<p id=\"Par30\">Beyond technical details, some general considerations may arise from the current experience.</p>", "<p id=\"Par31\">Versius can be used by surgeons from different disciplines and it heavily comprises diverse surgical specialties; the same occurs with the Da Vinci, even if major users have initially been urologists [##REF##37325289##1##]. It is representative that in a center owning three robotic systems such as the ASST Santi Paolo and Carlo (Da Vinci, Versius CMR and Hugo RAS), the majority of Versius cases have been accomplished by gynecologists and general surgeons.</p>", "<p id=\"Par32\">In line with these findings, laparoscopists seem to be those surgeons mostly advantaging from the introduction of Versius. The instruments are designed to mimic the articulation of the human arm and the wristed joints with seven degrees of freedom overcome the difficulties of laparoscopic surgery. The use of laparoscopic trocars—not robotic dedicated—raises opportunity for a hybrid procedure; thus, laparoscopists appreciate Versius features designed to simplify laparoscopic surgery.</p>", "<p id=\"Par33\">Severe malfunctioning of the Versius has been reported in few cases and has been fixed with the re-start of the system (2 cases). If compared to the Da Vinci, it should be recognized that the latter has reached its fourth generation, whereas the Versius system is at its very first one; technological improvement and optimization are awaited. Noticeably, similar considerations may apply also to other novel robotic platforms that entered the market within the last two years.</p>", "<p id=\"Par34\">As far as the Versius is concerned, technological updates are already developing: an improvement in arm clash recovery has been recently released—i.e., maintenance of arm engagement from the surgeon in case of clash—and other enhancements are yet to come, such as improvement of bipolar devices and the implementation of an energy sealer device [##REF##37334028##12##].</p>", "<p id=\"Par35\">The article is not devoid of limitations.</p>", "<p id=\"Par36\">First, the small sample size and short follow-up preclude any conclusion about mid- and long-term outcomes. The different kind of surgeries included make outcome assumptions weak; however, the report of long-term clinical follow-up was beyond the purpose of the paper, that merely aimed to address the feasibility of Versius surgery in the pelvic area.</p>", "<p id=\"Par37\">Second, no matched comparison with the Da Vinci nor a subjective comparison is herein provided. Some console surgeons (BR, DM, GG) and assistants (MCS, FT, MS) are currently operating on multiple platforms, but a subjective perception about systems is not herein provided. However, an objective comparison of the first RALP cases with the Versius and Hugo RAS has been described by our group throughout the metrics developed by the ERUS working group for radical prostatectomy [##UREF##1##17##]. Further comparative analysis of robotic systems is expected in the very next future, to highlight differences and peculiarity of each platform trying to draw definite outcomes.</p>" ]

[ "<title>Conclusions</title>", "<p id=\"Par38\">From the present series, pelvic surgery with the Versius system is feasible without severe intra- or peri-operative complications; long-term oncological and functional outcomes are yet to be defined. All steps of pelvic surgery are reproducible with the Versius, provided a proper surgical setup and trocar placement are pursued. Versius can be easily adopted by surgeons of different disciplines and background within the same institution; a further multi-specialty implementation is expected, and a cost analysis is awaited to highlight its future role into healthcare systems.</p>" ]

[ "<title>Introduction</title>", "<p id=\"Par1\">Versius CMR is a novel robotic system characterized by an open surgical console and independent bedside units. The system has potentials of flexibility and versatility, and has been used in urological, gynecological, and general surgical procedure. The aim is to depict a comprehensive analysis of the Versius system for pelvic surgery.</p>", "<title>Methods</title>", "<p id=\"Par2\">This is a study involving two Institutions, ASST Santi Paolo and Carlo, Milan, and Apuane Hospital, Massa, Italy. All interventions performed in the pelvic area with the Versius were included. Data about indications, intra-, and post-operative course were prospectively collected and analyzed.</p>", "<title>Results</title>", "<p id=\"Par3\">A total of 171 interventions were performed with the Versius. Forty-two of them involved pelvic procedures. Twenty-two had an oncological indication (localized prostate cancer), the remaining had a non-oncological or functional purpose. The mostly performed pelvic procedure was radical prostatectomy (22) followed by annexectomy (9). No intra-operative complication nor conversion to other approaches occurred. A Clavien II complication and one Clavien IIIb were reported. Malfunctioning/alarms requiring a power cycle of the system occurred in 2 different cases. An adjustment in trocar placement according to patients’ height was required in 2 patients undergoing prostatectomy, in which the trocar was moved caudally. In two cases, a pelvic prolapse was repaired concomitant with other gynecological procedures.</p>", "<title>Conclusions</title>", "<p id=\"Par4\">Pelvic surgery with the Versius is feasible without major complications; either dissection and reconstructive steps could be accomplished, provided a proper OR setup and trocar placement are pursued. Versius can be easily adopted by surgeons of different disciplines and backgrounds; a further multi-specialty implementation is presumed and long-term oncological and functional outcomes are awaited.</p>", "<title>Keywords</title>", "<p>Open access funding provided by Università degli Studi di Milano within the CRUI-CARE Agreement.</p>" ]

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[ "<title>Author contributions</title>", "<p>MCS: protocol and project development, manuscript writing. MDM: project development, data management. JM: data collection, data analysis. MF: project development, data management. APC: project development, data management, manuscript editing. LM: data management. CC: data management. AM: data collection, data analysis. TC: data collection, data analysis. EP: data collection, data analysis. MS: data management. FT: data management. ST: data analysis, manuscript editing. SA: data collection. LS: data collection. MA: data collection. AM: project development. PPB: project development, supervision. SM: supervision. BR: project development, supervision. GG: project development, data management, manuscript editing.</p>", "<title>Funding</title>", "<p>Open access funding provided by Università degli Studi di Milano within the CRUI-CARE Agreement.</p>", "<title>Data availability</title>", "<p>Data will be shared upon appropriate request.</p>", "<title>Declarations</title>", "<title>Conflict of interest</title>", "<p id=\"Par39\">Authors have no potential conflict of interest to disclose.</p>", "<title>The research involves human participants</title>", "<p id=\"Par40\">All patients signed an informed consent to the procedure they underwent; all interventions were planned and performed according to the current international guidelines and treatment recommendation.</p>" ]

[ "<fig id=\"Fig1\"><label>Fig. 1</label><caption><p>OR setup for prostatectomy (<bold>a</bold>), gynecologic surgery (<bold>b</bold>), rectopexy (<bold>c</bold>), signoidectomy (<bold>d</bold>)</p></caption></fig>" ]

[ "<table-wrap id=\"Tab1\"><label>Table 1</label><caption><p>Procedures stratified by Center</p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th align=\"left\">Procedure</th><th align=\"left\">Number</th><th align=\"left\">Center</th><th align=\"left\">Details</th></tr></thead><tbody><tr><td align=\"left\">Radical prostatectomy</td><td char=\".\" align=\"char\">22</td><td align=\"left\"><p>ASST Santi Paolo and Carlo (4)</p><p>Apuane Hospital (18)</p></td><td align=\"left\"/></tr><tr><td align=\"left\">Annexectomy</td><td char=\".\" align=\"char\">9</td><td align=\"left\">ASST Santi Paolo and Carlo</td><td align=\"left\">Bilateral in 8 cases</td></tr><tr><td align=\"left\">Surgery for Endometriosis</td><td char=\".\" align=\"char\">2</td><td align=\"left\">ASST Santi Paolo and Carlo</td><td align=\"left\"/></tr><tr><td align=\"left\">Hysterectomy</td><td char=\".\" align=\"char\">3</td><td align=\"left\">ASST Santi Paolo and Carlo</td><td align=\"left\">Concomitant POP repair in 1 case</td></tr><tr><td align=\"left\">Ovarian cyst enucleation</td><td char=\".\" align=\"char\">4</td><td align=\"left\">ASST Santi Paolo and Carlo</td><td align=\"left\">Bilateral in 2 cases</td></tr><tr><td align=\"left\">Dubuisson repair</td><td char=\".\" align=\"char\">1</td><td align=\"left\">ASST Santi Paolo and Carlo</td><td align=\"left\"/></tr><tr><td align=\"left\">Ventral rectopexy</td><td char=\".\" align=\"char\">1</td><td align=\"left\">ASST Santi Paolo and Carlo</td><td align=\"left\"/></tr><tr><td align=\"left\">Sigmoidectomy</td><td char=\".\" align=\"char\">1</td><td align=\"left\">ASST Santi Paolo and Carlo</td><td align=\"left\"/></tr></tbody></table></table-wrap>", "<table-wrap id=\"Tab2\"><label>Table 2</label><caption><p>Peri-operative outcomes stratified by procedure</p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th align=\"left\">Procedure</th><th align=\"left\">N° robotic arms</th><th align=\"left\">Console time</th><th align=\"left\">EBL</th><th align=\"left\">Complications</th><th align=\"left\">LOS</th></tr></thead><tbody><tr><td align=\"left\">Radical prostatectomy</td><td align=\"left\">4</td><td align=\"left\">201 (130–242)</td><td align=\"left\">140 (100–550)</td><td align=\"left\"><p>1 pelvic hematoma</p><p>1 UTI</p><p>1 port-site herniation</p></td><td align=\"left\">4 (3.75–7)</td></tr><tr><td align=\"left\">Annexectomy</td><td align=\"left\">4</td><td align=\"left\">58 (31–77)</td><td align=\"left\">0 (0–0)</td><td align=\"left\">No</td><td align=\"left\">2</td></tr><tr><td align=\"left\">Hysterectomy</td><td align=\"left\">4</td><td align=\"left\">142 (122–176)</td><td align=\"left\">133 (50–200)</td><td align=\"left\">No</td><td align=\"left\">2</td></tr><tr><td align=\"left\">Ovarian cyst enucleation</td><td align=\"left\">3 in 2 cases, 4 in 1</td><td align=\"left\">115 (76–147)</td><td align=\"left\">37 (0–62)</td><td align=\"left\">No</td><td align=\"left\">2</td></tr><tr><td align=\"left\">Surgery for Endometriosis</td><td align=\"left\">4</td><td align=\"left\">217 (210–225)</td><td align=\"left\">50 (0–100)</td><td align=\"left\">No</td><td align=\"left\">2</td></tr><tr><td align=\"left\">Dubuisson Lateral suspension</td><td align=\"left\">4</td><td align=\"left\">170</td><td align=\"left\">0</td><td align=\"left\">No</td><td align=\"left\">3</td></tr><tr><td align=\"left\">Ventral rectopexy</td><td align=\"left\">4</td><td align=\"left\">94</td><td align=\"left\">100</td><td align=\"left\">No</td><td align=\"left\">3</td></tr><tr><td align=\"left\">Sigmoidectomy</td><td align=\"left\">4</td><td align=\"left\">80</td><td align=\"left\">100</td><td align=\"left\">No</td><td align=\"left\">5</td></tr></tbody></table></table-wrap>" ]

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[ "<fn-group><fn><p><bold>Publisher's Note</bold></p><p>Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.</p></fn><fn><p>Maria Chiara Sighinolfi and Maurizio De Maria shared the first-authorship.</p></fn></fn-group>" ]

[ "<graphic xlink:href=\"345_2023_4730_Fig1_HTML\" id=\"MO1\"/>" ]

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{ "acronym": [], "definition": [] }

[ "<title>Introduction</title>", "<p id=\"Par5\">Breast cancer in men (male breast cancer; MBC) is a rare entity accounting for 0.5–1% of all breast cancer diagnoses and an estimated lifetime risk of 1:1000 [##REF##30620402##1##]. Due to its rarity and lack of prospective dedicated MBC trials, the treatment guidelines are mainly based on extrapolation from randomized evidence from trials including mainly women with breast cancer (female breast cancer; FBC) [##REF##32058842##2##].</p>", "<p id=\"Par6\">The utilization of neoadjuvant chemotherapy (NAC) has steadily increased over the years due to its potential to de-escalate breast and axillary surgery as well as to provide an opportunity for response-based tailored adjuvant therapy [##REF##33507815##3##]. There is solid evidence supporting the use of NAC in FBC, while the utilization and effectiveness of NAC in MBC is less studied [##REF##33507815##3##, ##REF##34933868##4##]. In fact, two retrospective studies have shown conflicting results regarding potential sex disparities on the effectiveness of NAC [##REF##33997921##5##, ##REF##36069365##6##], whereas one of them also showed a lower utilization of NAC in MBC compared to FBC [##REF##33997921##5##].</p>", "<p id=\"Par7\">Considering the limited and conflicting evidence on the role of NAC in MBC, the aims of the present nationwide, register-based, retrospective cohort study were to investigate the utilization of NAC in men and women with early breast cancer, and to compare the effectiveness of NAC between men and women in terms of pathologic complete response (pCR).</p>" ]

[ "<title>Patients and methods</title>", "<title>Study design, data sources, participants and data collection</title>", "<p id=\"Par8\">For this nationwide, register-based retrospective <italic>cohort study</italic>, all patients with stage I–III invasive breast cancer diagnosed in Sweden between January 1, 2008 and December 31, 2019 were identified through the National Quality Registry for breast cancer (Nationellt Kvalitetsregister för bröstcancer; NKBC). NKBC has a high coverage (99.8%) and data completeness ensuring the validity of data and the generalizability of the study results [##REF##31046737##7##]. Using the ten-digit personal identity number, data from NKBC was linked to other national databases of interest to build the research database BCBaSe 3.0 (<ext-link ext-link-type=\"uri\" xlink:href=\"https://cancercentrum.se/samverkan/regional-cancer-centres/research-and-innovation/register-based-research-databases/)\">https://cancercentrum.se/samverkan/regional-cancer-centres/research-and-innovation/register-based-research-databases/)</ext-link>. The study was performed in line with the principles of the Declaration of Helsinki. Approval was granted by the Regional Ethics Committee, Stockholm (Approval number: 2019–02610).</p>", "<p id=\"Par9\">All patients with stage I-III breast cancer in NKBC were included except those with lacking information on estrogen-receptor (ER) status, progesterone-receptor (PgR) status, HER2-status or information on preoperative or postoperative TNM stage. Patients who did not undergo surgery were also excluded. The number of patients treated with NAC but not underwent surgery were 91 females (1.4% among females with NAC) and 1 male (4.0% among males with NAC). The decision of no surgery has been considered unrelated to disease progression since all patients had a registered adjuvant therapy and none of the patients was registered with metastatic disease during the first three months from diagnosis (Fig. ##FIG##0##1##).</p>", "<p id=\"Par10\">Patient demographics (age, educational level, household income, health care region at diagnosis), tumor characteristics (tumor size, histological grade, clinical stage, pathological stage, morphological type and surrogate molecular subtypes based on immunohistochemistry (IHC) status) and treatment characteristics (data on breast and axillary surgery, chemotherapy, radiotherapy and endocrine therapy) were collected.</p>", "<title>Outcomes and definitions</title>", "<p id=\"Par11\">IHC-subtyping was used to classify tumors into three surrogate subtypes, namely luminal (ER or PgR-status ≥ 10%, HER2-negative), HER2-positive (any ER and PgR status, HER2-positive), and triple negative (ER and PgR status &lt; 10% and HER2-negative) breast cancer.</p>", "<p id=\"Par12\">For the research question on NAC utilization, patients were classified as treated with NAC if there was a treatment strategy including NAC, irrespective of chemotherapeutic agent used.</p>", "<p id=\"Par13\">For the research question on pCR, we defined pCR as the absence of invasive breast cancer in the surgical specimens from breast and axillary lymph nodes.</p>", "<title>Statistical analysis</title>", "<p id=\"Par14\">Data were summarized using descriptive statistics including numbers and percentages for categorical variables and median with range for continuous variables. Comparisons of patient characteristics between males and females were made by Pearson’s Chi-squared test, Fisher´s exact test, or Mann–Whitney test as appropriate. For the first research question on NAC utilization, a multivariable logistic regression analysis was performed to analyze the association between sex and the likelihood of receiving NAC while adjusting for prespecified patient- and tumor characteristics including age at diagnosis, educational level, household income, health care region at diagnosis, clinical T and N stage, morphological type, histological grade, and IHC-based subtype. For the second research question on pCR, a multivariable logistic regression analysis was conducted to evaluate the impact of sex on the odds of pCR, adjusted for prespecified parameters including age at diagnosis, clinical T and N stage, histological grade, and IHC-subtype. All <italic>p</italic> values reported were two-sided, and <italic>p</italic> values of &lt; 0.05 were considered statistically significant. All statistical analyses were performed using the SPSS statistical package (IBM Corp. Released 2021. IBM SPSS Statistics for Windows, Version 28.0. Armonk, NY: IBM Corp).</p>" ]

[ "<title>Results</title>", "<title>Characteristics of study cohort</title>", "<p id=\"Par15\">In total, 114,290 patients with breast cancer were registered in NKBC between 2008 and 2020. Applying the inclusion and exclusion criteria, 82,888 patients (82,401 women and 428 men) with stage I-III breast cancer who underwent surgery and had adequate information for IHC-subtyping were identified. This cohort comprised the NAC utilization cohort. When the cohort was restricted only to patients who received NAC, 6487 breast cancer patients (6463 women and 24 men) were available for analyses related to the effectiveness of NAC. A flowchart diagram of patients’ selection process is shown in Fig. ##FIG##0##1##.</p>", "<p id=\"Par16\">Table ##TAB##0##1## summarizes patient, tumor and treatment characteristics in the NAC utilization cohort, by sex. Men with breast cancer were older, had a lower educational level, more advanced anatomical stage at diagnosis (both T and N stage), higher histological grade, fewer lobular carcinomas (1.5% vs. 13.8% in women) and a different IHC-subtype distribution (luminal 86.9% vs. 77.3%; HER2-positive 12.5% vs. 13.4%, triple negative 0.6% vs. 9.2%, respectively). Treatment patterns, including breast and axillary surgery as well as adjuvant therapeutic approaches followed the statistically significant differences in anatomical staging and IHC-subtype. Regarding NAC effectiveness cohort, 6463 women (7.8%) and 24 men (4.9%) received neoadjuvant chemotherapy, respectively. When comparing baseline characteristics of women and men treated with NAC, statistically significant differences regarding educational level, IHC-subtype, use of adjuvant endocrine therapy, as well as type of breast surgery were seen (Table ##TAB##1##2##). The utilization of NAC seems to be steadily increased over time for both males and females in Sweden as shown in Table ##TAB##1##2##.</p>", "<title>Factors associated with NAC utilization patterns</title>", "<p id=\"Par17\">Using multivariate logistic regression model, using the complete case analysis method, no statistically significant difference in NAC utilization between women and men was observed (Odds Ratio (OR): 1.135; 95% Confidence Interval (CI): 0.606–2.128). The total number of patients included in this model was 78,760. Factors associated with higher likelihood of NAC were: young age, high educational level, high household income, treatment in certain healthcare regions Stockholm/Gotland, South, or Southeast), high clinical T and N stage, high histological grade, HER2-positive and triple negative IHC-subtype and ductal histology (Table ##TAB##2##3##).</p>", "<title>Factors associated with pCR after NAC</title>", "<p id=\"Par18\">No statistically significant difference in pCR rates were observed between women and men in the multivariate logistic regression analysis, using the complete case analysis method (OR: 1.141; 95% CI 0.141–9.238). The total number of patients included in this model was 6215. Factors associated with higher pCR rates were young age, high histologic grade, and HER2-positive IHC-subtype (Table ##TAB##3##4##).</p>" ]

[ "<title>Discussion</title>", "<p id=\"Par19\">Using nationwide, register-based data from Sweden, we found no evidence of sex disparities regarding utilization of NAC in breast cancer patients when analyses were adjusted for patient- and tumor characteristics. In addition, the effectiveness of NAC in terms of pCR seems to be similar between men and women with breast cancer, supporting the current recommendations on treating men with breast cancer with the same principles as women with regard to NAC indications.</p>", "<p id=\"Par20\">Due to the rarity of MBC and subsequently the lack of prospective trials, potential differences in utilization and effectiveness of NAC between men and women have been investigated only through register-based studies.</p>", "<p id=\"Par21\">Regarding utilization of NAC in men with breast cancer, Cao et al. analyzed data from the United States National Cancer Database (NCDB) between 2004 and 2016, and found that men with node positive (N +) disease were less likely to be treated with NAC when compared to women. Interestingly, Cao et al. found an underutilization of oncological treatment in men with breast cancer in general, a pattern not seen in the present study cohort. Breast cancer treatment practices may vary between countries, and may also have changed over time. Our study cohort included patients diagnosed during more recent years, when sex disparities in breast cancer treatment strategies have been acknowledged [##REF##19996029##8##], thus leading to efforts to mitigate these disparities. Differences between study cohorts with regard to age (a higher proportion of older adults in our cohort) and stage (only patients with N + disease in NCDB cohort) could also explain the partly conflicting study results.</p>", "<p id=\"Par22\">Also, with regard to NAC effectiveness in men compared to women with breast cancer, the current literature shows somewhat conflicting results. Cao et al. found similar pCR rate between men and women treated with neoadjuvant chemotherapy, as well as a comparable overall survival. Leone et al. found the odds for pCR in women compared to men to be nearly twice as high when studying patients diagnosed 2010 to 2016 from the same database as Cao. Interestingly, the difference in pCR rates observed in the latter study was mostly driven by differences in pCR within the luminal HER2-negative and luminal HER2-positive subgroups. Our results are in accordance with Cao et al., but differ from Leone et al. Although the number of included patients in certain subgroups in our study cohort was not large enough to enable subgroup analyses, we included this potential confounding factor into the multivariate model when the impact of sex on NAC effectiveness was analyzed. The lack of difference in pCR rates between men and women with breast cancer, in spite of the higher percentage of luminal breast cancer in men, could possibly be explained by different distribution of Luminal A/B tumors between men and women, i.e. a higher proportion of more high risk Luminal B tumors in men [##REF##32579768##9##, ##REF##28984296##10##]. Luminal B breast cancer is associated with higher pCR rates than Luminal A [##REF##22508812##11##], possibly balancing the chance of pCR between the two cohorts. Another potential explanation of similar NAC effectiveness between men and women despite the dominance of luminal tumors in men could be a higher presence of an immunological-enriched tumor microenvironment in male luminal tumors [##REF##36614261##12##], a condition that has been associated with improved pCR rates in all breast cancer subtypes [##REF##29233559##13##].</p>", "<p id=\"Par23\">The comparison of patient- and tumor-related characteristics between men and women with breast cancer confirmed some well-established differences between the sexes; older age at diagnosis, more advanced N-stage at diagnosis, the dominance of luminal subtype and the rarity of TNBC and lobular histology in men with breast cancer.</p>", "<p id=\"Par24\">Considering NAC utilization in general, some study results deserve attention. Our study results confirm a tumor-driven approach regarding NAC utilization with a higher use in more advanced and biologically more aggressive disease, which is in line with the current evidence and clinical practice. On the other hand, our study results imply some socioeconomic and geographic inequalities with lower odds to receive NAC among patients with lower income, lower education and among those from specific regions. Although similar inequalities have been reported previously [##REF##35658495##14##, ##REF##30232683##15##], such disparities are not acceptable and a deeper understanding is necessary in an effort to eliminate healthcare-related inequalities.</p>", "<p id=\"Par25\">An interesting finding in terms of factors associated with higher pCR rates was the association between young age and a higher pCR rate. This finding is in accordance with a pooled analysis from eight randomized trials, where younger patients had higher odds of pCR, thus supporting the notion that tumor biology in younger patients might be more aggressive also within subtypes and therefore more susceptible to chemotherapy [##REF##29632634##16##].</p>", "<p id=\"Par26\">The study has several limitations that need to be addressed. First, data on planned treatment was used rather than actual treatment given, as NKBC data on planned treatment have a higher validity than given treatment for the studied time period. As a result, however, a risk of misclassification between NAC and primary surgery for some patients does exist. Second, the duration of planned NAC is lacking. However, the Swedish guidelines have steadily and throughout the years, recommended the use of six cycles (q3w) of chemotherapy, similar to the recommendation for adjuvant chemotherapy. In our study cohort, we lacked information about dose-dense chemotherapeutic regimens. However, this strategy has been rather uncommon in Sweden during the study period and there is no reason to believe that there would be any sex disparity in using dose dense regimens that could impact the prognosis. The limited sample size for men with breast cancer in some specific subtypes is also a limitation as it precludes from relevant subgroup analysis. To mitigate this source of bias, we tried to adjust the multivariate analyses using parameters of potential interest as breast cancer subtype. One could argue that the exclusion of patients who did not undergo surgery can result in immortal-time bias if disease progression during NAC is the reason for no surgery. However, within the group of patients treated with NAC, the proportion of patients who did not undergo surgery was extremely low in both sexes and was considered unrelated to disease progression, thus eliminating the risk for immortal-time bias. Finally, the nature of collected data for the present study does not allow any information about the role of patient or clinician in treatment decision regarding the type and sequence of treatment strategy between sexes.</p>", "<p id=\"Par27\">Acknowledging the relative limited number of men with breast cancer included in the analyses, the current study did not find any sex disparities either in the NAC utilization or effectiveness supporting the current recommendations on treating men with breast cancer similar to women with regard to indications for NAC. The observed socioeconomical and geographical disparities in NAC utilization deserves a deeper understanding before designing strategies to eliminate these inequalities towards an equitable access to breast cancer care.</p>" ]

[]

[ "<title>Purpose</title>", "<p id=\"Par1\">Evidence supporting the use of neoadjuvant chemotherapy (NAC) in early breast cancer is based on studies mainly including women, whereas the utilization and effectiveness of NAC in men is less studied. The present study aimed to investigate the utilization and effectiveness of NAC in men and women with early breast cancer.</p>", "<title>Methods</title>", "<p id=\"Par2\">Eligible patients were identified through the Swedish National Breast Cancer Quality Register, that includes all newly diagnosed breast cancer cases in Sweden from 2008 and onwards. For the treatment utilization analysis, all patients with stage I–III between 2008 and 2020 were included (n = 82,888), whereas for the effectiveness analysis the cohort was restricted to patients receiving NAC (n = 6487). For both analyses, multivariate logistic regression models were applied to investigate potential sex disparities in NAC utilization and effectiveness, adjusted for patient- and tumor characteristics.</p>", "<title>Results</title>", "<p id=\"Par3\">In the NAC utilization analysis, 487 men and 82,401 women with stage I–III were included. No statistically significant difference between sexes in terms of NAC utilization was observed (adjusted Odds Ratio (adjOR): 1.135; 95% Confidence Interval (CI) 0.606–2.128) with an overall utilization rate of 4.9% in men compared to 7.8% in women. Among the 24 men and 6463 women who received NAC, the pathologic complete response (pCR) rates were 16.7% and 21.2%, respectively (adjOR: 1.141; 95% CI 0.141–9.238).</p>", "<title>Conclusion</title>", "<p id=\"Par4\">The present study did not find any sex disparities in NAC utilization or effectiveness in terms of pCR. This supports the current recommendations of treating men with breast cancer with the same indications for NAC as women.</p>", "<title>Keywords</title>", "<p>Open access funding provided by Uppsala University.</p>" ]

[]

[ "<title>Author contributions</title>", "<p>AV conceived the original idea about the study and study’s design. Material preparation, data collection and analysis were performed by AV. The first draft of the manuscript was written by AS and all authors commented on previous versions of the manuscript. All authors read and approved the final manuscript.</p>", "<title>Funding</title>", "<p>Open access funding provided by Uppsala University. This study did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.</p>", "<title>Data availability</title>", "<p>Data are available from register holders (Statistics Sweden, Swedish National Board of Health and Welfare, the Regional Cancer Center Stockholm Gotland) for researchers with relevant ethical approvals and who meet the criteria for access to confidential data. The data are not publicly available due to restrictions by Swedish and European law, in order to protect patient privacy.</p>", "<title>Declarations</title>", "<title>Conflict of interest</title>", "<p id=\"Par28\">AS, MS and AV have no competing interests. IF has received institutional research grants from MSD unrelated to the current work.</p>" ]

[ "<fig id=\"Fig1\"><label>Fig. 1</label><caption><p>CONSORT diagram of study population. Cohort I: All patients registered in NKBC with stage I-III breast cancer who underwent surgery and had adequate information for IHC-subtyping. Cohort II: restricted only to patients who received neoadjuvant chemotherapy (NAC) (BcBASE 3.0, NKBC, 2008-2019).</p></caption></fig>" ]

[ "<table-wrap id=\"Tab1\"><label>Table 1</label><caption><p>Summary of demographics, tumor-related variables, and treatment patterns for the utilization cohort; women and men with stage I-III breast cancer who underwent surgery and had adequate information for IHC-subtyping (BCBaSe 3.0/NKBC, 2008–2019)</p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th align=\"left\">Factors</th><th align=\"left\">Female (N = 82,401), n (%)</th><th align=\"left\">Male (N = 487), n (%)</th><th align=\"left\">p value</th></tr></thead><tbody><tr><td align=\"left\">Age in yrs, median (range)</td><td align=\"left\">64 (19–99)</td><td align=\"left\">69 (29–94)</td><td char=\".\" align=\"char\"><bold> &lt; 0.001</bold></td></tr><tr><td align=\"left\">Calendar year at diagnosis</td><td align=\"left\"/><td align=\"left\"/><td char=\".\" align=\"char\"/></tr><tr><td align=\"left\"> 2008–2011</td><td align=\"left\">22,001 (26.7)</td><td align=\"left\">108 (22.2)</td><td char=\".\" align=\"char\"><bold>0.026</bold></td></tr><tr><td align=\"left\"> 2012–2015</td><td align=\"left\">26,780 (32.5)</td><td align=\"left\">154 (21.6)</td><td char=\".\" align=\"char\"/></tr><tr><td align=\"left\"> 2016–2020</td><td align=\"left\">33,620 (40.8)</td><td align=\"left\">225 (46.2)</td><td char=\".\" align=\"char\"/></tr><tr><td align=\"left\">Education Level</td><td align=\"left\"/><td align=\"left\"/><td char=\".\" align=\"char\"/></tr><tr><td align=\"left\"> Low ≦9 years</td><td align=\"left\">17,330 (21.3)</td><td align=\"left\">149 (30.8)</td><td char=\".\" align=\"char\"><bold> &lt; 0.001</bold></td></tr><tr><td align=\"left\"> Intermediate 10–12 years</td><td align=\"left\">34,310 (42.1)</td><td align=\"left\">200 (41.3)</td><td char=\".\" align=\"char\"/></tr><tr><td align=\"left\"> High ≧13 years</td><td align=\"left\">29,790 (36.6)</td><td align=\"left\">135 (27.9)</td><td char=\".\" align=\"char\"/></tr><tr><td align=\"left\">Household Income</td><td align=\"left\"/><td align=\"left\"/><td char=\".\" align=\"char\"/></tr><tr><td align=\"left\"> Q1</td><td align=\"left\">20,166 (24.6)</td><td align=\"left\">114 (23.5)</td><td char=\".\" align=\"char\">0.602</td></tr><tr><td align=\"left\"> Q2</td><td align=\"left\">20,792 (25.4)</td><td align=\"left\">127 (26.2)</td><td char=\".\" align=\"char\"/></tr><tr><td align=\"left\"> Q3</td><td align=\"left\">20,545 (25.1)</td><td align=\"left\">132 (27.2)</td><td char=\".\" align=\"char\"/></tr><tr><td align=\"left\"> Q4</td><td align=\"left\">20,383 (24.9)</td><td align=\"left\">112 (23.1)</td><td char=\".\" align=\"char\"/></tr><tr><td align=\"left\">Regions</td><td align=\"left\"/><td align=\"left\"/><td char=\".\" align=\"char\"/></tr><tr><td align=\"left\"> Northern</td><td align=\"left\">6822 (8.3)</td><td align=\"left\">47 (9.7)</td><td char=\".\" align=\"char\">0.519</td></tr><tr><td align=\"left\"> Stockholm-Gotland</td><td align=\"left\">19,185 (23.4)</td><td align=\"left\">99 (20.4)</td><td char=\".\" align=\"char\"/></tr><tr><td align=\"left\"> Uppsala-Örebro</td><td align=\"left\">17,143 (20.9)</td><td align=\"left\">102 (21.0)</td><td char=\".\" align=\"char\"/></tr><tr><td align=\"left\"> South</td><td align=\"left\">12,965 (15.8)</td><td align=\"left\">79 (16.3)</td><td char=\".\" align=\"char\"/></tr><tr><td align=\"left\"> Southeast</td><td align=\"left\">8803 (10.7)</td><td align=\"left\">48 (9.9)</td><td char=\".\" align=\"char\"/></tr><tr><td align=\"left\"> Western (Halland)</td><td align=\"left\">17,142 (20.9)</td><td align=\"left\">11 (22.8)</td><td char=\".\" align=\"char\"/></tr><tr><td align=\"left\">Clinical T stage</td><td align=\"left\"/><td align=\"left\"/><td char=\".\" align=\"char\"/></tr><tr><td align=\"left\"> T 0–1</td><td align=\"left\">54,155 (65.7)</td><td align=\"left\">263 (54.0)</td><td char=\".\" align=\"char\"><bold> &lt; 0.001</bold></td></tr><tr><td align=\"left\"> T 2–4</td><td align=\"left\">27,893 (33.9)</td><td align=\"left\">221 (45.4)</td><td char=\".\" align=\"char\"/></tr><tr><td align=\"left\">Clinical N stage</td><td align=\"left\"/><td align=\"left\"/><td char=\".\" align=\"char\"/></tr><tr><td align=\"left\"> cN + </td><td align=\"left\">10,246 (12.4)</td><td align=\"left\">108 (22.2)</td><td char=\".\" align=\"char\"><bold> &lt; 0.001</bold></td></tr><tr><td align=\"left\"> cN-</td><td align=\"left\">71,881 (87.2)</td><td align=\"left\">378 (77.6)</td><td char=\".\" align=\"char\"/></tr><tr><td align=\"left\">Histological grade</td><td align=\"left\"/><td align=\"left\"/><td char=\".\" align=\"char\"/></tr><tr><td align=\"left\"> Well differentiated (G1)</td><td align=\"left\">16,013 (19.4)</td><td align=\"left\">48 (9.9)</td><td char=\".\" align=\"char\"><bold> &lt; 0.001</bold></td></tr><tr><td align=\"left\"> Moderately differentiated (G2)</td><td align=\"left\">40,137 (48.7)</td><td align=\"left\">246 (50.5)</td><td char=\".\" align=\"char\"/></tr><tr><td align=\"left\"> Poorly differentiated (G3)</td><td align=\"left\">21,346 (25.9)</td><td align=\"left\">171 (35.1)</td><td char=\".\" align=\"char\"/></tr><tr><td align=\"left\">Subtype according to IHC</td><td align=\"left\"/><td align=\"left\"/><td char=\".\" align=\"char\"/></tr><tr><td align=\"left\"> Luminal</td><td align=\"left\">63,728 (77.3)</td><td align=\"left\">423 (86.9)</td><td char=\".\" align=\"char\"><bold> &lt; 0.001</bold></td></tr><tr><td align=\"left\"> Her2 positive</td><td align=\"left\">11,059 (13.4)</td><td align=\"left\">61 (12.5)</td><td char=\".\" align=\"char\"/></tr><tr><td align=\"left\"> TNBC</td><td align=\"left\">7614 (9.2)</td><td align=\"left\">3 (0.6)</td><td char=\".\" align=\"char\"/></tr><tr><td align=\"left\">Morphological subtype</td><td align=\"left\"/><td align=\"left\"/><td char=\".\" align=\"char\"/></tr><tr><td align=\"left\"> Ductal</td><td align=\"left\">61,414 (79.1)</td><td align=\"left\">429 (91.9)</td><td char=\".\" align=\"char\"><bold> &lt; 0.001</bold></td></tr><tr><td align=\"left\"> Lobular</td><td align=\"left\">10,719 (13.8)</td><td align=\"left\">7 (1.5)</td><td char=\".\" align=\"char\"/></tr><tr><td align=\"left\"> Other</td><td align=\"left\">5503 (7.1)</td><td align=\"left\">31 (7.0)</td><td char=\".\" align=\"char\"/></tr><tr><td align=\"left\">Breast surgery</td><td align=\"left\"/><td align=\"left\"/><td char=\".\" align=\"char\"/></tr><tr><td align=\"left\"> Breast conserving surgery</td><td align=\"left\">49,891 (60.5)</td><td align=\"left\">480 (98.6)</td><td char=\".\" align=\"char\"><bold> &lt; 0.001</bold></td></tr><tr><td align=\"left\"> Mastectomy</td><td align=\"left\">32,510 (39.5)</td><td align=\"left\">7 (1.5)</td><td char=\".\" align=\"char\"/></tr><tr><td align=\"left\">Axillary surgery</td><td align=\"left\"/><td align=\"left\"/><td char=\".\" align=\"char\"/></tr><tr><td align=\"left\"> Sentinel lymph node dissection (SLND)</td><td align=\"left\">55,566 (70.2)</td><td align=\"left\">265 (57.7)</td><td char=\".\" align=\"char\"><bold> &lt; 0.001</bold></td></tr><tr><td align=\"left\"> Axillary lymph node dissection (ALND)</td><td align=\"left\">12,253 (15.5)</td><td align=\"left\">113 (24.6)</td><td char=\".\" align=\"char\"/></tr><tr><td align=\"left\"> SLND =  &gt; ALND</td><td align=\"left\">11,363 (14.4)</td><td align=\"left\">81 (17.6)</td><td char=\".\" align=\"char\"/></tr><tr><td align=\"left\">Adjuvant chemotherapy</td><td align=\"left\"/><td align=\"left\"/><td char=\".\" align=\"char\"/></tr><tr><td align=\"left\"> Yes</td><td align=\"left\">23,029 (27.9)</td><td align=\"left\">163 (33.5)</td><td char=\".\" align=\"char\"><bold>0.007</bold></td></tr><tr><td align=\"left\"> No</td><td align=\"left\">59,372 (72.1)</td><td align=\"left\">324 (66.5)</td><td char=\".\" align=\"char\"/></tr><tr><td align=\"left\">Adjuvant radiοtherapy</td><td align=\"left\"/><td align=\"left\"/><td char=\".\" align=\"char\"/></tr><tr><td align=\"left\"> Yes</td><td align=\"left\">50,513 (61.3)</td><td align=\"left\">156 (32.0)</td><td char=\".\" align=\"char\"><bold> &lt; 0.001</bold></td></tr><tr><td align=\"left\"> No</td><td align=\"left\">31,888 (38.7)</td><td align=\"left\">331 (68.0)</td><td char=\".\" align=\"char\"/></tr><tr><td align=\"left\">Adjuvant endocrine therapy</td><td align=\"left\"/><td align=\"left\"/><td char=\".\" align=\"char\"/></tr><tr><td align=\"left\"> Yes</td><td align=\"left\">53,368 (64.8)</td><td align=\"left\">383 (78.6)</td><td char=\".\" align=\"char\"><bold> &lt; 0.001</bold></td></tr><tr><td align=\"left\"> No</td><td align=\"left\">29,033 (35.2)</td><td align=\"left\">104 (21.4)</td><td char=\".\" align=\"char\"/></tr><tr><td align=\"left\">Neoadjuvant chemotherapy</td><td align=\"left\"/><td align=\"left\"/><td char=\".\" align=\"char\"/></tr><tr><td align=\"left\"> Yes</td><td align=\"left\">6463 (7.8)</td><td align=\"left\">24 (4.9)</td><td char=\".\" align=\"char\"><bold>0.017</bold></td></tr><tr><td align=\"left\"> No</td><td align=\"left\">75,938 (92.2)</td><td align=\"left\">463 (95.1)</td><td char=\".\" align=\"char\"/></tr></tbody></table></table-wrap>", "<table-wrap id=\"Tab2\"><label>Table 2</label><caption><p>Summary of demographic and clinical variables as well as treatment modalities for female and male patients with breast cancer who received NAC (BCBaSe 3.0/NKBC, 2008–2019)</p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th align=\"left\">Factors</th><th align=\"left\">Female (N = 6463), n (%)</th><th align=\"left\">Male (N = 24), n (%)</th><th align=\"left\">p value</th></tr></thead><tbody><tr><td align=\"left\">Age in yrs, median (range)</td><td align=\"left\">54 (21–83)</td><td align=\"left\">64.5 (41–84)</td><td char=\".\" align=\"char\">0.054</td></tr><tr><td align=\"left\">Calendar year at diagnosis</td><td align=\"left\"/><td align=\"left\"/><td char=\".\" align=\"char\"/></tr><tr><td align=\"left\"> 2008–2011</td><td align=\"left\">866 (13.4)</td><td align=\"left\">3 (12.5)</td><td char=\".\" align=\"char\">0.462</td></tr><tr><td align=\"left\"> 2012–2015</td><td align=\"left\">1706 (26.4)</td><td align=\"left\">9 (37.5)</td><td char=\".\" align=\"char\"/></tr><tr><td align=\"left\"> 2016–2020</td><td align=\"left\">3891 (60.2)</td><td align=\"left\">12 (50.0)</td><td char=\".\" align=\"char\"/></tr><tr><td align=\"left\">Education Level</td><td align=\"left\">N = 6344</td><td align=\"left\">N = 23</td><td char=\".\" align=\"char\"><bold>0.001</bold></td></tr><tr><td align=\"left\"> Low ≦9 years</td><td align=\"left\">965 (15.2)</td><td align=\"left\">10 (43.5)</td><td char=\".\" align=\"char\"/></tr><tr><td align=\"left\"> Intermediate 10–12 years</td><td align=\"left\">2576 (40.6)</td><td align=\"left\">8 (34.8)</td><td char=\".\" align=\"char\"/></tr><tr><td align=\"left\"> High ≧13 years</td><td align=\"left\">2803 (44.2)</td><td align=\"left\">5 (21.7)</td><td char=\".\" align=\"char\"/></tr><tr><td align=\"left\">Household Income</td><td align=\"left\">N = 6401</td><td align=\"left\">N = 24</td><td char=\".\" align=\"char\">0.405</td></tr><tr><td align=\"left\"> Q4</td><td align=\"left\">2162 (33.8)</td><td align=\"left\">9 (37.5)</td><td char=\".\" align=\"char\"/></tr><tr><td align=\"left\"> Q3</td><td align=\"left\">1611 (25.2)</td><td align=\"left\">8 (33.3)</td><td char=\".\" align=\"char\"/></tr><tr><td align=\"left\"> Q2</td><td align=\"left\">1435 (22.4)</td><td align=\"left\">2 (8.3)</td><td char=\".\" align=\"char\"/></tr><tr><td align=\"left\"> Q1</td><td align=\"left\">1193 (18.6)</td><td align=\"left\">5 (20.8)</td><td char=\".\" align=\"char\"/></tr><tr><td align=\"left\">Regions</td><td align=\"left\">N = 6420</td><td align=\"left\">N = 24</td><td char=\".\" align=\"char\">0.715</td></tr><tr><td align=\"left\"> Northern</td><td align=\"left\">408 (6.4)</td><td align=\"left\">1 (4.2)</td><td char=\".\" align=\"char\"/></tr><tr><td align=\"left\"> Stockholm-Gotland</td><td align=\"left\">2476 (38.6)</td><td align=\"left\">10 (41.7)</td><td char=\".\" align=\"char\"/></tr><tr><td align=\"left\"> Uppsala-Örebro</td><td align=\"left\">794 (12.4)</td><td align=\"left\">1 (4.2)</td><td char=\".\" align=\"char\"/></tr><tr><td align=\"left\"> South</td><td align=\"left\">1224 (19.1)</td><td align=\"left\">7 (29.2)</td><td char=\".\" align=\"char\"/></tr><tr><td align=\"left\"> Southeast</td><td align=\"left\">615 (9.6)</td><td align=\"left\">2 (8.3)</td><td char=\".\" align=\"char\"/></tr><tr><td align=\"left\"> Western (Halland)</td><td align=\"left\">903 (14.1)</td><td align=\"left\">3 (12.5)</td><td char=\".\" align=\"char\"/></tr><tr><td align=\"left\">Clinical T stage</td><td align=\"left\">N = 6463</td><td align=\"left\">N = 24</td><td char=\".\" align=\"char\">0.951</td></tr><tr><td align=\"left\"> T 0–1</td><td align=\"left\">1066 (16.5)</td><td align=\"left\">4 (16.7)</td><td char=\".\" align=\"char\"/></tr><tr><td align=\"left\"> T 2–4</td><td align=\"left\">5370 (83.1)</td><td align=\"left\">20 (83.3)</td><td char=\".\" align=\"char\"/></tr><tr><td align=\"left\">Clinical N stage</td><td align=\"left\">N = 6463</td><td align=\"left\">N = 24</td><td char=\".\" align=\"char\">0.407</td></tr><tr><td align=\"left\"> cN + </td><td align=\"left\">3444 (53.3)</td><td align=\"left\">16 (66.7)</td><td char=\".\" align=\"char\"/></tr><tr><td align=\"left\"> cN-</td><td align=\"left\">2978 (46.1)</td><td align=\"left\">8 (33.3)</td><td char=\".\" align=\"char\"/></tr><tr><td align=\"left\">Histological grade</td><td align=\"left\">N = 2394</td><td align=\"left\">N = 11</td><td char=\".\" align=\"char\">0.054</td></tr><tr><td align=\"left\"> Well differentiated (G1)</td><td align=\"left\">266 (11.1)</td><td align=\"left\">1 (9.1)</td><td char=\".\" align=\"char\"/></tr><tr><td align=\"left\"> Moderately differentiated (G2)</td><td align=\"left\">1357 (56.7)</td><td align=\"left\">10 (90.9)</td><td char=\".\" align=\"char\"/></tr><tr><td align=\"left\"> Poorly differentiated (G3)</td><td align=\"left\">771 (32.2)</td><td align=\"left\">0 (0)</td><td char=\".\" align=\"char\"/></tr><tr><td align=\"left\">Subtype according to IHC</td><td align=\"left\">N = 6463</td><td align=\"left\">N = 24</td><td char=\".\" align=\"char\"><bold>0.001</bold></td></tr><tr><td align=\"left\"> Luminal</td><td align=\"left\">2942 (45.4)</td><td align=\"left\">20 (83.3)</td><td char=\".\" align=\"char\"/></tr><tr><td align=\"left\"> Her2 positive</td><td align=\"left\">2147 (33.2)</td><td align=\"left\">4 (16.7)</td><td char=\".\" align=\"char\"/></tr><tr><td align=\"left\"> TNBC</td><td align=\"left\">1374 (21.3)</td><td align=\"left\">0 (0.0)</td><td char=\".\" align=\"char\"/></tr><tr><td align=\"left\">Morphological</td><td align=\"left\">N = 2337</td><td align=\"left\">N = 11</td><td char=\".\" align=\"char\">0.349</td></tr><tr><td align=\"left\"> Ductal</td><td align=\"left\">1961 (83.9)</td><td align=\"left\">11 (100)</td><td char=\".\" align=\"char\"/></tr><tr><td align=\"left\"> Lobular</td><td align=\"left\">262 (11.2)</td><td align=\"left\">0 (0.0)</td><td char=\".\" align=\"char\"/></tr><tr><td align=\"left\"> Other</td><td align=\"left\">114 (4.9)</td><td align=\"left\">0 (0.0)</td><td char=\".\" align=\"char\"/></tr><tr><td align=\"left\">Breast surgery</td><td align=\"left\">N = 6463</td><td align=\"left\">N = 24</td><td char=\".\" align=\"char\"><bold>0.001</bold></td></tr><tr><td align=\"left\"> Breast conserving surgery</td><td align=\"left\">2132 (33)</td><td align=\"left\">0 (0.0)</td><td char=\".\" align=\"char\"/></tr><tr><td align=\"left\"> Mastectomy</td><td align=\"left\">4331 (67)</td><td align=\"left\">24 (100)</td><td char=\".\" align=\"char\"/></tr><tr><td align=\"left\">Axillary surgery</td><td align=\"left\">N = 6271</td><td align=\"left\">N = 23</td><td char=\".\" align=\"char\">0.756</td></tr><tr><td align=\"left\"> Sentinel lymph node dissection (SLND)</td><td align=\"left\">1575 (25.1)</td><td align=\"left\">5 (21.7)</td><td char=\".\" align=\"char\"/></tr><tr><td align=\"left\"> Axillary lymph node dissection (ALND)</td><td align=\"left\">3912 (62.4)</td><td align=\"left\">16 (69.6)</td><td char=\".\" align=\"char\"/></tr><tr><td align=\"left\"> SLND =  &gt; ALND</td><td align=\"left\">784 (12.5)</td><td align=\"left\">2 (8.7)</td><td char=\".\" align=\"char\"/></tr><tr><td align=\"left\">Type of chemotherapy</td><td align=\"left\"/><td align=\"left\"/><td char=\".\" align=\"char\"/></tr><tr><td align=\"left\"> Anthracycline- and taxane-based</td><td align=\"left\">4858 (75.2)</td><td align=\"left\">18 (75.0)</td><td char=\".\" align=\"char\">0.817</td></tr><tr><td align=\"left\"> Anthracycline-based</td><td align=\"left\">1004 (15.5)</td><td align=\"left\">3 (12.5)</td><td char=\".\" align=\"char\"/></tr><tr><td align=\"left\"> Taxane-based</td><td align=\"left\">601 (9.3)</td><td align=\"left\">3 (12.5)</td><td char=\".\" align=\"char\"/></tr><tr><td align=\"left\">Anti-HER2 treatment</td><td align=\"left\"/><td align=\"left\"/><td char=\".\" align=\"char\"/></tr><tr><td align=\"left\"> Trastuzumab</td><td align=\"left\">1917 (29.7)</td><td align=\"left\">4 (16.7)</td><td char=\".\" align=\"char\">0.101</td></tr><tr><td align=\"left\">Adjuvant radiotherapy</td><td align=\"left\">N = 6463</td><td align=\"left\">N = 24</td><td char=\".\" align=\"char\">0.629</td></tr><tr><td align=\"left\"> Yes</td><td align=\"left\">4854 (75.1)</td><td align=\"left\">17 (70.8)</td><td char=\".\" align=\"char\"/></tr><tr><td align=\"left\"> No</td><td align=\"left\">1609 (24.9)</td><td align=\"left\">7 (29.2)</td><td char=\".\" align=\"char\"/></tr><tr><td align=\"left\">Adjuvant endocrine therapy</td><td align=\"left\">N = 6463</td><td align=\"left\">N = 24</td><td char=\".\" align=\"char\"><bold> &lt; 0.001</bold></td></tr><tr><td align=\"left\"> Yes</td><td align=\"left\">3853 (55.4)</td><td align=\"left\">22 (91.7)</td><td char=\".\" align=\"char\"/></tr><tr><td align=\"left\"> No</td><td align=\"left\">2880 (44.6)</td><td align=\"left\">2 (8.3)</td><td char=\".\" align=\"char\"/></tr></tbody></table></table-wrap>", "<table-wrap id=\"Tab3\"><label>Table 3</label><caption><p>In utilization cohort: multivariable logistic regression analyzing the association between receipt of NAC and clinical factors among patients with breast cancer (BCBaSe 3.0/NKBC, 2008–2019)</p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th align=\"left\"/><th align=\"left\">OR</th><th align=\"left\">95% CI</th><th align=\"left\">95% CI</th><th align=\"left\">P-value</th></tr><tr><th align=\"left\"/><th align=\"left\"/><th align=\"left\">Low</th><th align=\"left\">High</th><th align=\"left\"/></tr></thead><tbody><tr><td align=\"left\">Female versus male</td><td align=\"left\">1.135</td><td align=\"left\">0.606</td><td align=\"left\">2.128</td><td char=\".\" align=\"char\">0.692</td></tr><tr><td align=\"left\">Age</td><td align=\"left\">0.967</td><td align=\"left\">0.963</td><td align=\"left\">0.970</td><td char=\".\" align=\"char\"><bold> &lt; 0.001</bold></td></tr><tr><td align=\"left\">Educational Level</td><td align=\"left\"/><td align=\"left\"/><td align=\"left\"/><td char=\".\" align=\"char\"/></tr><tr><td align=\"left\"> High (≧13 years)</td><td align=\"left\">Ref</td><td align=\"left\"/><td align=\"left\"/><td char=\".\" align=\"char\"/></tr><tr><td align=\"left\"> Intermediate (10–12 years)</td><td align=\"left\">0.732</td><td align=\"left\">0.628</td><td align=\"left\">0.853</td><td char=\".\" align=\"char\"><bold> &lt; 0.001</bold></td></tr><tr><td align=\"left\"> Low (≦9 years)</td><td align=\"left\">0.890</td><td align=\"left\">0.801</td><td align=\"left\">0.989</td><td char=\".\" align=\"char\"><bold>0.030</bold></td></tr><tr><td align=\"left\">Household Income</td><td align=\"left\"/><td align=\"left\"/><td align=\"left\"/><td char=\".\" align=\"char\"/></tr><tr><td align=\"left\"> Q4 (High)</td><td align=\"left\">Ref</td><td align=\"left\"/><td align=\"left\"/><td char=\".\" align=\"char\"/></tr><tr><td align=\"left\"> Q3</td><td align=\"left\">0.708</td><td align=\"left\">0.624</td><td align=\"left\">0.803</td><td char=\".\" align=\"char\"><bold> &lt; 0.001</bold></td></tr><tr><td align=\"left\"> Q2</td><td align=\"left\">0.676</td><td align=\"left\">0.591</td><td align=\"left\">0.773</td><td char=\".\" align=\"char\"><bold> &lt; 0.001</bold></td></tr><tr><td align=\"left\"> Q1 (Low)</td><td align=\"left\">0.585</td><td align=\"left\">0.504</td><td align=\"left\">0.680</td><td char=\".\" align=\"char\"><bold> &lt; 0.001</bold></td></tr><tr><td align=\"left\">Healtcare regions</td><td align=\"left\"/><td align=\"left\"/><td align=\"left\"/><td char=\".\" align=\"char\"/></tr><tr><td align=\"left\"> Northern</td><td align=\"left\">Ref</td><td align=\"left\"/><td align=\"left\"/><td char=\".\" align=\"char\"/></tr><tr><td align=\"left\"> Stockholm Gotland</td><td align=\"left\">2.707</td><td align=\"left\">2.197</td><td align=\"left\">3.335</td><td char=\".\" align=\"char\"><bold> &lt; 0.001</bold></td></tr><tr><td align=\"left\"> Uppsala-Örebro</td><td align=\"left\">0.826</td><td align=\"left\">0.657</td><td align=\"left\">1.038</td><td char=\".\" align=\"char\">0.101</td></tr><tr><td align=\"left\"> South</td><td align=\"left\">2.119</td><td align=\"left\">1.698</td><td align=\"left\">2.645</td><td char=\".\" align=\"char\"><bold> &lt; 0.001</bold></td></tr><tr><td align=\"left\"> Southeast</td><td align=\"left\">1.664</td><td align=\"left\">1.313</td><td align=\"left\">2.108</td><td char=\".\" align=\"char\"><bold> &lt; 0.001</bold></td></tr><tr><td align=\"left\"> Western</td><td align=\"left\">0.813</td><td align=\"left\">0.647</td><td align=\"left\">1.023</td><td char=\".\" align=\"char\">0.077</td></tr><tr><td align=\"left\">Clinical T stage</td><td align=\"left\"/><td align=\"left\"/><td align=\"left\"/><td char=\".\" align=\"char\"/></tr><tr><td align=\"left\"> T1-2</td><td align=\"left\">Ref</td><td align=\"left\"/><td align=\"left\"/><td char=\".\" align=\"char\"/></tr><tr><td align=\"left\"> T3-4</td><td align=\"left\">4.151</td><td align=\"left\">1.782</td><td align=\"left\">9.667</td><td char=\".\" align=\"char\"><bold> &lt; 0.001</bold></td></tr><tr><td align=\"left\">Clinical N stage</td><td align=\"left\"/><td align=\"left\"/><td align=\"left\"/><td char=\".\" align=\"char\"/></tr><tr><td align=\"left\"> cN-</td><td align=\"left\">Ref</td><td align=\"left\"/><td align=\"left\"/><td char=\".\" align=\"char\"/></tr><tr><td align=\"left\"> cN + </td><td align=\"left\">1.888</td><td align=\"left\">1.039</td><td align=\"left\">3.432</td><td char=\".\" align=\"char\"><bold>0.037</bold></td></tr><tr><td align=\"left\">Histological grade</td><td align=\"left\"/><td align=\"left\"/><td align=\"left\"/><td char=\".\" align=\"char\"/></tr><tr><td align=\"left\"> I</td><td align=\"left\">Ref</td><td align=\"left\"/><td align=\"left\"/><td char=\".\" align=\"char\"/></tr><tr><td align=\"left\"> II</td><td align=\"left\">4.305</td><td align=\"left\">3.587</td><td align=\"left\">5.167</td><td char=\".\" align=\"char\"><bold> &lt; 0.001</bold></td></tr><tr><td align=\"left\"> III</td><td align=\"left\">4.049</td><td align=\"left\">3.575</td><td align=\"left\">4.586</td><td char=\".\" align=\"char\"><bold> &lt; 0.001</bold></td></tr><tr><td align=\"left\">Subtype according to IHC</td><td align=\"left\"/><td align=\"left\"/><td align=\"left\"/><td char=\".\" align=\"char\"/></tr><tr><td align=\"left\"> Luminal</td><td align=\"left\">Ref</td><td align=\"left\"/><td align=\"left\"/><td char=\".\" align=\"char\"/></tr><tr><td align=\"left\"> HER2-positive</td><td align=\"left\">3.077</td><td align=\"left\">2.714</td><td align=\"left\">3.489</td><td char=\".\" align=\"char\"><bold> &lt; 0.001</bold></td></tr><tr><td align=\"left\"> Triple-negative</td><td align=\"left\">5.876</td><td align=\"left\">5.077</td><td align=\"left\">6.802</td><td char=\".\" align=\"char\"><bold> &lt; 0.001</bold></td></tr><tr><td align=\"left\">Morphological type</td><td align=\"left\"/><td align=\"left\"/><td align=\"left\"/><td char=\".\" align=\"char\"/></tr><tr><td align=\"left\"> Ductal</td><td align=\"left\">Ref</td><td align=\"left\"/><td align=\"left\"/><td char=\".\" align=\"char\"/></tr><tr><td align=\"left\"> Lobular</td><td align=\"left\">0.778</td><td align=\"left\">0.669</td><td align=\"left\">0.906</td><td char=\".\" align=\"char\"><bold> &lt; 0.001</bold></td></tr><tr><td align=\"left\"> Other</td><td align=\"left\">0.667</td><td align=\"left\">0.530</td><td align=\"left\">0.838</td><td char=\".\" align=\"char\"><bold> &lt; 0.001</bold></td></tr></tbody></table></table-wrap>", "<table-wrap id=\"Tab4\"><label>Table 4</label><caption><p>Multivariable logistic regression analyzing factors associated with complete pathologic response, among patients who received NAC for breast cancer (NKBC, 2008–2019)</p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th align=\"left\"/><th align=\"left\">OR</th><th align=\"left\">95% CI</th><th align=\"left\">95% CI</th><th align=\"left\">P-value</th></tr><tr><th align=\"left\"/><th align=\"left\"/><th align=\"left\">Low</th><th align=\"left\">High</th><th align=\"left\"/></tr></thead><tbody><tr><td align=\"left\">Female versus male</td><td align=\"left\">1.141</td><td align=\"left\">0.141</td><td align=\"left\">9.238</td><td char=\".\" align=\"char\">0.902</td></tr><tr><td align=\"left\">Age</td><td align=\"left\">0.989</td><td align=\"left\">0.981</td><td align=\"left\">0.997</td><td char=\".\" align=\"char\"><bold>0.010</bold></td></tr><tr><td align=\"left\">Clinical T stage</td><td align=\"left\"/><td align=\"left\"/><td align=\"left\"/><td char=\".\" align=\"char\"/></tr><tr><td align=\"left\"> T1-2</td><td align=\"left\">Ref</td><td align=\"left\"/><td align=\"left\"/><td char=\".\" align=\"char\"/></tr><tr><td align=\"left\"> T3-4</td><td align=\"left\">1.227</td><td align=\"left\">0.142</td><td align=\"left\">10.612</td><td char=\".\" align=\"char\">0.853</td></tr><tr><td align=\"left\">Clinical N stage</td><td align=\"left\"/><td align=\"left\"/><td align=\"left\"/><td char=\".\" align=\"char\"/></tr><tr><td align=\"left\"> cN− </td><td align=\"left\">Ref</td><td align=\"left\"/><td align=\"left\"/><td char=\".\" align=\"char\"/></tr><tr><td align=\"left\"> cN + </td><td align=\"left\">0.693</td><td align=\"left\">0.180</td><td align=\"left\">2.675</td><td char=\".\" align=\"char\">0.595</td></tr><tr><td align=\"left\">Histological grade</td><td align=\"left\"/><td align=\"left\"/><td align=\"left\"/><td char=\".\" align=\"char\"/></tr><tr><td align=\"left\"> I</td><td align=\"left\">Ref</td><td align=\"left\"/><td align=\"left\"/><td char=\".\" align=\"char\"/></tr><tr><td align=\"left\"> II</td><td align=\"left\">1.989</td><td align=\"left\">1.191</td><td align=\"left\">3.320</td><td char=\".\" align=\"char\"><bold>0.009</bold></td></tr><tr><td align=\"left\"> III</td><td align=\"left\">4.623</td><td align=\"left\">2.743</td><td align=\"left\">7.793</td><td char=\".\" align=\"char\"><bold> &lt; 0.001</bold></td></tr><tr><td align=\"left\">Subtype according to IHC</td><td align=\"left\"/><td align=\"left\"/><td align=\"left\"/><td char=\".\" align=\"char\"/></tr><tr><td align=\"left\"> Luminal</td><td align=\"left\">Ref</td><td align=\"left\"/><td align=\"left\"/><td char=\".\" align=\"char\"/></tr><tr><td align=\"left\"> HER2-positive</td><td align=\"left\">2.774</td><td align=\"left\">2.134</td><td align=\"left\">3.607</td><td char=\".\" align=\"char\"><bold> &lt; 0.001</bold></td></tr><tr><td align=\"left\"> Triple-negative</td><td align=\"left\">1.046</td><td align=\"left\">0.753</td><td align=\"left\">1.451</td><td char=\".\" align=\"char\">0.790</td></tr></tbody></table></table-wrap>" ]

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[ "<table-wrap-foot><p>Bold text indicates a statistically significant difference with a p-value less than 0.05</p></table-wrap-foot>", "<table-wrap-foot><p>Bold text indicates a statistically difference with a p-value less than 0.05</p></table-wrap-foot>", "<table-wrap-foot><p>The multivariate models were complete case analyses</p><p>Bold text indicates a statistically significant difference with a p-value less than 0.05</p></table-wrap-foot>", "<table-wrap-foot><p>The multivariate models were complete case analyses</p><p>Bold text indicates a statistically significant difference with a p-value less than 0.05</p></table-wrap-foot>", "<fn-group><fn><p><bold>Publisher's Note</bold></p><p>Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.</p></fn></fn-group>" ]

[ "<graphic xlink:href=\"10549_2023_7129_Fig1_HTML\" id=\"MO1\"/>" ]

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{ "acronym": [], "definition": [] }

[ "<title>Introduction</title>", "<p id=\"Par2\">Robotic-assisted surgery (RAS) has gained widespread diffusion over the last years, demonstrating the ability to overcome the technical limitations of conventional laparoscopy. Enhancements provided by robotic assistance include three-dimensional view and magnification, increased dexterity with 7-degrees of freedom of robotic instruments, tremor filtering, and improved surgeons’ ergonomics [##REF##14685095##1##, ##REF##21815795##2##]. Some major drawbacks must be considered before using RAS in children: anesthesia, placement of trocars, and technical difficulties related to small space [##REF##28889938##3##]. Nevertheless, RAS has been described as safe and feasible option for a wide range of surgical indications in children, including urological, oncological, and gastrointestinal pathologies [##UREF##0##4##–##REF##36865692##8##]. Several reports have investigated safety and feasibility of RAS in pediatric population, compared with different approaches (open or laparoscopic) [##REF##29530407##9##–##REF##31555861##12##].</p>", "<p id=\"Par3\">To the current state, the field of pediatric gynecology remains the least explored, with only few pediatric reports of application of RAS for gynecological indications [##REF##23158752##13##–##REF##15313068##16##].</p>", "<p id=\"Par4\">This descriptive, retrospective study aimed to report a multicenter experience regarding the application of RAS for gynecological indications in pediatric patients.</p>" ]

[ "<title>Materials and methods</title>", "<p id=\"Par5\">All children and adolescents up to 18 years of age, operated using RAS for gynecological indications in 4 different institutions over a 3-year period, were included. The exclusion criteria were patients over 18 years old as well as all gynecological surgical conditions not treated with RAS.</p>", "<p id=\"Par6\">The surgical centers were contacted via mail and those accepting to participate to the study, were required to fill a study form, with requested information, for each enrolled patient.</p>", "<p id=\"Par7\">Patient baseline, including age at time of surgery, weight, possible comorbidities, clinical presentation, pre-operative diagnosis, and side of pathology, were reported in the first section.</p>", "<p id=\"Par8\">Details of operative technique, such as type of procedure, number of robotic and/or accessory ports, use of sealing device, method for specimen extraction, were reported in the second section.</p>", "<p id=\"Par9\">Operative results, including robot docking time, total operative time, length of stay (LOS), requirement time of pain medication, complication rate, conversion rate, pathology, and follow-up results, were analyzed in the third section.</p>", "<p id=\"Par10\">All data were elaborated using the statistical software Microsoft Excel, Windows vers.11. Descriptive statistics were used to present findings, and quantitative variables were expressed as median (range) to report the data.</p>", "<p id=\"Par11\">The study received appropriate Institute Review Board (IRB) approval.</p>" ]

[ "<title>Results</title>", "<title>Patient baseline</title>", "<p id=\"Par12\">Twenty-three girls, with median age at surgery of 12.3 years (range 0.6–17.8) and median weight of 47.2 kg (range 9–73), received RAS for gynecological indication in the study period and were included. Associated comorbidities were reported in 5/23 (21.7%). Most patients (16/23, 69.5%) were symptomatic at time of diagnosis, with non-specific abdominal/pelvic pain being the most frequent presentation. Pre-operative work-up included abdominal ultrasonography (US), pelvic computed tomography (CT) and/or magnetic resonance imaging (MRI), and voiding cystourethrogram (VCUG) in selected cases. Serum tumor markers, such as beta human chorionic gonadotropin (β-HCG), alfa-fetoprotein (α-FP), Cancer Antigen 125 (Ca125), lactic dehydrogenase (LDH), and human epididymis secretory protein 4 (HE-4), were performed in all patients with adnexal mass. Pre-operative diagnosis was ovarian cyst (<italic>n</italic> = 4), ovarian “complex” mass (<italic>n</italic> = 12), fallopian tube lesion (<italic>n</italic> = 3), uterine cyst (<italic>n</italic> = 1), gonadal dysgenesis (<italic>n</italic> = 1), pelvic paravaginal mass (<italic>n</italic> = 1), cloaca malformation (<italic>n</italic> = 1), and high-confluence urogenital sinus (UGS) (<italic>n</italic> = 1). One patient (4.3%) presented concomitant ovarian “complex” mass and paratubal cyst.</p>", "<title>Installation and operative technique</title>", "<p id=\"Par13\">All RAS procedures were carried out using the da Vinci Xi Surgical System (Intuitive Surgical, Sunnyvale, CA, USA). The patient was placed supine on the operative table and appropriate age-sized Foley catheter was inserted using sterile precautions pre-operatively.</p>", "<p id=\"Par14\">Three robotic arms, one 12-mm with 12–8 mm reducer, for the 3D, 0-degree, robotic optic, and two 8-mm ports to accommodate the robotic instruments, were placed on the umbilical line in all procedures. A fourth 5-mm accessory port was also placed. The robot was finally docked over the patient’s feet. Robotic vessel sealer was adopted in all procedures. Indocyanine green (ICG) near-infrared fluorescence (NIRF) was adopted in ovarian mass to check the resection margins and guide intra-operative decision making and in paratubal lesion to check the vascular permeability of the fallopian tube following the removal of the lesion (Fig. ##FIG##0##1##). A 10-mm bag-retrieval, introduced through the umbilical port, was adopted for specimen extraction.</p>", "<p id=\"Par15\">Video ##MEDIA##0##1## reproduces the technique of robotic-assisted resection of ovarian mass using ICG-NIRF.</p>", "<p>\n</p>", "<title>Operative results</title>", "<p id=\"Par16\">The RAS procedures included: ovarian cystectomy (<italic>n</italic> = 10), salpingo-oophorectomy (<italic>n</italic> = 6), bilateral gonadectomy (<italic>n</italic> = 1), salpingectomy (<italic>n</italic> = 1), paratubal cyst excision (<italic>n</italic> = 1), Gartner cyst excision (<italic>n</italic> = 1), paravaginal ganglioneuroma resection (<italic>n</italic> = 1), fistula closure in UGS (<italic>n</italic> = 1), and vaginoplasty using ileal flap in cloaca malformation (<italic>n</italic> = 1). Median operative time was 144.9 min (range 64–360), and median docking time was 17.3 min (range 7–50). Conversion to open or laparoscopy was not necessary in any case. Median LOS was 2.1 days (range 1–7), and median analgesic requirement was 2.2 days (range 1–6). One patient (4.3%) needed redo-surgery for recurrent Gartner cyst (Clavien 3b).</p>", "<p id=\"Par17\">The histopathology confirmed diagnosis of ovarian serous cystadenoma (<italic>n</italic> = 2), ovarian functional follicular cyst (<italic>n</italic> = 2), mature cystic teratoma (<italic>n</italic> = 6), immature teratoma (<italic>n</italic> = 4) (Fig. ##FIG##1##2##), ovotestis (<italic>n</italic> = 2), streak gonads in Turner syndrome SRY + (<italic>n</italic> = 1), paratubal cystadenoma (<italic>n</italic> = 2), Gartner cyst (<italic>n</italic> = 1), and ganglioneuroblastoma (<italic>n</italic> = 1).</p>", "<p id=\"Par18\">The median length of follow-up was 2.2 years (range 0.5–4.5). No patients required adjuvant chemotherapy following surgery and none reported recurrence of tumoral pathology.</p>", "<p id=\"Par19\">All results are summarized in Table ##TAB##0##1##.</p>" ]

[ "<title>Discussion</title>", "<p id=\"Par20\">Despite the growing number of indications in pediatric urology, the application of RAS remains still limited in other fields of pediatric surgery. Analyzing the pediatric literature, very few reports on RAS application in gynecology are available, with limited case series or single-case observations [##REF##23158752##13##–##REF##36061048##15##].</p>", "<p id=\"Par21\">Our study collected the number of 4 pediatric surgery units with high volume robotic activity and 24 patients, operated over a 3-year period, were enrolled. Despite the small number of patients included, our preliminary results were promising and showed that RAS may be fully applicable even to gynecological indications in pediatric patients. Improved dexterity, coordination, and visualization were provided by robot assistance. The absence of intra- and post-operative complications also confirmed the safety and feasibility of this approach in children.</p>", "<p id=\"Par22\">Moreover, our study added new elements to the current knowledge. First, the previous reports have described benign ovarian pathology as main indication to RAS [##REF##23158752##13##–##REF##36061048##15##]. Our study introduced further undescribed indications, such as tubal, uterine, and vaginal malformations, and demonstrated the feasibility of complex reconstructive procedures of internal genitalia such as vaginoplasty using ileal flap using robotic approach.</p>", "<p id=\"Par23\">Based on our experience, we believe that the key-points for an optimal management of such pathology using RAS are correct trocar placement, use of sealing device and ICG-NIRF technology, use of endobag for specimen extraction, and teamwork. The most critical step is the placement of trocars, especially in children [##REF##24548088##17##]. Improper placement of the trocars would limit robotic manipulation in the abdominal cavity and increase the chances of instrument conflicts due to the small surface area of the abdominal wall of children and the relatively small space in the abdominal cavity. As described by Xie et al. [##REF##31559290##14##], we always adopted three robotic arms and a fourth accessory laparoscopic port for the bedside surgeon. We placed the robotic ports on the umbilical line to keep proper distance from the pelvic area and have enough working space for manipulation of giant masses or insertion of retrieval bag. Our standard operation order was to insert the trocar for the scope first and then insert the trocars for the robotic arms. Furthermore, if use of specimen retrieval bag is planned, our suggestion is to place 12-mm umbilical robotic port with 12–8 mm reducer to use the 8-mm robotic scope. At time of specimen extraction, the optic is moved to one working arm and the 10-mm retrieval bag is inserted into the abdominal cavity through the umbilical robotic port and the specimen extraction is finally done under direct vision.</p>", "<p id=\"Par24\">Use of robotic vessel sealer is very helpful to perform a bloodless dissection of anatomic structures or resection of giant tumors. In some indications, such as ovarian tumors, use of ICG-NIRF was very helpful to visualize the resection margins of the mass and help guide intra-operative decision between salpingo-oophorectomy and ovarian-sparing surgery. This technology required an intra-operative administration of ICG (0.5 mg/kg) via intravenous route and in a matter of 60 s, fluorescence appeared in the target organs, allowing to identify the resection margins and the vascularization of the mass [##REF##32626676##18##–##REF##34616696##20##]. Recently, ICG-NIRF was also adopted during removal of paratubal lesion, to check the vascular permeability of the fallopian tube following the resection of the lesion.</p>", "<p id=\"Par25\">Use of endobag is needed for extraction of resected tumors with high suspicion of malignancy. We suggest adopting large endobags (volume up to 1000 mL) and extract the specimen by enlarging the umbilical incision and, whenever possible, aspirating the liquid content of cystic masses before extraction, to avoid additional large Pfannestiel incision.</p>", "<p id=\"Par26\">Finally, the teamwork is essential to perform a smooth operation, shorten the learning curve for docking, and ultimately reduce total operative time and anesthetic times.</p>", "<p id=\"Par27\">Limitations of the presented study are the small number series, the limited follow-up period, and the multi-institutional participation, that made the data hardly comparable. The number of accumulated procedures in children is difficult to compare with that in adults. Thus, it was necessary collect the data of different pediatric surgery units to collect more conspicuous evidence.</p>" ]

[ "<title>Conclusion</title>", "<p id=\"Par28\">This preliminary experience showed that RAS is safe and feasible for surgical treatment of pediatric gynecological pathology, although no conclusive data are available to confirm its superiority over traditional laparoscopy. Case–control and comparative prospective studies will help to delineate better the advantages of this new technology as well as its optimal use in pediatrics. The primary focus for future studies should therefore be on quality management, optimization of patient outcomes for the largest number of patients, and surgical and team training.</p>" ]

[ "<p id=\"Par1\">Robotic-assisted surgery (RAS) is increasingly adopted in the pediatric population. This retrospective multicenter study aimed to report application of RAS for gynecological indications in pediatric patients. The medical records of all girls with gynecological pathology, operated in 4 different institutions over a 3-year period, were retrospectively collected. Robot docking time, total operative time, length of stay (LOS), requirement time of pain medication, complication rate, conversion rate, and pathology were analyzed. Twenty-three girls, with median age of 12.3 years (range 0.6–17.8) and median weight of 47.2 kg (range 9–73), received the following RAS procedures: ovarian cystectomy for ovarian cyst/mass (<italic>n</italic> = 10), salpingo-oophorectomy for ovarian complex mass (<italic>n</italic> = 6), bilateral gonadectomy for Turner syndrome SRY + (<italic>n</italic> = 1), salpingectomy for fallopian tube lesion (<italic>n</italic> = 1), paratubal cyst excision (<italic>n</italic> = 1), Gartner cyst excision (<italic>n</italic> = 1), paravaginal ganglioneuroma resection (<italic>n</italic> = 1), fistula closure in urogenital sinus (<italic>n</italic> = 1), and vaginoplasty using ileal flap in cloaca malformation (<italic>n</italic> = 1). Median operative time was 144.9 min (range 64–360), and median docking time was 17.3 min (range 7–50). Conversion to open or laparoscopy was not necessary in any case. Median LOS was 2.1 days (range 1–7), and median analgesic requirement was 2.2 days (range 1–6). One patient (4.3%) needed redo-surgery for recurrent Gartner cyst (Clavien 3b). This preliminary experience showed that RAS is safe and feasible for surgical treatment of gynecological pathology in pediatric patients, although no conclusive data are available to confirm its superiority over traditional laparoscopy. Randomized, prospective, comparative studies are needed to identify the gold standard approach for such indication.</p>", "<title>Supplementary Information</title>", "<p>The online version contains supplementary material available at 10.1007/s11701-023-01767-9.</p>", "<title>Keywords</title>", "<p>Open access funding provided by Università degli Studi di Napoli Federico II within the CRUI-CARE Agreement.</p>" ]

[]

[ "<title>Author contributions</title>", "<p>All authors contributed to the study conception and design. All authors performed material preparation, data collection, and data analysis. M.E. and C.E. wrote the first draft of the manuscript. G.E. and C.D.M. prepared Fig. ##FIG##0##1##. A.C. and M.E. prepared video ##MEDIA##0##1##. All authors read and approved the final manuscript.</p>", "<title>Funding</title>", "<p>Open access funding provided by Università degli Studi di Napoli Federico II within the CRUI-CARE Agreement. The authors declare that no funds, grants, or other supports were received during the preparation of this manuscript.</p>", "<title>Data availability</title>", "<p>All data will be available on request.</p>", "<title>Declarations</title>", "<title>Conflict of interest</title>", "<p id=\"Par29\">The authors have no relevant financial or non-financial interests to disclose.</p>", "<title>Ethics approval</title>", "<p id=\"Par30\">This is a retrospective study. No ethical approval is required.</p>", "<title>Consent to participate</title>", "<p id=\"Par31\">Written informed consent was obtained from all individual participants included in the study and their parents.</p>" ]

[ "<fig id=\"Fig1\"><label>Fig. 1</label><caption><p>After removal of giant paratubal cyst (<bold>a</bold>, <bold>b</bold>), ICG-guided fluorescence was helpful to check the vascular permeability of fallopian tube (<bold>c</bold>)</p></caption></fig>", "<fig id=\"Fig2\"><label>Fig. 2</label><caption><p>Pre-operative MRI and specimen of right ovarian immature teratoma</p></caption></fig>" ]

[ "<table-wrap id=\"Tab1\"><label>Table 1</label><caption><p>Outcomes of RAS in pediatric gynecological indications</p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th align=\"left\">Patient</th><th align=\"left\">Age (years)</th><th align=\"left\">Side</th><th align=\"left\">Comorbidity</th><th align=\"left\">Pre-operative diagnosis</th><th align=\"left\">Surgical procedure</th><th align=\"left\">OT (min)</th><th align=\"left\">LOS (days)</th><th align=\"left\">Post-operative complications</th><th align=\"left\">Pathology results</th></tr></thead><tbody><tr><td align=\"left\">1</td><td align=\"left\">10</td><td align=\"left\">Right</td><td align=\"left\">No</td><td align=\"left\">Ovarian mass</td><td align=\"left\">Salpingo-oophorectomy</td><td char=\".\" align=\"char\">85</td><td align=\"left\">1</td><td align=\"left\">0</td><td align=\"left\">Mature cystic teratoma</td></tr><tr><td align=\"left\">2</td><td align=\"left\">16</td><td align=\"left\">Left</td><td align=\"left\">No</td><td align=\"left\">Giant ovarian cyst (12 cm)</td><td align=\"left\">Ovarian cystectomy</td><td char=\".\" align=\"char\">72</td><td align=\"left\">1</td><td align=\"left\">0</td><td align=\"left\">Serous cystadenoma</td></tr><tr><td align=\"left\">3</td><td align=\"left\">17</td><td align=\"left\">Right</td><td align=\"left\">No</td><td align=\"left\">Giant ovarian cyst (10 cm)</td><td align=\"left\">Ovarian cystectomy</td><td char=\".\" align=\"char\">75</td><td char=\".\" align=\"left\">1</td><td align=\"left\">0</td><td align=\"left\">Serous cystadenoma</td></tr><tr><td align=\"left\">4</td><td align=\"left\">12</td><td align=\"left\">Right</td><td align=\"left\">Type I neurofibromatosis</td><td align=\"left\">Ovarian mass</td><td align=\"left\">Salpingo-oophorectomy</td><td char=\".\" align=\"char\">124</td><td char=\".\" align=\"left\">3</td><td align=\"left\">0</td><td align=\"left\">Immature teratoma</td></tr><tr><td align=\"left\">5</td><td align=\"left\">12</td><td align=\"left\">Right</td><td align=\"left\">Rheumatoid arthritis</td><td align=\"left\">Ovarian mass</td><td align=\"left\">Salpingo-oophorectomy</td><td char=\".\" align=\"char\">95</td><td char=\".\" align=\"left\">2</td><td align=\"left\">0</td><td align=\"left\">Immature teratoma</td></tr><tr><td align=\"left\">6</td><td align=\"left\">13</td><td align=\"left\">Left</td><td align=\"left\">No</td><td align=\"left\">Ovarian mass + paratubal cyst</td><td align=\"left\">Salpingo-oophorectomy</td><td char=\".\" align=\"char\">78</td><td align=\"left\">2</td><td align=\"left\">0</td><td align=\"left\">Immature teratoma + paratubal cyst</td></tr><tr><td align=\"left\">7</td><td align=\"left\">3</td><td align=\"left\">Bilateral</td><td align=\"left\">Turner syndrome SRY + </td><td align=\"left\">Gonadal dysgenesis</td><td align=\"left\">Bilateral gonadectomy</td><td char=\".\" align=\"char\">64</td><td char=\".\" align=\"left\">2</td><td align=\"left\">0</td><td align=\"left\">Streak gonads</td></tr><tr><td align=\"left\">8</td><td align=\"left\">13</td><td align=\"left\">Right</td><td align=\"left\">No</td><td align=\"left\">Fallopian tube lesion</td><td align=\"left\">Salpingectomy</td><td char=\".\" align=\"char\">96</td><td align=\"left\">2</td><td align=\"left\">0</td><td align=\"left\">Tubal cystadenoma</td></tr><tr><td align=\"left\">9</td><td align=\"left\">17</td><td align=\"left\">N/A</td><td align=\"left\">No</td><td align=\"left\">Uterine cyst</td><td align=\"left\">Excision</td><td char=\".\" align=\"char\">200</td><td char=\".\" align=\"left\">3</td><td align=\"left\">Cyst recurrence (Clavien 3b)</td><td align=\"left\">Gartner cyst</td></tr><tr><td align=\"left\">10</td><td align=\"left\">7</td><td align=\"left\">Right</td><td align=\"left\">No</td><td align=\"left\"><p>Ovarian</p><p>Mass</p></td><td align=\"left\">Ovarian cystectomy</td><td char=\".\" align=\"char\">155</td><td align=\"left\">2</td><td align=\"left\">0</td><td align=\"left\">Mature cystic teratoma</td></tr><tr><td align=\"left\">11</td><td align=\"left\">14</td><td align=\"left\">Left</td><td align=\"left\">No</td><td align=\"left\">Pelvic paravaginal mass</td><td align=\"left\">Excision</td><td char=\".\" align=\"char\">198</td><td char=\".\" align=\"left\">2</td><td align=\"left\">0</td><td align=\"left\">Ganglioneuroblastoma</td></tr><tr><td align=\"left\">12</td><td align=\"left\">10</td><td align=\"left\">N/A</td><td align=\"left\">No</td><td align=\"left\">Cloaca</td><td align=\"left\">Vaginoplasty using ileal flap</td><td char=\".\" align=\"char\">360</td><td char=\".\" align=\"left\">7</td><td align=\"left\">0</td><td align=\"left\">N/A</td></tr><tr><td align=\"left\">13</td><td align=\"left\">0.6</td><td align=\"left\">N/A</td><td align=\"left\">No</td><td align=\"left\">Urogenital sinus</td><td align=\"left\">Fistula closure</td><td char=\".\" align=\"char\">155</td><td char=\".\" align=\"left\">3</td><td align=\"left\">0</td><td align=\"left\">N/A</td></tr><tr><td align=\"left\">14</td><td align=\"left\">17</td><td align=\"left\">Right</td><td align=\"left\">DSD</td><td align=\"left\">Ovarian mass</td><td align=\"left\">Ovarian cystectomy</td><td char=\".\" align=\"char\">145</td><td align=\"left\">1</td><td align=\"left\">0</td><td align=\"left\">Ovotestis</td></tr><tr><td align=\"left\">15</td><td align=\"left\">13</td><td align=\"left\">Left</td><td align=\"left\">DSD</td><td align=\"left\">Ovarian mass</td><td align=\"left\">Salpingo-oophorectomy</td><td char=\".\" align=\"char\">120</td><td char=\".\" align=\"left\">1</td><td align=\"left\">0</td><td align=\"left\">Ovotestis</td></tr><tr><td align=\"left\">16</td><td align=\"left\">13</td><td align=\"left\">Right</td><td align=\"left\">Epilepsy, mild cystic fibrosis</td><td align=\"left\">Ovarian mass</td><td align=\"left\">Salpingo-oophorectomy</td><td char=\".\" align=\"char\">80</td><td char=\".\" align=\"left\">1</td><td align=\"left\">0</td><td align=\"left\">Immature teratoma</td></tr><tr><td align=\"left\">17</td><td align=\"left\">17.8</td><td align=\"left\">Right</td><td align=\"left\">Epilepsy, Hashimoto thyroiditis</td><td align=\"left\">Paratubal cyst</td><td align=\"left\">Paratubal cystectomy</td><td char=\".\" align=\"char\">115</td><td char=\".\" align=\"left\">2</td><td align=\"left\">0</td><td align=\"left\">Serous papillary cystadenoma</td></tr><tr><td align=\"left\">18</td><td align=\"left\">13.2</td><td align=\"left\">Left</td><td align=\"left\">No</td><td align=\"left\">Ovarian mass</td><td align=\"left\">Ovarian cystectomy</td><td char=\".\" align=\"char\">228</td><td align=\"left\">3</td><td align=\"left\">0</td><td align=\"left\">Mature cystic teratoma</td></tr><tr><td align=\"left\">19</td><td align=\"left\">12.9</td><td align=\"left\">Right</td><td align=\"left\">No</td><td align=\"left\">Ovarian mass</td><td align=\"left\">Ovarian cystectomy</td><td char=\".\" align=\"char\">218</td><td align=\"left\">2.1</td><td align=\"left\">0</td><td align=\"left\">Mature cystic teratoma</td></tr><tr><td align=\"left\">20</td><td align=\"left\">8</td><td align=\"left\">Right</td><td align=\"left\">No</td><td align=\"left\">Ovarian mass</td><td align=\"left\">Ovarian cystectomy</td><td char=\".\" align=\"char\">185</td><td align=\"left\">2.7</td><td align=\"left\">0</td><td align=\"left\">Mature cystic teratoma</td></tr><tr><td align=\"left\">21</td><td align=\"left\">14.8</td><td align=\"left\">Right</td><td align=\"left\">No</td><td align=\"left\">Ovarian mass</td><td align=\"left\">Ovarian cystectomy</td><td char=\".\" align=\"char\">198</td><td align=\"left\">2.3</td><td align=\"left\">0</td><td align=\"left\">Mature cystic teratoma</td></tr><tr><td align=\"left\">22</td><td align=\"left\">14.7</td><td align=\"left\">Bilateral</td><td align=\"left\">No</td><td align=\"left\">Ovarian cyst (6 cm)</td><td align=\"left\">Ovarian cystectomy</td><td char=\".\" align=\"char\">155</td><td align=\"left\">1.5</td><td align=\"left\">0</td><td align=\"left\">Functional follicular cyst</td></tr><tr><td align=\"left\">23</td><td align=\"left\">15</td><td align=\"left\">Left</td><td align=\"left\">No</td><td align=\"left\"><p>Ovarian cyst</p><p>(7 cm)</p></td><td align=\"left\">Ovarian cystectomy</td><td char=\".\" align=\"char\">132</td><td align=\"left\">1</td><td align=\"left\">0</td><td align=\"left\">Functional follicular cyst</td></tr></tbody></table></table-wrap>" ]

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[ "<table-wrap-foot><p><italic>RAS</italic>  robotic-assisted surgery, <italic>OT</italic>  operative time, <italic>LOS</italic> length of stay, <italic>DSD</italic> disorder of sex development</p></table-wrap-foot>", "<fn-group><fn><p><bold>Publisher's Note</bold></p><p>Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.</p></fn></fn-group>" ]

[ "<graphic xlink:href=\"11701_2023_1767_Fig1_HTML\" id=\"MO1\"/>", "<graphic xlink:href=\"11701_2023_1767_Fig2_HTML\" id=\"MO2\"/>" ]

[ "<media xlink:href=\"11701_2023_1767_MOESM1_ESM.mp4\" id=\"MOESM1\"><caption><p>Supplementary file1 (MP4 46006KB)</p></caption></media>" ]

{ "acronym": [], "definition": [] }

[ "<title>Introduction</title>", "<p id=\"Par2\">In the DSM-5, a “with Limited Prosocial Emotions” specifier was added to the Conduct Disorder diagnostic criteria to differentiate those children and adolescents who also display a callous absence of empathy and remorse, shallow or narrow affect, and deficient concern regarding performance in activities (American Psychiatric Association, ##UREF##0##2013##). This addition was based on research extending the affective dimension of the adult psychopathy personality construct downward to advance developmental theories (Frick, ##REF##35878505##2022##; Frick et al., ##REF##24117854##2014##). Greater knowledge on how these traits manifest earlier in development, when they may be more malleable, can inform clinical prevention and intervention efforts. In research settings, this specifier is commonly referred to as callous-unemotional (CU) traits and, like psychopathy, these traits are associated with several unique behavioral, emotional, and cognitive correlates relative to other externalizing and antisocial constructs (Frick et al., ##REF##24117854##2014##). However, empirical work on CU traits has relied on panel (or cross-sectional) designs that primarily investigate within-person change. Such panel designs do not sufficiently assess within-person fluctuations and have comparatively low ecological validity–two important limitations of this line of research that can be circumvented through intensive longitudinal methodologies.</p>", "<title>Correlates of Callous-unemotional Traits</title>", "<p id=\"Par3\">Studies have pinpointed several distinguishing correlates of CU traits. Individuals with conduct problems and normative CU traits are often characterized by a defiant or difficult temperament with heightened levels of emotion dysregulation, negative affect, and impulsive aggression to provocation (Frick &amp; Viding, ##REF##19825260##2009##). In contrast, those with elevated conduct problems <italic>and</italic> CU traits do not tend to show the same types of emotional reactivity and are instead discerned by a fearless disposition reflected as emotional hypoactivity, reduced sensitivity to punishment, and often greater aggression severity (Fanti et al., ##REF##25916948##2016##; McMahon et al., ##REF##20939651##2010##). Despite the name, CU traits do not always confer risk for blunted emotional responsivity. For example, although differentiated on fear processing, one meta-analytic review (<italic>k</italic> = 20) found no differences across CU traits and control groups in processing angry, disgusted, or happy cues (Marsh &amp; Blair, ##REF##17915324##2008##). In addition to dysfunctional parenting practices, it is this fearlessness that may explain why individuals with CU traits have been less successfully served by treatment efforts (Hawes et al., ##REF##24748077##2014a##; McMahon et al., ##REF##33787407##2021##). Distinctions have also been made regarding the structure of CU traits, with studies having identified three (callousness, uncaring, unemotional; Ray &amp; Frick, ##REF##30142284##2020##) or two (callousness, uncaring; Hawes et al., ##REF##24188153##2014b##; Zheng et al., ##REF##34928648##2021##) dimensions. Callousness tends to capture deficient empathy and guilt, uncaring reflects an aloof attitude regarding others, and unemotionality is defined as impoverished emotional experiences. Regardless of the structure, CU traits discriminates an etiologically and clinically atypical group (Frick et al., ##REF##24117854##2014##).</p>", "<title>Traits and Symptoms in Adolescence</title>", "<p id=\"Par4\">Adolescence is a sensitive developmental period marked by profound cognitive, neurobiological, and interpersonal change (Dahl et al., ##REF##29469094##2018##). In a large meta-analysis including 192 epidemiological studies, Solmi et al. (##REF##34079068##2022##) found that the peak age of onset of any mental disorder was 14.5 years. Therefore, adolescence represents a prime time for understanding vulnerabilities and the development of psychopathological symptoms. Personality and psychopathology are typically considered distinct domains sometimes differentiated based on temporal stability. Specifically, some researchers have suggested that personality traits reflect long-term dispositions and that psychopathological symptoms are state-like instances of an experience or behavior (DeYoung et al., ##REF##32978977##2022##). In childhood and adolescence, however, personality traits may also change over time (Soto et al., ##REF##21171787##2011##). Indeed, some research has found little differences between mean-level change of several personality facets and internalizing symptoms across five 9-month follow-ups in a sample of female adolescents (Goldstein et al., ##REF##35241862##2022##). Other evidence has shown that personality pathology tends to decline across adolescence (Álvarez-Tomás et al., ##REF##30599336##2019##). In the CU traits literature, many panel studies have examined the long-term stability of CU traits with one review suggesting that CU traits show substantial rank-order stability and modest mean-level stability during childhood, adolescence, and into early adulthood (Frick et al., ##REF##24117854##2014##). However, what cannot be ascertained from this work is whether adolescents with CU traits are consistent in their manifestation of CU traits. In other words, do CU traits fluctuate day-to-day (or moment-by-moment)?</p>", "<title>Intensive Longitudinal Methods</title>", "<p id=\"Par5\">Innovations in technology (e.g., smart phones and wearables) have generated a surge in studies investigating dynamic processes on micro timescales using a range of intensive longitudinal methods (for recent reviews, see Russell &amp; Gajos, ##REF##31997358##2020##; Urben et al., ##UREF##10##2022##). Broadly, intensive longitudinal methods, such as daily diary, experience sampling, ecological momentary assessment, and other ambulatory assessment designs, are characterized by many measurement occasions over a brief observation window. These methods focus on within-person variability, such that the measured variables may show deviations or fluctuations relative to the individual mean at specific observation occasions, but the individual mean is relatively stable over time compared to developmental growth or decrease in conventional longitudinal designs. Major advantages of intensive longitudinal methods also include the capability to assess real-time experiences and memory compared to traditional longitudinal designs, which can result in retrospective and recall biases (Russell &amp; Gajos, ##REF##31997358##2020##). These methods also enhance ecological validity, such that assessments are conducted when participants are in real-world settings relative to artificial laboratory experiments. Hence, intensive longitudinal data provide a high level of temporal granularity and inform our understanding of real-life short-term within-person dynamics as opposed to long-term developmental change.</p>", "<title>Testing Personality and Psychopathology with Intensive Longitudinal Methods</title>", "<p id=\"Par6\">Studies are beginning to apply intensive longitudinal designs to elucidate the dynamics of personality and psychopathology. For example, across three samples of adults, Edershile and Wright (##REF##33090821##2021##) tested whether narcissism reflects a dynamic process and found variability in narcissistic states at a moment-to-moment level. Among 91 adults with a personality disorder, daily assessments of personality pathology including negative affect, detachment, impulsivity, and hostility predicted clinical characterizations of personality disorders and daily stress levels (Dotterer et al., ##REF##31597579##2019##). In a sample of adolescent girls, daily negative affect, detachment, disinhibition, and psychoticism were associated with daily socio-affective processes (Kaurin et al., ##UREF##3##2022##). Such studies enhanced our understanding of state-level fluctuations in personality and psychopathology. Yet, no research to date has investigated whether CU traits also vary day-to-day or moment-to-moment. Only two intensive longitudinal studies have included a measure of CU traits using justice-involved adolescent samples; however, both studies only assessed CU traits as a total score at a single timepoint (De Ridder et al., ##UREF##2##2016##; Suter et al., ##UREF##8##2017##). Another study examined callousness in the context of interpersonal antagonism in samples of undergraduates, community adults, and psychiatric outpatients, but, similarly, callousness was only assessed at one timepoint as a predictor (Vize et al., ##REF##35402088##2022##). Across these studies, higher levels of CU traits were associated with higher levels of perceived angry affect and self-reported antisocial behavior (De Ridder et al., ##UREF##2##2016##; Suter et al., ##UREF##8##2017##), and callousness was negatively associated with positive affect and empathy (Vize et al., ##REF##35402088##2022##).</p>", "<p id=\"Par7\">In line with other personality constructs (Wright &amp; Kaurin, ##REF##32375147##2020##), much of our current understanding of CU traits is founded on structural or trait-based models (Frick et al., ##REF##24117854##2014##). Research on dynamic modeling at micro timescales may help further inform functional narratives. As the affective domain of psychopathy, CU traits may demonstrate meaningful within-person fluctuations. Notably, the links between CU traits and conduct problems at the daily within-person level could possibly be instantiated by their intermediate and differential links involving daily positive and negative affect. Although most research places negatively valenced emotion processes at the center of personality and psychopathology, some research has implicated positive affect in the development of aggressive behaviors (Toro et al., ##REF##32952964##2020##). In addition, affect and emotion are best conceptualized as dynamic processes underpinning a range of psychopathological outcomes (Houben et al., ##REF##25822133##2015##). Other theoretical accounts suggest that certain forms of personality are derived and maintained by state-level fluctuations of the traits themselves (e.g., grandiosity and vulnerability in narcissism; Edershile &amp; Wright, ##REF##33090821##2021##). It may be the case that daily fluctuations in callousness and uncaring serves to perpetuate CU traits and associated conduct problems. Overall, state-level personality may be more malleable than traits, and thus, studying phenomena at the state-level can illuminate more effective treatment targets for this typically treatment-resistant group.</p>", "<title>The Present Study</title>", "<p id=\"Par8\">There is a lack of knowledge on CU traits on a micro timescale. In this exploratory work with data collected from a sample of adolescents via an intensive 30-day daily diary design, our aims were twofold. First, we aimed to explore whether CU traits (at the item level) demonstrated daily within-person fluctuations. In addition to the potential novel knowledge gained from investigating CU traits using intensive longitudinal designs, examining CU traits at the item level is congruent with recent research initiatives (e.g., Hierarchical Taxonomy of Psychopathology; Kotov et al., ##REF##33577350##2021##) that emphasize fine-grained trait or symptom level analyses to advance mental health science. Second, we aimed to investigate whether daily CU traits (including callousness and uncaring) were associated with daily positive and negative affect, as well as emotional and conduct problems. To address these aims, we applied dynamic structural equation modeling (DSEM) to parse both <italic>within-person</italic> state-level fluctuations including autoregressive and cross-lagged associations (i.e., whether participants’ level at time <italic>t</italic> was significantly predicted by their level of the same or different construct, respectively, at time <italic>t</italic>-1) from stable <italic>between-person</italic> trait-like differences (Asparouhov et al., ##UREF##1##2018##; Hamaker et al., ##REF##29624092##2018##). Given the exploratory nature of this study, we pose no specific a priori hypotheses.</p>" ]

[ "<title>Method</title>", "<title>Participants and Procedure</title>", "<p id=\"Par9\">Adolescents living in a western Canadian province were recruited between April 2019 and October 2020 to participate in an online study examining their psychological and behavioral adjustment (see Xu &amp; Zheng, ##REF##35583795##2022##). A total of 99 participants (12–17 years old, <italic>M</italic> = 14.60, <italic>SD</italic> = 1.76, 55.8% female, 51.5% white, 23.2% Asian, 8.1% multiracial, 4.0% Hispanic/Latinx, 2.0% Black, 6.1% Other) completed a baseline survey and at least one day of the 30-day daily diary surveys (2,108 total observations, <italic>M</italic> = 21.72 days, <italic>Range</italic> = 1–30, <italic>SD</italic> = 7.80). Among the adolescents, 81.1% reported living with both biological parents, 17.8% reported living with one biological parent, and 1.1% reported living with someone other than a biological parent. Most participants’ parents reported a personal annual income within the average (CAD $65,700) and median ($51,600) personal total incomes for groups of 25–54-year-olds in this Canadian province (17.2% had a personal annual income below $35,000, 9.1% were between $35,000–$45,000, 12.1% were between $45,000–$55,000, 17.2% were between $55,000–$65,000, and 38.4% had an annual income above $65,000).</p>", "<p id=\"Par10\">The research ethics committee at the University of Alberta approved the procedure and instruments for this study. Survey instruments were developed and administered via email through RedCap. Participants were recruited through newsletters, social media, and flyers posted or circulated in the western Canadian province. All adolescents were eligible for inclusion in the study. Interested participants were asked to contact the research team and were provided with information for the study. Participants received an online baseline survey (~45 min to complete) following the obtainment of assent online. Participants’ parents provided consent online for their children to participate in this study. Daily surveys (~10 min to complete) were sent out five days after completion of the baseline survey for 30 consecutive days. The daily survey was sent out at 5 pm each day and adolescents were asked to fill out the survey before going to sleep that night. Participants received a $45 e-gift card of their choosing as compensation for their participation in the baseline and daily surveys.</p>", "<title>Measures</title>", "<title>Callous-unemotional Traits</title>", "<p id=\"Par11\">Daily callous-unemotional traits were assessed with a shortened 12-item version of the 24-item Inventory of Callous Unemotional Traits (ICU; Hawes et al., ##REF##24188153##2014b##; Zheng et al., ##REF##34928648##2021##; see Table ##TAB##0##1##). Each item is scored on a 4-point Likert scale (0 = <italic>not at all true</italic>, 1 = <italic>somewhat true</italic>, 2 = <italic>mostly true</italic>, 3 = <italic>definitely true</italic>) in response to the stem “indicate how well the following statements described you today.” This ICU scale includes two subscales: callousness (6 items; ordinal ω_w = 0.48, ordinal ω_b = 0.91) and uncaring (5 items; ordinal ω_w = 0.71, ordinal ω_b = 0.91). Items that needed to be reverse-coded were done so prior to the analysis. Multilevel Confirmatory Factor Analyses (MLCFA) with the weighted least squares means and variances adjusted estimator demonstrated that the 2-factor model provided satisfactory and better model fit at within- and between-levels (CFI = 0.97, TLI = 0.97, RMSEA = 0.02, SRMR_within(w) = 0.08, SRMR_between(b) = 0.04), compared to the 1-factor model at both levels (CFI = 0.87, TLI = 0.83, RMSEA = 0.04, SRMR_w = 0.15, SRMR_b = 0.07). The estimated averaged standardized factor loading for the callousness factor was 0.60 and 0.84 at within- and between-levels, and for the uncaring factor was 0.70 and 0.80 at within- and between-levels. MLCFA using the maximum likelihood estimator with robust standard errors (MLR) demonstrated that the 2-factor model at within- and between-levels provided better model fit (Δ scaling correction = 3.68, Δχ² = 163.03, Δdfs = 2.00, <italic>p</italic> &lt; 0.001) compared to the 1-factor model at both levels. For the first aim, all 12-items were used. However, in line with several studies that have found that item 3 (<italic>does not show emotions</italic>), the only item left from the original unemotional subscale, exhibits low factor loading (Colins et al., ##REF##26493393##2016##; Zheng et al., ##REF##34928648##2021##), this item was not included in calculations of subscales for the second aim.</p>", "<title>Positive and Negative Affect</title>", "<p id=\"Par12\">Positive (5 items; e.g., <italic>active</italic>, <italic>attentive</italic>) and negative (5 items; e.g., <italic>ashamed</italic>, <italic>nervous</italic>) affect were assessed daily (i.e., “indicate to what extent you have felt the way as described by each of the following words today”) with the short-form Positive and Negative Affect Schedule (PANAS-SF; Cooke et al., ##REF##36174167##2022##; Thompson, ##UREF##9##2007##) on a 5-point Likert scale (1 = <italic>very slightly or not at all</italic> 2 = <italic>a little</italic>, 3 = <italic>moderately</italic>, 4 = <italic>quite a bit</italic>, 5 = <italic>extremely</italic>). MLCFA model fit for the 2-factor model are as follows: CFI = 0.91, TLI = 0.88, RMSEA = 0.04, SRMR_w = 0.13, SRMR_b = 0.14. The estimated averaged standardized factor loading for the positive affect factor was 0.58 and 0.71 at within- and between-levels, and for the negative affect factor was 0.70 and 0.90 at within- and between-levels. Items were averaged within days with higher scores indicative of higher levels of positive (ω_w = 0.63, ω_b = 0.88, intra-class correlation [ICC] = 0.71) and negative (ω_w = 0.74, ω_b = 0.94, ICC = 0.61) affect.</p>", "<title>Emotional and Conduct Problems</title>", "<p id=\"Par13\">Participants reported their daily emotional and conduct problems using 5 items each from the emotional problems (e.g., <italic>I worry a lot</italic>) and conduct problems (e.g., <italic>I take things that are not mine from home, school, or elsewhere</italic>) subscales of the Strengths and Difficulties Questionnaire (Goodman et al., ##REF##9826298##1998##). Items were rated on a 3-point Likert scale (0 = <italic>not true</italic>, 1 = <italic>somewhat true</italic>, 2 = <italic>certainly true</italic>) and averaged within days. MLCFA model fit for the 2-factor model are as follows: CFI = 0.98, TLI = 0.98, RMSEA = 0.01, SRMR_w = 0.10, SRMR_b = 0.05. The estimated averaged standardized factor loading for the emotional problems factor was 0.57 and 0.88 at within- and between-levels, and for the conduct problems factor was 0.48 and 0.84 at within- and between-levels. Higher scores represented more emotional (ordinal ω_w = 0.70, ordinal ω_b = 0.94, ICC = 0.76) and conduct (ordinal ω_w = 0.63, ordinal ω_b = 0.92, ICC = 0.60) problems that day (i.e., “mark how the following items described you on the basis of how things have been for you today”).</p>", "<title>Analytic Approach</title>", "<p id=\"Par14\">In M<italic>plus</italic> version 8.6 (Muthén &amp; Muthén, ##UREF##5##2021##), DSEM (Asparouhov et al., ##UREF##1##2018##) was conducted to examine both research aims (see Fig. ##FIG##0##1## for DSEM schematic of aim 2). Combining time-series and structural equation modeling, this multilevel modeling approach tests temporal relations between measured variables by decomposing the data into within-person (Level 1) and between-person (Level 2) components (Asparouhov et al., ##UREF##1##2018##; Hamaker et al., ##REF##29624092##2018##). At the within-person level, <italic>inertia</italic>, otherwise known as autoregression or carryover, is an estimate of the extent to which scores return to equilibrium (or persist) over measurement occasions, with higher scores indicating greater resistance to change (Kuppens et al., ##REF##20501521##2010##). Regarding cross-lagged associations, <italic>augmentation</italic> describes positive cross-lagged associations, such that an increase in the level of one construct predicts an increase in another construct at a successive time point (Pe &amp; Kuppens, ##REF##22642355##2012##). By contrast, <italic>blunting</italic> characterizes negative cross-lagged associations, in which an increase in the level of one construct predicts a decrease in another construct at the next time point. In other words, these within-person components assess whether participants’ level at time <italic>t</italic> were significantly predicted by their level of the same or different construct, respectively, at time <italic>t</italic>-1. Contemporaneously, <italic>covariation</italic> specifies whether constructs at the same time point are associated with one another (Krone et al., ##REF##29265839##2018##). Accordingly, at the within-person level (person-level centered), both autoregressive and cross-lagged parameters are estimated. Covariances among residuals between constructs within the same day after controlling for the previous day’s effect (i.e., autoregression and cross-lagged effects) were also estimated. At the between-person level, DSEM models means and (co)variances to examine between-person differences aggregated over the 30-day survey period (i.e., random intercepts). Bayesian Markov Chain Monte Carlo (MCMC) is used for estimation resulting in an entire distribution of possible values for each unknown parameter. Models were estimated using default priors and convergence was established by examining several diagnostic criteria including the Potential Scale Reduction (PSR) statistic, autocorrelation plots, and trace plots. Significance is determined by whether the 95% credible interval (CI) contains value 0 for each parameter. M<italic>plus</italic> code and output are supplied on the Open Science Framework: <ext-link ext-link-type=\"uri\" xlink:href=\"https://osf.io/n2rdf/?view_only=357ec650bfac4a63905e7ed34efd9255\">https://osf.io/n2rdf/?view_only=357ec650bfac4a63905e7ed34efd9255</ext-link>.</p>" ]

[ "<title>Results</title>", "<title>Callous-unemotional Traits at the Item Level</title>", "<p id=\"Par15\">Descriptive statistics, ICCs, and within-level and between-level CU item correlations are shown in ##SUPPL##0##Supplementary Table S1##. ICC is the proportion of the total variance that is accounted for by stable or trait-like between-person differences plus measurement error (Hamaker et al., ##REF##29624092##2018##). Accordingly, lower ICCs would indicate more within-person fluctuations. At the within-level, almost all items were significantly positively correlated with each other. At the between-level, items 2 (<italic>feels bad or guilty when I have done something wrong</italic>) and 3 (<italic>does not show emotions</italic>) showed few significant associations with other items and the highest ICC values (0.77 and 0.78, respectively).</p>", "<p id=\"Par16\">After controlling for the previous day’s autoregression and cross-lagged effects, item-level within-day residual correlations are shown in Table ##TAB##1##2##. Similar to the within-level item correlations presented above, the majority of items were still positively correlated with each other (although magnitudes were smaller) after controlling for the previous day’s effect.</p>", "<p id=\"Par17\">Significant item-level autoregressive effects are shown in Fig. ##FIG##1##2## (as indicated by curved arrows) and all item-level autoregressive estimates are described in ##SUPPL##0##Supplementary Table S2##. Except for items 6 (<italic>does not care about doing things well</italic>) and 10 (<italic>shows no remorse when I have done something wrong</italic>), all items showed significant autoregressive effects (βs = 0.08–0.23; 95% CI [0.02–0.18, 0.13–0.28]), suggesting that adolescents experiencing high levels of CU traits on the previous day were more likely to experience high levels the next day compared to their average level. More specifically, items 1 (<italic>does not care who I hurt to get what I want</italic>; β = 0.08; 95% CI [0.02, 0.14]), 4 (<italic>concerned about the feelings of others</italic>; β = 0.08; 95% CI [0.02, 0.15]), 8 (<italic>apologizes to persons I have hurt</italic>; β = 0.08; 95% CI [0.02, 0.14]), and 11 (<italic>feelings of others are unimportant</italic>; β = 0.08; 95% CI [0.02, 0.13]) showed the lowest levels of inertia and item 3 (<italic>does not show emotions</italic>; β = 0.23; 95% CI [0.18, 0.28]) revealed the largest inertia.</p>", "<p id=\"Par18\">Significant item-level cross-lagged effects are shown in Fig. ##FIG##1##2## (as indicated by straight arrows) and all item-level cross-lagged estimates are described in ##SUPPL##0##Supplementary Table S3##. All items, except for item 10, demonstrated cross-lagged effects with at least one other item. However, it is notable that most cross-lagged associations involved only four items as outcomes: items 2 (<italic>feels bad or guilty when I have done something wrong</italic>; βs = -0.07–0.06 95% CIs [-0.12–0.01, -0.02–0.12]), 7 (<italic>seems very cold and uncaring;</italic> βs = 0.06–0.10; 95% CIs [0.01–0.05, 0.11–0.14]), 9 (<italic>tries not to hurt others’ feelings</italic>; βs = -0.06–0.10; 95% CIs [-0.12–0.05, -0.00–0.14]), and 12 (<italic>does things to make others feel good</italic>; βs = 0.06–0.13; 95% CIs [0.00–0.07, 0.12–0.18]). These findings highlight several CU traits that adolescents reported as endorsing higher levels the next day compared to their average level. In addition, most items only showed two significant cross-lagged associations, with items 4, 8, and 9 displaying three cross-lagged effects. All significant cross-lagged associations were augmented, apart from four effects (i.e., 3 → 9, 7 → 2, 9 → 6, and 12 → 6), which revealed blunting effects.</p>", "<p id=\"Par19\">Between-level item correlations, means, and variances are shown in Table ##TAB##1##2##. Overall, consistent with the between-level descriptive statistics, all items displayed significant positive associations with almost all other items, except items 2 (<italic>feels bad or guilty when I have done something wrong</italic>) and 3 (<italic>does not show emotions</italic>), which showed few significant associations.</p>", "<title>Callous-unemotional Traits and Associations with Subscales</title>", "<p id=\"Par20\">Descriptive statistics, ICCs, and within-level and between-level subscale correlations are shown in ##SUPPL##0##Supplementary Table S4##. Notably, at the within-level, callousness was significantly positively associated with all variables except positive affect, whereas uncaring was positively associated with negative affect and conduct problems, and negatively with positive affect. At the between-level, callousness was significantly positively associated with all variables and negatively associated with positive affect; uncaring was positively associated with conduct problems and negatively with positive affect. Also, emotional problems displayed the highest ICC (0.76) and uncaring had the lowest ICC (0.60). Table ##TAB##2##3## shows within-day residual correlations after controlling for the previous day’s effects. Findings were similar to the within-level item correlations presented above.</p>", "<p id=\"Par21\">Significant autoregressive (curved arrows) and cross-lagged (straight arrows) effects are displayed in Fig. ##FIG##2##3## (all estimates are shown in ##SUPPL##0##Supplementary Tables S5 and S6##, respectively). All variables showed significant autoregressive effects (βs = 0.13–0.35; 95% CIs [0.08–0.29, 0.18–0.41]). Thus, adolescents experiencing high levels of each variable on the previous day were more likely to experience high levels of each variable, respectively, the next day compared to their average level. Regarding cross-lagged effects, all significant associations were positive (i.e., augmentation). Specifically, adolescents who reported higher than their average levels of callousness also reported higher than their average levels of uncaring the next day (β = 0.08; 95% CI [0.03, 0.13]). Similar cross-day associations were observed between conduct problems and uncaring (β = 0.05; 95% CI [0.00, 0.10]), positive affect and callousness (β = 0.05; 95% CI [0.00, 0.10]), negative affect and emotional problems (β = 0.07; 95% CI [0.01, 0.13]), and, finally, emotional problems and negative affect (β = 0.09; 95% CI [0.03, 0.15]). Notably, uncaring did not show any significant cross-lagged effects.</p>", "<p id=\"Par22\">Between-level correlations, means, and variances are shown in Table ##TAB##2##3##. Notably, adolescents with higher levels of callousness, on average, tended to also have higher average levels of uncaring, negative affect, conduct problems, and emotional problems and lower average levels of positive affect over the month. Those with higher average levels of uncaring also had higher average levels of conduct problems and lower average levels of positive affect over the month.</p>" ]

[ "<title>Discussion</title>", "<p id=\"Par23\">By collecting intensive longitudinal data and applying analytic approaches that elucidate temporal dynamics, we can expand our understanding of CU traits in a more ecologically valid way. Accordingly, this exploratory study examined whether adolescent CU traits manifest fluctuations across a 1-month observation window. Many CU traits items showed within-person autoregressive and cross-lagged links over the timeframe (i.e., participants’ levels at time <italic>t</italic> were significantly predicted by their levels at time <italic>t</italic>-1), and we described these effects using terminology from the affective dynamics literature. We also illustrated how daily CU traits were associated with daily affect and emotional and conduct problems. Overall, we observed several within- and between-person effects that serve as a starting point for further understanding of temporal dynamics and functional accounts of CU traits and related emotional and behavioral functioning.</p>", "<title>Callous-unemotional Traits in Daily Life</title>", "<p id=\"Par24\">In line with recent research calling for lower-order precision level analyses (Goulter et al., ##REF##36074612##2022##; Kotov et al., ##REF##33577350##2021##), we assessed CU traits at the item-level. Almost all items displayed within-person autoregressive effects (inertia). Of those items, two items that typically load onto callousness (<italic>does not care who I hurt to get what I want</italic>, <italic>feelings of others are unimportant</italic>) and two items that represent uncaring (<italic>concerned about the feelings of others</italic>, <italic>apologizes to persons I have hurt</italic>) had lower inertia and ICC scores (i.e., greater within-person fluctuations) relative to other items. Notably, these items all reflect CU traits related to behaviors directed toward others or others’ emotions or feelings. These items demonstrate greater variability likely because they tap interpersonal interactions with others, which could be less salient and stable relative to items regarding one’s own motivations and behaviors (e.g., <italic>does not care if I am in trouble</italic>, <italic>does not care about doing things well</italic>). Conceptualizations of CU traits point to deficits in both affiliative capacity and emotional responding, and intensive longitudinal designs may prove especially useful in informing socio-emotional accounts of CU traits.</p>", "<p id=\"Par25\">Conversely, <italic>does not show emotions</italic> showed the largest inertia, ICC (i.e., the least within-person fluctuations), and highest between-level mean. This is a notable finding because this item is the only item retained from the 24-item ICU to the brief 12-item version presumably loading onto the unemotional subscale. Although this 12-item adaptation has been well-validated in child, adolescent, and young adult samples (e.g., Hawes et al., ##REF##24188153##2014b##; Zheng et al., ##REF##34928648##2021##), others have emphasized the importance of unemotional items for understanding the broader CU construct (Ray &amp; Frick, ##REF##30142284##2020##). For example, one meta-analytic study found that although the unemotional subscale was only modestly associated with aggression (<italic>r</italic> = 0.06), it was more highly associated with lower empathy (<italic>r</italic> = -0.22; Cardinale &amp; Marsh, ##REF##29239206##2020##). These findings suggest that the unemotional scale may be more consequential for delineating “trait-like” impoverished emotion and empathy aspects of CU traits (Ray &amp; Frick, ##REF##30142284##2020##). Furthermore, both <italic>does not care about doing things well</italic> and <italic>shows no remorse when I have done something wrong</italic> did not show significant autoregressive effects. These results are somewhat expected as it is unlikely that our community sample committed transgressions every day, thus necessitating further research with higher-risk and clinical samples.</p>", "<p id=\"Par26\">A greater understanding of the short-term fluctuations in CU traits can also help inform ethical considerations in this field regarding labelling, stigmatization, and nomenclature. As a construct, CU traits was initially developed by downwardly extending the affective component of adult psychopathy to children and adolescents. Although definitions of psychopathy typically also comprise interpersonal and impulsive/antisocial dimensions, an extensive child and adult literature has revealed that the affective or CU dimension tends to distinguish those individuals with distinct neurocognitive correlates, as well as heightened risk for persistent antisocial behavior (Frick et al., ##REF##24117854##2014##). However, there are reasonable concerns regarding the stigmatization of labelling children and adolescents with the term CU “traits” (Prasad &amp; Kimonis, ##UREF##6##2018##). Several scholars have accordingly advocated for alternative language, such as CU <italic>behaviors</italic>, <italic>features</italic>, or <italic>symptoms</italic>, particularly in younger children when these characteristics may be less stable (e.g., Schuberth et al., ##REF##30155686##2019##; Waller &amp; Hyde, ##REF##28824706##2017##). Others have made clear at the outset that although they use the term CU “traits” in young samples, they are not suggesting that these indicators are immutable (e.g., Fleming et al., ##REF##36229121##2022##). As well-stated by Kaurin et al. (##UREF##3##2022##, p. 23) “A stricter focus on dysregulated interpersonal and affective processes in daily life [therefore] may also reform the professional and public view on personality pathology, which is dominated by perceptions of destiny rather than manageable risk.” Overall, our item-level findings provide preliminary granular and temporal evidence of within-person “state” fluctuations of CU traits in real-life.</p>", "<title>Daily Callous-unemotional Traits and Emotional and Behavioral Functioning</title>", "<p id=\"Par27\">We also evaluated whether daily CU traits, including callousness and uncaring, were associated with several indicators of emotional and behavioral functioning. Both callousness and conduct problems augmented uncaring over time. One explanation for these findings may relate to how adolescents with elevated callousness appraise certain situations. It may be the case that adolescents high on callousness with a cognitive predisposition underpinned by fearlessness and deficits in emotional processing interpret certain situational features (e.g., negative interactions with parents or peers) in a way that results in higher levels of uncaring. Also noteworthy, uncaring did not predict any other construct. Structural differences in the ICU have been suggested to be a spurious by-product of method variance (Ray &amp; Frick, ##REF##30142284##2020##); however, our findings contribute some evidence pointing to meaningful differences between CU subscales. Taken together, these findings suggest that uncaring may be better conceptualized as an outcome rather than an antecedent, and callousness could represent a critical treatment target.</p>", "<p id=\"Par28\">An interesting finding was that positive affect was linked with callousness. Previous research found that callousness at baseline negatively predicted positive affect assessed six times per day across a 1-week period (Vize et al., ##REF##35402088##2022##); however, callousness was assessed with a personality measure only at baseline. One explanation for our findings may be due to the specific measure of positive affect assessing high valence constructs, including being active, inspired, and determined (Cooke et al., ##REF##36174167##2022##; Thompson, ##UREF##9##2007##). It may be the case that callousness also designates resolute individuals characterized by ambition. This notion goes against the DSM-5 “with Limited Prosocial Emotions” criterion specifying an absence of interest in school or work performance (American Psychiatric Association, ##UREF##0##2013##). However, the current uncaring subscale from the 12-item ICU is operationalized as a lack of concern for others (not activities). In addition, echoing Vize et al. (##REF##35402088##2022##), such traits may be strongly associated with extroversion, which is linked to heightened positive affect (Watson et al., ##REF##25751628##2015##)–although these findings are derived from adult samples. Future studies should consider a broader range of positive affect items to replicate and extend current findings, as well as further research on personality in adolescent samples.</p>", "<title>Strengths and Limitations</title>", "<p id=\"Par29\">Notable strengths of the present study include the naturalistic 30-day diary design, which advances understanding of CU traits under higher ecological validity relative to traditional longitudinal studies. In addition, we applied an analytic approach that disentangles within-person fluctuations from between-person differences. We also evaluated whether daily CU traits were associated with multiple forms of daily emotional and behavioral functioning. However, current findings should be interpreted within the context of some limitations. First, the current sample was primarily comprised of community adolescents from families with higher annual income than the provincial median income. Participants also endorsed lower levels of CU traits (<italic>M</italic> = 6.88; <italic>SD</italic> = 4.26) relative to other studies using the 12-item ICU scale. For example, Colins et al. (##REF##26493393##2016##) found higher levels of CU traits in a sample of justice-involved adolescent girls (<italic>M</italic> = 9.14; <italic>SD</italic> = 5.88). Thus, our results may not generalize to other populations, and as highlighted, it would be particularly important to examine daily CU traits in higher-risk or clinical samples.</p>", "<p id=\"Par30\">Second, all data are solely based on adolescent self-reports–we did not collect data from other informants (e.g., parents) on adolescent daily CU traits. Past research has revealed distinct associations between CU traits and conduct problems across adolescent- and parent-reports (e.g., Goulter &amp; Moretti, ##REF##33825099##2021##). Meta-analytic work has also shown less discrepancies in cross-informant correspondence when informants are reporting on observable behaviors (e.g., externalizing vs. internalizing) or when the behaviors are measured within the same context (De Los Reyes et al., ##REF##25915035##2015##). Regarding CU traits, it may be more difficult for other informants to report on CU traits given these traits are not necessarily readily observable. </p>", "<p id=\"Par31\">Third, our sample was relatively small, although simulation studies have shown that sufficient power can be achieved for DSEM with a sample size of 100 and a minimum of 25 measured occasions (e.g., Schultzberg &amp; Muthén, ##UREF##7##2018##). In particular, power at Level 2 is relatively small compared to conventional longitudinal studies, and thus, these between-person effects should be interpreted with caution. However, with over 2,000 observations at the daily level, power at Level 1 is sufficient for the current approach. Because of our sample size, we were not able to optimally investigate gender differences–an important direction for future research given most studies examining CU traits have relied predominantly on male samples.</p>", "<title>Future Directions and Implications</title>", "<p id=\"Par32\">By decomposing “trait” and “state”-like aspects of CU traits, current findings inform understanding of fluctuations in CU traits and lay a foundation for future intensive longitudinal designs to test critical questions in the field. For instance, adolescence represents a sensitive period marked by distinct change including within neural systems involved in social, emotional, and motivational processes (Dahl et al., ##REF##29469094##2018##). Because of this, studies testing daily dynamics among adolescents may be particularly well-suited to reveal important information regarding socio-behavioral functioning. Atypical levels of CU traits have also been observed in samples as young as preschool age (Kimonis et al., ##REF##26344015##2016##), and developmental models describe early childhood as the period in which the emergence of conscience and empathy typically occurs (Kochanska, ##UREF##4##1993##). Thus, future research should also investigate daily CU traits in younger samples. Ideally, CU traits would be examined at multiple developmental stages across multiple timescales, and measurement burst designs that combine micro and macro longitudinal methods could identify changes in short-term CU dynamics and their relations with long-term outcomes. Future research should also apply multiple measurement approaches (e.g., self-report, observational, psychophysiological) to better characterize the emotional components of daily experiences–a particularly important avenue in CU traits research given the distinct psychophysiological profiles among these samples (Fanti et al., ##REF##30797946##2019##).</p>", "<p id=\"Par33\">It will also be crucial to investigate the role of specific situational features in CU traits fluctuations and employ different intensive longitudinal methods to test these relations. For example, the present study employed a time-based sampling scheme prompting participants at 5 pm each day. By contrast, other research may use a variable-interval scheme generating prompts at random throughout the day to further enhance ecological validity (Russell &amp; Gajos, ##REF##31997358##2020##). Future studies may also want to consider event contingent designs, which could be useful for examining negative interactions with parents, peers, or the broader environment. Theoretical and empirical works highlight the role of warm and supportive parent–child relationships in promoting empathy development and eliciting positive affect (Kochanska, ##UREF##4##1993##). Intensive longitudinal methods with parent–child dyads (e.g., Xu &amp; Zheng, ##REF##35583795##2022##) have the potential to further advance understanding of familial relations in the development of CU traits. Particularly, parenting behaviors, including parental warmth and harsh discipline, have been linked to child CU traits (Goulter et al., ##REF##31166154##2020##; Waller et al., ##REF##23583974##2013##; Zheng et al., ##REF##27179538##2017##), and recent daily diary studies have demonstrated substantial daily fluctuations in parental warmth (Xu &amp; Zheng, ##REF##36273075##2023##). Thus, examining how daily parenting behaviors are associated with CU traits may provide modifiable targets in daily family processes for future treatment efforts. In adolescence, research may want to examine the role of peer affiliation and interactions in daily CU traits fluctuations. Although the present study assessed negative affect, a greater understanding of the dynamics of stress and associated stressful experiences would also be particularly informative.</p>" ]

[ "<title>Conclusion</title>", "<p id=\"Par34\">In this exploratory study, we observed meaningful fluctuations of CU traits at both item- and subscale-levels. These findings suggest that CU traits may be more susceptible to change and malleable than previously thought. In addition, callousness and uncaring subscales demonstrated distinct cross-day dynamics in relation to emotional and behavioral problems. Here, we have laid a foundation for future research focus applying intensive longitudinal methods to advance understanding of CU traits in daily life. Further functional information gained from these methods can potentially inform intervention efforts by offering modifiable targets situated in daily lives to promote personalized and just-in-time interventions. By conducting sampling in real-time in real-world situations, we can further inform personalized models of treatment targeting daily processes.\n</p>" ]

[ "<p id=\"Par1\">Intensive longitudinal methods (e.g., daily diary) inform understanding of dynamic processes by parsing <italic>within-person</italic> state-like fluctuations from stable <italic>between-person</italic> trait-like differences. In this exploratory study, we investigated whether self-reported callous-unemotional (CU) traits (callousness, uncaring) demonstrated daily fluctuations, as well as whether daily CU traits were associated with multiple forms of daily emotional and behavioral functioning. A sample of 99 adolescents (55.8% female; <italic>M</italic>_age = 14.60 years) provided baseline information and completed a naturalistic 30-day diary reporting on CU traits, positive and negative affect, and emotional and conduct problems in their daily lives. Dynamic structural equation modeling revealed that many CU traits items showed within-person autoregressive and cross-lagged links; however, there was substantial between-person variation in within-person fluctuations across items. At the subscale level, cross-day associations were observed between callousness and uncaring, conduct problems and uncaring, positive affect and callousness, negative affect and emotional problems, and emotional problems and negative affect. By harnessing intensive longitudinal data, our findings provide preliminary state-level evidence of CU traits, as well as functional information with regards to CU traits and emotional and behavioral problems in daily life. We consider the implications of our findings in terms of informing future CU traits intensive longitudinal evaluations.</p>", "<title>Supplementary Information</title>", "<p>The online version contains supplementary material available at 10.1007/s10802-023-01077-6.</p>", "<title>Keywords</title>" ]

[ "<title>Supplementary Information</title>", "<p>Below is the link to the electronic supplementary material.</p>" ]

[ "<title>Acknowledgements</title>", "<p>We thank all the participating parents, adolescents, Elk Island and St. Albert public schools, and our research assistants. Study data were collected and managed using REDCap electronic data capture tools hosted and supported by the Women and Children’s Health Research Institute at the University of Alberta.</p>", "<title>Author Contribution</title>", "<p>Goulter, N: Conceptualization, Writing-Original Draft, Writing-Review/Editing; Cooke, EM: Formal Analysis, Writing-Original Draft, Writing-Review/Editing; Zheng, Y: Conceptualization, Funding Acquisition, Methodology, Writing-Review/Editing.</p>", "<title>Funding</title>", "<p>This research was supported partly with funding from the Social Sciences and Humanities Research Council (IDG 430-2018-00317 and 409-2020-00080) and Natural Sciences and Engineering Research Council (RGPIN-2020-04458 and DGECR-2020-00077) of Canada. EC was supported by a Mitacs Accelerate grant (IT18227) awarded to YZ.</p>", "<title>Data Availability</title>", "<p>Data are available by emailing the last author. Code and output are available on the Open Science Framework: <ext-link ext-link-type=\"uri\" xlink:href=\"https://osf.io/n2rdf/?view_only=357ec650bfac4a63905e7ed34efd9255\">https://osf.io/n2rdf/?view_only=357ec650bfac4a63905e7ed34efd9255</ext-link>.</p>", "<title>Compliance with Ethical Standards</title>", "<title>Ethical Approval</title>", "<p id=\"Par35\">The study was approved by the institutional research ethics committee and was performed in accordance with the ethical standards as laid down in the 1964 Declaration of Helsinki and its later amendments or comparable ethical standards.</p>", "<title>Informed Consent</title>", "<p id=\"Par36\">Informed consent and assent for study participation and research publication was obtained from all individual participants included in the study.</p>", "<title>Conflicts of Interest</title>", "<p id=\"Par37\">The authors have no relevant or non-financial interests to disclose.</p>" ]

[ "<fig id=\"Fig1\"><label>Fig. 1</label><caption><p>Schematic of DSEM. Note. Observed items are decomposed into time-varying within-level (w) and time-invariant between-level (b) components. CAL = callousness, UNC = uncaring, POS = positive affect, NEG = negative affect, CON = conduct problems, EMO = emotional problems</p></caption></fig>", "<fig id=\"Fig2\"><label>Fig. 2</label><caption><p>DSEM Standardized Estimates for Callous-Unemotional Traits Items Within-Person Autoregressive (curved arrows) and Cross-Lagged Effects (straight arrows). Note. Only significant effects based on 95% credible intervals are shown. CU = callous-unemotional</p></caption></fig>", "<fig id=\"Fig3\"><label>Fig. 3</label><caption><p>DSEM Standardized Estimates for Callous-Unemotional Traits Subscales, Positive Affect, Negative Affect, Conduct Problems, and Emotional Problems Within-Person Autoregressive (curved arrows) and Cross-Lagged Effects (straight arrows). Note. Only significant effects based on 95% credible intervals are shown. CAL = callousness, UNC = uncaring, POS = positive affect, NEG = negative affect, CON = conduct problems, EMO = emotional problems</p></caption></fig>" ]

[ "<table-wrap id=\"Tab1\"><label>Table 1</label><caption><p>Callous-Unemotional Traits Items and Subscales</p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th align=\"left\"><italic>Item Description</italic></th><th align=\"left\"><italic>Subscale</italic></th></tr></thead><tbody><tr><td align=\"left\">CU1: Does not care who I hurt to get what I want</td><td align=\"left\">Callousness</td></tr><tr><td align=\"left\">CU2: Feels bad or guilty when I have done something wrong (r)</td><td align=\"left\">Uncaring</td></tr><tr><td align=\"left\">CU3: Does not show emotions</td><td align=\"left\">Unemotional</td></tr><tr><td align=\"left\">CU4: Concerned about the feelings of others (r)</td><td align=\"left\">Uncaring</td></tr><tr><td align=\"left\">CU5: Does not care if I am in trouble</td><td align=\"left\">Callousness</td></tr><tr><td align=\"left\">CU6: Does not care about doing things well</td><td align=\"left\">Callousness</td></tr><tr><td align=\"left\">CU7: Seems very cold and uncaring</td><td align=\"left\">Callousness</td></tr><tr><td align=\"left\">CU8: Apologizes to persons I have hurt (r)</td><td align=\"left\">Uncaring</td></tr><tr><td align=\"left\">CU9: Tries not to hurt others’ feelings (r)</td><td align=\"left\">Uncaring</td></tr><tr><td align=\"left\">CU10: Shows no remorse when I have done something wrong</td><td align=\"left\">Callousness</td></tr><tr><td align=\"left\">CU11: Feelings of others are unimportant</td><td align=\"left\">Callousness</td></tr><tr><td align=\"left\">CU12: Does things to make others feel good (r)</td><td align=\"left\">Uncaring</td></tr></tbody></table></table-wrap>", "<table-wrap id=\"Tab2\"><label>Table 2</label><caption><p>Within-Person Same Day Residual Correlations and Between-Person Correlations for Callous-Unemotional Traits Items</p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th align=\"left\"><bold>Item</bold></th><th align=\"left\"><bold>CU1</bold></th><th align=\"left\"><bold>CU2</bold></th><th align=\"left\"><bold>CU3</bold></th><th align=\"left\"><bold>CU4</bold></th><th align=\"left\"><bold>CU5</bold></th><th align=\"left\"><bold>CU6</bold></th><th align=\"left\"><bold>CU7</bold></th><th align=\"left\"><bold>CU8</bold></th><th align=\"left\"><bold>CU9</bold></th><th align=\"left\"><bold>CU10</bold></th><th align=\"left\"><bold>CU11</bold></th><th align=\"left\"><bold>CU12</bold></th></tr></thead><tbody><tr><td align=\"left\">CU1</td><td char=\".\" align=\"char\">–</td><td char=\".\" align=\"char\">0.01</td><td char=\".\" align=\"char\">0.03</td><td char=\".\" align=\"char\">0.13^***</td><td char=\".\" align=\"char\">0.14^***</td><td char=\".\" align=\"char\">0.21^***</td><td char=\".\" align=\"char\">0.12^***</td><td char=\".\" align=\"char\">0.08^***</td><td char=\".\" align=\"char\">0.09^***</td><td char=\".\" align=\"char\">0.10^***</td><td char=\".\" align=\"char\">0.07^***</td><td char=\".\" align=\"char\">0.12^***</td></tr><tr><td align=\"left\">CU2</td><td char=\".\" align=\"char\">0.22^*</td><td char=\".\" align=\"char\">–</td><td char=\".\" align=\"char\">-0.08^***</td><td char=\".\" align=\"char\">0.24^***</td><td char=\".\" align=\"char\">0.12^***</td><td char=\".\" align=\"char\">0.04</td><td char=\".\" align=\"char\">-0.04^*</td><td char=\".\" align=\"char\">0.17^***</td><td char=\".\" align=\"char\">0.18^***</td><td char=\".\" align=\"char\">0.02</td><td char=\".\" align=\"char\">0.06^**</td><td char=\".\" align=\"char\">0.15^***</td></tr><tr><td align=\"left\">CU3</td><td char=\".\" align=\"char\">-0.01</td><td char=\".\" align=\"char\">0.46^***</td><td char=\".\" align=\"char\">–</td><td char=\".\" align=\"char\">-0.20^***</td><td char=\".\" align=\"char\">0.04^*</td><td char=\".\" align=\"char\">0.05^*</td><td char=\".\" align=\"char\">0.10^***</td><td char=\".\" align=\"char\">-0.12^***</td><td char=\".\" align=\"char\">-0.11^***</td><td char=\".\" align=\"char\">0.04^*</td><td char=\".\" align=\"char\">0.03</td><td char=\".\" align=\"char\">-0.06^**</td></tr><tr><td align=\"left\">CU4</td><td char=\".\" align=\"char\">0.52^***</td><td char=\".\" align=\"char\">0.23^*</td><td char=\".\" align=\"char\">-0.28^*</td><td char=\".\" align=\"char\">–</td><td char=\".\" align=\"char\">0.07^**</td><td char=\".\" align=\"char\">0.07^**</td><td char=\".\" align=\"char\">0.08^***</td><td char=\".\" align=\"char\">0.37^***</td><td char=\".\" align=\"char\">0.38^***</td><td char=\".\" align=\"char\">0.04</td><td char=\".\" align=\"char\">0.10^***</td><td char=\".\" align=\"char\">0.34^***</td></tr><tr><td align=\"left\">CU5</td><td char=\".\" align=\"char\">0.75^***</td><td char=\".\" align=\"char\">0.17</td><td char=\".\" align=\"char\">-0.04</td><td char=\".\" align=\"char\">0.57^***</td><td char=\".\" align=\"char\">–</td><td char=\".\" align=\"char\">0.31^***</td><td char=\".\" align=\"char\">0.09^***</td><td char=\".\" align=\"char\">0.04^*</td><td char=\".\" align=\"char\">0.06^**</td><td char=\".\" align=\"char\">0.09^***</td><td char=\".\" align=\"char\">0.07^**</td><td char=\".\" align=\"char\">0.05^*</td></tr><tr><td align=\"left\">CU6</td><td char=\".\" align=\"char\">0.73^***</td><td char=\".\" align=\"char\">0.13</td><td char=\".\" align=\"char\">0.03</td><td char=\".\" align=\"char\">0.49^***</td><td char=\".\" align=\"char\">0.87^***</td><td char=\".\" align=\"char\">–</td><td char=\".\" align=\"char\">0.19^***</td><td char=\".\" align=\"char\">0.07^**</td><td char=\".\" align=\"char\">0.05^*</td><td char=\".\" align=\"char\">0.11^***</td><td char=\".\" align=\"char\">0.10^***</td><td char=\".\" align=\"char\">0.09^***</td></tr><tr><td align=\"left\">CU7</td><td char=\".\" align=\"char\">0.35^**</td><td char=\".\" align=\"char\">0.09</td><td char=\".\" align=\"char\">0.34^**</td><td char=\".\" align=\"char\">0.40^**</td><td char=\".\" align=\"char\">0.42^***</td><td char=\".\" align=\"char\">0.61^***</td><td char=\".\" align=\"char\">–</td><td char=\".\" align=\"char\">0.05^*</td><td char=\".\" align=\"char\">0.07^**</td><td char=\".\" align=\"char\">0.11^***</td><td char=\".\" align=\"char\">0.09^***</td><td char=\".\" align=\"char\">0.10^***</td></tr><tr><td align=\"left\">CU8</td><td char=\".\" align=\"char\">0.50^***</td><td char=\".\" align=\"char\">0.18</td><td char=\".\" align=\"char\">-0.23^*</td><td char=\".\" align=\"char\">0.84^***</td><td char=\".\" align=\"char\">0.56^***</td><td char=\".\" align=\"char\">0.48^***</td><td char=\".\" align=\"char\">0.34^**</td><td char=\".\" align=\"char\">–</td><td char=\".\" align=\"char\">0.54^***</td><td char=\".\" align=\"char\">0.07^**</td><td char=\".\" align=\"char\">0.08^***</td><td char=\".\" align=\"char\">0.34^***</td></tr><tr><td align=\"left\">CU9</td><td char=\".\" align=\"char\">0.53^***</td><td char=\".\" align=\"char\">0.28^*</td><td char=\".\" align=\"char\">-0.19</td><td char=\".\" align=\"char\">0.93^***</td><td char=\".\" align=\"char\">0.56^***</td><td char=\".\" align=\"char\">0.51^***</td><td char=\".\" align=\"char\">0.40^***</td><td char=\".\" align=\"char\">0.82^***</td><td char=\".\" align=\"char\">–</td><td char=\".\" align=\"char\">0.00</td><td char=\".\" align=\"char\">0.08^***</td><td char=\".\" align=\"char\">0.34^***</td></tr><tr><td align=\"left\">CU10</td><td char=\".\" align=\"char\">0.67^***</td><td char=\".\" align=\"char\">0.19</td><td char=\".\" align=\"char\">-0.05</td><td char=\".\" align=\"char\">0.61^***</td><td char=\".\" align=\"char\">0.72^***</td><td char=\".\" align=\"char\">0.75^***</td><td char=\".\" align=\"char\">0.48^***</td><td char=\".\" align=\"char\">0.48^***</td><td char=\".\" align=\"char\">0.62^***</td><td char=\".\" align=\"char\">–</td><td char=\".\" align=\"char\">0.11^***</td><td char=\".\" align=\"char\">0.05^*</td></tr><tr><td align=\"left\">CU11</td><td char=\".\" align=\"char\">0.76^***</td><td char=\".\" align=\"char\">0.12</td><td char=\".\" align=\"char\">0.01</td><td char=\".\" align=\"char\">0.60^***</td><td char=\".\" align=\"char\">0.64^***</td><td char=\".\" align=\"char\">0.73^***</td><td char=\".\" align=\"char\">0.54^***</td><td char=\".\" align=\"char\">0.47^***</td><td char=\".\" align=\"char\">0.54^***</td><td char=\".\" align=\"char\">0.81^***</td><td char=\".\" align=\"char\">–</td><td char=\".\" align=\"char\">0.08^**</td></tr><tr><td align=\"left\">CU12</td><td char=\".\" align=\"char\">0.39^**</td><td char=\".\" align=\"char\">0.15</td><td char=\".\" align=\"char\">-0.29^**</td><td char=\".\" align=\"char\">0.90^***</td><td char=\".\" align=\"char\">0.44^***</td><td char=\".\" align=\"char\">0.41^***</td><td char=\".\" align=\"char\">0.36^**</td><td char=\".\" align=\"char\">0.79^***</td><td char=\".\" align=\"char\">0.84^***</td><td char=\".\" align=\"char\">0.49^***</td><td char=\".\" align=\"char\">0.40^***</td><td char=\".\" align=\"char\">–</td></tr></tbody></table><table frame=\"hsides\" rules=\"groups\"><tbody><tr><td align=\"left\"><bold><italic>M</italic></bold></td><td char=\".\" align=\"char\">0.12</td><td char=\".\" align=\"char\">1.00</td><td char=\".\" align=\"char\">1.50</td><td char=\".\" align=\"char\">0.53</td><td char=\".\" align=\"char\">0.29</td><td char=\".\" align=\"char\">0.24</td><td char=\".\" align=\"char\">0.42</td><td char=\".\" align=\"char\">0.63</td><td char=\".\" align=\"char\">0.51</td><td char=\".\" align=\"char\">0.25</td><td char=\".\" align=\"char\">0.17</td><td char=\".\" align=\"char\">0.66</td></tr><tr><td align=\"left\"><bold><italic>σ</italic></bold>^{<bold>2</bold>}</td><td char=\".\" align=\"char\">0.09</td><td char=\".\" align=\"char\">1.19</td><td char=\".\" align=\"char\">1.26</td><td char=\".\" align=\"char\">0.37</td><td char=\".\" align=\"char\">0.26</td><td char=\".\" align=\"char\">0.21</td><td char=\".\" align=\"char\">0.36</td><td char=\".\" align=\"char\">0.46</td><td char=\".\" align=\"char\">0.38</td><td char=\".\" align=\"char\">0.17</td><td char=\".\" align=\"char\">0.13</td><td char=\".\" align=\"char\">0.52</td></tr></tbody></table></table-wrap>", "<table-wrap id=\"Tab3\"><label>Table 3</label><caption><p>Within-Person Same Day Residual Correlations and Between-Person Correlations for Callous-Unemotional Traits Subscales, Positive Affect, Negative Affect, Conduct Problems, and Emotional Problems</p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th align=\"left\"><bold>Item</bold></th><th align=\"left\"><bold>1</bold></th><th align=\"left\"><bold>2</bold></th><th align=\"left\"><bold>3</bold></th><th align=\"left\"><bold>4</bold></th><th align=\"left\"><bold>5</bold></th><th align=\"left\"><bold>6</bold></th></tr></thead><tbody><tr><td align=\"left\">1. Callousness</td><td char=\".\" align=\"char\">–</td><td char=\".\" align=\"char\">0.17^***</td><td char=\".\" align=\"char\">-0.03</td><td char=\".\" align=\"char\">0.21^***</td><td char=\".\" align=\"char\">0.22^***</td><td char=\".\" align=\"char\">0.13^***</td></tr><tr><td align=\"left\">2. Uncaring</td><td char=\".\" align=\"char\">0.58^***</td><td char=\".\" align=\"char\">–</td><td char=\".\" align=\"char\">-0.10^***</td><td char=\".\" align=\"char\">0.09^***</td><td char=\".\" align=\"char\">0.21^***</td><td char=\".\" align=\"char\">0.03</td></tr><tr><td align=\"left\">3. Positive Affect</td><td char=\".\" align=\"char\">-0.24^*</td><td char=\".\" align=\"char\">-0.36^**</td><td char=\".\" align=\"char\">–</td><td char=\".\" align=\"char\">0.06^**</td><td char=\".\" align=\"char\">-0.06^**</td><td char=\".\" align=\"char\">-0.09^***</td></tr><tr><td align=\"left\">4. Negative Affect</td><td char=\".\" align=\"char\">0.52^***</td><td char=\".\" align=\"char\">0.08</td><td char=\".\" align=\"char\">-0.16</td><td char=\".\" align=\"char\">–</td><td char=\".\" align=\"char\">0.28^***</td><td char=\".\" align=\"char\">0.48^***</td></tr><tr><td align=\"left\">5. Conduct Problems</td><td char=\".\" align=\"char\">0.72^***</td><td char=\".\" align=\"char\">0.59^***</td><td char=\".\" align=\"char\">-0.29^**</td><td char=\".\" align=\"char\">0.52^***</td><td char=\".\" align=\"char\">–</td><td char=\".\" align=\"char\">0.17^***</td></tr><tr><td align=\"left\">6. Emotional Problems</td><td char=\".\" align=\"char\">0.41^***</td><td char=\".\" align=\"char\">0.09</td><td char=\".\" align=\"char\">-0.38^***</td><td char=\".\" align=\"char\">0.88^***</td><td char=\".\" align=\"char\">0.47^***</td><td char=\".\" align=\"char\">–</td></tr></tbody></table><table frame=\"hsides\" rules=\"groups\"><tbody><tr><td align=\"left\"><bold><italic>M</italic></bold></td><td char=\".\" align=\"char\">0.25</td><td char=\".\" align=\"char\">0.66</td><td char=\".\" align=\"char\">2.67</td><td char=\".\" align=\"char\">1.61</td><td char=\".\" align=\"char\">0.18</td><td char=\".\" align=\"char\">0.46</td></tr><tr><td align=\"left\"><bold><italic>σ</italic></bold>^{<bold>2</bold>}</td><td char=\".\" align=\"char\">0.12</td><td char=\".\" align=\"char\">0.29</td><td char=\".\" align=\"char\">0.74</td><td char=\".\" align=\"char\">0.46</td><td char=\".\" align=\"char\">0.05</td><td char=\".\" align=\"char\">0.24</td></tr></tbody></table></table-wrap>" ]

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[ "<supplementary-material content-type=\"local-data\" id=\"MOESM1\"></supplementary-material>" ]

[ "<table-wrap-foot><p><italic>r</italic> reverse coded items</p></table-wrap-foot>", "<table-wrap-foot><p>Correlations estimated using the Bayes estimator. Within-day residual correlations are shown above the diagonal. Between-level correlations, means (<italic>M</italic>), and variances (σ²<bold>)</bold> are shown below the diagonal. Items 2, 4, 8, 9, 12 are reverse coded</p><p>*<italic>p</italic> ≤ 0.05, **<italic>p</italic> ≤ 0.01, ***<italic>p</italic> ≤ 0.001</p></table-wrap-foot>", "<table-wrap-foot><p>Correlations estimated using the Bayes estimator. Within-day residual correlations are shown above the diagonal. Between-level correlations, means (<italic>M</italic>), and variances (σ²<bold>)</bold> are shown below the diagonal</p><p>*<italic>p</italic> ≤ 0.05, **<italic>p</italic> ≤ 0.01, ***<italic>p</italic> ≤ 0.001</p></table-wrap-foot>", "<fn-group><fn><p><bold>Publisher's Note</bold></p><p>Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.</p></fn></fn-group>" ]

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[ "<media xlink:href=\"10802_2023_1077_MOESM1_ESM.pdf\"><caption><p>Supplementary file1 (PDF 306 KB)</p></caption></media>" ]

{ "acronym": [], "definition": [] }

[ "<title>Introduction</title>", "<p id=\"Par3\">Social learning can be defined “as a change in behavior that follows the observation of another (typically a conspecific) perform a similar behavior, the products of the behavior, or even the products alone” (Zentall ##REF##21895354##2012##: 114). Which type of information is acquired through observation may differ strongly and is reflected in so-called social learning mechanisms. Motivational factors such as social facilitation (Zajonc ##REF##14300526##1965##) promote the acquisition of information or change in behaviour by the observer via the mere presence of a demonstrator (Zentall ##UREF##29##2001##). Perceptual factors such as local (Roberts ##UREF##21##1941##) or stimulus enhancement (Galef ##UREF##8##1988##) can facilitate information acquisition by drawing the attention of the observer to either the location or stimulus of importance (Zentall ##UREF##29##2001##). In emulation (Tomasello ##UREF##24##1998##) the results of a demonstrated behaviour affect the observer who may strive to generate the same effect on the environment/objects as the demonstration did, without necessarily understanding the actions or reproducing the behaviour. Imitation is a specific learning mechanism, defined by a high degree of copying fidelity/response matching (Whiten and Ham ##UREF##27##1992##; Whiten et al. ##UREF##28##2009##), and cannot be explained by motivational, perceptual, or attentional factors alone (Zentall ##REF##21895354##2012##), for a comprehensive overview of social learning mechanisms see Hoppitt and Laland (##UREF##11##2013##). Social learning is taxonomically widespread, ranging from insects to birds and mammals, possibly because it is a cost-effective way of acquiring information. Yet, social learning is not always advantageous (Giraldeau et al. ##UREF##9##2002##; Garcia-Nisa et al. ##REF##36670123##2023##) and different social learning mechanisms may have different thresholds in this respect.</p>", "<p id=\"Par4\">Great apes, for instance, seem to be less prone to show imitation than other forms of social learning like emulation (Horner and Whiten ##REF##15549502##2005##; Tennie et al. ##UREF##23##2006##; Clay and Tennie ##REF##28741660##2018##). On the other hand, Marmosets (<italic>Callithrix jacchus</italic>) (Bugnyar and Huber ##REF##9344436##1997##; Voelkl and Huber ##REF##10973721##2000##, ##REF##17622356##2007##) and dogs (<italic>Canis familiaris</italic>) (Huber et al. ##UREF##14##2009##, ##REF##29980941##2018##, ##REF##31975325##2020##) show high-fidelity imitation despite not being closely related to humans (<italic>Homo imitans</italic> as proposed by Meltzoff ##UREF##16##1988##). This suggests that high-fidelity imitation may be driven by natural ecology and social structure rather than phylogenetic relatedness. In fact, experimental evidence has illustrated that various avian species, namely budgerigars (<italic>Melopsittacus undulates</italic>) (Dawson and Foss ##REF##5882805##1965##; Heyes and Saggerson ##UREF##10##2002##), European starlings (<italic>Sturnus vulgaris</italic>) (Fawcett et al. ##UREF##5##2002##), Japanese Quail (<italic>Coturnix japonica</italic>) (Akins and Zentall ##UREF##0##1998##; Akins et al. ##REF##12391793##2002##), Common Ravens (<italic>Corvus corax</italic>) (Loretto et al. ##UREF##15##2020##) and Pigeons (<italic>Columba livia</italic>) (Nguyen et al. ##REF##16235638##2005##) show (simple forms) of motor imitation.</p>", "<p id=\"Par5\">Parrots are renowned for their technical intelligence, vocal mimicry and social learning capacities (Pepperberg and Funk ##UREF##20##1990##; Huber et al. ##UREF##13##2001##; Funk ##REF##12357289##2002##; Huber and Gajdon ##REF##16909237##2006##; Werdenich and Huber ##UREF##25##2006##; Auersperg et al. ##REF##19411271##2009##, ##REF##21687666##2011##, ##REF##23137681##2012##, ##UREF##1##2014##; Miyata et al. ##REF##20640911##2011##; Goodman et al. ##REF##30224791##2018##; Klump et al. ##REF##34437121##2021##; Smith et al. ##REF##36258015##2022##). Yet surprisingly few parrot species have been tested on their motor imitation skills (budgerigars: Dawson and Foss ##REF##5882805##1965##; Galef et al. ##UREF##7##1986##; Heyes and Saggerson ##UREF##10##2002##; grey parrots (<italic>Psittacus erithacus</italic>): Moore ##UREF##18##1992##; kea (<italic>Nestor notabilis</italic>): Huber et al. ##UREF##13##2001##; Suwandschieff et al. ##REF##37261570##2023##; Goffin cockatoos (<italic>Cacatua goffiniana</italic>): Auersperg et al. ##REF##23137681##2012##), revealing mixed results. Whereas most studies find evidence for motor imitation, the studies on kea remained inconclusive.</p>", "<p id=\"Par6\">Kea (<italic>Nestor notabilis</italic>) possess well-developed technical skills (Huber and Gajdon ##REF##16909237##2006##), have long lifespans with multiple reproductive cycles, extended juvenile periods accompanied by considerable in-group tolerance, are highly neophilic and exploratory (Diamond and Bond ##UREF##4##1999##). They also have a very large number of documented food sources (Brejaart ##UREF##2##1988##; Clarke ##UREF##3##1971##; O’Donnell and Dilks ##UREF##19##1994##) many of which need to be extracted, which strongly suggest transfer of knowledge between individuals. All these characteristics facilitate the development of social learning (Gajdon et al. ##UREF##6##2004##), yet experimental evidence for imitation in this species is still missing. Therefore, we tested kea, for their social learning skills in a demonstrated sequence task. Specifically, we aimed at exploring kea’s imitative social learning capacities. We hypothesised that when confronted with a relatively complex two-step task, kea would pay attention to, and copy the behaviour of, a skilled conspecific. We thus predicted that observers would preferentially use the demonstrated opening side, sequence and colour whereas non-observing control individuals would apply trial-and-error learning to solve the task.</p>" ]

[ "<title>Method</title>", "<title>Subjects</title>", "<p id=\"Par7\">Eighteen kea from the Haidlhof Research Station (Bad Vöslau) participated in this study. All individuals were group-housed in an outdoor aviary equipped with perches, nesting areas, ponds, and various enrichment. All birds were fed three times a day, had access to water ad libitum and were not food deprived for testing. All individuals had prior experience with experimental testing and participated on a voluntary basis in the task. The testing compartment at the Haidlhof Research Station can be visually separated from the rest of the aviary and can be further divided into two different areas. The individuals were assigned to test groups of three and five individuals and two control groups of five individuals each. The distribution was sex and age balanced, for details see Table 1 of the supplementary material.\n</p>", "<title>Apparatus</title>", "<p id=\"Par8\">A rectangular box (44 × 18 × 18 cm) with two aluminium sliding lids, two pins, two strings and two rings served as the test box, see Fig. ##FIG##0##1##. The test box was designed to provide a sequence function, requiring the subjects to pull a pin on top of the box, to then be able to open the opposite sliding lid by pulling a ring attached at the side of the box. The adjacent sliding lid remained locked. Hence, the only solving sequences for the test box were left pin–right ring or right pin–left ring. Electronics were added inside the box to provide the sequence function. If no pin or both pins were pulled the mechanism locked both lids and no rewards could be retrieved. Manual locking keys were added to provide a full and partial locking function for demonstrator training and demonstration sessions. The box was divided into equally sized reward sections underneath the sliding lids and a GoPro fixture was attached at the base.</p>", "<p id=\"Par9\">The pins were coloured in red and yellow, which is on the preferred spectrum for kea, and the strings and rings in green and blue (less preferred spectrum) respectively (Weser and Ross ##UREF##26##2013##). Each side and sequence had one preferred and one less preferred colour pairing, minimising the potential for a side bias based on colour preference alone. To increase salience, only those parts that needed direct manipulation were coloured, i.e., the pins, strings, and rings.</p>", "<title>Procedure</title>", "<p id=\"Par10\">To minimize the effects of individual learning over social learning, all test sessions consisted of one trial only. Each session/trial was terminated either after successful reward retrieval or a maximum of two minutes of exploration—i.e. pulling and touching different ring-pin combinations without opening success. A stopwatch was used to track the two minutes and all sessions were terminated by removing the individuals from the test compartment once either criterion (removal response or time-out) was met. Subjects from test groups received two experimental phases, whereas subjects from control groups received three phases (see below). In all phases, both sides of the box were baited and each sequence was rewarded. However, removing a pin and pulling on the wrong ring, pulling both pins or pulling on the rings without having removed the respective pin did not lead to any reward (as the doors were locked). Altogether four sessions (of one trial each) were tested on three consecutive days. On the first day, phase one (consisting of only one session) was tested in the morning and the first session of phase two (consisting of three sessions) in the afternoon. On the second and third day, each individual only received a morning test session of session two and three of phase two respectively.</p>", "<p id=\"Par11\">The experimenter wore mirrored sunglasses (as has been applied before by Bastos and Taylor ##REF##31911652##2020##; Suwandschieff et al. ##REF##37261570##2023##) and remained silent during testing (excluding direct commands such as “enter”, “exit” etc.) to avoid unintentional cueing.</p>", "<p><bold>Phase 1: Forced failure task to all birds</bold> Pins were removed on this occasion and the test box could not be opened. All individuals (of all experimental groups) received access to the test box and were allowed to try and open it for two minutes. As the pins were missing all individuals were forced to fail this task. After two minutes passed the individual exited the testing compartment and the session was completed. This phase was introduced to prime individuals on the non-functioning of the task, focusing the attention towards the essential pins and increasing the motivation to follow a demonstration.</p>", "<p id=\"Par13\"><bold>Phase 2: Non-demonstrated task Control Group (CG)</bold> Control group individuals were allowed to try and solve the task by trial-and-error for three sessions (on three consecutive days).</p>", "<p id=\"Par14\"><bold>Phase 2: Demonstrated task Test Groups (TG)</bold> Test group individuals received three demonstration sessions of three trials each followed by three test sessions of one trial each (test following demonstration session in direct succession) on three consecutive days. Side/sequence and demonstrator assignment were counterbalanced across the two groups (see Table 1 of the supplementary material).</p>", "<p id=\"Par15\"><bold>Phase 3: Demonstrated task Control Test (CGTest)</bold> In the third phase, former control group birds (minus the two demonstrator birds) were randomly assigned to the two demonstrators and were tested again, this time as observer individuals. The test setup was identical to the phase two demonstrated task of the test group.</p>", "<title>Data scoring and analysis</title>", "<p id=\"Par16\">All experiments were videotaped from two sides, behind the observation compartment and directly above the test box (GoPro) within the test compartment, for the exact setup see Fig. ##FIG##0##1## of the supplementary material. All GoPro videos of the test sessions were scored/coded with Solomon Coder (version beta 19.08.02) and one independent rater (blind to the study) scored 10% of all videos. Interobserver reliability was tested with Cohen’s Kappa for the categorical (<italic>k</italic> = 0.93) and Intraclass Correlation Coefficient for the numerical data (ICC = 0.711, approach duration; ICC = 0.999, response duration; ICC = 1, solving latency), for more information see the supplementary material.</p>", "<p id=\"Par17\">A total of 74 sessions were analysed. Two individuals did not participate in two sessions and one session respectively, all other individuals completed all three of their test sessions. A binomial test was applied to compare the success rate of the test versus the control group, using the proportion of successful control birds as the baseline chance level of solving the setup spontaneously without demonstration. All other analyses were strictly descriptive.</p>" ]

[ "<title>Results</title>", "<p id=\"Par18\">After all subjects had experienced a non-functional apparatus (phase 1), they were allowed to engage in the task with or without prior demonstration of a skilled conspecific (phase 2). In sum, four individuals from the test group (out of a total of 8) successfully solved the task and two individuals from the control group (out of a total of 10). The exact binomial test resulted in a nearly significant trend between the test and control group (<italic>x</italic> = 4, <italic>n</italic> = 8, <italic>p</italic> = 2/10; <italic>p-value</italic> = 0.056).</p>", "<p id=\"Par19\">The control group individuals participated in an additional round of testing (phase 3), in which they received a demonstration prior to getting access to the task again. Four individuals who had failed to solve the task without receiving a demonstration successfully solved the task after receiving a demonstration. In contrast, one of the individuals from the control test group, who had solved the task without receiving a demonstration (in the control group round), did not solve the task after receiving a demonstration (in the control test group round). Assessing the birds’ performance after receiving a demonstration (test group plus control test group) <italic>versus</italic> without a demonstration (control group), the exact binomial test revealed a significant difference between the groups with the test groups performing much better on average than the control group (<italic>x</italic> = 8, <italic>n</italic> = 16, <italic>p</italic> = 2/10; <italic>p-value</italic> = 0.007). When comparing the within-subject design of the control group <italic>versus</italic> the control test group (repeated measure) twice as many individuals successfully solved the task after having seen a demonstration (<italic>x</italic> = 4, <italic>n</italic> = 8, <italic>p</italic> = 2/10; <italic>p-value</italic> = 0.056). Half of the successful individuals (4/8) that saw a demonstration solved the task via the opposite (non-demonstrated) sequence and later reversed to the demonstrated sequence in subsequent sessions, see Table ##TAB##0##1##.</p>" ]

[ "<title>Discussion</title>", "<p id=\"Par20\">We show that kea were able to solve a rather difficult two-step sequence task and that receiving a demonstration of a skilled conspecific had a positive effect on solving success. However, successful birds showed a high variation in their response topography and often abandoned faithfully copying the task in favour of exploration. This is particularly interesting in the case of John, who followed the demonstrated sequence twice and then generalised in the other direction.</p>", "<p id=\"Par21\">The effect of demonstration becomes particularly clear when combining the results from phase two and three. Eight out of 10 subjects that came to solve the task did so after a demonstration. That kea can profit from social learning is in line with previous studies (Huber et al. ##UREF##13##2001##; Huber ##UREF##12##2002##; Suwandschieff et al. ##REF##37261570##2023##). However, the current study provides little evidence for motor imitation, despite various methodological differences from the other studies. For instance, previous experiments illustrated that the motivation to follow a demonstration was low. It was theorized that the complexity of the demonstrated actions could have contributed, as they likely were too simple to require a demonstration to solve the task (Suwandschieff et al. ##REF##37261570##2023##), or too complex to follow the demonstration from afar (Huber et al. ##UREF##13##2001##) and reproduce with high fidelity. Therefore, the task was made more difficult than the basic two-choice task, while avoiding the complexity of the multi-lock box, and the forced failure in phase one was introduced to prime individuals on the task difficulty and to increase their motivation to follow a demonstration. Yet, these measures did not result in a higher copying fidelity than the other studies. On the one hand, half of the solvers used the opposite sequence to the demonstrators. On the other hand, solving success appeared not to reference solving consistency, as successful birds continued to explore the solving potential by applying different opening methods (see supplementary material for details). This variation in solving behaviour unfortunately made a statistical comparison not possible in terms of actual mechanisms, or quantifiable behavioural differences with previous studies.</p>", "<p id=\"Par22\">As all individuals, regardless of the experimental group, participated in the forced failure task, it is improbable that our results can be solely explained by social facilitation or local enhancement. Although we cannot rule out the possibility that the mere presence of a demonstrator motivated individuals to engage in the task, we have established that all individuals will do so even in the absence of a conspecific. Therefore, the presence of another individual does not appear to be the primary factor explaining our results. Additionally, since the location of the test box remained constant throughout the different phases, no additional information could have been gained from the demonstration. Consequently, this does not explain the discrepancy between the experimental groups. While the success rate of observer birds and the varied response patterns exhibited by successful individuals indicate emulation, we cannot dismiss the potential influence of stimulus enhancement. A test setup that clearly distinguishes these two mechanisms would have to be devised, i.e., one including a ghost control (Whiten and Ham ##UREF##27##1992##). Consistent with previous findings kea show high behavioural flexibility (Werdenich and Huber ##UREF##25##2006##; Auersperg et al. ##REF##21687666##2011##; Laschober et al. ##REF##34110523##2021##) and preferentially engage in exploratory behaviour, being more interested in potential affordances than feeding success (Diamond and Bond ##UREF##4##1999##; Huber et al. ##UREF##13##2001##; Smith et al. ##REF##36258015##2022##; Suwandschieff et al. ##REF##37261570##2023##). These results are in accordance with kea’s natural feeding strategies, as opportunistic group foragers, with kea paying close attention to what others feed on while engaging in individual manipulation strategies to obtain the resources (Diamond and Bond ##UREF##4##1999##). In addition, it corresponds with the characteristics of island-dwelling parrots, as described by Mettke-Hofmann and colleagues (##UREF##17##2002##) who found that island species spend significantly more time on exploratory behaviour especially in areas of seasonally fluctuating food availability.</p>", "<p id=\"Par23\">In conclusion, we find strong evidence that observing a conspecific opening an apparatus via two steps affected the solving success of observer kea. They thus profit from social learning, which aligns well with other studies on parrots and songbirds showing social information transmission and the spread of novel foraging techniques within captive groups and wild populations (e.g. Slagsvold and Wiebe ##UREF##22##2011##; Auersperg et al. ##UREF##1##2014##; Aplin et al. ##REF##25470065##2015##; Klump et al. ##REF##34437121##2021##). However, in our study, the response topography of solvers was variable and the copying fidelity was at a very low level, providing no indication of motor imitation over emulation or stimulus enhancement in kea. Therefore, our findings corroborate that kea display strong behavioural variability when attempting to solve a complex motor task. They also keep on exploring options after a successful solution and may rapidly shift solving strategies. Taken together, this makes kea a great model system to study behavioural flexibility but not so much for imitation.</p>" ]

[]

[ "<p>Communicated by F. Bairlein.</p>", "<p id=\"Par1\">Social learning is an important aspect of dealing with the complexity of life. The transmission of information via the observation of other individuals is a cost-effective way of acquiring information. It is widespread within the animal kingdom but may differ strongly in the social learning mechanisms applied by the divergent species. Here we tested eighteen Kea (<italic>Nestor notabilis</italic>) parrots on their propensity to socially learn, and imitate, a demonstrated sequence of steps necessary to open an apparatus containing food. The demonstration by a conspecific led to more successful openings by observer birds, than control birds without a demonstration. However, all successful individuals showed great variation in their response topography and abandoned faithfully copying the task in favour of exploration. While the results provide little evidence for motor imitation they do provide further evidence for kea’s propensity towards exploration and rapidly shifting solving strategies, indicative of behavioural flexibility.</p>", "<title>Supplementary Information</title>", "<p>The online version contains supplementary material available at 10.1007/s10336-023-02127-y.</p>", "<p id=\"Par2\"><bold>Keas, Vögel der Vielseitigkeit. Kea-Papageien (</bold><bold><italic>Nestor notabilis</italic></bold><bold>) zeigen eine hohe Verhaltensflexibilität bei der Lösung einer demonstrierten Sequenzaufgabe.</bold>\n</p>", "<p>Soziales Lernen ist ein wichtiger Aspekt im Umgang mit der Komplexität des Lebens. Das Erlangen von Informationen durch die Beobachtung Anderer ist eine effiziente Möglichkeit der Informationsbeschaffung. Diese Art des Lernens ist im Tierreich weit verbreitet, kann sich jedoch hinsichtlich der unterschiedlichen sozialen Lernmechanismen, welcher sich die verschiedenen Arten bedienen, stark unterscheiden. Achtzehn Kea-Papageien (<italic>Nestor notabilis</italic>) wurden auf soziales Lernen, im Speziellen auf imitatives Lernen einer demonstrierten Abfolge von Schritten zum Öffnen eines mit Futter gefüllten Apparats, getestet. Die Resultate ergaben, dass eine Demonstration durch einen Artgenossen bei Beobachtervögeln zu erfolgreicheren Lösungsansätzen führten als bei Kontrollvögeln welche keine Demonstration erhielten. Allerdings zeigten alle erfolgreichen Individuen große Unterschiede in der Topographie ihrer Reaktion, der genauen Abfolge an Bewegungen. Ebenfalls gaben sie das getreue Kopieren der Aufgabe zugunsten von Erkundungs- und/oder individuellen Lösungsstrategien auf. Insgesamt zeigten unsere Ergebnisse, dass Keas starke Verhaltensvariabilität und Flexibilität zeigen, wenn sie eine komplexe motorische Aufgabe lösen. Damit stellen sie ein großartiges Modellsystem zur Untersuchung der Verhaltensflexibilität dar.</p>", "<title>Keywords</title>", "<p>Open access funding provided by Austrian Science Fund (FWF).</p>" ]

[ "<title>Supplementary Information</title>", "<p>Below is the link to the electronic supplementary material.</p>" ]

[ "<title>Acknowledgements</title>", "<p>We are thankful to Andràs Pèter for his help with devising the test box, Louise Mackie for her help with interobserver reliability check, Remco Folkertsma for his help with data analysis, the Austrian Science Fund (FWF) for funding the project (P 33507-B) and to the entire staff at the research station Haidlhof for their hard work and ongoing support.</p>", "<title>Author contributions</title>", "<p>ES: design, data collection, data curation and analysis, manuscript original draft, revisions, final manuscript; LH: conceptualisation, design, supervision, manuscript review; TB: conceptualisation, design, supervision, manuscript review; RS: conceptualisation, funding acquisition, design, supervision, manuscript review and editing. All authors read and approved the final manuscript.</p>", "<title>Funding</title>", "<p>Open access funding provided by Austrian Science Fund (FWF). This research was funded in whole, or in part, by the Austrian Science Fund (FWF) [P 33507-B]. For the purpose of open access, the author has applied a CC BY public copyright licence to any Author Accepted Manuscript version arising from this submission.</p>", "<title>Data availability</title>", "<p>All data generated or analysed during this study are included in this published article [supplementary material file: raw data].</p>", "<title>Declarations</title>", "<title>Conflict of interest</title>", "<p id=\"Par24\">All authors have read the submitted version of this manuscript and declare no conflicts of interest. The authors have no relevant non-financial interests to disclose.</p>", "<title>Ethical approval</title>", "<p id=\"Par25\">The study is in accordance with the Good Scientific Practice guidelines and national legislation (ETK-10/11/2016) and has been approved by the institutional ethics and animal welfare committee at the University of Veterinary Medicine, Vienna.</p>" ]

[ "<fig id=\"Fig1\"><label>Fig. 1</label><caption><p>Test box <bold>a</bold> front view of two pins <bold>b</bold> side view of green and blue ring respectively</p></caption></fig>" ]

[ "<table-wrap id=\"Tab1\"><label>Table 1</label><caption><p>Individuals that successfully solved the task per experimental group</p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th align=\"left\">Name</th><th align=\"left\">Sex</th><th align=\"left\">Age</th><th align=\"left\">Raised</th><th align=\"left\">Group</th><th align=\"left\">No of successful sessions (out of 3)</th><th align=\"left\">Demonstrator</th><th align=\"left\">Applied demonstrated sequence</th><th align=\"left\">Applied non demonstrated sequence</th><th align=\"left\">Stayed consistent for no of sessions</th><th align=\"left\">Switched in session no</th></tr></thead><tbody><tr><td align=\"left\">Frowin</td><td align=\"left\">Male</td><td align=\"left\">17</td><td align=\"left\">Parent</td><td align=\"left\">Control</td><td align=\"left\">3</td><td align=\"left\">NA</td><td align=\"left\">NA</td><td align=\"left\">NA</td><td align=\"left\">3/3</td><td align=\"left\">NA</td></tr><tr><td align=\"left\">Papu</td><td align=\"left\">Female</td><td align=\"left\">8</td><td align=\"left\">Hand</td><td align=\"left\">Control</td><td align=\"left\">1</td><td align=\"left\">NA</td><td align=\"left\">NA</td><td align=\"left\">NA</td><td align=\"left\">1/1</td><td align=\"left\">NA</td></tr><tr><td align=\"left\">John*</td><td align=\"left\">Male</td><td align=\"left\">22</td><td align=\"left\">Parent</td><td align=\"left\">Test</td><td align=\"left\">3</td><td align=\"left\">Frowin</td><td align=\"left\">in Session 1 and 2</td><td align=\"left\">in Session 3</td><td align=\"left\">2/3</td><td align=\"left\">3</td></tr><tr><td align=\"left\">Tai*</td><td align=\"left\">Female</td><td align=\"left\">3</td><td align=\"left\">Parent</td><td align=\"left\">Test</td><td align=\"left\">2</td><td align=\"left\">Frowin</td><td align=\"left\">in Session 1 and 3</td><td align=\"left\">Never</td><td align=\"left\">2/2</td><td align=\"left\">NA</td></tr><tr><td align=\"left\">Coco</td><td align=\"left\">Female</td><td align=\"left\">14</td><td align=\"left\">Hand</td><td align=\"left\">Test</td><td align=\"left\">2</td><td align=\"left\">Paul</td><td align=\"left\">in Session 2</td><td align=\"left\">In Session 1</td><td align=\"left\">1/2</td><td align=\"left\">2</td></tr><tr><td align=\"left\">Pick</td><td align=\"left\">Male</td><td align=\"left\">17</td><td align=\"left\">Hand</td><td align=\"left\">Test</td><td align=\"left\">2</td><td align=\"left\">Paul</td><td align=\"left\">Never</td><td align=\"left\">In Session 1 and 2</td><td align=\"left\">2/2</td><td align=\"left\">NA</td></tr><tr><td align=\"left\">Sunny</td><td align=\"left\">Female</td><td align=\"left\">14</td><td align=\"left\">Hand</td><td align=\"left\">Control Test</td><td align=\"left\">1</td><td align=\"left\">Paul</td><td align=\"left\">Never</td><td align=\"left\">In Session 1</td><td align=\"left\">1/1</td><td align=\"left\">NA</td></tr><tr><td align=\"left\">Skipper</td><td align=\"left\">Male</td><td align=\"left\">4</td><td align=\"left\">Hand</td><td align=\"left\">Control Test</td><td align=\"left\">2</td><td align=\"left\">Paul</td><td align=\"left\">in Session 2 and 3</td><td align=\"left\">Never</td><td align=\"left\">2/2</td><td align=\"left\">NA</td></tr><tr><td align=\"left\">Fay</td><td align=\"left\">Female</td><td align=\"left\">5</td><td align=\"left\">Parent</td><td align=\"left\">Control Test</td><td align=\"left\">1</td><td align=\"left\">Frowin</td><td align=\"left\">in Session 1</td><td align=\"left\">Never</td><td align=\"left\">1/1</td><td align=\"left\">NA</td></tr><tr><td align=\"left\">Plume</td><td align=\"left\">Female</td><td align=\"left\">14</td><td align=\"left\">Hand</td><td align=\"left\">Control Test</td><td align=\"left\">2</td><td align=\"left\">Frowin</td><td align=\"left\">in Session 3</td><td align=\"left\">In Session 1</td><td align=\"left\">1/2</td><td align=\"left\">3</td></tr></tbody></table></table-wrap>" ]

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[ "<supplementary-material content-type=\"local-data\" id=\"MOESM1\"></supplementary-material>", "<supplementary-material content-type=\"local-data\" id=\"MOESM2\"></supplementary-material>", "<supplementary-material content-type=\"local-data\" id=\"MOESM3\"></supplementary-material>", "<supplementary-material content-type=\"local-data\" id=\"MOESM4\"></supplementary-material>" ]

[ "<table-wrap-foot><p>Table indicates name, sex, age, rearing, and experimental group. Number of successful sessions out of three test sessions, assigned demonstrator, application of demonstrated sequence and according session information, application of non-demonstrated sequence and according session information, count of consistent opening sequence out of three test sessions, and number of switches in opening sequence out of three test sessions</p><p>*Received two additional trials</p></table-wrap-foot>", "<fn-group><fn><p>This article is a contribution to the Topical Collection ‘50 years anniversary of the Nobel Prize in Physiology or Medicine to Karl von Frisch, Konrad Lorenz and Niko Tinbergen in 1973’.</p></fn><fn><p><bold>Publisher's Note</bold></p><p>Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.</p></fn></fn-group>" ]

[ "<graphic xlink:href=\"10336_2023_2127_Fig1_HTML\" id=\"MO1\"/>" ]

[ "<media xlink:href=\"10336_2023_2127_MOESM1_ESM.pdf\"><caption><p>Supplementary file1 (PDF 574 KB)</p></caption></media>", "<media xlink:href=\"10336_2023_2127_MOESM2_ESM.xlsx\"><caption><p>Supplementary file2 (XLSX 60 KB)</p></caption></media>", "<media xlink:href=\"10336_2023_2127_MOESM3_ESM.mov\"><caption><p>Supplementary file3 (MOV 212979 KB)</p></caption></media>", "<media xlink:href=\"10336_2023_2127_MOESM4_ESM.mov\"><caption><p>Supplementary file4 (MOV 115808 KB)</p></caption></media>" ]

{ "acronym": [], "definition": [] }

[ "<title>Introduction</title>", "<p id=\"Par4\">Glucuronoyl esterases cleave ester-linked lignin-carbohydrate complexes (LCCs) in lignocellulosic biomass (Mosbech et al. ##REF##29321810##2018##). In this way, their catalytic action assists in decomposing the plant cell wall matrix by removing some of the covalent interpolymeric linkages that sustain lignocellulose recalcitrance.</p>", "<p id=\"Par5\">As literature sees it today, ester-linked LCCs exist in vivo between the α-1,2-linked-D-glucuronoyl substitutions of glucuronoxylan and the aliphatic hydroxyl groups (Balakshin et al. ##REF##21298285##2011##; Yuan et al. ##REF##21879769##2011##). An important additional feature observed in many wood and cereal types of biomass, is the 4-<italic>O</italic>-methylation on said glucuronoyl, which creates a unique plant-based trademark for these aldouronic acids. We commonly agree on the carbohydrate side of the LCC, but the exact nature of the aliphatic alcohol on lignin remains a matter of debate (Giummarella et al. ##REF##31294799##2019##; Sapouna &amp; Lawoko ##UREF##3##2021##). In the most abundant lignin-substructures, two alcohol moieties present themselves as possible esterification points, namely the α- and the γ-positioned ones (Fig. ##FIG##0##1##, structure 1 vs. structure 2), and whichever is most dominant in vivo is difficult to verify, because the occurrence is low, and because they prove difficult to isolate and annotate exactly. Ester bound acids tend to migrate (Puchart et al. ##REF##31952591##2020##) and extensive sample handling in order to enrich these linkages may introduce artifacts. Hence, de novo ester synthesis in the plant cell may start in one position and migrate to a different one, as the cell wall matures (Li &amp; Helm ##UREF##1##1995##).</p>", "<p id=\"Par6\">Glucuronoyl esterases of both fungal and bacterial origin has received much attention in research during the past ten to fifteen years since their discovery (Agger et al. ##REF##37256329##2023##; Biely ##UREF##0##2016##; Larsbrink and Lo Leggio ##REF##36651189##2023##). Activity has been demonstrated widely on both synthetic and natural substrates, and several crystal structures reveal the catalytic mechanism of these canonical α/β-hydrolases of the serine-esterase type (Baath et al. ##REF##30814248##2019##; Charavgi et al. ##REF##23275164##2013##; Ernst et al. ##REF##31911652##2020##; Mazurkewich et al. ##UREF##2##2019##; Topakas et al. ##REF##20473662##2010##). We know that fungal variants are generally more restricted in their substrate preferences than the bacterial enzymes, because most of the fungal ones are dependent on the 4-<italic>O</italic>-methyl-modification on the glucuronoyl moiety for recognition. Bacterial enzymes are broader in their substrate preferences in terms of the sugar-moiety, but it is still uncertain for all types of glucuronoyl esterases, if they have specificity towards the configuration of the lignin-alcohol and if so, which of the two possible variants they prefer.</p>", "<p id=\"Par7\">We have investigated the substrate specificity of a fungal glucuronoyl esterase from <italic>Cerrena unicolor</italic> (<italic>Cu</italic>GE) by exploiting the fact that esterases often display transesterification capacities under the right reaction conditions. We hypothesize that the enzyme will preferably perform transesterification with the type of alcohol (α- or γ-positioned) that fits its substrate specificities best. We performed transesterifications with either methyl-glucuronate or methyl-4-<italic>O</italic>-methyl-glucuronate as donor substrates, and benzyl-alcohol or 3-phenyl-1-propanol as the alcohol-acceptors according to the four reactions outlined in Fig. ##FIG##0##1##.</p>" ]

[ "<title>LC–MS method</title>", "<p id=\"Par17\">Reaction product profiles were analyzed by LC–MS. 5 µL of reaction mixture was injected onto a Hypercarb column (150 mm × 2.1 mm; 3μm, Thermo Fischer Scientific, Sunnyvale, CA, USA). The chromatography was performed on a Dionex UltiMate 3000 UPLC (Thermo Fischer Scientific, Sunnyvale, CA, USA) at 0.4 mL min⁻¹ and 70 °C with a two-eluent system consisting of eluent A (acetonitrile) and eluent B (water). The elution was performed as follows (time indicated in min): 0–3, 0% A 100% B; 3–10, isocratic 40% A 60% B; 10–19, isocratic 0% A 100% B. Water was used as eluent B to avoid excessive spontaneous hydrolysis of either donor substrates or products on the LC-column.</p>", "<p id=\"Par18\">For the analysis of the transesterification of PhPrOH and Me-4-<italic>O</italic>-MeGlcA the chromatography was performed on the same system and condition of flow, temperature and eluent system but the elution was performed as follows (time indicated in min): 0–3, 0% A 100% B; 3–10, isocratic 70% A 30% B; 10–19, isocratic 0% A 100% B. The differences in elution system for this particular reaction was in order to elute the stronger retained transesterification product of this reaction compared to the other reactions within the same period.</p>", "<p id=\"Par19\">The HPLC was connected to an ESI-iontrap (model Amazon SL from Bruker Daltonics, Bremen, Germany) and the electrospray was operated in positive ultra scan mode using a target mass of 300 m/z. A scan range from 100 to 2000 m/z was selected and capillary voltage was set to 4.5 kV, end plate offset 0.5 kV, nebulizer pressure at 3.0 bar, dry gas flow at 12.0 L min⁻¹, and dry gas temperature at 280 °C.</p>" ]

[ "<title>Results and discussion</title>", "<p id=\"Par26\">Initial hydrolytic experiments with either Me-4-<italic>O</italic>-MeGlcA or Me-GlcA as substrate and in the absence of alcohol acceptors show substrate conversion as expected (Fig. ##FIG##1##2##), with a significantly higher level of conversion for the 4-<italic>O</italic>-methylated ester compared to the non-derivatized glucuronate ester within the first 100 min of reaction (Table ##TAB##0##1##). Enzyme concentrations were dosed to allow similar rates of catalysis in all experiments, independent of substrate, and consequently <italic>Cu</italic>GE was dosed 50 times less in the experiments where Me-4-<italic>O</italic>-MeGlcA was the substrate compared to the experiments with Me-GlcA as substrate. The fact that <italic>Cu</italic>GE prefers its substrates to have the 4-<italic>O</italic>-methyl-modification on the glucuronoyl is well-establised (d’Errico et al. ##REF##25425346##2015##; Ernst et al. ##REF##31911652##2020##; Monrad et al. ##REF##29739247##2018##). As the 100 min reaction time progresses, it becomes evident that the hydrolytic reaction slows down for both substrates (Table ##TAB##0##1##), which again is not surprising as the affinity (K_M) for these synthetic model substrates is known to be relatively high (d’Errico et al. ##REF##25425346##2015##).</p>", "<p id=\"Par27\">It is known, that glucuronoyl esterases prefer bulky alcohols and in that respect, the methyl-esters are poor substrates. Experiments with either benzyl-alcohol or 3-phenyl-1-propanol as the primary solvent and thereby dominating acceptor molecule (14% v/v water), show that the transesterification reaction occurring between Me-4-<italic>O</italic>-MeGlcA and PhPrOH (Fig. ##FIG##0##1##, reaction 1) is significantly higher than any of the other combinations (Fig. ##FIG##2##3##). The reaction between Me-4-<italic>O</italic>-MeGlcA and BnzOH is second best (Fig. ##FIG##0##1##, reaction 2) with about 3–4 fold less product compared to the PhPr-GlcA ester. Quantifications of the ester products are done relative to the standard of Bnz-GlcA dissolved either in BnzOH or PhPrOH depending on the relevant acceptor alcohol, and the calibrations indicate a certain level of ion suppression when PhPrOH is the solvent (Supplementary standards curves), hence leading to a potential underestimation in that respect. PhPrOH has similar retention time as the transesterification products. At the same time, compounds carrying the 4-<italic>O</italic>-methylation tends to ionize better, and therefore give stronger responses. In the reactions performed here, enzyme loadings vary according to formation of transesterification products, as the goal is to quantify transesterification products. Hence, at low enzyme loadings, hydrolysis is not observed for reactions containing Me-4-<italic>O</italic>-MeGlcA as donor (and relatively low water concentration), whereas increasing the enzyme concentration yields hydrolytic reactions in parallel to the transesterification (data not shown). Ultimately, the extent of hydrolysis is a competition for acceptor substrate, which appears favored in the case of the alternative alcohol when enzyme concentrations are low. Hydrolytic reactions are certainly present but becomes un-quantifiable at conditions where transesterification is dominating.</p>", "<p id=\"Par28\">Interestingly, the 4-<italic>O</italic>-methylation is a more important feature for catalysis than the nature of the acceptor-alcohol, since the Bnz-4-<italic>O</italic>-MeGlcA-ester forms faster than the PhPr-GlcA ester. A cautious estimation of the effect of the donor versus the effect of the acceptor demonstrates that the ratio between concentrations of products formed during the reaction time is higher when the donor is changed, compared to when the acceptor is changed (Fig. ##FIG##3##4##).</p>", "<p id=\"Par29\">Hence, the effect of changing the donor substrate between the 4-<italic>O</italic>-methylated and non-methylated counterparts is larger than the effect of changing the alcohol acceptor. This observation may not be valid for all glucuronoyl esterases, as it is common knowledge that not all GEs favor the 4-<italic>O</italic>-methylation.</p>", "<p id=\"Par30\">Recent QM/MM studies of the fungal <italic>Thermothelomyces thermophila</italic> glucuronoyl esterase (<italic>Tt</italic>GE) show that the acylation step is the most energy demanding and hence rate-limiting step (Viegas et al. ##REF##35925549##2022##), whereas similar studies with a bacterial <italic>Ot</italic>CE15A from <italic>Opitutus terrae</italic> show that the deacylation-step is rate-limiting (Zong et al. ##REF##34983933##2022##). Without having investigated the energetic landscape of catalysis by <italic>Cu</italic>GE, the transesterification reactions we observe support the observation that the acylation-step is determining for reaction speed. However, it is well-known that the differences between fungal and bacterial GEs are quite large in terms of overall structure (Larsbrink and Lo Leggio ##REF##36651189##2023##), and transesterification reactions of the kind presented here, may turn out different for bacterial GEs with respect to energetic fingerprints.</p>", "<p id=\"Par31\">The aim of this study is to investigate the preference for either an α- or a γ-positioned alcohol, and the results show that the γ-positioned alcohol is most favorable, yet the enzyme can perform the reaction with the α-alcohol. There are currently no univocal structural explanations for how the GEs interact with the alcohol moiety of the substrate. However, we have recently performed docking simulations with these exact two esters; Bnz-4-<italic>O</italic>-MeGlcA and PhPr-4-<italic>O</italic>-MeGlcA as ligands in <italic>Cu</italic>GE and guided by the sugar-moiety, which show steric obstacles when the α-benzyl constitutes the alcohol part of the ligand in contrary to the γ-linked ester (Agger et al. ##REF##37256329##2023##).</p>", "<p id=\"Par32\">Exploiting the transesterification capacities of these enzymes potentially opens for other biotechnological applications than biomass deconstruction, such as functionalization of hemicellulose or smaller aldouronic acids with alcohols of different properties (hydrophobicity, other functional groups etc.). Future studies may explore the field of potential acceptors more broadly than here, and thereby determine the diversity of alcohols and properties that could be relevant to investigate further.</p>" ]

[ "<title>Results and discussion</title>", "<p id=\"Par26\">Initial hydrolytic experiments with either Me-4-<italic>O</italic>-MeGlcA or Me-GlcA as substrate and in the absence of alcohol acceptors show substrate conversion as expected (Fig. ##FIG##1##2##), with a significantly higher level of conversion for the 4-<italic>O</italic>-methylated ester compared to the non-derivatized glucuronate ester within the first 100 min of reaction (Table ##TAB##0##1##). Enzyme concentrations were dosed to allow similar rates of catalysis in all experiments, independent of substrate, and consequently <italic>Cu</italic>GE was dosed 50 times less in the experiments where Me-4-<italic>O</italic>-MeGlcA was the substrate compared to the experiments with Me-GlcA as substrate. The fact that <italic>Cu</italic>GE prefers its substrates to have the 4-<italic>O</italic>-methyl-modification on the glucuronoyl is well-establised (d’Errico et al. ##REF##25425346##2015##; Ernst et al. ##REF##31911652##2020##; Monrad et al. ##REF##29739247##2018##). As the 100 min reaction time progresses, it becomes evident that the hydrolytic reaction slows down for both substrates (Table ##TAB##0##1##), which again is not surprising as the affinity (K_M) for these synthetic model substrates is known to be relatively high (d’Errico et al. ##REF##25425346##2015##).</p>", "<p id=\"Par27\">It is known, that glucuronoyl esterases prefer bulky alcohols and in that respect, the methyl-esters are poor substrates. Experiments with either benzyl-alcohol or 3-phenyl-1-propanol as the primary solvent and thereby dominating acceptor molecule (14% v/v water), show that the transesterification reaction occurring between Me-4-<italic>O</italic>-MeGlcA and PhPrOH (Fig. ##FIG##0##1##, reaction 1) is significantly higher than any of the other combinations (Fig. ##FIG##2##3##). The reaction between Me-4-<italic>O</italic>-MeGlcA and BnzOH is second best (Fig. ##FIG##0##1##, reaction 2) with about 3–4 fold less product compared to the PhPr-GlcA ester. Quantifications of the ester products are done relative to the standard of Bnz-GlcA dissolved either in BnzOH or PhPrOH depending on the relevant acceptor alcohol, and the calibrations indicate a certain level of ion suppression when PhPrOH is the solvent (Supplementary standards curves), hence leading to a potential underestimation in that respect. PhPrOH has similar retention time as the transesterification products. At the same time, compounds carrying the 4-<italic>O</italic>-methylation tends to ionize better, and therefore give stronger responses. In the reactions performed here, enzyme loadings vary according to formation of transesterification products, as the goal is to quantify transesterification products. Hence, at low enzyme loadings, hydrolysis is not observed for reactions containing Me-4-<italic>O</italic>-MeGlcA as donor (and relatively low water concentration), whereas increasing the enzyme concentration yields hydrolytic reactions in parallel to the transesterification (data not shown). Ultimately, the extent of hydrolysis is a competition for acceptor substrate, which appears favored in the case of the alternative alcohol when enzyme concentrations are low. Hydrolytic reactions are certainly present but becomes un-quantifiable at conditions where transesterification is dominating.</p>", "<p id=\"Par28\">Interestingly, the 4-<italic>O</italic>-methylation is a more important feature for catalysis than the nature of the acceptor-alcohol, since the Bnz-4-<italic>O</italic>-MeGlcA-ester forms faster than the PhPr-GlcA ester. A cautious estimation of the effect of the donor versus the effect of the acceptor demonstrates that the ratio between concentrations of products formed during the reaction time is higher when the donor is changed, compared to when the acceptor is changed (Fig. ##FIG##3##4##).</p>", "<p id=\"Par29\">Hence, the effect of changing the donor substrate between the 4-<italic>O</italic>-methylated and non-methylated counterparts is larger than the effect of changing the alcohol acceptor. This observation may not be valid for all glucuronoyl esterases, as it is common knowledge that not all GEs favor the 4-<italic>O</italic>-methylation.</p>", "<p id=\"Par30\">Recent QM/MM studies of the fungal <italic>Thermothelomyces thermophila</italic> glucuronoyl esterase (<italic>Tt</italic>GE) show that the acylation step is the most energy demanding and hence rate-limiting step (Viegas et al. ##REF##35925549##2022##), whereas similar studies with a bacterial <italic>Ot</italic>CE15A from <italic>Opitutus terrae</italic> show that the deacylation-step is rate-limiting (Zong et al. ##REF##34983933##2022##). Without having investigated the energetic landscape of catalysis by <italic>Cu</italic>GE, the transesterification reactions we observe support the observation that the acylation-step is determining for reaction speed. However, it is well-known that the differences between fungal and bacterial GEs are quite large in terms of overall structure (Larsbrink and Lo Leggio ##REF##36651189##2023##), and transesterification reactions of the kind presented here, may turn out different for bacterial GEs with respect to energetic fingerprints.</p>", "<p id=\"Par31\">The aim of this study is to investigate the preference for either an α- or a γ-positioned alcohol, and the results show that the γ-positioned alcohol is most favorable, yet the enzyme can perform the reaction with the α-alcohol. There are currently no univocal structural explanations for how the GEs interact with the alcohol moiety of the substrate. However, we have recently performed docking simulations with these exact two esters; Bnz-4-<italic>O</italic>-MeGlcA and PhPr-4-<italic>O</italic>-MeGlcA as ligands in <italic>Cu</italic>GE and guided by the sugar-moiety, which show steric obstacles when the α-benzyl constitutes the alcohol part of the ligand in contrary to the γ-linked ester (Agger et al. ##REF##37256329##2023##).</p>", "<p id=\"Par32\">Exploiting the transesterification capacities of these enzymes potentially opens for other biotechnological applications than biomass deconstruction, such as functionalization of hemicellulose or smaller aldouronic acids with alcohols of different properties (hydrophobicity, other functional groups etc.). Future studies may explore the field of potential acceptors more broadly than here, and thereby determine the diversity of alcohols and properties that could be relevant to investigate further.</p>" ]

[ "<title>Conclusion</title>", "<p id=\"Par33\"><italic>Cu</italic>GE prefers methyl-4-<italic>O</italic>-methyl-glucuronate as donor substrate and the γ-positioned acceptor according to the experiments conducted here, and evaluated based on yield and rate of product formation. It is important to emphasize that the results reported here illustrate the substrate preferences of <italic>Cu</italic>GE, which is not to be generalized for all GEs. These results exemplify a methodology for investigating substrate preferences, and future studies may include reverse hydrolytic reactions.</p>", "<p id=\"Par34\">We observe large differences in product formation rates during the first 100 min of reaction with the formation of the 4-<italic>O</italic>-methyl-glucuronoyl-3-propane-phenyl as fastest, and the results clearly indicate that <italic>Cu</italic>GE prefers the γ-positioned alcohol during transesterification. The most important factor for reaction though, continues to be the presence of the 4-<italic>O</italic>-methylation derivatization of the glucuronoyl moiety.</p>", "<p id=\"Par35\">These experiments demonstrate a methodology where the enzyme reveals its substrate preferences immediately by favoring product formation from the most suitable substrates. These results do not inform about the prevalence of either α- or γ-LCC esters in lignocellulosic biomass, but it certainly informs about the preferences of this particular enzyme. Given that enzymes evolve to tackle the linkages present in biomass, it is tempting to speculate that the γ-esters LCC is the more prevalent of the two types in biomass.</p>" ]

[ "<title>Purpose</title>", "<p id=\"Par1\">Glucuronoyl esterases (GE, family CE15) catalyse the cleavage of ester linkages in lignin-carbohydrate complexes (LCCs), and this study demonstrate how transesterification reactions with a fungal GE from <italic>Cerrena unicolor</italic> (<italic>Cu</italic>GE) can reveal the enzyme’s preference for the alcohol-part of the ester-bond.</p>", "<title>Methods</title>", "<p id=\"Par2\">This alcohol-preference relates to where the ester-LCCs are located on the lignin molecule, and has consequences for how the enzymes potentially interact with lignin. It is unknown exactly what the enzymes prefer; either the α-benzyl or the γ-benzyl position. By providing the enzyme with a donor substrate (the methyl ester of either glucuronate or 4-<italic>O</italic>-methyl-glucuronate) and either one of two acceptor molecules (benzyl alcohol or 3-phenyl-1-propanol) we demonstrate that the enzyme can perform transesterification and it serves as a method for assessing the enzyme’s alcohol preferences.</p>", "<title>Conclusion</title>", "<p id=\"Par3\"><italic>Cu</italic>GE preferentially forms the γ-ester from the methyl ester of 4-<italic>O</italic>-methyl-glucuronate and 3-phenyl-1-propanol and the enzyme’s substrate preferences are primarily dictated by the presence of the 4-<italic>O</italic>-methylation on the glucuronoyl donor, and secondly on the type of alcohol.</p>", "<title>Supplementary Information</title>", "<p>The online version contains supplementary material available at 10.1007/s10529-023-03456-x.</p>", "<title>Keywords</title>", "<p>Open access funding provided by Technical University of Denmark</p>" ]

[ "<title>Materials and enzyme preparation</title>", "<p id=\"Par8\">Methyl 4-<italic>O</italic>-methyl-D-glucopyranosyluronate (Me-4-O-MeGlcA) was purchased from Institute of Chemistry, Slovak Academy of Sciences, Bratislava, Slovakia. Glucuronic acid methyl ester (Me-GlcA) and benzyl D-glucuronate (Bnz-GlcA) were purchased from Carbosynth. Benzyl alcohol (BnzOH), 3-phenyl-1-propanol (PhPrOH) and all other chemicals were purchased from Sigma.</p>", "<p id=\"Par9\">The construct containing the gene encoding for the CE15 glucuronoyl esterase from <italic>Cerrena unicolor</italic> (<italic>Cu</italic>GE) was produced in <italic>P. pastoris</italic> as previously described (Mosbech et al. ##REF##29321810##2018##). After fermentation the cells were separated by centrifugation and the fermentation broth (3 L) was sterile filtered and concentrated by ultrafiltration on a 10 kDa cut-off membrane to a final volume of approx. 80 mL. <italic>Cu</italic>GE was purified by affinity chromatography on an IMAC-column (HisTrap HP 5 mL column, GE Healthcare) using an Äkta Purifier 100 (GE Healthcare, Uppsala Sweden).</p>", "<title>Enzyme reactions</title>", "<p id=\"Par10\">The transesterification activity of <italic>Cu</italic>GE was tested in four different experimental setups using either Me-4-<italic>O</italic>-MeGlcA or Me-GlcA as donor substrates and benzyl alcohol (BnzOH) or 3-phenyl-1-propanol (PhPrOH) as acceptor substrates, allowing the assessment of the transesterification ability and preferences of <italic>Cu</italic>GE in either the α and γ position (Fig. ##FIG##0##1##).</p>", "<p id=\"Par11\">Transesterification with PhPrOH as acceptor molecule was performed in a 100 µL reaction mixture containing 0.45 mM of Me-4-<italic>O</italic>-MeGlcA or 0.48 mM of Me-GlcA (dissolved in 10 mM Na acetate buffer pH 6) and 6.32 M of PhPrOH in order to have an acceptor–donor ratio of approx. 14,000 and of 13,000 with Me-4-<italic>O</italic>-MeGlcA and Me-GlcA, respectively. The reaction was started in a HPLC vial by addition of 0.02 µM or 0.87 µM of <italic>Cu</italic>GE for the reaction with Me-4-<italic>O</italic>-MeGlcA and Me-GlcA, respectively.</p>", "<p id=\"Par12\">Transesterification with BnzOH as acceptor molecule was performed in a 100 µL reaction mixture containing 0.45 mM of Me-4-<italic>O</italic>-MeGlcA or 0.48 mM of Me-GlcA (dissolved in 10 mM Na acetate buffer pH 6) and 8.31 M of BnzOH in order to have an acceptor donor ratio of approx.. 18,500 and of 17,500 with Me-4-<italic>O</italic>-MeGlcA and Me-GlcA, respectively. The reaction was started in a HPLC vial by addition of 0.11 µM or 4.33 µM of <italic>Cu</italic>GE for the reaction with Me-4-<italic>O</italic>-MeGlcA and Me-GlcA, respectively. All transesterification reactions were run in triplicates.</p>", "<p id=\"Par13\">The reaction samples were placed in the UHPLC’s autosampler at 40 °C and the reaction evolution was followed directly on LC–MS by injecting 5 µL of the reaction mixture onto the column every 20 min followed by the LC–MS method described below.</p>", "<p id=\"Par14\">Control experiments containing 0.45 mM of Me-4-<italic>O</italic>-MeGlcA or 0.48 mM of Me-GlcA (dissolved in 10 mM Na acetate buffer pH 6) and 6.32 M of PhPrOH or 8.31 M of BnzOH and replacing the enzyme volume with an equal amount of 10 mM Na acetate buffer pH 6 were run to assess auto-hydrolysis during the reaction time.</p>", "<p id=\"Par15\">Chromatograms of all transesterification reactions and auto-hydrolytic control experiments are found in Supplementary Figs. S1-S8.</p>", "<p id=\"Par16\">Control experiments showing <italic>Cu</italic>GE’s hydrolytic activity were performed in 100 µL reaction mixture containing 0.45 mM of Me-4-<italic>O</italic>-MeGlcA or 0.48 mM of Me-GlcA dissolved in 10 mM Na acetate buffer pH 6. The reactions were started in the HPLC vial by the addition of 0.08 µM or 4.33 µM of <italic>Cu</italic>GE for the reaction with Me-4-<italic>O</italic>-MeGlcA and Me-GlcA, respectively. The hydrolytic reactions were performed in triplicates. Direct quantification results of substrate depletion are provided in Supplementary Table S1.</p>", "<title>Quantification method</title>", "<p id=\"Par20\">Quantification of all precursor ions was performed using Bruker TASQ software (Bruker Daltonics, Bremen, Germany). All ions were observed as [M + Na]⁺.</p>", "<p id=\"Par21\">Quantification of the Me-4-<italic>O</italic>-MeGlcA hydrolysis by <italic>Cu</italic>GE was performed by defining an Extracted Ion Chromatogram (EIC) of <italic>m/z</italic> 244.95 and <italic>m/z</italic> 467.03 (pseudo-double ion with one sodium), with a width of ± 0.5 and retention time 6.5 min ± 0.2 min, and reaction extend was quantified as substrate depletion relative to calibration with Me-4-<italic>O</italic>-MeGlcA.</p>", "<p id=\"Par22\">Quantification of the Me-GlcA hydrolysis by <italic>Cu</italic>GE was performed by defining an EIC of <italic>m/z</italic> 230.92 and <italic>m/z</italic> 439.07 (pseudo-double ion with one sodium), with a width of ± 0.5, retention time 3.0 min ± 0.2 min, and reaction extent was quantified as substrate depletion relative to calibration with Me-GlcA.</p>", "<p id=\"Par23\">Quantification of product formations of both Bnz-4-<italic>O</italic>-MeGlcA (EIC on <italic>m/z</italic> 321.08 with a width of ± 0.5, retention time 9.8 min) and Bnz-GlcA (EIC on <italic>m/z</italic> 307.04 and <italic>m/z</italic> 591.01 with a width of ± 0.5, retention time 8.0 min), were performed relative to a Bnz-GlcA standard dissolved in BnzOH (EIC on <italic>m/z</italic> 307.04 and <italic>m/z</italic> 591.01 with a width of ± 0.5 and retention time 8.0 min with a window of 0.2 min).</p>", "<p id=\"Par24\">Estimations of PhPr-4-<italic>O</italic>-MeGlcA (EIC on <italic>m/z</italic> 349.14, retention time 9.3 min) and PhPr-GlcA (EIC on <italic>m/z</italic> 335.12, retention time 7.7 min); products of <italic>Cu</italic>GE transesterification of PhPrOH with Me-4-<italic>O</italic>-MeGlcA or Me-GlcA, respectively, were performed using Bnz-GlcA diluted in PhPrOH as standard.</p>", "<p id=\"Par25\">Calibration curves was performed using 6–8 levels of concentrations and fitting the data with a quadratic curve for the Me-4-<italic>O</italic>-MeGlcA and Me-GlcA calibrations and with a linear curve for the Bnz-GlcA (Supplementary Figure S9).</p>", "<title>Supplementary Information</title>", "<p>Below is the link to the electronic supplementary material.</p>" ]

[ "<title>Author contributions</title>", "<p>Author VP performed the experiments and wrote parts of the paper. Author JWA designed the experiments and wrote the paper.</p>", "<title>Funding</title>", "<p>Open access funding provided by Technical University of Denmark. The work was funded by the Novo Nordisk Foundation “Emerging Investigator 2022 – Research within Industrial Biotechnology and Environmental Biotechnology” (grant number NNF22OC0074634) Lignin for the Future granted to JWA.</p>", "<title>Declarations</title>", "<title>Competing interests</title>", "<p id=\"Par36\">The author’s declare no competing interests.</p>" ]

[ "<fig id=\"Fig1\"><label>Fig. 1</label><caption><p>Four transesterification reactions catalyzed by <italic>Cu</italic>GE performed by combining either one of the two different donor substrates, methyl-4-<italic>O</italic>-methyl-glucuronate (1 and 2) and methyl-glucuronate (3 and 4) with either 3-phenyl-1-propanol (PhPrOH, reaction 1 and 3) or benzyl alcohol (BnzOH, reaction 2 and 4) as acceptor substrates</p></caption></fig>", "<fig id=\"Fig2\"><label>Fig. 2</label><caption><p>Comparison between the hydrolysis of Me-4-<italic>O</italic>-MeGlcA and Me-GlcA by <italic>Cu</italic>GE. Reaction evolution over 100 min of the Me-GlcA hydrolysis by (4.3 µM) <italic>Cu</italic>GE is showed in violet at retention time (RT) 3.0 min. MS2 spectrum of Me-GlcA m/z 230.92 is reported in the upper right corner (violet). Reaction evolution over 100 min of the Me-4-<italic>O</italic>-MeGlcA hydrolysis by (0.087 µM) <italic>Cu</italic>GE is showed in blue at RT 6.5 min. MS2 spectrum of Me-4-<italic>O</italic>-MeGlcA m/z 244.95 is reported in the lower right corner (blue). All ions are shown as [M + Na]⁺. The two hydrolysis were run separately</p></caption></fig>", "<fig id=\"Fig3\"><label>Fig. 3</label><caption><p>Product formation after transesterification reactions catalyzed by <italic>Cu</italic>GE. Reactions are combinations of Me-GlcA or Me-4-<italic>O</italic>-MeGlcA with either benzyl-alcohol or 3-phenyl-1-propanol according to reactions in Fig. ##FIG##0##1##. Blue circle: PhPr-4-<italic>O</italic>-MeGlcA ester (product reaction 1, Fig. ##FIG##0##1##). <italic>Cu</italic>GE dosage; 0.02 µM. Green triangle: Bnz-4-<italic>O</italic>-MeGlcA ester (product reaction 2, Fig. ##FIG##0##1##). <italic>Cu</italic>GE dosage; 0.11 µM. Red square: PhPr-GlcA ester (product reaction 3, Fig. ##FIG##0##1##). <italic>Cu</italic>GE dosage; 0.87 µM. Grey diamond: Bnz-GlcA ester (product reaction 4, Fig. ##FIG##0##1##). <italic>Cu</italic>GE dosage; 4.3 µM. Individual plots of each reaction is found in Supplementary Figure S10</p></caption></fig>", "<fig id=\"Fig4\"><label>Fig. 4</label><caption><p>Ratios between concentrations of product formations, which illustrates the effect of changing the donor (left hand side), or changing the acceptor (right hand side). The ratios are calculated based on product concentrations at 100 min reaction time. On the left hand side, the donor substrate is changed, and the acceptor is kept constant. In the right hand side, the acceptor is changed and the donor is kept constant</p></caption></fig>" ]

[ "<table-wrap id=\"Tab1\"><label>Table 1</label><caption><p>Comparison of substrate conversion of either Me-4-<italic>O</italic>-MeGlcA or Me-GlcA relative to enzyme dosage within 100 min reaction time</p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th align=\"left\" rowspan=\"2\">Reaction time</th><th align=\"left\" colspan=\"2\">Accumulated substrate conversion</th><th align=\"left\" colspan=\"2\">Substrate conversion within each time interval</th></tr><tr><th align=\"left\">Me-4-<italic>O</italic>-MeGlcA</th><th align=\"left\">Me-GlcA</th><th align=\"left\">Me-4-<italic>O</italic>-MeGlcA</th><th align=\"left\">Me-GlcA</th></tr><tr><th align=\"left\">min</th><th align=\"left\">mmol/µmol <italic>Cu</italic>GE</th><th align=\"left\">mmol/µmol <italic>Cu</italic>GE</th><th align=\"left\">mmol/µmol <italic>Cu</italic>GE</th><th align=\"left\">mmol/µmol <italic>Cu</italic>GE</th></tr></thead><tbody><tr><td align=\"left\">20</td><td align=\"left\">2.3 ± 0.1</td><td align=\"left\">0.05 ± 0.00</td><td align=\"left\">2.31 ± 0.1</td><td align=\"left\">0.05 ± 0.00</td></tr><tr><td align=\"left\">40</td><td align=\"left\">3.0 ± 0.08</td><td align=\"left\">0.07 ± 0.00</td><td align=\"left\">0.91 ± 0.03</td><td align=\"left\">0.02 ± 0.00</td></tr><tr><td align=\"left\">60</td><td align=\"left\">3.2 ± 0.03</td><td align=\"left\">0.08 ± 0.00</td><td align=\"left\">0.91 ± 0.05</td><td align=\"left\">0.02 ± 0.01</td></tr><tr><td align=\"left\">80</td><td align=\"left\">3.6 ± 0.13</td><td align=\"left\">0.09 ± 0.01</td><td align=\"left\">0.52 ± 0.00</td><td align=\"left\">0.02 ± 0.01</td></tr><tr><td align=\"left\">100</td><td align=\"left\">3.6 ± 0.1</td><td align=\"left\">0.10 ± 0.01</td><td align=\"left\">0.43 ± 0.12</td><td align=\"left\">0.01 ± 0.00</td></tr></tbody></table></table-wrap>" ]

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[ "<supplementary-material content-type=\"local-data\" id=\"MOESM1\"></supplementary-material>" ]

[ "<table-wrap-foot><p>Accumulated substrate conversion signifies how much substrate is converted at a given time point relative to the enzyme concentration. Substrate conversion within each time interval describes how much additional substrate is converted at a given time point since the previous, relative to the enzyme concentration</p></table-wrap-foot>", "<fn-group><fn><p><bold>Publisher's Note</bold></p><p>Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.</p></fn></fn-group>" ]

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[ "<title>Introduction</title>", "<p id=\"Par5\">Adolescent and young adult cancer survivors (AYAs), those aged 15–39 years at initial cancer diagnosis, are considered a unique group with age-specific challenges, from cancer diagnosis until end of life [##REF##25228567##1##, ##UREF##0##2##]. This age range is, however, flexibly applied depending on the research question of interest, country, and health care system [##REF##26849003##3##]. In the Netherlands, pediatric oncology is centralized and includes patients until 18 years of age at diagnosis. The Dutch AYA definition applies to all cancer patients initially diagnosed between 18 and 39 years. As the overall cancer incidence of AYAs has increased over the last decades and the 5-year relative survival is now exceeding 80%, a large part of this growing population will eventually become long-term survivors [##REF##33218178##4##]. As a result of the cancer diagnosis and treatment, these AYAs are at increased risk of long-term (e.g., infertility) and late effects (e.g., secondary malignancies), and experience unmet (age-specific) needs related to finances and mental health for example [##REF##34638332##5##–##UREF##1##7##]. AYAs are in a particularly exposed position for these risk factors due to their often invasive and long-lasting treatments [##UREF##2##8##], and being diagnosed during a complex phase of life, including many physical, emotional, and social transitions [##REF##17613877##9##].</p>", "<p id=\"Par6\">AYAs can have a long life ahead in which suffering from these long-term and late effects can have a significant impact on their health-related quality of life (HRQoL). HRQoL is defined by the survivor’s own perception of one’s health or well-being, including physical, mental, and social aspects [##UREF##3##10##]. Several studies have focused on the HRQoL issues among AYAs: literature shows that AYAs are at increased risk of fatigue [##REF##28917270##11##], cognitive impairment [##REF##34118000##12##], work and financial problems [##REF##34118000##12##, ##REF##35772237##13##], psychological distress [##REF##30636177##14##], and body image issues [##REF##33275802##15##], which can result in a diminished HRQoL. In addition, studies described impacted physical health and functioning [##UREF##4##16##–##REF##29572734##19##], and lower mental health [##REF##33332603##17##–##REF##29572734##19##] in AYAs compared to the general population/older cancer survivors [##REF##29572734##19##]. In line, the systematic review of Quinn showed that AYAs are more likely to have impaired HRQoL compared to the general population, although QoL was difficult to measure due to their age-specific needs [##UREF##5##20##].</p>", "<p id=\"Par7\">Many factors can independently impact HRQoL, yet they often co-occur in cancer patients [##REF##26187660##21##–##REF##26645947##25##]. To study the interconnectedness between these factors, a network analysis can be performed. It provides insight into the relationships among symptoms, risk factors, and protective factors. Network approaches involve the identification of symptoms and factors (network nodes) and the relations among them (positive or negative associations between nodes) [##UREF##6##26##]. Taking into account the dependence of factors, this type of analysis is more likely reflecting reality compared to focusing on these factors independently [##REF##31435892##27##].</p>", "<p id=\"Par8\">Although network analyses have been performed previously among cancer survivors in general [##REF##31435892##27##–##REF##26370099##29##], they are lacking among AYAs, especially when focusing on HRQoL outcomes. Gaining these insights is of importance to optimize supportive care and provide targeted interventions for this unique long-term surviving population. Within a large population-based sample of long-term AYAs, using an exploratory approach, we want to (1) assess HRQoL in long-term AYAs, (2) identify the most central nodes in a HRQoL network, and (3) determine how these nodes are linked to one another ((strengths of) interconnectedness).</p>" ]

[ "<title>Methods</title>", "<title>Study population and data collection</title>", "<p id=\"Par9\">Data of the population-based, cross-sectional SURVAYA study was used, which was approved by the Institutional Review Board (IRBd18122) and registered within clinical trial registration (NCT05379387). The study population is extensively described previously [##REF##36005166##30##]. In short, the SURVAYA study was performed among AYAs (18–39 years old at time of initial cancer diagnosis) diagnosed with cancer between 1999 and 2015 (5–20 years post diagnosis at study invitation), treated at the Netherlands Cancer Institute or one of the University Medical Centers in the Netherlands, and registered within the Netherlands Cancer Registry (NCR). Survivors were invited to complete a one-time questionnaire within PROFILES (Patient Reported Outcomes Following Initial treatment and Long-term Evaluation of Survivorship) [##REF##21621408##31##].</p>", "<title>Measures</title>", "<p id=\"Par10\">Sociodemographic characteristics were obtained through a one-time questionnaire including age at time of questionnaire, sex at birth, marital status, and educational level. Clinical characteristics, obtained by the NCR, include tumor type, stage, primary treatment received, and time since diagnosis. Tumor type was classified according to the third International Classification of Diseases for Oncology (ICDO-3) [##UREF##7##32##]. Cancer stage was classified according to TNM or Ann Arbor Code (Hodgkin lymphoma and Non-Hodgkin lymphoma) [##UREF##8##33##].</p>", "<p id=\"Par11\">The EORTC QLQ-SURV111 [##UREF##9##34##], a cancer core survivorship questionnaire that is currently being developed by the European Organization for Research and Treatment of Cancer (EORTC), was used for our network analysis. This questionnaire assesses long-term HRQoL outcomes, including physical, mental, and social HRQoL issues specifically relevant to cancer survivors. We selected 8 functioning scales (physical functioning, cognitive functioning, emotional functioning, role functioning, body image, symptom awareness, sexual functioning, and overall quality of life), 9 symptoms scales (fatigue, sleep problems, pain, social interference, health distress, negative health outlook, social isolation, symptom checklist, and sexual problems), and 3 single items (financial difficulties, worry cancer risk family, and treated differently). Scales and items measuring positive factors of HRQoL and items that were not applicable for all participants were excluded from the analysis. The rationale for this selection is that including these positive scales/items and optional items would result in difficulties with respectively interpreting network associations to intervene on and the development of a network structure. Participants scored the items on a 4-point Likert scale from 1 (not at all) to 4 (very much). Overall quality of life scores ranged from 1 (very poor) to 7 (excellent). All scales and single items were linearly transformed to a “0–100” scale [##UREF##10##35##]. A higher score on the functioning scales indicates better HRQoL/functioning, while a higher score for symptoms indicates more complaints. For our analysis, we transformed the scores of the symptom scales once again, so a higher score on the symptom scale indicates fewer complaints. Now, a higher score on all scales corresponds with better functioning, fewer complaints, and a better overall quality of life.</p>", "<title>Data analysis</title>", "<p id=\"Par12\">Data were analyzed using SPSS Statistics (IBM Corporation, version 26.0, Armonk, NY, USA) and R version 4.2.1 packages MVN, huge, qgraph, and bootnet. Descriptive statistics were calculated and presented as frequencies, percentages, means, and standard deviations.</p>", "<p id=\"Par13\">We used listwise deletion to exclude participants with missing items on the EORTC QLQ-SURV111 questionnaire, except for the items related to sexuality. This is because sexual functioning and problems can be an important aspect of HRQoL, and missing responses to sensitive questions like sexuality are common [##REF##21134739##36##]. If half of the items from the two sexuality scales were answered, we assumed that the missing item had a value equal to the item that was present for that respondent according to the QLQ-SURV111 scoring manual. In case both items for the scale were missing, we used a copy mean imputation of the study population.</p>", "<p id=\"Par14\">We assessed the assumption of multivariate normality with Mardia’s test [##UREF##11##37##], which needs to be fulfilled prior to estimating the network [##REF##28342071##38##]. Mardia’s multivariate skewness and kurtosis coefficients of the numeric scales were calculated [##UREF##11##37##]. In the case of multivariate normality, both <italic>p</italic>-values of skewness and kurtosis should be greater than 0.05 [##UREF##11##37##]. As the data were not multivariate normally distributed according to Mardia’s test (<italic>p</italic>&lt;0.05), a nonparanormal transformation was applied to relax the normality assumption [##UREF##12##39##].</p>", "<p id=\"Par15\">In our network model, HRQoL is conceptualized as a network of mutually interrelated factors. Because data are continuous scales or items, we used Guassian Graphical model (GGM) [##UREF##6##26##, ##REF##29595293##40##]. Nodes represent the selected HRQoL scales and items, and edges (links connecting two nodes) represent the regularized partial correlation coefficients after controlling for all other nodes. The thickness of the edge visualizes the strength, and the color a positive (red) or negative (blue) partial correlation. The partial correlation is indicated as very small (<italic>r</italic>&lt;0.1), small (0.1≤<italic>r</italic>&lt;0.3), moderate (0.3≤<italic>r</italic>&lt;0.5), and large (<italic>r</italic>&gt;0.5) [##UREF##13##41##].</p>", "<p id=\"Par16\">We applied graphical lasso tuned with the Extended Bayesian Information Criterion (EBIC) [##REF##28342071##38##, ##UREF##14##42##]. The EBIC hyperparameter, used to set the preferred simplicity of the model, was set to 0.5 to minimize spurious connections [##REF##28342071##38##]. Graphical lasso is a form of lasso regularization to prevent that edges between two nodes are spurious because of other nodes (i.e., conditional independence association) and small edges were shrinked to zero by dropping them from the model [##UREF##14##42##]. In this way, the estimated network is not over fitted and interpretable.</p>", "<p id=\"Par17\">We estimated node strength (i.e., number and strength of edges between nodes), betweenness (i.e., how often a node lies in shortest path between any combination of two nodes), and closeness (i.e., average distance from one node to all other nodes, which indicates how fast a node can be reached), which are indices of node centrality [##REF##28342071##38##, ##UREF##15##43##].</p>", "<p id=\"Par18\">Bootstrapping was performed to explore the accuracy and stability of the network [##REF##28342071##38##]. To estimate the accuracy of edge weights, 95% bootstrapped confidence intervals (CIs) around each edge in the network were calculated. Non-parametric bootstrapping (1000 bootstrap samples) was used to construct CIs. To estimate the stability of node centrality, we applied case-dropping bootstrap (1000 bootstrap samples) to calculate the correlational stability coefficient (CS coefficient) [##REF##28342071##38##]. This coefficient represents the maximum proportion of participants that can be dropped from the analysis with the correlation between the original centrality indices and the subset centrality indices of at least 0.7 with 95% probability [##UREF##16##44##]. The CS coefficients of at least above 0.25, but preferable above 0.5, are considered stable [##UREF##16##44##]. Additionally, the bootstrapped values were used to test the significance of edge weights and node strength [##REF##28342071##38##]. These bootstrapped difference tests indicate the difference between two different edge weights or node strengths. A bootstrapped CI around these difference scores was calculated [##REF##28342071##38##].</p>", "<p id=\"Par19\">Detection of communities was performed using the Louvain clustering method, a hierarchical clustering method based on multi-level modularity optimization algorithm [##UREF##17##45##].</p>" ]

[ "<title>Results</title>", "<title>Characteristics AYA cancer survivors</title>", "<p id=\"Par20\">In total, 11296 AYAs were invited to participate in the study, of whom 4010 (36%) responded. After excluding 414 records with missing data, we included 3596 AYAs in our final analysis; their sociodemographic and clinical characteristics are described in Table ##TAB##0##1##. AYAs were on average 31.5 years old at diagnosis and mostly female (61%). The average time since diagnosis was 12.4 years and the most common cancer types were breast cancer (24%), germ cell tumors (18%), lymphoid hematological malignancies (15%), and tumors of female genitalia (11%).\n</p>", "<p id=\"Par21\">The mean scores of the functioning and symptom scales of the QLQ-SURV111 are shown in Table ##TAB##1##2##. The overall global quality of life score of AYAs was on average 77.3 (SD 18.7). The functioning scale with the highest score was physical functioning (mean 91.7; SD 13.9), whereas sexual functioning was the scale with the lowest score (mean 43.7; SD 25.5). AYAs scored the lowest on the symptom scales social isolation (mean 69.1; SD 30.7), fatigue (mean 70.3; SD 26.0), and negative health outlook (mean 74.2; SD 20.0).\n</p>", "<title>Network analysis</title>", "<title>Overall network</title>", "<p id=\"Par22\">The partial correlation network model is shown in Fig. ##FIG##0##1##. In our sample, health distress had a strong partial correlation with negative health outlook (<italic>r</italic> = 0.71) and moderate partial correlation with worries about family getting cancer (<italic>r</italic> =0.48). Symptom checklist had a strong partial correlation with pain (<italic>r</italic> =0.67). Role functioning was strongly partially correlated to physical functioning (<italic>r</italic> =0.70) and to social interference (<italic>r</italic> =0.72), and there was a moderate partial correlation between sexual functioning and sexual problems (<italic>r</italic> =0.43).</p>", "<p id=\"Par23\">The bootstrapped confidence intervals of estimated edge weights (Fig. ##FIG##1##2##) show that the previously described strong/moderate correlations in our network are robust. The negative correlations in our network, on the other hand, are not reliable. To test if a correlation between two nodes was significantly different from other correlations, we used the edge difference test (Supplementary material Figure ##SUPPL##0##S1##). This plot showed that the correlation between health distress and negative health outlook was significantly different from all other correlations.</p>", "<title>Cluster analysis</title>", "<p id=\"Par24\">Within our network model, we identified four clusters (Fig. ##FIG##0##1##); (1) the worriment cluster (orange), (2) the daily functioning cluster (yellow), (3) the psychological cluster (pink), and (4) sexual cluster (green). The worriment cluster consists of the nodes: health distress, negative health outlook, symptom awareness, social isolation, worried about family getting cancer, and people treating you differently. Role functioning, physical functioning, pain, symptom checklist, social interference, and financial difficulties were all part of the daily functioning cluster. Emotional functioning, fatigue, sleep problems, cognitive functioning, and overall quality of life formed the psychological cluster. The sexual cluster consists of sexual functioning, sexual problems, and body image.</p>", "<title>Network stability and centrality</title>", "<p id=\"Par25\">The CS coefficient for strength, closeness, and betweenness were 0.75, 0.75, and 0.75 respectively, indicating a stable and reliable network (Fig. ##FIG##2##3##). Regarding centrality, the nodes with the highest strength are the most central, and therefore the most important nodes of the network model. In our model, nodes with the highest strength were negative health outlook (standardized centrality estimates (SCE) = 1.60), role functioning (SCE = 1.40), health distress (SCE = 1.40), and emotional functioning (SCE = 1.30). In addition, health distress and negative health outlook had the highest betweenness, and emotional functioning and health distress had the highest closeness (Fig. ##FIG##3##4##). The centrality difference plot (Supplementary material Figure ##SUPPL##0##S2##) demonstrates that the strength of the node negative health outlook significantly differed from the other nodes in the network model.</p>" ]

[ "<title>Discussion</title>", "<p id=\"Par26\">This study shows the results of a stable and reliable network analysis based on HRQoL data of 3596 long-term AYAs, as the first to our knowledge. Although the overall global quality of life score was 77.3 on average, the lowest functioning scale score was sexual functioning and the lowest symptom scale scores were social isolation, fatigue, and negative health outlook. This network showed several strong/moderate partial correlations, with the partial correlation between health distress and negative health outlook being the only one significantly different from all other. Also, the strength of the node negative health outlook was significantly different from all other nodes. In total, four clusters of negative symptom and functioning HRQoL scales were identified, including a worriment cluster, daily functioning cluster, psychological cluster, and sexual cluster.</p>", "<p id=\"Par27\">As this study is the first to apply a network analysis based on a wide range of HRQoL data of AYAs using the relatively new QLQ-SURV111 questionnaire, findings are difficult to compare with other studies. Although previous studies had similar aims, they mostly used the EORTC QLQ-C30 to assess HRQoL and studied adult cancer survivors [##REF##34387856##28##, ##REF##35472477##46##]. This was also the case in the network analysis conducted by Rooij et al. in a heterogeneous sample of adult cancer survivors where the EORTC QLQ-C30 was used [##REF##34387856##28##]. In their analysis, fatigue was consistently central and had moderate direct relationships with emotional symptoms, cognitive symptoms, appetite loss, dyspnea, and pain. Fatigue being the most central symptom was in contrast with our results, which might be explained by the much younger population in our study (31.5 years vs 61 years) and the difference in time since diagnosis, which was considerably longer in our study (12.4 years vs 4.2 years). In our network, negative health outlook was the node with the highest strength, which had a strong correlation with health distress within the worriment cluster. This suggests that psychological and emotional issues remain more of relevance to AYA cancer survivors also after long-term follow-up, and are better picked up by the QLQ-SURV111 questionnaire that covers a more complete range of relevant survivorship issues.</p>", "<p id=\"Par28\">Other network studies focused specifically on a construct (e.g., fear of recurrence) or predefined clusters of symptoms, for example, the network analysis on fear of cancer recurrence, anxiety, and depression in breast cancer patients of Yang et al. [##REF##35472477##46##]. In their network, “having trouble relaxing” was the most central node, anxiety and depression were well-connected, and fear of cancer recurrence formed a distinct cluster. The use of the broad range of survivorship issues of the QLQ-SURV111, including psychological, social, physical symptoms and functioning, allowed us to explore clusters within our network as well. An interesting cluster that emerged was the worriment cluster, which consisted of negative health outlook, health distress, symptom awareness, social isolation, worried about family getting cancer, and people treating you differently. Although our questions regarding worries were different, one other network analysis on AYAs focused on the construct of fear of cancer recurrence [##REF##35565220##47##]. Here, the researchers found fear of serious medical interventions as the most central symptom in their network, with the highest node strengths for fear of pain, fear of relying on strangers for activities of daily living, and fear of severe medical treatments. Based on their results, which emphasize the centrality of emotional issues among AYA patients, they stress the importance of prioritizing these symptoms for interventions. However, as stated previously, comparing the results of these studies with our results should be done cautiously as study populations, study designs, aims, and used questionnaires differ. The lack of studies among AYAs to compare our findings with stresses the need for more AYA-specialized research.</p>", "<p id=\"Par29\">The results of a network analysis can provide more insight in the HRQoL of AYAs, in which symptoms and functioning can influence each other, instead of perceiving them as individual factors [##UREF##18##48##]. Identifying the most important factors in a network can help to address these problems with targeted interventions and healthcare, and lead to novel research ideas. First, we recommend to replicate network analysis studies in other groups of AYAs to be able to make comparisons between studies with similar study populations and draw conclusions with more certainty—changing the exploratory approach into a confirmative approach. Ideally, longitudinal data needs to be collected to draw conclusions over time and adapt healthcare to these time-related changes where needed. This could even be specific to longitudinal ecological momentary assessment (EMA) in which participants are asked to complete (parts of) the expected momentary dynamic items of the HRQoL questionnaire multiple times a day during a study period. In this way, variations over time can be taken into account to a more detailed and individual level. Inter- and intra-individual differences over time might result in changes in prevalence (scores of the scales or items), partial correlations of nodes, centrality, and clusters formed.</p>", "<p id=\"Par30\">In addition, subgroup analysis should be performed to make healthcare interventions even more tailored. AYAs subgroup analysis could focus on age, gender, stage, treatment, and type of cancer. In order to establish tailored care, it is important to make the subgroups as specific as possible while remaining a stable and reliable network. However, it should also be noted that these results represent means and thus differences may exist on an individual level in clinical practice.</p>", "<p id=\"Par31\">Future research can lead to and optimize supportive care and targeted interventions, like psycho-oncological care and psychosocial, behavioral, and supportive interventions [##REF##33675909##49##]. For example, the strong significant correlation between health distress and negative health outlook, which is part of the worriment cluster in this study, might be intervened on by distress screening and renewed/tailored, age-specific psycho-oncological aftercare [##REF##35602702##50##]. For healthcare providers (HCPs) involved in AYA healthcare, and in general, visualization of the nodes, correlations, and clusters may help to understand the cohesion between different factors/symptoms and the influence they might have on each other, and more important, how a targeted intervention can influence several HRQoL outcomes simultaneously. This advocates for holistic and age-specific psycho-oncological aftercare in which multiple factors are targeted simultaneously by a multidisciplinary team of HCPs, to be as efficient and effective as possible.</p>" ]

[ "<title>Conclusions</title>", "<p id=\"Par33\">This innovative network analysis provides insight in the nodes, correlations, and clusters that could be targeted to improve the HRQoL outcomes of AYAs. Future studies with longitudinal data and subgroup analyses can tailor the interventions and provided healthcare even more, specifically for those at risk of poor HRQoL outcomes. With these insights, more targeted interventions and healthcare can be provided and developed.</p>" ]

[ "<title>Purpose</title>", "<p id=\"Par1\">Adolescent and young adult cancer survivors (AYAs) are at increased risk of long-term and late effects, and experience unmet needs, impacting their health-related quality of life (HRQoL). In order to provide and optimize supportive care and targeted interventions for this unique population, it is important to study HRQoL factors’ interconnectedness on a population level. Therefore, this network analysis was performed with the aim to explore the interconnectedness between HRQoL factors, in the analysis described as nodes, among long-term AYAs.</p>", "<title>Methods</title>", "<p id=\"Par2\">This population-based cohort study used cross-sectional survey data of long-term AYAs, who were identified by the Netherlands Cancer Registry (NCR). Participants completed a one-time survey (SURVAYA study), including the EORTC survivorship questionnaire (QLQ-SURV111) to assess their long-term HRQoL outcomes and sociodemographic characteristics. The NCR provided the clinical data. Descriptive statistics and a network analysis, including network clustering, were performed.</p>", "<title>Results</title>", "<p id=\"Par3\">In total, 3596 AYAs (on average 12.4 years post diagnosis) were included in our network analysis. The network was proven stable and reliable and, in total, four clusters were identified, including a worriment, daily functioning, psychological, and sexual cluster. Negative health outlook, part of the worriment cluster, was the node with the highest strength and its partial correlation with health distress was significantly different from all other partial correlations.</p>", "<title>Conclusion</title>", "<p id=\"Par4\">This study shows the results of a stable and reliable network analysis based on HRQoL data of long-term AYAs, and identified nodes, correlations, and clusters that could be intervened on to improve the HRQoL outcomes of AYAs.</p>", "<title>Supplementary Information</title>", "<p>The online version contains supplementary material available at 10.1007/s00520-023-08295-0.</p>", "<title>Keywords</title>" ]

[ "<title>Strengths and limitations</title>", "<p id=\"Par32\">This explorative study represents the very first network analysis using data on a range of HRQoL outcomes of long-term AYAs to our knowledge. Strengths include the large sample size, the establishment of a stable and reliable network, and the inclusion of a wide range of survivorship issues. However, the results should be interpreted with caution as the EORTC QLQ-SURV111 is not yet finalized and validated. In addition, our study included mostly females and over 40% was diagnosed with a stage I tumor. With a response rate of 36%, there are several subgroups (males, AYAs with a more aggressive disease, and AYAs diagnosed at the age of 18–24) underrepresented in this analysis who might have different HRQoL outcomes [##REF##36005166##30##]. Results may therefore not be generalizable to the total AYAs population. Also, we have not taken a closer look at the outcomes of specific subgroups (between groups), as the study group as a whole was analyzed. As mentioned previously, the subgroup analyses should be part of future research to tailor interventions with a risk-based approach. In line with this, due to the methodology of this study, i.e., a cross-sectional questionnaire study, no causal pathways or changes in HRQoL factors over time can be assessed. In the future, this might be tackled by using longitudinal data instead of cross-sectional data.</p>", "<title>Supplementary information</title>", "<p>\n</p>" ]

[ "<title>Acknowledgements</title>", "<p>The authors wish to thank all the patients for their participation in the study and the registration team of the Netherlands Comprehensive Cancer Organisation (IKNL) for the collection of data for the Netherlands Cancer Registry.</p>", "<title>Author contribution</title>", "<p>Conceptualization: C.V., S.H.M.J., T.I.B., D.W., D.C.R., C.D., and O.H.; data curation: C.V. and T.I.B.; formal analysis: T.I.B., D.W., and C.D.; funding acquisition: W.T.A.G. and O.H.; investigation: T.I.B., D.W., C.V., D.C.R., C.D., and S.H.M.J.; methodology: T.I.B., D.W., C.V., D.C.R., C.D., and S.H.M.J.; project administration: C.V.; resources: T.I.B., D.W., C.V., D.C.R., C.D., and S.H.M.J.; software: T.I.B., D.W., C.V., D.C.R., C.D., and S.H.M.J.; supervision: W.T.A.G. and O.H.; validation: T.I.B., D.W., C.V., D.C.R., C.D., and S.H.M.J.; visualization: T.I.B., D.W., C.V., D.C.R., C.D., and S.H.M.J.; writing—original draft preparation: C.V., D.C.R., and S.H.M.J.; writing—review and editing: T.I.B., D.W., C.V., D.C.R., C.D., R.T., R.B., S.E.J.K., J.M.K., J.M.T., M.E.M.M.B., T.H., R.I.L., J.N., M.C.M.K., W.T.A.G., S.H.M.J., and O.H.; all authors have read and approved the final manuscript.</p>", "<title>Funding</title>", "<p>This study was funded by a VIDI grant (VIDI198.007) and an investment grant (#480-08-009) from the Netherlands Organization for Scientific Research. This research was also supported by an institutional grant of the Dutch Cancer Society and of the Dutch Ministry of Health, Welfare and Sport.</p>", "<title>Data availability</title>", "<p>The data presented in this study are available on request from the corresponding author. The data are not publicly available due to privacy issues.</p>", "<title>Declarations</title>", "<title>Ethics approval</title>", "<p id=\"Par34\">The study was conducted in accordance with the Declaration of Helsinki, and approved by the Netherlands Cancer Institute Institutional Review Board (IRBd18122) on February 6, 2019.</p>", "<title>Consent to participate</title>", "<p id=\"Par35\">Informed consent was obtained from all subjects (responders) involved in the study.</p>", "<title>Competing interests</title>", "<p id=\"Par36\">The authors have no relevant financial or non-financial interests to disclose.</p>" ]

[ "<fig id=\"Fig1\"><label>Fig. 1</label><caption><p>The (cluster) network of HRQoL outcomes of long-term AYA cancer survivors. In this partial correlation network model, the nodes (PF: physical functioning, CF: cognitive functioning, EF: emotional functioning, RF: role functioning, BI: body image, SA: symptom awareness, SF: sexual functioning, FA: fatigue, SL: sleep problems, PA: pain, Sif: social interference, HD: health distress, NHO: negative health outlook, SI: social isolation, SC: symptom checklist, SP: sexual problems, FD: financial difficulties, WF: worried about family getting cancer, TD: people treating you differently, QL: overall quality of life) represent all the HRQoL scales of the QLQ-SURV111 and the edges (links connecting two nodes) represent the regularized partial correlation coefficients after controlling for all other nodes. The blue color indicates a positive partial correlation and a red color a negative partial correlation between two nodes. The thickness of the edge visualizes the strength of the partial correlation between two nodes. The four clusters that we identified within our network model include in orange the worriment cluster, in yellow the daily functioning cluster, in pink the psychological cluster, and in green the sexual cluster</p></caption></fig>", "<fig id=\"Fig2\"><label>Fig. 2</label><caption><p>Bootstrapped confidence intervals of estimated edge weights. Each horizontal line represents one edge of the network, ordered from the edge with the highest edge weight to the edge with the lowest edge weight. The red line indicates the sample values and the gray lines are the bootstrapped CIs. The larger the gray line the less certain the edge value is</p></caption></fig>", "<fig id=\"Fig3\"><label>Fig. 3</label><caption><p>Correlational stability plot of centrality indices by case-dropping subset bootstrap. Correlations between centrality indices of network sampled with persons dropped and the original sample. Lines indicate the means and areas indicate the range from the 2.5th quantile to the 97.5th quantile</p></caption></fig>", "<fig id=\"Fig4\"><label>Fig. 4</label><caption><p>Central indices of the network. Centrality indices are shown as standardized <italic>z</italic>-scores. PF: physical functioning, CF: cognitive functioning, EF: emotional functioning, RF: role functioning, BI: body image, SA: symptom awareness, SF: sexual functioning, FA: fatigue, SL: sleep problems, PA: pain, Sif: social interference, HD: health distress, NHO: negative health outlook, SI: social isolation, SC: symptom checklist, SP: sexual problems, FD: financial difficulties, WF: worried about family getting cancer, TD: people treating you differently, QL: overall quality of life</p></caption></fig>" ]

[ "<table-wrap id=\"Tab1\"><label>Table 1</label><caption><p>Sociodemographic and clinical characteristics of the study cohort</p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th rowspan=\"2\" colspan=\"2\"/><th colspan=\"2\">Respondents N = 3596</th></tr><tr><th><italic>n</italic></th><th>%</th></tr></thead><tbody><tr><td rowspan=\"2\">Gender</td><td>Male</td><td>1420</td><td>39</td></tr><tr><td>Female</td><td>2176</td><td>61</td></tr><tr><td rowspan=\"4\">Age at diagnosis mean (SD)</td><td/><td colspan=\"2\">31.5 (5.9)</td></tr><tr><td>18–24 years</td><td>569</td><td>16</td></tr><tr><td>25–34 years</td><td>1584</td><td>44</td></tr><tr><td>35–39 years</td><td>1443</td><td>40</td></tr><tr><td>Age at completing questionnaire mean (SD)</td><td/><td colspan=\"2\">44.5 (7.5)</td></tr><tr><td rowspan=\"4\">Time since diagnosis</td><td/><td colspan=\"2\">12.4 (4.5)</td></tr><tr><td>5–10 years</td><td>1452</td><td>40</td></tr><tr><td>11–15 years</td><td>1247</td><td>35</td></tr><tr><td>16–20 years</td><td>897</td><td>25</td></tr><tr><td rowspan=\"12\">Type of cancer</td><td>Breast cancer</td><td>846</td><td>23.5</td></tr><tr><td>Germ cell tumors</td><td>637</td><td>17.7</td></tr><tr><td>Lymphoid hematological malignancies</td><td>540</td><td>15.0</td></tr><tr><td>Female genitalia tumors</td><td>383</td><td>10.7</td></tr><tr><td>Melanoma</td><td>252</td><td>7.0</td></tr><tr><td>Thyroid cancer</td><td>224</td><td>6.2</td></tr><tr><td>Bone or soft tissue sarcoma</td><td>161</td><td>4.5</td></tr><tr><td>Myeloid hematological malignancies</td><td>136</td><td>3.8</td></tr><tr><td>Central nervous system tumors</td><td>131</td><td>3.6</td></tr><tr><td>Head and neck cancer</td><td>107</td><td>3.0</td></tr><tr><td>Digestive tract tumors</td><td>104</td><td>2.9</td></tr><tr><td>Other*</td><td>75</td><td>2.1</td></tr><tr><td rowspan=\"5\">Tumor stage</td><td>I</td><td>1537</td><td>42.7</td></tr><tr><td>II</td><td>957</td><td>26.6</td></tr><tr><td>III</td><td>515</td><td>14.3</td></tr><tr><td>IV</td><td>165</td><td>4.6</td></tr><tr><td>Missing</td><td>422</td><td>11.7</td></tr><tr><td rowspan=\"6\">Primary treatment modality</td><td>Surgery</td><td>2799</td><td>77.8</td></tr><tr><td>Chemotherapy</td><td>2030</td><td>56.5</td></tr><tr><td>Radiotherapy</td><td>1716</td><td>47.7</td></tr><tr><td>Hormonal therapy</td><td>435</td><td>12.1</td></tr><tr><td>Targeted therapy</td><td>282</td><td>7.8</td></tr><tr><td>Stem cell therapy</td><td>130</td><td>3.6</td></tr><tr><td>Marital status (at time of questionnaire)</td><td>In a relation</td><td>3001</td><td>83.5</td></tr><tr><td rowspan=\"4\">Educational level</td><td>Primary school or equivalent</td><td>16</td><td>0.4</td></tr><tr><td>Secondary school or equivalent</td><td>1514</td><td>42.1</td></tr><tr><td>College/university or equivalent</td><td>2060</td><td>57.3</td></tr><tr><td>Missing</td><td>6</td><td>0.2</td></tr></tbody></table></table-wrap>", "<table-wrap id=\"Tab2\"><label>Table 2</label><caption><p>The long-term HRQoL outcomes from the cancer survivorship core questionnaire (QLQ-SURV111) of the AYA cancer survivors, arranged by scale and total score mean</p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th>Scales</th><th>Short names</th><th>Functional/symptom</th><th>Number of items (<italic>n</italic>)</th><th>Item numbers</th><th>Raw scores<break/>mean (SD)</th><th>Total scores<break/>mean (SD)</th></tr></thead><tbody><tr><td>Physical functioning</td><td>PF</td><td>Functional</td><td>5</td><td>1–5</td><td>1.2 (0.4)</td><td>91.7 (13.9)</td></tr><tr><td>Role functioning</td><td>RF</td><td>Functional</td><td>3</td><td>69–71</td><td>1.5 (0.7)</td><td>83.7 (24.5)</td></tr><tr><td>Emotional functioning</td><td>EF</td><td>Functional</td><td>7</td><td>52–58</td><td>1.6 (0.6)</td><td>81.0 (19.9)</td></tr><tr><td>Cognitive functioning</td><td>CF</td><td>Functional</td><td>4</td><td>47, 48, 50, 51</td><td>1.6 (0.7)</td><td>79.6 (22.6)</td></tr><tr><td>Body image</td><td>BI</td><td>Functional</td><td>2</td><td>40, 41</td><td>1.7 (0.7)</td><td>77.8 (24.6)</td></tr><tr><td>Overall quality of life</td><td>QL</td><td>Functional</td><td>1</td><td>121</td><td>5.6 (1.1)</td><td>77.3 (18.7)</td></tr><tr><td>Symptom awareness</td><td>SA</td><td>Functional</td><td>2</td><td>76, 77</td><td>2.2 (0.8)</td><td>60.9 (25.4)</td></tr><tr><td>Sexual functioning</td><td>SF</td><td>Functional</td><td>2</td><td>111, 112</td><td>2.3 (0.8)</td><td>43.7 (25.5)</td></tr><tr><td>Financial difficulties</td><td>FD</td><td>Symptom</td><td>1</td><td>68</td><td>1.3 (0.7)</td><td>89.4 (23.8)</td></tr><tr><td>Social interference</td><td>Sif</td><td>Symptom</td><td>2</td><td>73, 74</td><td>1.4 (0.7)</td><td>88.1 (22.0)</td></tr><tr><td>Worry cancer risk family</td><td>WF</td><td>Symptom</td><td>1</td><td>61</td><td>1.4 (0.7)</td><td>86.6 (23.1)</td></tr><tr><td>Symptom checklist</td><td>SC</td><td>Symptom</td><td>17</td><td>20, 21, 23–26, 28, 30–38, 75</td><td>1.4 (0.4)</td><td>85.8 (13.5)</td></tr><tr><td>Treated differently</td><td>TD</td><td>Symptom</td><td>1</td><td>107</td><td>1.4 (0.7)</td><td>85.2 (22.6)</td></tr><tr><td>Pain</td><td>PA</td><td>Symptom</td><td>2</td><td>22, 72</td><td>1.5 (0.7)</td><td>84.2 (23.0)</td></tr><tr><td>Sexual problems</td><td>SP</td><td>Symptom</td><td>2</td><td>113, 117</td><td>1.5 (0.8)</td><td>82.8 (26.3)</td></tr><tr><td>Health distress</td><td>HD</td><td>Symptom</td><td>3</td><td>63–65</td><td>1.6 (0.7)</td><td>79.0 (22.3)</td></tr><tr><td>Sleep problems</td><td>SL</td><td>Symptom</td><td>4</td><td>16–19</td><td>1.8 (0.7)</td><td>74.5 (23.6)</td></tr><tr><td>Negative health outlook</td><td>NHO</td><td>Symptom</td><td>7</td><td>60, 62, 64, 81, 82, 98, 99</td><td>1.8 (0.6)</td><td>74.2 (20.0)</td></tr><tr><td>Fatigue</td><td>FA</td><td>Symptom</td><td>4</td><td>6–9</td><td>1.9 (0.8)</td><td>70.3 (26.0)</td></tr><tr><td>Social isolation</td><td>SI</td><td>Symptom</td><td>2</td><td>89, 90</td><td>1.9 (0.9)</td><td>69.1 (30.7)</td></tr></tbody></table></table-wrap>" ]

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[ "<table-wrap-foot><p>*Other includes urinary tract, respiratory tract, male genitalia, neuroblastoma, adrenal, paraganglioma, and eyes</p></table-wrap-foot>", "<fn-group><fn><p>Tom I. Bootsma and Deborah van de Wal share first authorship.</p></fn><fn><p><bold>Publisher’s note</bold></p><p>Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.</p></fn></fn-group>" ]

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[ "<title>Introduction</title>", "<p id=\"Par11\">Bioactive lipids are important for brain function. In particular, the homeostasis of the sphingolipids ceramide-1-phosphate, sphingosine, and sphingosine-1-phosphate (S1P) has been described to regulate proliferation, differentiation, cell growth and inflammation in the cells of the central nervous system (CNS) [##REF##24459205##1##]. S1P is produced from ceramide by ceramidase followed by phosphorylation by sphingosine kinase (SphK1/2) [##REF##18216770##2##]. S1P is abundantly produced by erythrocytes, platelets and endothelial cells [##REF##19268560##3##, ##REF##7795224##4##]. Although, S1P synthesis in the CNS has also been reported, namely by astrocytes after fibroblast growth factor stimulation [##REF##16470810##5##].</p>", "<p id=\"Par12\">S1P in the extracellular space binds to its five specific G-coupled receptors S1PR1-5, which signal through diverse downstream pathways [##REF##24459205##1##]. All S1P receptors are expressed in the CNS [##REF##35781853##6##]. They participate in multiple important functions during CNS development, and further contribute to the development and/or resolution of neurodegeneration in pathological conditions such as ischemic stroke [##REF##31165772##7##], multiple sclerosis [##REF##27825807##8##], hearing loss [##REF##27080739##9##] and seizures [##UREF##0##10##, ##REF##20930159##11##]. S1P receptors have received attention in the field of multiple sclerosis, for which immunomodulation is achieved by the oral drug Fingolimod (FTY720) that activates S1PR1,3–5 [##REF##20061941##12##]. S1PR1 is the most studied S1P receptor with important roles in angiogenesis and neurogenesis [##REF##16314531##13##]. S1PR2 is a modulator of neuronal excitability during neuronal development [##REF##11553273##14##] and controls spontaneous activity of cultured neurons [##REF##35781853##6##]. S1PR3 receptor controls activity of microglia and their participation in neuroinflammation [##REF##23813380##15##–##REF##27098703##17##], and is highly expressed in astrocytes, where it regulates astrogliosis via activation of the small GTPase RhoA [##REF##28577576##18##]. Until recently, S1PR4 has received little attention in the CNS [##REF##25309325##19##]. Our recent work proposed S1PR4 presence in synapses, and a role in neuronal activity control [##REF##35781853##6##]. Furthermore, previous studies have demonstrated loss of sphingosine kinase activity and lowering of brain region-specific S1P levels early in the progression of Alzheimer’s disease [##REF##24456642##20##, ##REF##29615132##21##].</p>", "<p id=\"Par13\">Recent evidence also suggests an important role for S1P and its generating enzymes SphK1/2 in the development of diabetes [##REF##18458870##22##] through adverse effects on endothelial function [##REF##27098703##17##, ##REF##16179586##23##], hepatic insulin sensitivity and secretion [##REF##30448236##24##], regulating insulin secretion in pancreatic β-cell [##REF##22389505##25##], and contributing to diabetes-associated inflammation [##REF##16179586##23##].</p>", "<p id=\"Par14\">The sphingosine rheostat has been extensively studied in diabetes and insulin resistance, with special focus on ceramide and ceramide-1-phosphate [##REF##18458870##22##]. Because S1P is involved in a plethora of cellular functions, a tight regulation of S1P homeostasis seems important for proper brain functioning, which would be disrupted in metabolic syndrome and diabetes. Diabetes impacts synapses with negative consequences for brain function, including memory performance [##REF##32265637##26##]. Furthermore, rodent models of diabetes show altered neuromodulation systems operated by adenosine [##REF##16256246##27##–##REF##30686981##29##], ATP [##REF##17869435##30##], or endocannabinoids [##REF##17222407##31##, ##REF##32428627##32##], which control synaptic activity and brain energy metabolism, and might interact with brain insulin signaling [##REF##36629507##33##]. These systems can also afford neuroprotection. For example, pharmacologically targeting the altered adenosinergic system components confers neuroprotection and improves memory performance in diabetes models [##REF##22514596##28##, ##REF##30686981##29##, ##REF##19694901##34##]. It is hitherto unknown whether the neuromodulation system operated by S1P in the CNS is altered in diabetes and metabolic syndrome.</p>", "<p id=\"Par15\">In this study, we set out to test the hypothesis that T2D impacts the neuromodulation system operated by S1P in the CNS. Since S1PRs are present in nerve terminals of the cortex [##REF##35781853##6##], we determined changes induced by T2D in the density of S1PR1-4 in cortical synaptosomes from insulin-resistant Goto-Kakizaki (GK) rats and from diet-induced obese mice, which are models with well-established brain dysfunction (e.g., [##REF##30686981##29##, ##REF##36222315##35##, ##REF##30670942##36##]).</p>" ]

[ "<title>Methods</title>", "<title>Animals</title>", "<p id=\"Par16\">Experiments were performed according to EU Directive 2010/63/EU under approval of the Malmö/Lund Committee for Animal Experiment Ethics (permit numbers 994/2018 and 9987/2020) and are reported following the ARRIVE guidelines (Animal Research: Reporting In Vivo Experiments, NC3Rs initiative, UK). Male and female GK rats were obtained from a local colony, and male and female age-matched Wistar rats from Janvier (Saint Berthevin, France) were used as controls [##REF##30686981##29##]. Wistar rats were housed in the facility for at least a month before experimentation. Eight weeks-old C57BL/6 J mice were purchased from Taconic (Ry, Denmark). Only male mice were used due to sex differences in responses to diet-induced obesity (see [##REF##36222315##35##], and references therein). Animals were housed in ventilated cages enriched with a cylinder, wood toys and nesting material, at controlled temperature of 22 °C, 50–60% humidity and a 12:12-h light–dark cycle. Food and water were provided ad libitum. Rats were kept on a regular chow and were euthanized at 6 months of age (Fig. ##FIG##0##1##A). Mice were fed a lard-based high-fat diet (HFD; 60% calories from fat) or a control diet (CD, 10% calories from fat) from Research diets (New Brunswick, NJ-USA), as previously described [##REF##36222315##35##]. Mice were held on the diet for 1 week, 1 or 2 months, starting from 9 weeks of age (Fig. ##FIG##0##1##B).</p>", "<title>Glucose Tolerance Test and Insulin Determination</title>", "<p id=\"Par17\">A glucose tolerance test (GTT) was performed 1–3 days before sacrifice. Food was removed at 08:00 for 6 h, and mice were put into clean cages to avoid coprophagy before and during the test. Before each test, a blood sample was collected from vena saphena into a heparinized tube for determination of plasma insulin concentration. Glycemia was measured from tail tip blood with the Accu-Chek Aviva glucometer (Roche, Solna, Sweden). Mice were then administered 2 g/kg glucose i.p. from a 20%(w/v) solution in saline, followed by determination of glucose levels after 15, 30, 60, 90 and 120 min.</p>", "<p id=\"Par18\">Plasma insulin was measured with ELISA kits from Mercodia (Uppsala, Sweden; #10-1250-01 for rats; #10-1247-10 for mice).</p>", "<title>Preparation of Nerve Terminal-Enriched Membranes</title>", "<p id=\"Par19\">After weighing and measuring tail-tip blood glucose, animals were anesthetized with isoflurane and quickly decapitated. Trunk blood was collected, and plasma was stored at − 80 °C for insulin and S1P determination. Brains were quickly dissected, frozen in N₂ (l) and stored at − 80 °C until further experiments. Synaptosomes were prepared as described previously [##REF##30670942##36##]. Briefly, cortical tissue was homogenized (12 strokes at 800 rotations/min) with a glass-teflon Potter–Elvehjem homogenizer (rotor head InterMed/STIR20) in Sucrose-HEPES buffer (0.32 mol/L sucrose, 1 mmol/L EDTA, 10 mmol/L HEPES, 1 mg/mL bovine serum albumin, pH 7.4) at 4 °C. Homogenates were centrifuged at 3000 <italic>g</italic> for 10 min at 4 °C (Beckman Coulter/Avanti J-20 XP). After, supernatants were centrifuged at 14,000 <italic>g</italic> for 12 min at 4 °C. The pellet was re-suspended in 1 mL of 45% (v/v) Percoll (GE Healthcare, Uppsala, Sweden) solution prepared in Krebs-HEPES buffer (in mmol/L: 140 NaCl, 5 KCl, 10 HEPES, 1 EDTA, 5 glucose, pH 7.4), and then centrifuged at 21,000 <italic>g</italic> for 2 min at 4 °C. The top layer (rich in synaptosomes) was washed by re-suspending in Krebs-HEPES buffer and centrifuging again. The resulting pellets were re-suspended in Krebs-HEPES solution containing protease inhibitors (cOmplete cocktail from Roche, Mannheim, Germany), and stored at − 80 °C until immunoblotting.</p>", "<title>Immunoblotting</title>", "<p id=\"Par20\">Total protein content of the samples was measured with the bicinchoninic acid assay (kit from Pierce, ThermoFisher Scientific, Uppsala, Sweden). Then, Western blotting was carried out as detailed by Lizarbe et al. [##REF##30670942##36##]. Briefly, samples were dissolved in sample buffer (#NP0007, Invitrogen, ThermoFisher), boiled at 95 °C for 5 min, and then separated by SDS-PAGE in 4–12% Bis–Tris gradient gels (#NP0336, Invitrogen), followed by transfer onto nitrocellulose membranes of 0.45-μm pore size (#GE10600002, GE Healthcare). The membranes were blocked for 1–2 h in Tris-buffered saline (in mmol/L: 20 Tris, 150 NaCl, pH 7.6) containing 5% (w/v) skim milk, 1% (v/v) Tween-20, and then sequentially incubated with primary and secondary antibodies (Table ##TAB##0##1##) diluted in this blocking solution. Immunoblots were developed with the chemiluminescence Super-Signal kit (#34,580, ThermoFisher). Whenever necessary, sensitivity was enhanced with the biotin-streptavidin kit VectaStain ABC-HRP according to manufacturer’s instructions (#PK-4000, Vectorlabs, CA-USA). Luminescence was detected using the Chemidoc XRS + interfaced to Image Lab 5.2.1 (Biorad, Stockholm, Sweden).</p>", "<title>Mass Spectrometry</title>", "<p id=\"Par21\">Plasma samples obtained from trunk blood, and cortical homogenates in phosphate-buffered saline (PBS; in mmol/L: 137 NaCl, 2.7 KCl, 1.5 KH₂PO₄, 8.1 Na₂HPO₄, pH 7.4) were spiked with deuterated S1P as internal standard (S1P-D7 &gt; 99% deuterated; Avanti Polar Lipids/Merck, Darmstadt, Germany), and S1P was extracted as described in [##REF##22528437##37##]. Extracts were dried under a nitrogen stream, dissolved in methanol, and subjected to liquid chromatography-coupled tandem mass spectrometry as previously described [##REF##33993723##38##, ##REF##35055052##39##].</p>", "<title>Statistics</title>", "<p id=\"Par22\">Results are shown as mean ± SD, and were analyzed with Prism 9.4.1 (GraphPad Software, San Diego, CA). Normal distribution was assessed with the Kolmogorov–Smirnov test. In the presence of normality deviations, results were analyzed using Mann–Whitney test or Kruskal–Wallis test followed by Dunn’s multiple comparisons. Normally distributed data was analyzed with unpaired, two-tailed Student <italic>t</italic>-test or ANOVA followed by independent comparisons with the Fisher’s least significant difference (LSD) test. Significance was accepted for <italic>P</italic> &lt; 0.05.</p>" ]

[ "<title>Results</title>", "<p id=\"Par23\">We have previously detailed metabolic phenotypes of insulin-resistant GK rats [##REF##30686981##29##] and HFD-induced obese mice [##REF##29027041##40##]. In the present study, we have confirmed that GK rats displayed lower body weight, increased glycemia, and similar fed plasma insulin concentration, when compared to controls (Table ##TAB##1##2##). HFD-fed mice showed increased body weight, a tendency for increased fasting glycemia and plasma insulin concentrations, reduced glucose tolerance, and increased insulin resistance, when compared to controls (Table ##TAB##2##3##).</p>", "<p id=\"Par24\">Plasma concentrations of S1P were higher in GK than Wistar rats (+ 60%, <italic>P</italic> = 0.008, Fig. ##FIG##0##1##A). However, S1P levels in homogenates from the cortex of GK rats were not significantly different from those in Wistar rats (<italic>P</italic> &gt; 0.05, Fig. ##FIG##1##2##B). Nerve terminal-enriched membranes from GK rats showed lower immunoreactivity of S1PR1 (− 21%, <italic>P</italic> = 0.004), S1PR2 (− 26%, <italic>P</italic> = 0.006) and S1PR4 (− 40%, <italic>P</italic> = 0.046,) than those from control Wistar rats (Fig. ##FIG##1##2##C–D). S1PR3 immunoreactivity was similar between the groups (<italic>P</italic> &gt; 0.05).</p>", "<p id=\"Par25\">Plasma concentrations of S1P were increased by both HFD exposure and age (interaction F(2,64) = 1.4, <italic>P</italic> &gt; 0.05; diet F(1,64) = 7.2, <italic>P</italic> = 0.009; time F(2,64) = 17, <italic>P</italic> &lt; 0.001). In particular, HFD-feeding increased plasma S1P levels by 11–16%, most prominently at 1 month after diet intervention (Fig. ##FIG##2##3##A). In the cortex, S1P levels were also increased with HFD, and showed a variation with age that was unrelated to plasma S1P levels (interaction F(2,27) = 0.12, <italic>P</italic> &gt; 0.05; diet F(1,27) = 5.5, <italic>P</italic> = 0.026; time F(2,27) = 4.4, <italic>P</italic> = 0.022, Fig. ##FIG##2##3##B). The density of S1PRs was analyzed in nerve-terminal membranes from the cortex of mice fed a CD or HFD for 1 week, 1 month, and 2 months (Fig. ##FIG##2##3##C–E). SNAP25 analyzed as constitutive protein showed no HFD-induced changes (Fig. ##FIG##2##3##F). Relative to CD, HFD exposure reduced S1PR1 immunoreactivity (interaction F(2,30) = 0.64, <italic>P</italic> &gt; 0.05; diet F(1,30) = 13, <italic>P</italic> = 0.001; time F(2,30) = 0.64, <italic>P</italic> &gt; 0.05; Fig. ##FIG##2##3##G), which is particularly prominent at 1 week (− 52%, <italic>P</italic> = 0.011) and 2 months of HFD feeding (− 50%, <italic>P</italic> = 0.014). The density of S1PR2 and S1PR3 was not modified by HFD feeding (Fig. ##FIG##2##3##G). S1PR4 immunoreactivity in cortical nerve terminal membranes was lower in HFD-fed mice than controls (interaction F(2,31) = 1.6, <italic>P</italic> &gt; 0.05; diet F(1,31) = 7.4, <italic>P</italic> = 0.011; time F(2,31) = 1.6, <italic>P</italic> &gt; 0.05; Fig. ##FIG##2##3##G), most prominently after 1 week of HFD feeding (− 54%, <italic>P</italic> = 0.005), and recovering to control levels at 2 months of HFD.</p>" ]

[ "<title>Discussion</title>", "<p id=\"Par26\">This study demonstrates for the first time that S1PR density in cortical nerve terminals is reduced in lean and obese models of insulin resistance, along with possible increases of S1P concentrations in the cortex, implying a role for S1PR signaling in the neuropathological process that occurs in T2D. Since S1PR activation generally attenuates neuronal activity [##REF##35781853##6##], lower S1PR density might lead to overall cortical hyper-excitability.</p>", "<p id=\"Par27\">From those analyzed, S1PR1 is the receptor with the highest expression level in the CNS. Its density was decreased in cortical synapses of both insulin-resistant GK rats and HFD-fed mice, relative to their respective controls. Previous studies have reported similar S1PR1 level reductions in the hypothalamus of HFD-fed rats or mice, as well as in the leptin deficient ob/ob mice [##REF##25255053##41##]. Silva et al. proposed that S1P is an appetite inhibitor through S1PR1 signaling, and reduced S1PR1 levels in the hypothalamus of obese models were associated with increased food intake. S1PR1 activation inhibits neuronal activity [##REF##35781853##6##] and, therefore, the S1PR1 reduction observed in cortical synapses is likely to result in loss of S1P-dependent synaptic control. However, S1PR1 is ubiquitously expressed, and exert a multitude of regulatory actions, not being specific to synapses. Thus, a potential approach of pharmacologically targeting S1PR1 for synaptic modulation might also trigger a plethora of non-synaptic actions. Adding to the complexity of S1PR1 signaling, others have reported that specific S1PR1 activation increases excitability of some sensory neurons in the rat dorsal root ganglia [##REF##20844107##42##]. Given the ubiquitous actions of S1P, any systemic interventions on S1PR1 should be taken with caution.</p>", "<p id=\"Par28\">The density of both S1PR2 and S1PR4 was decreased in cortical synapses from GK rats compared to controls. Considering the role of S1PR2 as neuromodulator in excitatory neurons [##REF##11553273##14##], and its preferential localization at the pre-synaptic level in the cortex [##REF##35781853##6##], dampening of S1PR2 signaling in GK rats is likely to contribute to the synaptic dysfunction and impaired synaptic plasticity [##REF##30686981##29##]. Our previous study [##REF##35781853##6##] showed that specifically S1PR2 and S1PR4 activation results in dampening of spontaneous spiking frequency of cultured primary neurons. A reduction in levels of synaptic S1PR2/4 could thus result in uncontrolled, excessive synaptic activity that might lead to excitotoxicity and synaptic damage [##REF##32265637##26##]. We thus speculate that S1PR2/4 activation might allow for synaptic protection in diabetic conditions.</p>", "<p id=\"Par29\">HFD-fed mice did not reproduce the T2D-induced S1PR2 alterations observed in GK rats. Moreover, S1PR4 density was reduced in short- but not long-term HFD, when compared to the respective controls. Besides obesity, a key difference between the two models is the hyperglycemia in GK rats [##REF##23855509##43##, ##REF##29220408##44##] that is negligible in HFD-fed mice [##REF##32265637##26##, ##REF##30670942##36##]. On the other hand, both models become insulin resistant. In this regard, it needs to be noted that S1P signaling through S1PR2 has been shown to interact with insulin signaling, and might participate in the development of insulin resistance in peripheral cells [##REF##26943364##45##]. Given the ability of S1PR2 to inhibit insulin-mediated signals, it is plausible that reduced S1PR2 in synaptic membranes results in enhanced insulin signaling. Thus, S1PR2 might contribute to the development of central insulin resistance. In turn, putatively increased S1PR2 signaling due to higher S1P concentrations in HFD-fed mice could also dampen synaptic insulin signaling. Loss of tonic insulin receptor signaling in nerve terminals is believed to contribute to memory impairment [##REF##36629507##33##, ##REF##27881773##46##].</p>", "<p id=\"Par30\">Although S1PR5 is present in the CNS, its relevance for neuromodulation at the synaptic level remains to be established ([##REF##35781853##6##], and references therein). Therefore, we have not investigated diabetes-induced alterations of S1PR5 density in the present study.</p>", "<p id=\"Par31\">Concentrations of S1P measured by ELISA were found to increase in the liver of patients with T2D as well as in streptozotocin-induced T2D rats [##REF##30972941##47##]. Plasma S1P concentration also increases in the ob/ob mouse model, mice exposed to HFD for 6 weeks, and in a population of young obese humans [##REF##24039766##48##]. Interestingly, plasma S1P in humans was found to be associated with body fat, insulin levels, insulin resistance (by HOMA-IR), as well with cholesterol [##REF##24039766##48##]. The same study reported an increase in concentrations of S1P in plasma after 12 h of fasting, suggesting a relation between S1P levels and increased inter-organ lipid flux. In our study, plasma samples for S1P determination were collected under fed state, which we therefore expect to depict diabetes-induced S1P changes rather than acute fasting-induced mobilization of lipid stores.</p>", "<p id=\"Par32\">While plasma S1P levels increase in non-diabetic young adults with obesity (&lt; 30 years old; body mass index ~ 37 kg/m²) relative to lean controls [##REF##24039766##48##], others have reported that T2D is associated with a decrease of plasma S1P concentrations [##REF##29543843##49##, ##REF##31602237##50##]. Namely, these studies found lower plasma S1P concentrations in individuals with T2D relative to healthy controls matched for age and body mass index (average: ~ 60 years old and ~ 29 kg/m² in Vaisar et al. [##REF##29543843##49##]; 44–49 years old and 25–26 kg/m² in Sui et al. [##REF##31602237##50##]). While reported effects of obesity and T2D on plasma S1P levels are opposite, these studies differ in the studied populations, age of the subjects, and methods for S1P extraction and detection.</p>", "<p id=\"Par33\">S1P levels in the cortex are reported to be lower than in other brain regions [##REF##30805275##51##], and we now found S1P concentrations increased in the cortex of HFD-fed mice. To our knowledge, this is the first study investigating S1P concentration in the cerebral cortex of T2D/pre-diabetic animal models. We report an HFD-induced increase of S1P in cortical tissue that seems to be independent of HFD-induced increases in circulating S1P levels. In turn, in GK rats, the large variance of S1P concentrations measured in the cortex precluded determining the expected diabetes-induced increase in S1P levels. This contrasts the concurrent increases of brain and plasma S1P that have previously been reported in a mouse model of hypertension [##REF##29856066##52##].</p>", "<p id=\"Par34\">The assessment of S1PR density in mouse cortical synapses across the HFD treatment is limited to the effects of the exposure to the diet, and not the effect of age. This is due to the semi-quantitative nature of immunoblotting methods, and the fact that we have decided to analyze all age-matched samples in parallel. Nevertheless, mice were treated for a relatively short period of time (only 2 months), at an age range from about 2 to 4 months, during which changes of S1PR density might not be as important as effects of the diet. Another limitation of this study is that despite including both sexes in the GK rat part of the study, analysis of plasma S1P levels was only available for males. We are not able to determine sex effects on plasma S1P concentrations, although they were not apparent in cortical levels of S1P. Finally, another important limitation of this study is that the nerve-terminal enriched membranes obtained with our protocol also contain non-synaptic components, such as the perisynaptic astrocytic processes. In particular, S1PR1 is abundant in astrocytes, and the observed S1PR1 reduction could be due to alterations in astrocytic rather than synaptic compartments. However, in total membranes obtained as previously described [##REF##16256246##27##], we have not observed any S1PR1 reduction in GK vs Wistar rats, or HFD-fed mice vs controls (data not shown). Thus, S1PR density alterations in the present study most likely take place within the synaptic membranes.</p>", "<p id=\"Par35\">In conclusion, we have observed a decrease in the protein levels of three S1P receptors in the cortex of the GK model that were likely unrelated to cortical concentrations of S1P. Two of the altered receptors, namely S1PR2 and S1PR4, have been proposed to control neuronal activity and synaptic transmission ([##REF##35781853##6##], and references therein). Furthermore, S1PR4 levels were also decreased in the cortex of a diet-induced obese mouse model with metabolic syndrome, along with increased cortical levels of S1P. Altogether, these results point towards T2D-induced alterations of the neuromodulation system to which S1P signaling contributes. Our results open the door for testing brain-specific S1P-S1PR modulation as a potential neuroprotection strategy in diabetes.</p>" ]

[]

[ "<p id=\"Par1\">Sphingosine-1-phosphate (S1P) is a phosphosphingolipid with pleiotropic biological functions. S1P acts as an intracellular second messenger, as well as extracellular ligand to five G-protein coupled receptors (S1PR1-5). In the brain, S1P regulates neuronal proliferation, apoptosis, synaptic activity and neuroglia activation. Moreover, S1P metabolism alterations have been reported in neurodegenerative disorders. We have previously reported that S1PRs are present in nerve terminals, exhibiting distinct sub-synaptic localization and neuromodulation actions. Since type 2 diabetes (T2D) causes synaptic dysfunction, we hypothesized that S1P signaling is modified in nerve terminals. In this study, we determined the density of S1PRs in cortical synaptosomes from insulin-resistant Goto-Kakizaki (GK) rats and Wistar controls, and from mice fed a high-fat diet (HFD) and low-fat-fed controls. Relative to their controls, GK rats showed similar cortical S1P concentration despite higher S1P levels in plasma, yet lower density of S1PR1, S1PR2 and S1PR4 in nerve-terminal-enriched membranes. HFD-fed mice exhibited increased plasma and cortical concentrations of S1P, and decreased density of S1PR1 and S1PR4. These findings point towards altered S1P signaling in synapses of insulin resistance and diet-induced obesity models, suggesting a role of S1P signaling in T2D-associated synaptic dysfunction.</p>", "<title>Keywords</title>", "<p>Open access funding provided by Lund University.</p>" ]

[]

[ "<title>Acknowledgements</title>", "<p>The authors thank Nadiia Kravchenko for technical assistance with insulin determination, and Eugenia Cordero Concha for handling the GK rats.</p>", "<title>Author Contributions</title>", "<p>AM, JMND designed the study. CS, HE, LV, JPPV, FM conducted experiments, CS, JMND analyzed data. CS wrote the manuscript. AM, LE and JMND revised the manuscript. CS and JMND verified all data. All authors approved the final version of the manuscript.</p>", "<title>Funding</title>", "<p>Open access funding provided by Lund University. This work was supported by the Swedish foundation for International Cooperation in Research and Higher education (#BR2019- 8508), Swedish Research Council (#2019-01130 #2019-01406), Diabetesfonden (#Dia2019-440, #Dia2021-637, #Dia2022-723), Dementiafonden, and Direktör Albert Påhlssons Foundation. The authors acknowledge support from the Lund University Diabetes Center, which is funded by the Swedish Research Council (Strategic Research Area EXODIAB; #2009-1039) and the Swedish Foundation for Strategic Research (#IRC15-0067).</p>", "<title>Data Availability</title>", "<p>The datasets from the current study are available from the corresponding author on reasonable request.</p>", "<title>Declarations</title>", "<title>Conflict of interest</title>", "<p id=\"Par37\">The authors declare no competing interests.</p>" ]

[ "<fig id=\"Fig1\"><label>Fig. 1</label><caption><p>Study design. Samples from Wistar and GK rats were collected at 6 months of age (<bold>A</bold>). Mice were kept on CD or HFD for 1, 4 or 8 weeks before tissue sampling (<bold>B</bold>)</p></caption></fig>", "<fig id=\"Fig2\"><label>Fig. 2</label><caption><p>S1P concentrations and cortical nerve-terminal density of S1PRs of GK and Wistar rats. S1P concentration in <bold>A</bold> plasma and <bold>B</bold> cortical homogenates. <bold>C</bold> Immunoblots for S1PR1-4 and β-actin (protein loading control) after SDS-PAGE separation of 15 µg of protein from nerve terminal-enriched cortical membranes. <bold>D</bold> relative immunoreactivity estimated from the relative signal intensity in the immunoblots. Data shown as mean ± SD overlaid on individual data points (<italic>n</italic> = 3–7; circles = male rats, triangles = female rats). Symbols over data-points indicate significant differences between Wistar and GK rats (*<italic>P</italic> &lt; 0.05, **<italic>P</italic> &lt; 0.01) based on Mann–Whitney (for cortical S1P concentration and S1PR1 immunoreactivity) or Student t-tests (remaining comparisons)</p></caption></fig>", "<fig id=\"Fig3\"><label>Fig. 3</label><caption><p>S1P concentrations and cortical nerve-terminal density of S1PRs of mice fed HFD and CD. S1P concentration in <bold>A</bold> plasma and <bold>B</bold> cortical homogenates. <bold>C</bold> Immunoblots for S1PR1-4 and SNAP25 (protein loading control) after SDS-PAGE separation of 15 µg of protein from nerve terminal-enriched cortical membranes. <bold>D–G</bold> Scatter plots show relative immunoreactivity estimated from the signal intensity in the immunoblots after normalization to mean of CD samples. Data shown as mean ± SD overlaid on individual data points (<italic>n</italic> = 5–20). Circles, triangles, and squares represent 1 week, 1 month, and 2 months of diet intervention, respectively. Symbols over data-points (*<italic>P</italic> &lt; 0.05, **<italic>P</italic> &lt; 0.01) indicate significant mean differences from post-tests comparing CD vs HFD within each age, or age within each diet, using Kruskal–Wallis test followed by Dunn’s multiple comparisons (in panel B), or ANOVA followed by independent comparisons with the Fisher’s least significant difference (LSD) test (in panels A, F–G). Samples indicated by <italic>ex</italic> were excluded from analysis</p></caption></fig>" ]

[ "<table-wrap id=\"Tab1\"><label>Table 1</label><caption><p>Antibodies used for Western blot (WB) and molecular weight of the analyzed band</p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th align=\"left\">Antibody</th><th align=\"left\">Dilution</th><th align=\"left\">Source</th><th align=\"left\">Molecular weight (kDa)</th></tr></thead><tbody><tr><td align=\"left\">Rabbit anti-S1PR1</td><td align=\"left\">1:1,000</td><td align=\"left\">ThermoFisher (PA1-1040)</td><td align=\"left\">47 kDa</td></tr><tr><td align=\"left\">Rabbit anti-S1PR2</td><td align=\"left\">1:500</td><td align=\"left\">Origene (AP01311PU-N)</td><td align=\"left\">42 kDa</td></tr><tr><td align=\"left\">Rabbit anti-S1PR3</td><td align=\"left\">1:1,000</td><td align=\"left\">Origene (TA329055)</td><td align=\"left\">50 kDa</td></tr><tr><td align=\"left\">Rabbit anti-S1PR4</td><td align=\"left\">1:1,000</td><td align=\"left\">Novus (NBP2-24,500)</td><td align=\"left\">39 kDa</td></tr><tr><td align=\"left\">Rabbit anti-SNAP25</td><td align=\"left\">1:5,000</td><td align=\"left\">Abcam (ab109105)</td><td align=\"left\">25 kDa</td></tr><tr><td align=\"left\">HRP-tagged anti-β-actin</td><td align=\"left\">1:10,000</td><td align=\"left\">Sigma-Aldrich (A3854)</td><td align=\"left\">42 kDa</td></tr><tr><td align=\"left\">HRP-tagged anti-rabbit IgG</td><td align=\"left\">1:5,000</td><td align=\"left\">Abcam (ab6802)</td><td align=\"left\">–</td></tr><tr><td align=\"left\">Biotinylated anti-rabbit IgG</td><td align=\"left\">1:5,000</td><td align=\"left\">Vectorlabs (BA-1000)</td><td align=\"left\">–</td></tr></tbody></table></table-wrap>", "<table-wrap id=\"Tab2\"><label>Table 2</label><caption><p>Metabolic parameters of GK and Wistar rats</p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th align=\"left\"/><th align=\"left\">Wistar (<italic>n</italic> = 3)</th><th align=\"left\" colspan=\"2\">GK (<italic>n</italic> = 3)</th><th align=\"left\">F(DFn,DFd), <italic>P</italic>-value</th></tr></thead><tbody><tr><td align=\"left\" colspan=\"4\">Body weight (g)</td><td align=\"left\">Interaction F(1,8) = 38, <italic>P</italic> &lt; 0.001</td></tr><tr><td align=\"left\"> Male</td><td char=\"±\" align=\"char\">582 ± 37</td><td char=\"±\" align=\"char\">353 ± 44</td><td align=\"left\"><italic>P</italic> &lt; 0.001</td><td align=\"left\">Diabetes F(1,8) = 47, <italic>P</italic> &lt; 0.001</td></tr><tr><td align=\"left\"> Female</td><td char=\"±\" align=\"char\">279 ± 7</td><td char=\"±\" align=\"char\">266 ± 20</td><td align=\"left\">n.s.</td><td align=\"left\">Sex F(1,8) = 122, <italic>P</italic> &lt; 0.001</td></tr><tr><td align=\"left\" colspan=\"4\">Glycemia (mmol/L)</td><td align=\"left\">Interaction F(1,8) = 18, <italic>P</italic> = 0.003</td></tr><tr><td align=\"left\"> Male</td><td char=\"±\" align=\"char\">6.5 ± 0.8</td><td char=\"±\" align=\"char\">18.2 ± 3.4</td><td align=\"left\"><italic>P</italic> &lt; 0.001</td><td align=\"left\">Diabetes F(1,8) = 39, <italic>P</italic> &lt; 0.001</td></tr><tr><td align=\"left\"> Female</td><td char=\"±\" align=\"char\">6.3 ± 0.6</td><td char=\"±\" align=\"char\">8.7 ± 1.1</td><td align=\"left\"><italic>P</italic> = 0.035</td><td align=\"left\">Sex F(1,8) = 19, <italic>P</italic> = 0.002</td></tr><tr><td align=\"left\" colspan=\"4\">Plasma insulin (µg/L)</td><td align=\"left\">Interaction F(1,8) = 0.096, n.s.</td></tr><tr><td align=\"left\"> Male</td><td char=\"±\" align=\"char\">2.0 ± 0.6</td><td char=\"±\" align=\"char\">2.8 ± 1.1</td><td align=\"left\">n.s.</td><td align=\"left\">Diabetes F(1,8) = 3.6, n.s.</td></tr><tr><td align=\"left\"> Female</td><td char=\"±\" align=\"char\">0.9 ± 0.3</td><td char=\"±\" align=\"char\">2.0 ± 1.1</td><td align=\"left\">n.s.</td><td align=\"left\">Sex F(1,8) = 4.0, n.s.</td></tr></tbody></table></table-wrap>", "<table-wrap id=\"Tab3\"><label>Table 3</label><caption><p>Metabolic parameters of HFD-fed mice compared to CD-fed mice</p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th align=\"left\"/><th align=\"left\">CD (<italic>n</italic> = 9–16)</th><th align=\"left\" colspan=\"2\">HFD (<italic>n</italic> = 10–16)</th><th align=\"left\">F(DFn,DFd), <italic>P</italic>-value</th></tr></thead><tbody><tr><td align=\"left\" colspan=\"5\">Body weight (g)</td></tr><tr><td align=\"left\"> 1 week</td><td char=\".\" align=\"char\">23.6 ± 1.4</td><td char=\".\" align=\"char\">26.0 ± 1.8</td><td align=\"left\"><italic>P</italic> &lt; 0.001</td><td align=\"left\">Interaction F(2,70) = 2.2, n.s.</td></tr><tr><td align=\"left\"> 1 month</td><td char=\".\" align=\"char\">28.3 ± 1.5</td><td char=\".\" align=\"char\">31.9 ± 2.1</td><td align=\"left\"><italic>P</italic> &lt; 0.001</td><td align=\"left\">Diet F(1,70) = 50, <italic>P</italic> &lt; 0.001</td></tr><tr><td align=\"left\"> 2 months</td><td char=\".\" align=\"char\">30.6 ± 2.3</td><td char=\".\" align=\"char\">35.2 ± 3.4</td><td align=\"left\"><italic>P</italic> = 0.017</td><td align=\"left\">Time F(2,70) = 91, <italic>P</italic> &lt; 0.001</td></tr><tr><td align=\"left\" colspan=\"5\">Fasting glycemia (mmol/L)</td></tr><tr><td align=\"left\"> 1 week</td><td char=\".\" align=\"char\">8.6 ± 1.3</td><td char=\".\" align=\"char\">10.0 ± 0.9</td><td align=\"left\">n.s.</td><td align=\"left\">Interaction F(2,70) = 1.67, n.s.</td></tr><tr><td align=\"left\"> 1 month</td><td char=\".\" align=\"char\">8.2 ± 1.5</td><td char=\".\" align=\"char\">11.0 ± 3.0</td><td align=\"left\"><italic>P</italic> &lt; 0.001</td><td align=\"left\">Diet F(1,70) = 17, <italic>P</italic> &lt; 0.001</td></tr><tr><td align=\"left\"> 2 months</td><td char=\".\" align=\"char\">8.4 ± 1.1</td><td char=\".\" align=\"char\">9.2 ± 1.1</td><td align=\"left\">n.s.</td><td align=\"left\">Time F(2,70) = 1.9, n.s.</td></tr><tr><td align=\"left\" colspan=\"5\">Fasting plasma insulin (µg/L)</td></tr><tr><td align=\"left\"> 1 week</td><td char=\".\" align=\"char\">1.7 ± 1.1</td><td char=\".\" align=\"char\">1.8 ± 0.8</td><td align=\"left\">n.s.</td><td align=\"left\">Interaction F(2,62) = 0.85, n.s.</td></tr><tr><td align=\"left\"> 1 month</td><td char=\".\" align=\"char\">0.6 ± 0.3</td><td char=\".\" align=\"char\">1.0 ± 0.6</td><td align=\"left\">n.s.</td><td align=\"left\">Diet F(1,70) = 95, <italic>P</italic> = 0.023</td></tr><tr><td align=\"left\"> 2 months</td><td char=\".\" align=\"char\">0.8 ± 0.7</td><td char=\".\" align=\"char\">1.5 ± 0.8</td><td align=\"left\"><italic>P</italic> = 0.026</td><td align=\"left\">Time F(2,70) = 10, <italic>P</italic> &lt; 0.001</td></tr><tr><td align=\"left\" colspan=\"5\">2-h glycemia in GTT (mmol/L)</td></tr><tr><td align=\"left\"> 1 week</td><td char=\".\" align=\"char\">8.3 ± 1.5</td><td char=\".\" align=\"char\">12.0 ± 1.8</td><td align=\"left\"><italic>P</italic> &lt; 0.001</td><td align=\"left\">Interaction F(2,70) = 0.66, n.s.</td></tr><tr><td align=\"left\"> 1 month</td><td char=\".\" align=\"char\">7.7 ± 1.0</td><td char=\".\" align=\"char\">11.5 ± 2.2</td><td align=\"left\"><italic>P</italic> &lt; 0.001</td><td align=\"left\">Diet F(1,70) = 95, <italic>P</italic> &lt; 0.001</td></tr><tr><td align=\"left\"> 2 months</td><td char=\".\" align=\"char\">7.3 ± 1.1</td><td char=\".\" align=\"char\">12.1 ± 2.5</td><td align=\"left\"><italic>P</italic> &lt; 0.001</td><td align=\"left\">Time F(2,70) = 0.76, n.s.</td></tr><tr><td align=\"left\" colspan=\"5\">HOMA-IR*</td></tr><tr><td align=\"left\"> 1 week</td><td char=\".\" align=\"char\">13.8 ± 9.2</td><td char=\".\" align=\"char\">17.7 ± 7.3</td><td align=\"left\">n.s.</td><td align=\"left\">Interaction F(2,62) = 0.42, n.s.</td></tr><tr><td align=\"left\"> 1 month</td><td char=\".\" align=\"char\">4.7 ± 2.4</td><td char=\".\" align=\"char\">11.2 ± 7.4</td><td align=\"left\"><italic>P</italic> = 0.011</td><td align=\"left\">Diet F(1,62) = 12, <italic>P</italic> &lt; 0.001</td></tr><tr><td align=\"left\"> 2 months</td><td char=\".\" align=\"char\">6.7 ± 6.2</td><td char=\".\" align=\"char\">14.8 ± 9.1</td><td align=\"left\"><italic>P</italic> = 0.011</td><td align=\"left\">Time F(2,62) = 6.8, <italic>P</italic> = 0.002</td></tr></tbody></table></table-wrap>" ]

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[ "<table-wrap-foot><p>Data is mean ± SD (<italic>n.s.</italic> non-significant difference, <italic>P</italic> &gt; 0.05)</p></table-wrap-foot>", "<table-wrap-foot><p>Data is mean ± SD (<italic>n.s.</italic> non-significant difference, <italic>P</italic> &gt; 0.05)</p><p>*Homeostatic model assessment for insulin resistance calculated from fasting insulinemia and glycemia</p></table-wrap-foot>", "<fn-group><fn><p><bold>Publisher's Note</bold></p><p>Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.</p></fn></fn-group>" ]

[ "<graphic xlink:href=\"11064_2023_4033_Fig1_HTML\" id=\"MO1\"/>", "<graphic xlink:href=\"11064_2023_4033_Fig2_HTML\" id=\"MO2\"/>", "<graphic xlink:href=\"11064_2023_4033_Fig3_HTML\" id=\"MO3\"/>" ]

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{ "acronym": [ "CNS", "HFD", "GK", "GTT", "S1P", "S1PR", "Sph", "SphK", "T2D" ], "definition": [ "Central nervous system", "High-fat diet", "Goto-Kakizaki", "Glucose tolerance test", "Sphingosine 1-phosphate", "Sphingosine-1-phosphate receptors", "Sphingosine", "Sphingosine kinase", "Type 2 diabetes" ] }

[ "<title>Introduction</title>", "<p id=\"Par5\">Antibiotic resistance, a globalized public health peril in this new era of modern medicine, prevailed from the pathogenic microbial development of novel resistance mechanisms of genetic and epigenetic origins against the available antibiotics by circumventing the therapeutic actions of these drugs, leading to the failure of antimicrobial drugs curbing the menace of infection (Dar et al. ##UREF##3##2017##; Soares et al. ##UREF##9##2021##). The discovery, research, and clinical applications of immuno-peptides known as antimicrobial peptides (AMPs), such as defensins, LL-37, gramicidins D, histatins, Renalexin, and others with antibacterial properties, have been considered and proven as alternative antimicrobial agents to antibiotics (Perez-Perez et al. ##REF##34680851##2021##; Peters et al. ##REF##21060861##2010##; Zhang et al. ##REF##34496967##2021##). Antimicrobial peptides are mostly cationic, amphipathic, and composed of 10–100 amino acid residues. These biological agents have shown novel therapeutic potencies against resistant bacterial pathogens elicited via the disruption of the cell wall and plasma membrane, inducing pore formation, upon the establishment of electrostatic interactions with the negatively charged membrane phospholipids, or inhibiting DNA replication and protein synthesis, thereby exhibiting a broad spectrum of therapeutic activities (Jindal et al. ##REF##26046345##2015##; Ołdak and Zielińska ##UREF##8##2017##). Many AMPs possess a net charge of +2 to +9, confirming strong polarity to bacterial cell membrane surface structures, thereby conceding antibacterial activity against bacterial pathogens that pose global public health threats (Wei and Zhang ##UREF##10##2022##). AMPs are known for modulation and orchestration and as indispensable components of the innate immune system functioning as the first line of defense against bacterial attack in eukaryotes and often synthesized as a competitive strategy in prokaryotes to limit and outcompete the cellular growth of competitive microbes (Seyedjavadi et al. ##REF##37744143##2021##). The mechanisms of therapeutic action of AMPs, including antibacterial peptides, have been extensively exploited over the years. Studies in both in vitro, in vivo, and model plasma membranes have confirmed novel and distinct modes of action compared to clinically prescribed antibiotics by extensively provoking plasma membrane incision and permeability, leading to membrane disruption and cell death (Erdem Büyükkiraz and Kesmen ##REF##34606679##2022##; Wei and Zhang ##UREF##10##2022##).</p>", "<p id=\"Par6\">The recent applications of AMPs as disease control therapeutics, coupled with their growing interest as antimicrobial agents produced via recombinant expression systems, have provided a promising and safer platform for the expression and the clinical applications of AMPs <italic>(</italic>Jindal et al. ##REF##26046345##2015##; Montfort-Gardeazabal et al. ##REF##33129981##2021##; Nuti et al. ##REF##28814242##2017##). This study focused on two antimicrobial peptides, LL-37 and Renalexin. LL−37 is the only α-helix cathelicidin-based AMP with therapeutic action against bacterial, viral, and fungal infections (Kang et al. ##REF##31170191##2019##). The biosynthesis of LL−37 occurs in most immune cells, including the mast cell, neutrophils, mucosal epithelial cell, keratinocytes, adipocytes, and the T and B lymphocytes. Cathelicidins (hCAP18) are distinct mammalian immune proteins that act as precursor molecules that undergo proteolytic cleavage at the C-terminal to release a short peptide with antimicrobial and immune-modulatory activity commonly known as LL-37 (Erdem Büyükkiraz and Kesmen ##REF##34606679##2022##). The antibacterial activity of naturally and recombinantly purified LL−37 against multi-drug resistant (MDR) bacteria pathogens has been elucidated and published <italic>(</italic>Dürr et al. ##UREF##4##2006##; Erdem Büyükkiraz and Kesmen ##REF##34606679##2022##; Perez-Perez et al. ##REF##34680851##2021##; Scott et al. ##REF##12244186##2002##). Antimicrobial activity of single-peptide LL-37 showed promising results against <italic>Staphylococcus aureus</italic> (69% sensitivity) and <italic>Escherichia coli</italic> (64% sensitivity) at 50 μM peptide concentration (Perez-Perez et al. ##REF##34680851##2021##). A study by Douglas Clark and colleagues in early 1994 led to the discovery of a novel antimicrobial peptide Renalexin (sometimes referred to as Ranalexin), isolated from the skin of the American Bullfrog <italic>Rana catesbeiana</italic>. It contains a single intramolecular disulfide bond between two cysteine amino acids at positions 14 and 20, which forms a heptapeptide ring within the molecule similar to that seen in the antibiotic Polymyxin B. Renalexin is initially synthesized as a precursor peptide with a putative signal sequence and an acidic amino acid–rich region at its N-terminal <italic>(</italic>Aleinein et al. ##REF##23053091##2013##; Clark et al. ##REF##8144672##1994##). Renalexin has shown therapeutic actions against both gram-positive <italic>S. aureus</italic> and gram-negative <italic>E. coli</italic> bacteria at concentrations ranging from 50 μM and above by interacting with phospholipid membrane via electrostatic binding; in cases where the peptide traverses the membrane, it inhibits protein expression which leads to cell death (Dar et al. ##UREF##3##2017##; Nuti et al. ##REF##28814242##2017##).</p>", "<p id=\"Par7\">Here, we report on an effective and reliable design and expression of soluble hybrid antimicrobial peptide LL-37_Renalexin in <italic>Escherichia coli</italic>. We started with synthetic DNA encoding for the target peptide and using CusF3H+ and SmbP as carrier proteins. The recombinant LL-37_Renalexin gene was cloned into the expression vector pET30a+. The fusion peptides CusF3H+_LL-37_Renalexin and SmbP_LL-37_Renalexin were expressed in <italic>E. coli</italic> with isopropyl β-D-1-thiogalactopyranoside induction under optimized conditions. The recombinant tag-free LL-37_Renalexin was purified via immobilized metal affinity chromatography after being released from the carrier proteins by enterokinase treatment. The in vitro antibacterial activities of the hybrid peptide were ascertained and evaluated.</p>" ]

[ "<title>Materials and methods</title>", "<title>Reagents, plasmid, enzymes, and bacteria strains</title>", "<p id=\"Par8\">\n<italic>Escherichia coli</italic> DH5α cells employed for routine plasmid propagation and subcloning experiments were supplied by New England Biolabs (NEB) (Ipswich MA, USA). Protease-deficient bacteria strains <italic>Escherichia coli</italic> BL21(DE3) and <italic>Escherichia coli</italic> SHuffle T7(DE3) also supplied by NEB were used as microbial expression hosts. The plasmid pET30a+ purchased from EMD Biosciences (Darmstadt, Germany) was chosen for the design and construction of plasmid expression vectors. The MEGAquick-spin DNA and plasmid purification kits were bought from iNtRON Biotechnology (Seoul, South Korea). Restriction enzymes <italic>Nde</italic>I and <italic>Xho</italic>I were purchased from NEB. T4 DNA ligase, <italic>Vent</italic>, and <italic>Taq</italic> DNA polymerases used for molecular cloning and amplification were provided from NEB as well. The synthetic DNA encoding for the hybrid peptide and the protease enterokinase were purchased from GenScript Inc. (Centennial, Piscataway, USA). Isopropyl-β-D-1-thiogalactopyranoside (IPTG) used for induction of expression was purchased from A.G. Scientific Inc. (San Diego, CA, USA). Kanamycin employed as a selective marker for transformed cells and as a positive control was supplied by Sigma-Aldrich (Darmstadt, Germany). Standard protein markers used for peptide size characterization were purchased from NEB and Bio Basic Inc. (Amherst, NY, USA). All culture media including Luria-Bertani, Tryptic Soy Broth, bacteriological agar, and Mueller-Hinton agar used for microbial cultivation, expression, enrichment, and antimicrobial activity were provided by Sigma-Aldrich (Darmstadt, Germany), and Legacy Biologicals (Mount Prospect, IL, USA). All clinical bacteria pathogens including <italic>S. aureus</italic> and <italic>E. coli</italic> used for the antimicrobial activities were supplied from the University Hospital, Department of Pathology, Clinical Microbiology Laboratory (UANL, Monterrey, Mexico).</p>", "<title>Design of the hybrid AMP LL-37_Renalexin</title>", "<p id=\"Par9\">The mature amino acid sequences of the single antimicrobial peptide LL-37 and Renalexin were retrieved from the AMP database (<ext-link ext-link-type=\"uri\" xlink:href=\"https://APD3.unmc.edu/structure\">https://APD3.unmc.edu/structure</ext-link>) with the accession number AP0030/2K60 and AP00513/P39084, respectively, and were used for the design of a novel hybrid peptide LL-37_Renalexin via the application of our newly designed GS peptide linker and the protein tags CusF3H+ and SmbP (Fig. ##FIG##0##1##) that allow for expression and immobilized metal-affinity chromatography (IMAC) purification. The designed hybrid peptide amino acid sequence was flanked at respective positions with the amino acid sequences of protein tags SmbP or CusF3H+, enterokinase site, GS linker, <italic>Nde</italic>I, <italic>Kpn</italic>I, and <italic>Xho</italic>I restriction enzyme sites. The entire amino acid sequence of the designed complete hybrid peptide was optimized (<ext-link ext-link-type=\"uri\" xlink:href=\"http://genomes.urv.es/CAIcal\">http://genomes.urv.es/CAIcal</ext-link>) for efficient expression based on codon usage in <italic>E. coli</italic> as the microbial expression host. The biochemical properties and molecular structure of the designed hybrid peptide were elucidated and analyzed using Expasy and I-TASSER tools (<ext-link ext-link-type=\"uri\" xlink:href=\"https://web.expasy.org/cgi-bin/protparam/protparam\">https://web.expasy.org/cgi-bin/protparam/protparam</ext-link>, <ext-link ext-link-type=\"uri\" xlink:href=\"https://zhanggroup.org/I-TASSER/\">https://zhanggroup.org/I-TASSER/</ext-link>) to access the efficiency and the reliability of production in microbial systems.</p>", "<title>Construction of pET30a+ expression vectors</title>", "<p id=\"Par10\">A 492- and 480-mer oligonucleotide DNA (gene) sequences (Fig. ##SUPPL##0##S1##) encoding for SmbP_LL-37_Renalexin and CusF3H+_LL-37_Renalexin with the accession numbers OR356112 and OR356113, respectively, (<ext-link ext-link-type=\"uri\" xlink:href=\"https://www.ncbi.nlm.nih.gov/Genbank.html\">https://www.ncbi.nlm.nih.gov/Genbank.html</ext-link>) were synthesized based on the optimized hybrid peptide sequence with reference to codon usage in <italic>E. coli</italic>. The gene was cloned into pUC57 and supplied to the protein expression and purification laboratory, UANL, Mexico. Both the pUC57 and pET30a+ plasmid vectors were digested with <italic>Nde</italic>I and <italic>Xho</italic>I restriction enzymes (Fig. ##SUPPL##0##S2##). The restriction digestion products were visualized on a 1% agarose gel stained with 0.5 μg/ml ethidium bromide. The target DNA fragment was cut from the gel and purified using the MEGAquick-spin DNA purification kit. Purified DNA fragments were ligated with the T4 DNA ligase for the design of two plasmid expression vectors pET30a+_CusF3H+_LL-37_Renalexin and pET30a+_SmbP_LL-37_Renalexin. The resulting plasmid construct was transformed for propagation into <italic>E. coli</italic> DH5α by heat shock at 42 °C for 45 s. Transformant cells were selected on Luria-Bertani–kanamycin (30 μg/ml) agar plates. To confirm the presence of the gene in the designed plasmid constructs, the 5′-T7 promoter forward primer (5′-TAATACGACTCACTATAGGG-3′) and the 3′-T7 terminator reserve primer (3′-GCTAGTTATTGCTCACGG-5′) pairs were used for polymerase chain reaction (DNase-free water, 5 mM dNTPs, 10 ng DNA template, 0.5 μM primer, 5U <italic>Taq</italic> polymerase) under the following conditions: initial denaturation at 95 °C for 1 min, denaturation at 95 °C for 30 s, primer annealing at 55 °C for 45 s, elongation at 72 °C for 50 s, and final elongations at 72 °C for 5 min, 32 reaction cycles of amplification. The PCR amplicons were analyzed on a 1% agarose gel and visualized under a UV transilluminator (Fig. ##SUPPL##0##S3##). The plasmid constructs were sequenced by STARSEQ GmbH (Instituto de Biotecnologia, UNAM, Mexico). FinchTV version 1.4 was employed to analyze and confirm the sequenced nucleotides using EMBOSS (data not shown) (Fig. ##SUPPL##0##S4##).</p>", "<title>Expression and purification of recombinant fusion peptides CusF3H+_LL-37_Renalexin and SmbP_LL-37_Renalexin</title>", "<p id=\"Par11\">For expression, the designed recombinant plasmid DNA construct was transformed by heat shock at 42 °C for 45 s into <italic>E. coli</italic> BL21(DE3) and <italic>E. coli</italic> SHuffle T7(DE3) calcium competent cells. An inoculum of 5 ml LB broth (30 μg/ml kanamycin) was made, inoculated with a fresh single colony of transformed cells, and incubated at 37 °C overnight with shaking at 220 rpm. Overnight cultures confirming cell viability and plasmid stability were used to inoculate 1000 ml LB broth (30 μg/ml kanamycin) in a baffled flask and incubated at 37 °C, 220 rpm for about 3–4 h until optical cell density (OD_600nm) of 0.4–0.6 was obtained. Recombinant fusion peptide expression was induced with 1 M isopropyl β-D-1-thiogalactopyranoside (IPTG) at a final concentration of 1 mM. Induced culture flasks were incubated under the optimized condition of 25 °C, 220 rpm for 16 h. Cell pellets harboring the expressed recombinant fusion peptide were harvested by centrifugation at 8500 × g, 4 °C for 15 min into sterilized 50-ml Eppendorf tubes. Collected pellets were resuspended in ice-cold lysis buffer (500 mM NaCl, 50 mM Tris-HCl pH 8.0) and lysed by vortexing on ice with 0.1 mm glass beads. Clear soluble cell lysate was obtained after centrifugation at 8500 × g, 4 °C for 15 min. Purification of recombinant fusion peptide was performed via IMAC using the ÄKTA Prime Plus System (GE Healthcare) for fast protein liquid chromatography (FPLC). Briefly, a 1-ml HisTrap FF agarose-resin Ni(II) charged column was employed for the IMAC purification. The column was equilibrated with 5 column volumes (CV) of equilibrating buffer (500 mM NaCl, 50 mM Tris-HCl pH 8.0). The clarified soluble lysate was loaded onto the column under the conditions of 0.5 MPa pressure and 0.5 ml/min flow rate. Subsequently, the column was washed with 3 CVs of washing buffer (500 mM NaCl, 2.5 mM Imidazole, 50 mM Tris-HCl pH 8.0). The recombinant fusion peptide was eluted by gradient elution with elution buffer (500 mM NaCl, 200 mM Imidazole, 50 mM Tris-HCl pH 8.0). A 10 μl cell lysate, column flow-through, and elution fractions were analyzed on a 15% sodium dodecyl sulfate-polyacrylamide gel electrophoresis (SDS-PAGE), and the purity of protein bands (Tab. ##SUPPL##0##S1##) was measured by densitometry using ImageJ software (v.2.0) (Adamíková et al. ##REF##30810853##2019##).</p>", "<title>Enterokinase cleavage and purification of LL-37_Renalexin</title>", "<p id=\"Par12\">To induce enterokinase treatment, the purified fusion peptide elution fractions were pooled together in a 6-cm dialysis membrane and dialyzed against dialysis buffer (1× PBS pH 7.2) to desalt the purified peptide and exchange the elution buffer. In brief, the dialysis setup was incubated at room temperature for 1 h with gentle shaking on a magnetic stirrer, followed by overnight incubation at 4 °C. The concentration of the dialyzed fusion peptide was estimated by Bradford analysis using the Bovine Serum Albumin (BSA) (Tab. ##SUPPL##0##S2##, Fig. ##SUPPL##0##S5## and Tab. ##SUPPL##0##S3##, Fig. ##SUPPL##0##S6##) as standard protein (Colyer and Walker ##UREF##2##1996##). A 1 mg purified fusion peptide was exposed to 20 U enterokinase cleavage (5 U/μl) for the release of hybrid peptide LL-37_Renalexin (tag-free). The cleavage reaction was incubated at room temperature for 16 h, followed by enzyme inactivation at −20 °C for 3 h. 10 μl of cleavage mixture was analyzed on a 15% and 18% Tricine SDS-PAGE.</p>", "<p id=\"Par13\">In purifying the tag-free hybrid peptide by IMAC, a 1.0 × 10-cm chromatographic syringe column was loaded with 1 ml of agarose resins charged with 1 M Ni(II) ions. The column was equilibrated with 10 CV of 1× PBS pH 7.2, and the enterokinase cleavage mixture was loaded into the column, mixed, and incubated at 4 °C for 1.5 h for efficient binding of protein tags to the affinity column. Column flow-through was collected as tag-free hybrid AMP LL-37_Renalexin. The protein tags were released with elution buffer (500 mM NaCl, 200 mM Imidazole, 50 mM Tris-HCl pH 8.0). The purified tag-free hybrid peptide was characterized on a 15% Tricine SDS-PAGE, and the concentration was estimated by Bradford analysis and Nanodrop absorbance readings at 280 nm (Tab. ##SUPPL##0##S4##, Fig. ##SUPPL##0##S7## and Tab. ##SUPPL##0##S5##, Fig ##SUPPL##0##S8##).</p>", "<title>Antimicrobial activity assay of recombinant hybrid peptide LL-37_Renalexin</title>", "<title>Culture media</title>", "<p id=\"Par14\">Tryptic soy broth (TSB): tryptone (pancreatic digest of casein) 17.0 g, soytone (peptic digest of soybean) 3.0 g, glucose (dextrose) 2.5 g, sodium chloride 5.0 g, and dipotassium phosphate 2.5 g, pH 7.3. Bacteriological agar: Mueller–Hinton agar (MHA), beef extract 2.0 g, acid hydrolysate of casein 17.5 g, starch 1.5 g, agar 17.0 g, pH 7.3. Phosphate-buffered saline (PBS); sodium chloride 8 g, potassium chloride 0.2 g, sodium phosphate dibasic 1.44 g, potassium phosphate monobasic 0.245 g, pH 7.4. All culture media and buffers were used for microbial cultivation of bacteria clinical isolates, inoculum suspension preparations, and antimicrobial activity tests.</p>", "<title>Inoculum preparation</title>", "<p id=\"Par15\">Single colonies of test bacteria clinical isolates were cultured in 5 ml TSB and incubated at 37 °C, 220 rpm for 16 h. A 20 μl overnight culture was inoculated into 5 ml TSB and incubated at 37 °C with shaking until an optical cell density (OD_600nm) of 0.8–1.0 was obtained (mid-logarithmic growth). Cells at the log growth phase were serially diluted in 1× PBS pH 7.2 buffer and the dilution suspensions at 1 × 10⁵ CFU/ml equivalent to 0.5 McFarland turbidity standard were employed for antimicrobial assay.</p>", "<title>Antimicrobial activity</title>", "<title>Dose-response assay: minimum inhibition concentration (MIC) determination</title>", "<p id=\"Par16\">The antimicrobial activity of the hybrid peptide was evaluated against gram-positive and gram-negative bacteria clinical isolates of <italic>Staphylococcus aureus</italic>, <italic>Escherichia coli</italic>, methicillin-resistant <italic>Staphylococcus aureus</italic>, and <italic>Klebsiella pneumoniae</italic> from the University Hospital. The minimum inhibition concentrations against the test pathogens were ascertained and estimated in accordance with the modification of the National Committee for Clinical Laboratory Standards (NCCLS) for the broth microdilution method (CLS ##UREF##1##2022##). Briefly, bacteria cell cultures in the mid-log growth phase in TSB were serially diluted in 1× PBS pH 7.2 buffer to 1 × 10⁵ CFU/ml. A 2-fold broth microdilution assay was performed in a 96-well microtiter plate with a 200 μl assay volume per well composed of 100 μl diluted peptide at concentrations ranging from 0.5–33 μM, 80 μl bacteria suspensions, and 20 μl TSB medium. The plate was incubated at 37°C for 3 h with shaking. After 3 h incubation, 0.1 ml aliquot was taken per well and a 10-fold dilution was made from which a 100 μl aliquot was spread on tryptic soy agar (TSA). Inoculated plates were incubated at 37 °C for 20 h and the remaining colony-forming units were evaluated (Tab. ##SUPPL##0##S6##, ##SUPPL##0##S7##, ##SUPPL##0##S8##, ##SUPPL##0##S9##, ##SUPPL##0##S14##, and ##SUPPL##0##S15##), and MICs were calculated as the lowest peptide concentration that obviate visible turbidity using a modified B. Gompertz function for the line of best fit in dose-response analysis (Lambert and Pearson ##REF##10792538##2000##).</p>", "<title>Time-killing assay</title>", "<p id=\"Par17\">The antibacterial killing kinetics of the hybrid peptide was evaluated against two bacterial isolates by ascertaining the time course to kill test bacteria isolates suspension of <italic>S. aureus</italic> (gram+), and <italic>E. coli</italic> (gram−). In brief, bacterial cultures in the mid-logarithmic growth phase were incubated as described previously with the peptide LL-37_Renalexin at a concentration of approximately 2× MIC in a TSB medium. The 10 μl aliquot suspensions were taken at every 20 min interval until a period of 3 h incubation was observed. A 10-fold dilution in 1× PBS pH 7.2 was made, and 0.1 ml aliquots were inoculated on TSA medium. Inoculated plates were incubated at 37 °C for 20 h. Log remaining CFU/ml of test pathogens were taken and plotted against time (Tab. ##SUPPL##0##S10##, and ##SUPPL##0##S11##). Two control samples were made, positive control (bacterial suspension and kanamycin at the same peptide concentration) and negative control (bacterial suspension and 1X PBS buffer). The average of the total remaining CFU/ml from each treatment was evaluated (Tab. ##SUPPL##0##S12## and ##SUPPL##0##S13##) and analyzed via one-way analysis of variance (ANOVA). The hybrid peptide with a single disulfide linkage (S–S bond) expressed in <italic>E. coli</italic> SHuffle T7(DE3) showing relatively lower minimum inhibitory concentrations (MICs) was employed for time-killing kinetic assay.</p>" ]

[ "<title>Results</title>", "<title>Design of hybrid peptide and construction of recombinant plasmids</title>", "<p id=\"Par18\">In designing the hybrid peptide with the molecular gene map shown (Fig. ##FIG##1##2##A), we employed the mature amino acid sequences that encode for peptides LL-37 and Renalexin retrieved from the antimicrobial peptide database <ext-link ext-link-type=\"uri\" xlink:href=\"http://aps.unmc.edu/AP/prediction/\">http://aps.unmc.edu/AP/prediction/</ext-link> with the accession number AP00310/2K60 and AP00513/P39084, respectively. The amino acid GS (glycine and serine) between LL-37 and Renalexin in the gene construct was employed as a novel, simple, flexible peptide linker allowing for the construction of the hybrid peptide LL-37_Renalexin. DDDDK amino acid sequence at the N-terminal of the hybrid peptide serves as an enterokinase site that allows for the molecular cleavage of the protein tags SmbP and CusF3H+ yielding a tag-free hybrid AMP LL-37_Renalexin. For efficient production and affinity purification, the carrier proteins CusF3H+ and SmbP were inserted between the LL-37_Renalexin gene sequence and the start codon. The molecular Riben structure (Fig. ##FIG##1##2##B) of the designed hybrid peptide predicted in i-TASSER (<ext-link ext-link-type=\"uri\" xlink:href=\"https://zhanggroup.org/I-TASSER/\">https://zhanggroup.org/I-TASSER/</ext-link>) suggests a secondary structural motif with efficient production in <italic>E. coli</italic> (Kesidis et al. ##REF##32534131##2020##). The structure predicted and modeled in i-Tasser was confirmed in the Phyre2 server (Fig. ##SUPPL##0##S9##) (<ext-link ext-link-type=\"uri\" xlink:href=\"http://www.sbg.bio.ic.ac.uk/phyre2\">http://www.sbg.bio.ic.ac.uk/phyre2</ext-link>) with 99.9–99.3% confidence level under the model template identifier (i.d) c2K6oA (LL-37) and c2fcgF (Renalexin) (Kelley et al. ##UREF##6##2016##). The application of GS peptide linker facilitates the design of a novel hybrid peptide by maintaining the molecular α-helix structure in LL-37 and the heptapeptide ring of Renalexin (<ext-link ext-link-type=\"uri\" xlink:href=\"https://APD3.unmc.edu/structure\">https://APD3.unmc.edu/structure</ext-link>). The target hybrid peptide has the following biochemical properties: 59 amino acid length, +9 net charge, pI of 10.3, 44% hydrophobicity and 56% hydrophilicity, GRAVY index of 0.00, instability index of 21.3, aliphatic index of 102.37, and a molecular weight of 6.740 kDa theoretically predicted in <ext-link ext-link-type=\"uri\" xlink:href=\"http://www.expasy.org/tools/proparameter\">http://www.expasy.org/tools/proparameter</ext-link>. The physicochemical properties of the hybrid peptide suggest strong electrostatic cationic polarity, hydropathicity, stability, and good isoelectric potentials which provide theoretical evidence of strong antimicrobial activity of this hybrid peptide at a varying pH range compared to its counterpart single peptides. Cellular toxicity of the hybrid peptide against healthy human cells was ascertained using ToxIBTL advanced bioinformatic online server (<ext-link ext-link-type=\"uri\" xlink:href=\"https://server.wei-group.net/ToxIBTL/server.html\">https://server.wei-group.net/ToxIBTL/server.html</ext-link>). ToxIBTL is an online in silico peptide and protein toxicity prediction tool that operates using evolutionary information and the physicochemical properties of peptide sequence via the integration of bottleneck principle to predict peptide or protein toxicity level. Our in silico cytotoxicity analysis (Fig. ##SUPPL##0##S10##) showed that the hybrid peptide has no toxic effect (zero toxicity) on normal human cells at the niche of infection at a toxicity score of 3.7139784e-05 (0.0000371) which is by far below the standard threshold of 0.5.</p>", "<p id=\"Par19\">The DNA nucleotide (gene) of the coding sequence (CDS) was chemically synthesized by GenScript based on the optimized amino acid sequence that encodes for the recombinant fusion proteins CusF3H+_LL-37_Renalexin and SmbP_LL-37_Renalexin. The gene map of the synthetic DNA constituted restriction sites for <italic>Nde</italic>I, <italic>Kpn</italic>I, <italic>Xho</italic>I, and enterokinase, a protein tag site at the N-terminal, with the hybrid peptide site between the protein tag site and a stop codon. The 492-bp (SmbP tag construct) and 480-bp (CusF3H+ tag construct) synthetic DNAs were molecularly cloned into pET30a+ at <italic>Nde</italic>I and <italic>Xho</italic>I restriction sites under the control of T7 promoter using the T4-DNA ligase for the design of two plasmid expression vectors pET30a+_CusF3H+_LL-37_Renalexin and pET30a+_SmbP_LL-37_Renalexin. The plasmid construct was confirmed by colony-based PCR screening using the T7 promoter and terminator-specific primers. After colony-based PCR screening, the correct recombinant plasmid sequence was further confirmed by DNA sequencing (data not shown).</p>", "<title>Expression and IMAC purification of recombinant fusion protein</title>", "<p id=\"Par20\">The protease-deficient bacterial strains <italic>E. coli</italic> BL21(DE3) and <italic>E. coli</italic> SHuffle T7(DE3) competent cells used as expression hosts encode for the T7 RNA polymerase allowing for the expression of the recombinant fusion peptides CusF3H+_LL-37_Renalexin and SmbP_LL-37_Renalexin under the influence of T7 promoter. IPTG induction and expression of fusion peptides successfully showed an efficient insertion of DNA under the T7 promoter in the designed plasmid constructs mentioned above. The carrier proteins CusF3H+ and SmbP show significant advantages in the production and purification of the recombinant fusion peptides expressed as soluble proteins with no formation of inclusion bodies (Perez-Perez et al. ##REF##34680851##2021##; Vargas-Cortez et al. ##REF##26494603##2016##). In evaluating small-scale expression of recombinant fusion proteins CusF3H+_LL-37_Renalexin and SmbP_LL-37_Renalexin (17 kDa) in <italic>E. coli</italic> BL21(DE3) and <italic>E. coli</italic> SHuffle T7(DE3), soluble cell lysates were prepared by lysing cell pellets collected from 2 ml of 1 mM IPTG-induced overnight cultures to access the presence of the target protein in soluble cell lysate. Analysis of 5 μl aliquot of clear lysate on a 15% SDS-PAGE showed successful evidence of expression of the target fusion proteins with 17–18 kDa-expected protein band compared to non-induced cells as negative control samples (Fig. ##FIG##2##3##). For the large-scale production of recombinant fusion proteins, <italic>E. coli</italic> strains <italic>E. coli</italic> BL21(DE3) and <italic>E. coli</italic> SHuffle T7(DE3) were used as expression hosts. From a 1-L expression volume, cell pellets were collected from 1 mM IPTG-induced overnight cultures and lysed by mechanical vortexing on ice with 0.1 mm glass beads. After cell lysis, clear soluble lysate was collected and employed as a protein source for the immobilized metal affinity chromatography.</p>", "<p id=\"Par21\">For the purification of fusion protein SmbP_LL-37_Renalexin and CusF3H+_LL-37_Renalexin, the ÄKTA Prime Plus system (GE Healthcare Systems) was employed for fast protein liquid chromatography (FPLC) by metal affinity chromatography. A 1-ml HisTrap FF column charged with Ni(II) was used to isolate the target recombinant peptide from the pool of cellular proteins present in the soluble lysate fractions collected. Analysis of IMAC purification fractions of the recombinant fusion protein on a 15% SDS PAGE (Fig. ##FIG##3##4##) showed evidence of target peptide in the elution fractions (200 mM imidazole) without any trace of peptide indications in the column flow-through. This result confirms the high affinity of CusF3H+ and SmbP to agarose-resin Ni(II)-charged column that facilitates the binding of the target proteins unto the column while the untargeted cellular proteins exit the column as flow-through (Montfort-Gardeazabal et al. ##REF##33129981##2021##; Perez-Perez et al. ##REF##34680851##2021##; Vargas-Cortez et al. ##REF##28087367##2017##). Our Bradford quantification analysis using standard Bovine Serum Albumin (BSA) calibration equations (data not shown) indicated a peptide concentration of 3.136 mg/L for CusF3H+_LL-37_Renalexin produced in <italic>E. coli</italic> SHuffle T7(DE3) and 1.523 mg/L for SmbP_LL-37_Renalexin produced in <italic>E. coli</italic> BL21(DE3) with a purity of 90–95% matching the purity standard of commercially available synthetic therapeutic peptides. We observed a 2-fold higher recombinant fusion peptide yield in <italic>E. coli</italic> SHuffle T7(DE3) compared to production in <italic>E. coli</italic> BL21(DE3).</p>", "<title>Enterokinase cleavage and purification recombinant LL-37_Renalexin</title>", "<p id=\"Par22\">Our previous studies (Montfort-Gardeazabal et al. ##REF##33129981##2021##; Perez-Perez et al. ##REF##34680851##2021##) reported the abrogative effect on the bioactivity of recombinant antimicrobial peptides with attached protein tags. We employed the restriction enzyme enterokinase for the selective cleavage of protein tags CusF3H+ and SmbP from the above-purified fusion peptides. Electrophoretic analysis of inactivated enterokinase cleavage mixture revealed three protein bands of size 18 kDa (uncleaved fusion peptide), ≈13 kDa (CusF3H+ or SmbP), and ≈10 kDa (LL-37_Renalexin, tag-free) on Tricine SDS-PAGE (Fig. ##FIG##4##5##). The uncleaved fusion peptide detected was due to an incomplete enzymatic cleavage reaction. Finally, the tag-free LL-37_Renalexin was purified via a one-step IMAC purification using an agarose resin Ni(II)-charged syringe column unto which the uncleaved fusion peptide and the carrier proteins CusF3H+ and SmbP remain bounded, allowing for the elution of the target hybrid peptide LL-37_Renalexin as column flowthrough. A 2.16 mg/L for tag-free LL-37_Renalexin expressed in <italic>E. coli</italic> SHuffle T7(DE3) and 0.72 mg/L expressed in <italic>E. coli</italic> BL21(DE3) were obtained after Bradford and Nanodrop spectrometry (A280nm) quantification analysis.</p>", "<title>Antimicrobial activity of recombinant LL-37_Renalexin</title>", "<p id=\"Par23\">The antimicrobial potency of the purified hybrid peptide against clinical isolates of <italic>S. aureus</italic>, <italic>E. coli</italic>, MRSA, and <italic>K. pneumoniae</italic> was evaluated, and the minimum inhibitory concentrations (MICs) were determined via the broth microdilution assay as described by NCCLSI (Lacy et al. ##UREF##7##2004##). Data on remaining colony-forming units (CFU/ml) of test pathogens were taken and analyzed after overnight culture with the peptide. The MIC, defined as the minimum peptide concentration that prevented visible turbidity in the test pathogen, was calculated using a modified Benjamin Gompertz sigmoid function (Lambert and Pearson ##REF##10792538##2000##) from the plot of peptide concentrations against the remaining CFU/ml (dose-response plot). The dose-response plots (Fig. ##FIG##5##6##) show the antibacterial activity of the hybrid AMP LL-37_Renalexin at MIC levels of 10–27 μM, much lower than the reported MICs of the single-peptide LL-37 and Renalexin (50–100 μM) (Aleinein et al. ##REF##23053091##2013##; Kang et al. ##REF##31170191##2019##; Perez-Perez et al. ##REF##34680851##2021##). The MIC result we observed in this study affirms with data reported on related hybrid and dimeric peptides tested against the same bacterial pathogens (Cheng et al. ##REF##34885732##2021##; Dürr et al. ##UREF##4##2006##; Kang et al. ##REF##31170191##2019##; Wei and Zhang ##UREF##10##2022##; Seyedjavadi et al. ##REF##37744143##2021##). The dose-response antimicrobial activity results indicated that the test bacterial pathogens were sensitive to the recombinant hybrid peptide at active peptide concentrations as low as 10 μM and 33 μM. Interestingly, we evaluated the bioactivity of the hybrid peptide without disulfide-linkage expressed in <italic>E. coli</italic> BL21(DE3); the results (Table ##TAB##0##1##) suggested no significant difference in the MICs compared to the hybrid peptide with disulfide-linkage expressed in <italic>E. coli</italic> SHuffle T7(DE3).</p>", "<title>Time-kill kinetics analysis</title>", "<p id=\"Par24\">The time-kill kinetic assay (Fig. ##FIG##6##7##) disclosed that the hybrid peptide shows a multifunctional antibacterial activity within 1.5 h via disruption of membrane integrity and membrane traversing, demonstrating a strong but relatively slow antibacterial potency against all investigated gram-positive and gram-negative bacterial pathogens as compared to the classical antibiotic kanamycin that exhibited its antibacterial activity within less than an hour. We observed that the antibacterial activity of the hybrid peptide in comparison to the known antibiotic kanamycin analyzed against total remaining CFU/ml showed a significant difference at <italic>p</italic>-values &lt; 0.05 (Fig. ##FIG##7##8##), with the hybrid peptide showing relatively similar antibacterial actions as exhibited by kanamycin (Aleinein et al. ##REF##23053091##2013##; Hanafiah et al. ##REF##33281340##2020##; Montfort-Gardeazabal et al. ##REF##33129981##2021##; Perez-Perez et al. ##REF##34680851##2021##).</p>" ]

[ "<title>Discussion</title>", "<p id=\"Par25\">Antimicrobial peptides (AMPs) as drug candidates are being considered the new hope for the biomedical and pharmaceutical industries in conjunction with their multifunctional antibacterial pharmacological actions against infectious agents (Montfort-Gardeazabal et al. ##REF##33129981##2021##; Moretta et al. ##REF##34195099##2021##; Nuti et al. ##REF##28814242##2017##). The production of AMPs as therapeutic peptides via the applications of recombinant DNA technology (rDNA) and the use of cost-effective microbial expression systems have facilitated the large-scale acquisition of bioactive AMPs, enhancing clinical and research applications (Akhtar et al. ##REF##22675369##2012##; Dar et al. ##UREF##3##2017##). In addition to the successful application of rDNA, the efficient, fast, and easy growth of microbial expression hosts like <italic>E. coli</italic> BL21(DE3) and <italic>E. coli</italic> SHuffle T7(DE3) has allowed for commercial production of bioactive cationic AMPs independent of posttranslational modifications (Aleinein et al. ##REF##23053091##2013##; Klubthawee et al. ##REF##32499514##2020##; Montfort-Gardeazabal et al. ##REF##33129981##2021##). Novel AMPs like Brevinin-2R, Renalexin, Cecropin A, and LL-37 as single peptides have been produced as recombinant protein-based drug candidates in microbial systems and have shown a promising antimicrobial effect (Aleinein et al. ##REF##23053091##2013##; Chhetri et al. ##REF##26629417##2015##; Zhang et al. ##UREF##11##2018##). The thriving commercial production of AMPs in <italic>E. coli</italic> largely depends on the design and use of protein tags, including but not limited to poly-histidine, maltose-binding protein (MBP), thioredoxin protein (THX), and glutathione S-transferase (GST), that allow for the expression of recombinant proteins as fusion proteins in benign forms (Akhtar et al. ##REF##22675369##2012##; El-Gayar ##UREF##5##2015##; Gddoa Al-sahlany et al. ##REF##32168785##2020##; Mo et al. ##REF##30267582##2018##; Riguero et al. ##REF##32861092##2020##). Recently, our newly designed small metal-binding proteins SmbP and CusF3H+ have been exploited as protein tags which aided in the production and purification of AMPs like LL-37 and Bin1b, and green fluorescence protein (GFP) in different <italic>E. coli</italic> strains (Montfort-Gardeazabal et al. ##REF##33129981##2021##; Perez-Perez et al. ##REF##34680851##2021##; Vargas-Cortez et al. ##REF##26494603##2016##; Vargas-Cortez et al. ##REF##28087367##2017##). Our previous study has unraveled the capabilities of these protein tags that facilitate the secretion, folding, and purification of expressed recombinant LL-37 and Bin1b as single peptides with intact bioactivity at purity above 80% (Montfort-Gardeazabal et al. ##REF##33129981##2021##; Perez-Perez et al. ##REF##34680851##2021##; Santos et al. ##REF##30997666##2019##). Peptide hybridization has been considered an advanced technique for the design of novel AMPs having reliable peptide stability, long half-life with intact therapeutic activity with hybrid peptides like Cecropin A_Thanatin, and Indolicidin_Renalexin showing broad-spectrum antimicrobial activity against multidrug-resistant bacterial pathogens (Bayarbat et al. ##UREF##0##2016##; Seyedjavadi et al. ##REF##37744143##2021##; Wade et al. ##REF##31067436##2019##).</p>", "<p id=\"Par26\">The broad-spectrum antibacterial activity of LL-37 as a single peptide although at a higher peptide concentration has led us to design a recombinant hybrid peptide production strategy via the application of GS flexible peptide linker and a carrier proteins CusF3H+ and SmbP. The simple GS peptide linker enhances the expression of the hybrid peptide LL-37_Renalexin and maintains the spatial configuration within the hybrid peptide with intact and advanced bioactivity. This data strongly suggests that the GS peptide linker can be employed as a reliable, simple, flexible linker for the design and expression of recombinant therapeutic peptides as compared to the RGGPDGSGPDESGPDE flexible linker employed in the design of hybrid and dimeric peptides with primary structural modifications (Klubthawee et al. ##REF##32499514##2020##; Seyedjavadi et al. ##REF##37744143##2021##).</p>", "<p id=\"Par27\">In this study, we have efficiently employed the mature amino acids of LL-37 and Renalexin for the design of a novel hybrid peptide LL-37_Renalexin with zero cytotoxicity against healthy human cells. We reliably cloned the cDNA that encodes for the target peptide into pET30a+ under the T7 promoter and terminator regions and obtained a successful expression in <italic>E. coli</italic> BL21(DE3) and <italic>E. coli</italic> SHuffle T7(DE3) under the condition of 25 °C, for 16 h with 1 mM IPTG. In other studies, the microbial strains, induction, and temperature conditions demonstrated profound effect on the expression of both single, hybrid, and dimeric peptides with inclusion bodies indications (Chhetri et al. ##REF##26629417##2015##; Montfort-Gardeazabal et al. ##REF##33129981##2021##; Seyedjavadi et al. ##REF##37744143##2021##; Shang et al. ##REF##33042031##2020##; Wade et al. ##REF##31067436##2019##; Xu et al. ##REF##24945359##2014##). The expression condition and peptide isolation protocols observed in this study made it possible for efficient production coupled with higher yield.</p>", "<p id=\"Par28\">Recombinant fusion peptides CusF3H+_LL-37_Renalexin and SmbP_LL-37_Renalexin expression level and its presence in soluble cell lysate provide an evidential advantage of protein tag CusF3H+ and SmbP over others like glutathione S-transferase, maltose-binding protein, amyloid-β peptide, and thioredoxin tag (Aleinein et al. ##REF##23053091##2013##; Chhetri et al. ##REF##26629417##2015##; Zhang et al. ##UREF##11##2018##) yielding up to 95% peptide purity which matched the purity standard of commercially available synthetic therapeutic peptides (Zhongxuan et al. ##REF##33255863##2020##). In this present study, Bradford quantification of purified and PBS-dialyzed protein elution fractions revealed a total recombinant peptide yield of 1.5–3.1 mg/L fusion proteins SmbP_LL-37_Renalexin and CusF3H+_LL-37_Renalexin expressed in BL21(DE3) and SHuffle T7(DE3), respectively. We observed a 2-fold higher peptide yield in <italic>E. coli</italic> SHuffle T7(DE3) for both SmbP and CusF3H+ tagged fusion proteins than in <italic>E. coli</italic> BL21(DE3), providing relevant supportive data on the usage of <italic>E. coli</italic> SHuffle T7(DE3) as microbial host for the production of either single or hybrid recombinant hybrid peptides with or without disulfide bonds (Montfort-Gardeazabal et al. ##REF##33129981##2021##). Our result showed a higher expression level in recombinant hybrid peptide as soluble protein than reported from other studies, 0.9 mg/L by Seyedjavadi et al. (##REF##37744143##2021##); 900 μg/L by Clement et al. (##REF##25587248##2015##), and 0.3 mg/L by Cheng et al. (##REF##34885732##2021##).</p>", "<p id=\"Par29\">We employed the enzyme enterokinase for the selective cleavage of protein tags CusF3H+ and SmbP from the purified fusion peptides due to the presence of an enterokinase site between the hybrid peptide and the protein tag. Analysis of inactivated enterokinase cleavage mixture revealed three protein bands of an approximate molecular size of 18 kDa (uncleaved fusion peptide), 13 kDa (CusF3H+ or SmbP), and 10 kDa (LL-37_Renalexin, tag-free) on Tricine SDS-PAGE with tag-free recombinant hybrid peptide showing slightly higher band size than the theoretically expected size (6.740 kDa). This result can be attributed to the inefficient enterokinase cleavage observed, which may be associated with the presence of phenylalanine, leucine, and isoleucine residues and heptapeptide motif (Rana box) in Renalexin that forms a cyclic disulfide bond aiding a molecular structural loop formation folded unto the N-terminal that is known to influence enzymatic cleavage and peptide reduction. Also, the high hydrophobicity of the peptide which prevents complete reduction with SDS and mercaptoethanol reagents, and the peptide molecular folding in aqueous systems all of which influence poor peptide mobility. These findings agree with previous studies reported where the single peptides LL-37 and Renalexin show higher protein band sizes than the theoretically determined sizes (Aleinein et al. ##REF##23053091##2013##; Perez-Perez et al. ##REF##34680851##2021##). A 0.7–2.1 mg/L LL-37_Renalexin (tag-free) peptide was obtained upon spectroscopic and Bradford quantification of the second IMAC purification elution fractions (Montfort-Gardeazabal et al. ##REF##33129981##2021##; Perez-Perez et al. ##REF##34680851##2021##; Vargas-Cortez et al. ##REF##28087367##2017##).</p>", "<p id=\"Par30\">The relatively slow induction of the antibacterial activity of LL-37_Renalexin compared to the known antibiotic kanamycin we observed in the time-killing kinetic assay results can be related to the high hydrophobicity of the hybrid peptide which may cause partial exposure of hydrophilic regions to bacterial membrane (Wei and Zhang ##UREF##10##2022##) coupled with the presence of monovalent Na(I) and K(I) cations in the assay medium that are known to cause shielding effects between the cationic peptides and anionic bacterial membrane surface, hence, the delay in eliciting antibacterial activity in the case of the hybrid peptide as compared to kanamycin (Huan et al. ##REF##33178164##2020##; Nuti et al. ##REF##28814242##2017##). Our findings show that the hybrid peptide LL-37_Renalexin with 44% hydrophobicity and 56% hydrophilicity elicited near-microbicidal activity against all tested pathogens with above 85% reduction in bacteria colony-forming units at 33 μM peptide concentration. The reduction in CFU/ml observed suggests a bactericidal activity since the level of reduction (<italic>R</italic>_L) in CFU/ml is more significant than three times the logarithm CFU/ml of bacteria (Klubthawee et al. ##REF##32499514##2020##; Zhang et al. ##UREF##11##2018##). All <italic>S. aureus</italic>, <italic>E. coli</italic>, MRSA, and <italic>K. pneumoniae</italic> clinical isolates showed 85% sensitivity at 33 μM as minimum peptide bactericidal concentration with about 25% increment in sensitivity indicative of higher antibacterial potency of this novel hybrid peptide compared to its counterpart single-peptide LL-37 as reported from our previous study with 64% sensitivity against <italic>E. coli</italic> and 69% against <italic>S. aureus</italic> (Perez-Perez et al. ##REF##34680851##2021##). We also observed approximately a 2-fold reduction with respect to the minimum inhibitory hybrid peptide concentration required to inhibit bacterial growth as compared to its single-peptide LL-37. We envisioned that the hybrid peptide’s antimicrobial effects are brought about through its ability to disrupt cell membranes, thanks to its pronounced cationic polarity which enables it to create a strong electrostatic bond with the negatively charged bacteria membrane. Additionally, it is thought that the hybrid peptide’s v-shaped structure, made possible by the presence of the GS flexible linker, allows it to effectively engage with bacterial chromosomal DNA. This interaction can lead to the formation of supercoils, ultimately hindering DNA replication and transcription (Zhang et al. ##REF##34496967##2021##).</p>", "<p id=\"Par31\">In this study, we have designed, produced, and purified a novel multifunctional recombinant hybrid peptide LL-37_Renalexin for the first time via the application of newly designed flexible GS peptide linker and a characterized carrier proteins SmbP and CusF3H+. The small metal-binding protein tags SmbP and CusF3H+ provide an evidential advantage in cytoplasmic production and purification of the novel hybrid AMP LL-37_Renalexin with intact biochemical properties and can be applied as a new avenue to produce recombinant peptides and proteins. The purified tag-free hybrid peptide LL-37_Renalexin exhibited above 85% reduction in bacteria CFU/ml against <italic>S. aureus</italic>, <italic>E. coli</italic>, MRSA, and <italic>K. pneumoniae</italic> clinical isolates at lower minimum inhibition concentration levels of 10–33 μM as compared to its counterpart single-AMPs LL-37 and Renalexin of 50–100 μM reported (Aleinein et al. ##REF##23053091##2013##; Kang et al. ##REF##31170191##2019##; Perez-Perez et al. ##REF##34680851##2021##), making it a competitive antimicrobial agent.</p>", "<p id=\"Par32\">From our antibacterial bioassay findings, we proposed that the newly designed hybrid peptide LL-37_Renalexin can be classified as an antibacterial peptide that may have no toxic effects against normal cells at the niche of infection as theoretically predicted. Previous investigations (Aleinein et al. ##REF##23053091##2013##; Jindal et al. ##REF##26046345##2015##; Kang et al. ##REF##31170191##2019##; Perez-Perez et al. ##REF##34680851##2021##) have confirmed the exclusive antibacterial effect of recombinant single-peptides LL-37 and Renalexin, respectively, showing no toxic effects against normal human cells. Notably, we successfully express the hybrid peptide in <italic>E. coli</italic> with intact bioactivity conferred by the conserved secondary α-helical structures as seen in the single peptides supporting the predicted secondary structure of LL-37_Renalexin clearly showing the conserved helical domains of LL-37 and Renalexin. It is imperative to note, however, that this study lacks empirical data from wet laboratory circular dichroism (CD) spectrometry and 3-(4,5-dimthylthiazol-2-yl)-2,5-diphenyltetrazolium bromide (MTT) assay to support the theoretically predicted structure and toxicity. The recombinant DNA strategies used in the design, production, and purification of recombinant fusion proteins provide a reliable platform and protocol for the expression and purification of therapeutic recombinant proteins in <italic>E. coli</italic> BL21(DE3) and <italic>E. coli</italic> SHuffle T7(DE3) as microbial expression hosts.</p>" ]

[]

[ "<title>Abstract</title>", "<p id=\"Par1\">An alarming global public health and economic peril has been the emergence of antibiotic resistance resulting from clinically relevant bacteria pathogens, including <italic>Enterococcus faecium</italic>, <italic>Staphylococcus aureus</italic>, <italic>Klebsiella pneumonia</italic>, <italic>Acinetobacter baumannii</italic>, <italic>Pseudomonas aeruginosa</italic>, and <italic>Enterobacter</italic> species constantly exhibiting intrinsic and extrinsic resistance mechanisms against last-resort antibiotics like gentamycin, ciprofloxacin, tetracycline, colistin, and standard ampicillin prescription in clinical practices. The discovery and applications of antimicrobial peptides (AMPs) with antibacterial properties have been considered and proven as alternative antimicrobial agents to antibiotics. In this study, we have designed, produced, and purified a recombinant novel multifunctional hybrid antimicrobial peptide LL-37_Renalexin for the first time via the application of newly designed flexible GS peptide linker coupled with the use of our previously characterized small metal-binding proteins SmbP and CusF3H+ as carrier proteins that allow for an enhanced bacterial expression, using BL21(DE3) and SHuffle T7(DE3) <italic>Escherichia coli</italic> strains, and purification of the hybrid peptide via immobilized metal affinity chromatography. The purified tag-free LL-37_Renalexin hybrid peptide exhibited above 85% reduction in bacteria colony-forming units and broad-spectrum antimicrobial effects against <italic>Staphylococcus aureus</italic>, <italic>Escherichia coli</italic>, Methicillin-resistant <italic>Staphylococcus aureus</italic> (MRSA), and <italic>Klebsiella pneumoniae</italic> bacteria clinical isolates at a lower minimum inhibition concentration level (10–33 μM) as compared to its counterpart single-AMPs LL-37 and Renalexin (50–100 μM).</p>", "<title>Key points</title>", "<p id=\"Par2\">• <italic>The hybrid antimicrobial peptide LL-37_Renalexin has been designed using a GS linker.</italic></p>", "<p id=\"Par3\">• <italic>The peptide was expressed with the carrier proteins SmbP and CusF3H+.</italic></p>", "<p id=\"Par4\">• <italic>The hybrid peptide shows antibacterial potency against clinical bacterial isolates.</italic></p>", "<title>Supplementary Information</title>", "<p>The online version contains supplementary material available at 10.1007/s00253-023-12887-5.</p>", "<title>Keywords</title>" ]

[ "<title>Supplementary Information</title>", "<p>\n</p>" ]

[ "<title>Acknowledgements</title>", "<p>We thank Mexico’s Consejo Nacional de Humanidades Ciencias y Tecnologias (CONAHCYT) for the financial support to the graduate student JKN.</p>", "<title>Author contribution</title>", "<p>JKN made the DNA constructs, performed the experiments, and wrote the manuscript. NGC-V provided the fully characterized <italic>S. aureus</italic>, <italic>E. coli</italic>, MRSA, and <italic>K. pneumoniae</italic> strains from his laboratory at the University Hospital and assisted in the evaluation of the antimicrobial assays for MRSA and <italic>K. pneumoniae</italic>. XZ edited the manuscript, designed the experiments, and proposed the research as Principal Investigator. All authors read and approved the final manuscript.</p>", "<title>Funding</title>", "<p>This work was funded by grant UANL-PAICYT-330-CN-2022 awarded to XZ.</p>", "<title>Data availability</title>", "<p>All data supporting the findings of this study are available within the paper and its supplementary information.</p>", "<title>Declarations</title>", "<title>Ethics approval and consent to participate</title>", "<p id=\"Par33\">This in vitro research study does not involve human or animal models; hence, no ethical consent and approval were required.</p>", "<title>Conflict of interest</title>", "<p id=\"Par34\">The authors declare no competing interest.</p>" ]

[ "<fig id=\"Fig1\"><label>Fig. 1</label><caption><p>Amino acid sequence flowchart representation of the expression cassette encoding for the recombinant fusion peptides. <bold>A</bold> Expression cassette encoding for CusF3H+_LL-37_Renalexin. <bold>B</bold> Expression cassette encoding for SmbP_LL-37_Renalexin</p></caption></fig>", "<fig id=\"Fig2\"><label>Fig. 2</label><caption><p>Design of the hybrid antimicrobial peptide LL-37_Renalexin. <bold>A</bold> Representation of expression cassette gene map for LL-37_Renalexin expression in <italic>E. coli</italic>. <bold>B</bold> 3D Riben molecular secondary structure predicted for the hybrid peptide LL-37_Renalexin showing the components including LL-37 (long peptide on left), the flexible GS peptide linker (midway), and Renalexin (short peptide on far right). The peptide structure was predicted in i-Tasser server and confirmed in Phyre2 server with good positive <italic>z</italic>-score (1.00) and <italic>c</italic>-score (−2.09) suggesting an efficient sequence alignment with good, modelled confidence level supporting the predicted structure</p></caption></fig>", "<fig id=\"Fig3\"><label>Fig. 3</label><caption><p>Small-scale expression of recombinant fusion peptides in <italic>E. coli</italic>. <bold>A</bold> 15% SDS-PAGE analysis of CusF3H+_LL-37_Renalexin expressed in <italic>E. coli</italic> BL21(DE3): Lane 1—protein ladder; Lanes 2 and 3—soluble (SF) and insoluble (IF) fractions of untransformed control cells; Lanes 4 and 5—SF and IF of uninduced transformed cells; Lanes 6, 7, and 8—SF of induced transformed cells. <bold>B</bold> 15% SDS PAGE analysis of SmbP_LL-37_Renalexin expressed in <italic>E. coli</italic> BL21(DE3): Lane 1—protein ladder; Lanes 2 and 3—SF and IF of untransformed control cells; Lanes 4 and 5—SF and IF of uninduced transformed cells; Lane 6, 7, and 8—SF of induced transformed cells. <bold>C</bold> 15% SDS-PAGE analysis of CusF3H+_LL-37_Renalexin expressed in <italic>E. coli</italic> SHuffle T7(DE3): Lane 1—protein ladder; Lane 2—soluble fraction of uninduced transformed cells (control); Lanes 3, 4, and 5—SF of induced transformed cells</p></caption></fig>", "<fig id=\"Fig4\"><label>Fig. 4</label><caption><p>Large-scale expression and IMAC purification of recombinant fusion peptides. <bold>A</bold> IMAC purification of SmbP_LL-37_Renalexin (17 kDa) expressed in <italic>E. coli</italic> BL21(DE3). A 15% SDS-PAGE analysis of elution fractions. Lane 1, protein marker; Lane 2, cell lysate; Lane 3, column flow-through; Lane 4–10, elution fractions. <bold>B</bold> IMAC purification of CusF3H+_LL-37_Renalexin (17 kDa) expressed in <italic>E. coli</italic> SHuffle T7. A 15% SDS PAGE analysis of the elution fractions. Lane 1, protein marker; Lane 2, cell lysate; Lane 3, column flow-through; Lane 4-10, elution fractions</p></caption></fig>", "<fig id=\"Fig5\"><label>Fig. 5</label><caption><p>Enterokinase cleavage, tag removal, and second IMAC purification of tag-free LL-37_Renalexin analyzed on Tricine SDS-PAGE. <bold>A</bold> 18% Tricine gel—Lane 1, protein ladder; Lane 2, CusF3H+_LL-37_Renalexin (uncut) expressed in <italic>E. coli</italic> SHuffle T7(DE3); Lane 3, cut CusF3H+_LL-37_Renalexin (enterokinase mix). <bold>B</bold> 18% Tricine gel—Lane 1, protein ladder; Lane 2, SmbP_LL-37_Renalexin (uncut) expressed in <italic>E. coli</italic> BL21(DE3); Lane 3, cut SmbP_LL-37_Renalexin (enterokinase mix). <bold>C</bold> 15% Tricine gel—Lane 1, fusion peptide CusF3H+_LL-37_Renalexin (uncut); Lane 2, enerokinase mix (Protein tag and tag-free LL-37_Renalexin); Lane 3, second IMAC purified hybrid peptide LL-37_Renlexin (tag-free)</p></caption></fig>", "<fig id=\"Fig6\"><label>Fig. 6</label><caption><p>Antimicrobial activity of purified recombinant hybrid AMP LL-37_Renalexin (tag-free) against 1 × 10⁵ CFU/ml of bacteria pathogens. <bold>A</bold> Dose-response activity of LL-37_Renalexin expressed in <italic>E. coli</italic> SHuffle T7(DE3) against CFU/ml of <italic>S. aureus</italic>. <bold>B</bold> Dose-response activity of LL-37_Renalexin expressed in <italic>E. coli</italic> SHuffle T7(DE3) against CFU/ml of <italic>E. coli</italic>. <bold>C</bold> Dose-response activity of LL-37_Renalexin expressed in <italic>E. coli</italic> SHuffle T7(DE3) against CFU/ml of MRSA. <bold>D</bold> Dose-response activity of LL-37_Renalexin expressed in <italic>E. coli</italic> SHuffle T7(DE3) against CFU/ml of <italic>K</italic>. <italic>pneumoniae</italic>. <bold>E</bold> Dose-response activity of LL-37_Renalexin expressed in <italic>E. coli</italic> BL21(DE3) against CFU/ml of <italic>S. aureus</italic>. <bold>F</bold> Dose-response activity of LL-37_Renalexin expressed in <italic>E. coli</italic> BL21(DE3) against CFU/ml of <italic>E. coli</italic>. The data points represent the mean remaining CFU/ml of the test pathogen from three replica plates, and the error bar represents the standard deviation of the mean. Statistical analysis was performed using Gompertz sigmoid function for non-linear regression between the peptide concentration and the CFU/ml of test pathogen</p></caption></fig>", "<fig id=\"Fig7\"><label>Fig. 7</label><caption><p>Time-killing kinetics of LL-37_Renalexin (tag-free) expressed in <italic>E. coli</italic> SHuffle T7(DE3) at 2X MIC against the log 1 × 10⁵ CFU/ml of the test pathogens within 3 h time interval of treatment. <bold>A</bold> Time-kill assay of the hybrid peptide against <italic>S. aureus.</italic>\n<bold>B</bold> Time-kill assay of the hybrid peptide against <italic>E. coli</italic>. A 1 × PBS buffer (pH 7.2) and suspensions of bacteria inoculum were used as the negative control. Antibiotic kanamycin was employed as positive control. The data points represent the mean of log remaining CFU/ml of the test pathogens from three replica plates, and the error bars represent the standard deviation (SD) of the mean CFUs</p></caption></fig>", "<fig id=\"Fig8\"><label>Fig. 8</label><caption><p>Antimicrobial activity of the hybrid peptide LL-37_Renalexin (tag-free) expressed in <italic>E. coli</italic> SHuffle T7(DE3) against 1 × 10⁵ CFU/ml of test pathogens analyzed by one-way analysis of variance (ANOVA). <bold>A</bold> Antibacterial activity against the CFU/ml of <italic>S. aureus</italic>, <bold>B</bold> Antibacterial activity against the CFU/ml of <italic>E. coli</italic>. A 1 × PBS buffer (pH 7.2) and suspension of bacteria inoculum were used as negative control. The bars represent the mean remaining CFU/ml of the test pathogens from three replica plates and the error bars represent the standard deviation (SD) of the means. Asterisks indicate the statistical significance difference (all <italic>P</italic>-values &lt; 0.05) between the peptide, the negative control, and kanamycin (positive control)</p></caption></fig>" ]

[ "<table-wrap id=\"Tab1\"><label>Table 1</label><caption><p>MICs of LL-37_Renalexin expressed in BL21(DE3) and SHuffle T7(DE3) <italic>E. coli</italic> strains</p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th rowspan=\"2\">Pathogens</th><th colspan=\"2\">MICs: LL-37_Renalexin (μM)</th></tr><tr><th><italic>E. coli</italic> BL21(DE3)</th><th><italic>E. coli</italic> SHuffle T7(DE3)</th></tr></thead><tbody><tr><td><italic>Staphylococcus aureus</italic></td><td>20.1</td><td>18.2</td></tr><tr><td><italic>Escherichia coli</italic></td><td>21.4</td><td>19.7</td></tr><tr><td>MRSA</td><td>N/A</td><td>17.5</td></tr><tr><td><italic>Klebsiella pneumoniae</italic></td><td>N/A</td><td>27.8</td></tr></tbody></table></table-wrap>" ]

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[ "<supplementary-material content-type=\"local-data\" id=\"MOESM1\"></supplementary-material>" ]

[ "<table-wrap-foot><p><italic>N/A</italic> Not Analyzed (peptide expressed in <italic>E. coli</italic> SHuffle T7(DE3) was used for subsequent bioactivity)</p></table-wrap-foot>", "<fn-group><fn><p><bold>Publisher’s note</bold></p><p>Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.</p></fn></fn-group>" ]

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[ "<title>Background</title>", "<p id=\"Par5\">According to the National Cancer Institute (NCI), an individual is considered a cancer survivor from diagnosis to the end of life [##UREF##0##1##]. In the United States (US), it is estimated that by 2026, the number of cancer survivors will surpass 20 million, which can be attributed to the ongoing innovation of treatment and early disease detection [##REF##27253694##2##, ##REF##23678936##3##]. While this increase in survival is optimistic, this population requires additional healthcare services to prevent or manage chronic health conditions, sequelae of cancer treatment, and monitoring for cancer reoccurrence [##REF##23678936##3##]. Moreover, given that at least 50% of cancer survivors will experience physical and mental health consequences due to their disease or treatment [##REF##33033685##4##], is important to mitigate healthcare delays in this population.</p>", "<p id=\"Par6\">Overall, healthcare delays (due to the domains of accessibility, financial burden, and social support) can significantly impact cancer survivors, most notably individuals who are ethnic minorities, lower SES, or uninsured [##REF##1899012##5##]. Cancer survivors who experience healthcare delays can significantly suffer the consequences as their care is typically time sensitive [##REF##33315115##6##]. Because timely cancer care is associated with a favorable prognosis, barriers or delays to treatment can result in a more advanced stage of cancer at the time of eventual care, thus, resulting in poorer outcomes [##REF##20564623##7##]. However, physical health is not the only aspect affected; mental health may also suffer at the hands of delayed care [##REF##35393637##8##]. For instance, COVID-19 led to numerous appointment cancelations for cancer survivor patients, including self-cancelations that were caused by depression and anxiety symptoms surrounding the pandemic and safe access to care [##REF##36209390##9##]. These missed appointments, in turn, exacerbated patients’ fears of cancer recurrence as their follow-up care were halted (e.g., laboratory testing, imaging, and appointments), impacting their overall well-being and physical and mental health [##REF##33624572##10##, ##REF##32772225##11##].</p>", "<p id=\"Par7\">One factor associated with increases in healthcare delays among cancer survivors is lower SES status, which is associated with all-cause mortality risk, poorer mental (e.g., depression), and physical health outcomes [##REF##35448185##12##, ##REF##28408935##13##]. Similarly, SES has been linked to a range of cancer outcomes and higher SES (suggesting more financial resources and the ability to afford medical care) is associated with decreased in the length of healthcare delays [##REF##25591711##14##]. Because of the long-term and specialized care needed for cancer survivors, they are at risk of experiencing higher financial burden, which in turn, impacts receipt of survivorship care and increases the risk of mortality, and worsens quality of life [##REF##29596618##15##].</p>", "<p id=\"Par8\">A second factor associated with healthcare delays among cancer survivors is a lack of health literacy, which presents a barrier in properly understanding, communicating, and obtaining information required to navigate the complexity of the healthcare systems efficiently to obtain the care needed and to make educated health decisions [##REF##33494122##16##, ##REF##33938484##17##]. Among the general population, low health literacy has been positively associated with various care delays, including seeking treatment, forgoing care, and struggling with accessibility to needed care and providers [##REF##27043757##18##]. As cancer survivors have complex healthcare needs, mastering the skills required for their continual care is important.</p>", "<p id=\"Par9\">Being foreign-born (immigrant) brings an additional barrier to accessing healthcare and consequently promotes healthcare delays [##REF##33390160##19##]. These barriers among foreign-born may be partially attributed to a higher likelihood of lack of insurance, healthcare cultural perception, and English skills proficiency [##REF##21841297##20##]. Evidence suggests that cancer survivors who are immigrants have a lower quality of life and higher depression symptomology compared to native-born [##REF##23465493##21##]. Moreover, foreign-born individuals may experience unique barriers (e.g., language, discrimination, laws and regulations to qualify for services) that impact the quality of care they can receive. For instance, language barriers and insurance difficulties caused by laws and policies may make it much more difficult for an immigrant patient than a US-born patient to receive adequate needed care [##REF##26586971##22##].</p>", "<p id=\"Par10\">While it is well established that SES barriers and a lack of health literacy are associated with healthcare delays, there is limited and inconsistent knowledge of how these associations differ by nativity. To guide the selection of our variables for our model, we used the theoretical framework from Wafula and Snipes for barriers to healthcare among Black immigrants in the US [##REF##24006174##23##]. Additionally, we adapted their framework to focus on assessing the association between the barrier factors (i.e., SES barriers, health literacy) and general healthcare delays among cancer survivors and tested the moderating effects of nativity between SES barriers and health literacy with general healthcare delays (Fig. ##FIG##0##1##). Thus, this study aims to contribute to the current literature by examining whether nativity status modifies the relationship between a combination of SES and health literacy barriers, and healthcare delays in a large national cohort.</p>" ]

[ "<title>Methods</title>", "<title>Data collection and sample</title>", "<p id=\"Par11\">Cross-sectional data for this study were obtained from the “All of Us” research program collected by online survey between May 2018 and April 2021. Briefly, this program is open to individuals who are 18 and over and are living in the US. Participants signed a consent form following the Declaration of Helsinki for data collection. The participants’ data used are de-identified and available to approved researchers. The All of Us program was approved by the National Institutes of Health (NIH) Institutional Review Board (IRB).</p>", "<p id=\"Par12\">In this study, cancer survivors were defined as those participants who indicated that they had ever been diagnosed with cancer. Inclusion criteria for our cohort included participants who were ever told by their healthcare provider that had/have cancer. Skin cancer is one of the most prevalent cancers in the US with most cases being reasonably benign basal cells, not often tracked on most cancer registries, and having over 90% 5-year survival rate [##REF##28885696##24##]. In addition, previous studies have excluded these cancer survivors as their follow-up care is often reasonably minor [##REF##33942535##25##]. Thus, we excluded those with skin cancer and participants with missing data on the healthcare delay survey questions.</p>" ]

[ "<title>Results</title>", "<p id=\"Par18\">The median age of the study population (<italic>n</italic> = 10,020) was approximately 64 (interquartile range [IQR Q1, Q3] 55.5, 71.8) years. The majority of participants were female (66.1%), US-born (92%), and self-identified as White (82.3%). There was a higher distribution of females vs males and other sex, foreign-born vs US-born, and Black cancer survivors vs all other race/ethnicity categories that had three or more SES barriers (Table ##TAB##0##1##). While a higher proportion of females vs males and other sex, foreign-born vs US-born, and Hispanic cancer survivors vs all other race/ethnicity cancer survivors had one or more healthcare delays (Table ##TAB##1##2##).</p>", "<p id=\"Par19\">Results from the multivariable-adjusted model showed that neither nativity (OR 1.04, 95% CI [0.87, 1.25]) nor health literacy (OR 1.20, 95% CI [0.89, 1.59]) were statistically significantly associated with healthcare delays (see Table ##TAB##2##3##). However, when assessing for a p-trend for health literacy, for every one-unit increase in health literacy there was an 8% (OR 0.92, 95% CI 0.89, 0.95) decrease in the likelihood to experience healthcare delays. Furthermore, compared to those who did not have any SES barriers, those who reported two or three or more were 65% (OR 1.65, 95% CI [1.43, 1.90]) and 118% (OR 2.18, 95% CI [1.84, 2.58]) more likely to experience delays in healthcare, respectively. In addition, using SES barriers as a continuous measure to test for a p-trend, we found that for every one additional barrier increase, there was a 29% increase (OR 1.29, 95% CI [1.23, 1.36]) in the likelihood of experiencing healthcare delays. The association between SES barriers and healthcare delays differs by nativity status (<italic>p</italic>_interaction = 0.02). The stratified model by nativity showed that among those who were foreign-born, those who experienced two or three or more SES barriers were almost three times (OR 4.35, 95% CI [2.61, 7.34] vs OR 1.53, 95% CI [1.32, 1.78]) and two times (OR 3.83, 95% CI [2.14, 6.98] vs OR 2.10, 95% CI [1.76, 2.50]) as likely to experience healthcare delays as their US-born counterparts who experienced the same levels of SES barriers, respectively (see Table ##TAB##2##3##).</p>", "<p id=\"Par20\">Assessing for p-trend in the stratified model by nativity, we found that for every additional SES barrier experienced among foreign-born individuals, they were 72% (OR 1.72, 95% CI [1.43, 2.08] vs OR 1.27, 95% CI [1.21, 1.34]) more likely to experience healthcare delays compared to their US counterparts. Finally, in the stratified model, low health literacy was associated with a 41% (OR 1.41, 95% CI [1.02, 1.97]) increase in the likelihood of healthcare delays, and each one-point increase in health literacy score was associated with a 9% (OR 0.91, 95% CI [0.88, 0.94]) decrease in the odds of healthcare delays among US-born cancer survivors. While for foreign-born cancer survivors, low health literacy was not statistically significantly associated with experiencing healthcare delays (OR 0.66, 95% CI [0.34, 1.25]), and for every one-unit increase in health literacy score there was a 2% (OR 0.98, 95% CI [0.90, 1.07]) decrease in the odds of healthcare delays. Although this association did not reach statistical significance (see Table ##TAB##2##3##).</p>" ]

[ "<title>Discussion</title>", "<p id=\"Par21\">Using data from the All of Us research cohort, we aimed to investigate the associations between SES barriers and health literacy with healthcare delays. We also explored whether there were any differences in these associations by nativity. We found that among all cancer survivors, health literacy (binary) and nativity were not statistically significantly associated with healthcare delays. We also found that experiencing 2 or 3+ SES barriers was significantly associated with an increased likelihood of healthcare delays. Further, at equal levels of SES barriers, foreign-born individuals had significantly higher odds of healthcare delays when compared to US-born individuals. Lastly, in our separate models by nativity status assessing for a trend, we found that health literacy was inversely associated with healthcare delays among US-born cancer survivors only.</p>", "<title>Socioeconomic barriers</title>", "<p id=\"Par22\">Our multivariable model suggested that cancer survivors who experienced more than one SES barrier experienced an increase in the likelihood of healthcare delays compared to cancer survivors who experienced no SES barriers. These findings are consistent with previous research that found that low SES cancer survivors are more likely to not receive appropriate follow-up care [##REF##27006193##30##] and that those who experience financial, housing, and employment barriers have a greater likelihood of delaying needed care [##REF##35559872##31##–##REF##22412136##33##]. Similarly, in another study, cancer survivors from low SES who reported lower income and education levels were less likely to have follow-up care discussions with their medical providers [##REF##27006193##30##]. This lack of follow-up discussions can potentially contribute to prolonged delays in preventative care. Regarding education, in a previous study by Gonzalez and colleagues, cancer survivors who had a college degree or higher were more likely to have higher access to care but experienced more delays than those with less than or equal to a high school diploma [##REF##35576025##34##]. Perhaps belonging to a higher educational level improves health literacy, enabling cancer survivors to adequately make an informed decision when seeking the appropriate care needed.</p>", "<p id=\"Par23\">Among cancer survivors, those with low SES barriers are more likely to delay medical care, preventive care (dental and vision care), and not fill prescription medications due to cost-related concerns [##REF##23907958##32##, ##REF##33043464##35##, ##REF##20549763##36##]. Furthermore, as cancer survivors are met with the unexpected financial costs of cancer treatment, they may worry about struggling to meet their housing and household bills payments, food insecurities, and retirement [##UREF##2##37##], thus, potentially forgoing or delaying the crucial care they are required to enhance their survival. Similarly, housing insecurities are linked with negative health outcomes and poor access and quality of healthcare [##REF##35559872##31##, ##REF##29299816##38##]. Lastly, the impact of modifiable factors such as education and SES are inversely associated with experiencing more unmet healthcare needs [##REF##15836548##39##] and a lack of health insurance [##REF##15836548##39##, ##REF##28766209##40##]. Hence, uninsured cancer survivors have a higher risk for comorbidities, bearing a greater mortality risk than uninsured non-cancer survivors [##REF##30461610##41##].</p>", "<title>Nativity differences</title>", "<p id=\"Par24\">Previous research among cancer survivors has shown nativity to be a factor in cost-related barriers among US-born Hispanic cancer survivors [##REF##24904178##42##]. In our study, we found that foreign-born individuals who experienced the same level of SES barriers had higher likelihood of experiencing healthcare delays than their native-born counterparts when compared to those who experienced no SES barriers. Previous research that explored nativity differences among cancer survivors is limited and inconsistent. For example, although the results did not reach statistical significance, Diamant and colleagues reported the opposite in a sample of non-cancer survivors, showing that non-native-born were less likely to report healthcare delays compared to native-born individuals [##REF##15117701##43##]. Whereas, in a study of female cancer survivors that assessed disparities in healthcare access and utilization, they found that non-US-born females were less likely to report having a routine place to go to meet their healthcare compared to US-born cancer survivors [##REF##34661881##44##]. This is important, as not having a primary healthcare office to seek care or have routine services can promote delaying accessing the extended care cancer survivors need. In addition, a higher prevalence of sociodemographic and SES barriers (e.g., income, education) was found among foreign-born individuals than among native-born in two North American countries (US and Canada), and disparities in healthcare access were higher among foreign- compared to native-born [##REF##21841297##20##]. Thus, supporting the role of SES barriers among foreign-born individuals.</p>", "<title>Health literacy</title>", "<p id=\"Par25\">Our study found that, after adjusting for confounding sociodemographic, nativity, and SES barriers, health literacy was not statistically significantly associated with healthcare delays in our entire study cohort. However, we found that solely among US-born participants, limited health literacy was associated with an increased likelihood of healthcare delays compared to adequate health literacy. We also saw a statistically significant monotonic relationship between increased health literacy scores and decreased odds of healthcare delays among US-born. While our findings only reached statistical significance among US-born cancer survivors, the magnitude of our findings was in the same direction among foreign-born cancer survivors. This indicates that regardless of nativity, increasing health literacy could mitigate the impact of healthcare delays. While we controlled for nativity, sociodemographic, and SES barriers, our findings suggest that health literacy could also be associated with cultural differences and language that we were not able to control in our study. For example, cancer survivors with low health literacy may suffer difficulties when trying to decode their symptoms or understand their diagnoses through communication with providers, which could result in healthcare delays and a later stage of disease at diagnosis [##REF##12018928##45##]. In the case of cancer survivors, there is a great need for complex health services after they have completed their primary treatment for their illness and subsequent management of their health [##REF##30561774##46##]. These deficiencies in communicating efficiently with their providers could increase the risk of healthcare delays.</p>", "<title>Strengths and limitations</title>", "<p id=\"Par26\">An important strength of this study is using data from the All of Us research program, as it has a high proportion of enrolled underrepresented minority populations, which increases our sample size and access to geographically and ethnically diverse populations. Similarly, we were able to analyze a relatively large sample with complete socioeconomic and sociodemographic data of foreign-born cancer survivors, which helps to expand the current cancer survivor, cancer health disparities, and health disparities research. Lastly, our results can be generalized to cancer survivors and individuals with similar characteristics and settings that experienced similar SES barriers in the US.</p>", "<p id=\"Par27\">This study is not without limitations. The cross-sectional nature of the design does not allow us to establish a temporal or causal relationship. There is potential for misclassification for some of the factors included in our main independent variables as this data is from self-reported questionnaires. Our sample was also comprised of a higher distribution of individuals with a college degree or higher and a higher income. We were unable able to assess for acculturation; thus, future studies should account for it as it can be a potential confounder in these associations. Using a complete case (CC) analysis method may have introduced bias to our results. We addressed this concern by conducting a sensitivity analysis using multiple imputation (MI) method for our outcome variable. This analysis revealed that there was a slight overestimation in the relationship between SES barriers and healthcare delays among foreign-born cancer survivors, compared to their US-born counterparts with the same level of SES barriers. Despite the small differences observed in the effect estimates, the overall trend and interpretation of the results remained consistent across both the MI and CC analyses (Supplemental Table 3).</p>", "<p id=\"Par28\">Although the All of Us collects data nationwide, these results cannot be generalizable to all cancer survivors in the United States. A clear example of this limitation can be seen in that our sample reported roughly 90% some college or higher degree, whereas in 2021, those who had completed some college, or more were approximately 63% in the general US population [##UREF##3##47##]. Lastly, an additional limitation is that during our study period, the COVID-19 pandemic may have worsened SES barriers and healthcare delays that may have contributed to our findings and may need further exploring.</p>", "<title>Implications and future direction</title>", "<p id=\"Par29\">Our results can help guide policymakers to promote the development of policies that aim at eliminating SES barriers. For example, many of the variables that are part of our SES index are system-modifiable factors. Implementation of laws that make education equitable, job creation and training, housing affordability, and universal healthcare are ways in which policies can aid in mitigating these SES barriers. At the healthcare system level, practitioners and systems should recognize that these SES barriers exist and promote solutions. For example, systems can offer transportation services to those who are experiencing SES barriers to lessen healthcare delays [##REF##34723394##48##]. Similarly, providing adult- and childcare services can help avoid delays in seeking care [##REF##36261213##49##].</p>", "<p id=\"Par30\">As the All of Us continues to enroll participants and participants complete all surveys, future studies should reassess this association to determine if our findings remain true. Moreover, in future analyses, it is important to consider adjusting for acculturation, as well as other types of stressors such as discrimination, as these experiences can contribute to healthcare delays. Additionally, future studies should aim to explore racial differences between US-born and foreign-born cancer survivors. It is crucial to recognize that races and ethnicities such as Black, Hispanic, and Asian are heterogeneous, varying across cultural and socioeconomic aspects. Thus, understanding the SES barriers associated with healthcare delays among US-born ethnic minorities from different racial and ethnic backgrounds (e.g., Mexican-US-born, Guatemalan-US-born, Chinese-US-born, Nigerian-US-born) compared to foreign-born counterparts (e.g., Mexican-foreign-born, Guatemalan-foreign-born, Chinese-foreign-born, Nigerian-foreign-born) is of extreme importance.</p>" ]

[ "<title>Conclusion</title>", "<p id=\"Par31\">Using data from the All of Us research program, we found that SES-related barriers are significantly associated with healthcare delays in cancer survivors in our study. However, a greater impact was observed among those who were foreign-born. Similarly, we observed a possible protective effect of health literacy on healthcare delays among US-born only. Our study highlights that to mitigate the impact of delayed healthcare, both policymakers and healthcare providers must prioritize addressing the social determinants of health and promoting health literacy in these populations.</p>" ]

[ "<title>Purpose</title>", "<p id=\"Par1\">We aimed to assess whether nativity differences in socioeconomic (SES) barriers and health literacy were associated with healthcare delays among US cancer survivors.</p>", "<title>Methods</title>", "<p id=\"Par2\">“All of Us” survey data were analyzed among adult participants ever diagnosed with cancer. A binary measure of healthcare delay (1+ delays versus no delays) was created. Health literacy was assessed using the Brief Health Literacy Screen. A composite measure of SES barriers (education, employment, housing, income, and insurance statuses) was created as 0, 1, 2, or 3+. Multivariable logistic regression model tested the associations of (1) SES barriers and health literacy with healthcare delays, and (2) whether nativity modified this relationship.</p>", "<title>Results</title>", "<p id=\"Par3\">Median participant age was 64 years (<italic>n</italic> = 10,020), with 8% foreign-born and 18% ethnic minorities. Compared to survivors with no SES barriers, those with 3+ had higher likelihood of experiencing healthcare delays (OR 2.18, 95% CI 1.84, 2.58). For every additional barrier, the odds of healthcare delays were greater among foreign-born (1.72, 1.43, 2.08) than US-born (1.27, 1.21, 1.34). For every 1-unit increase in health literacy among US-born, the odds of healthcare delay decreased by 9% (0.91, 0.89, 0.94).</p>", "<title>Conclusion</title>", "<p id=\"Par4\">We found that SES barriers to healthcare delays have a greater impact among foreign-born than US-born cancer survivors. Higher health literacy may mitigate healthcare delays among US cancer survivors. Healthcare providers, systems and policymakers should assess and address social determinants of health and promote health literacy as a way to minimize healthcare delays among both foreign- and US-born cancer survivors.</p>", "<title>Supplementary Information</title>", "<p>The online version contains supplementary material available at 10.1007/s10552-023-01782-z.</p>", "<title>Keywords</title>", "<p>Open access funding provided by SCELC, Statewide California Electronic Library Consortium</p>" ]

[ "<title>Measures</title>", "<title>Demographics</title>", "<p id=\"Par13\">Personal level characteristics accounted for in this study were age (at survey completion), sex (male vs female), race (Asian, Black, Hispanic, White, multiracial/biracial, other [includes those who selected: none of this, another population, and prefer not to answer]), marital status (married [includes those who selected: married and living with a partner] vs single [includes those who selected: Single, divorced, widowed, and separated]), nativity (US- vs foreign-born), annual income (using quintiles, the lowest quintile were those who reported income of &lt; 35 K vs quintile 2–5 ≥ 35 K), education (college or more vs ≤ high school or equivalent), insured (yes vs no), housing status (own vs rent/other arrangements), employed (yes vs no), current treatment (yes vs no), and cancer type (range of multiple cancer sites).</p>", "<title>Health literacy</title>", "<p id=\"Par14\">Health Literacy was assessed using the three-item Brief Health Literacy Screen (BHLS) [##REF##23918160##26##, ##UREF##1##27##], which measures individual needs for help with filling out forms (“How confident are you filling out medical forms by yourself?”), reading health-related documents (“How often do you have someone help you read health-related materials?”), and difficulty learning due to a lack of understanding of written medical documents (“How often do you have problems learning about your medical condition because of difficulty understanding written information?”). Response options were on a five-point Likert scale, with options for reading and understanding health documents including “Always,” “Often,” “Sometimes,” Occasionally,” and “Never” and for the need for help to fill forms were, “Extremely,” “Quite a bit,” “Somewhat,” “A little bit,” “Not at all.” The survey question measuring if the participant required help with forms was reversed coded. All items were then summed to create a composite score with higher scores (max = 15), indicating fewer health literacy problems. Following a previous study that assessed health literacy using the BHLS scale by Willens and colleagues, we dichotomized this score, with those who scored ≤ 9 as having limited health literacy and scores &gt; 9 as having adequate health literacy [##REF##24093351##28##] (Supplemental Table 1).</p>", "<title>Socioeconomic barriers</title>", "<p id=\"Par15\">Five SES factors (education, income, insurance, housing, and employment status) were dichotomized to create a composite measure following a previous study [##REF##31472134##29##]. If individuals selected an income “ ≥ 35 K,” an education level of “college or more,” being insured, owning a home, and being employed, they were coded as having no SES barriers (0). Those who selected either one of the following: an income between “ &lt; 35 k,” an educational level of “ ≤ high school or equivalent,” not being insured, not being employed, and/or having a housing status as rent/another arrangement, they were given an additive score ranging from 1 to 5 [##REF##31472134##29##]. Due to sparse counts in categories of four and five SES barriers, scores were truncated to range from 0 to 3 or more SES barriers (Supplemental Table 2).</p>", "<title>Healthcare delays</title>", "<p id=\"Par16\">Nine questions were used to assess healthcare delays. These questions were obtained from the National Health Interview Survey asking participants if they experienced delays in any healthcare received due to various reasons in the past 12 months [##REF##33942535##25##]. These reasons include transportation, living in a rural area where healthcare providers are too far, nervousness about seeing a healthcare provider, could not get time off work, could not get childcare, cannot leave adult unattended due to being a caretaker, could not afford copays, deductible was too high, and could not afford it or had to pay out of pocket for some or all procedures. Response options were “no,” “yes,” or “don’t know.” A dichotomized measure was created with those who responded “yes” to one or more reasons as having experienced healthcare delays, and those who responded “no” to all reasons as having experienced no delays, those who reported “don’t know” were not counted in this measure.</p>", "<title>Statistical methods</title>", "<p id=\"Par17\">To characterize the study population, descriptive statistics were calculated for all demographic variables and variables of interest (nativity, SES barriers, and health literacy). Listwise deletion method was used to address missing data (15.9%). We conducted a post-hoc sensitivity analysis to determine the direction of the potential bias introduced from our listwise deletion method for a complete case analysis. We used a multiple imputation analysis using chained equations (MICE). This approach allowed us to impute missing values based on observed data and estimate relationships between variables. To satisfy the safe data sharing policy of “All of Us,” groups with less than 20 participants are reported in tables as ≤ 20 or &lt; x% with another category in the same column/row also showing ≥ (%) to ensure that another count value cannot be used to derive the exact count that is less than <italic>n</italic> = 20 in the suppressed group. First, a multivariable logistic regression model tested the hypothesized independent relationships between SES barriers, and health literacy with healthcare delays adjusted for covariates (age, sex, ethnicity/race, marital status, treatment status, and cancer type). Secondly, to assess nativity differences in the association between SES barriers and healthcare delays, a product interaction term for nativity and SES barriers (SES barriers*nativity) was included in the model. Furthermore, we assessed for a p-trend in the adjusted and stratified model by introducing SES barriers and health literacy as continuous measures in the model. All statistical analyses were performed using R Jupyter Notebooks embedded in the “All of Us” workbench, with a significance level at alpha 0.05. Odds ratios (ORs) with 95% confidence intervals (CI) and <italic>p</italic> value are reported.</p>", "<title>Supplementary Information</title>", "<p>Below is the link to the electronic supplementary material.</p>" ]

[ "<p>The authors would like to thank all of the participants of the All of Us Research Program, Drs. Cecilia Patino-Sutton, and Elizabeth Burner. The All of Us Research Program is supported by the National Institutes of Health, Office of the Director: Regional Medical Centers: 1 OT2 OD026549; 1 OT2 OD026554; 1 OT2 OD026557; 1 OT2 OD026556; 1 OT2 OD026550; 1 OT2 OD 026552; 1 OT2 OD026553; 1 OT2 OD026548; 1 OT2 OD026551; 1 OT2 OD026555; IAA #: AOD 16037; Federally Qualified Health Centers: HHSN 263201600085U; Data and Research Center: 5 U2C OD023196; Biobank: 1 U24 OD023121; The Participant Center: U24 OD023176; Participant Technology Systems Center: 1 U24 OD023163; Communications and Engagement: 3 OT2 OD023205; 3 OT2 OD023206; and Community Partners: 1 OT2 OD025277; 3 OT2 OD025315; 1 OT2 OD025337; 1 OT2 OD025276. In addition, the All of Us Research Program would not be possible without the partnership of its participants.</p>", "<title>Author contributions</title>", "<p>AA: Conception/design, assembly of data, data analysis, writing, and approval of the final version. SN: Conception/design, writing and editing of the article, and approval of the final version. CYO: Conception/design, writing and editing the article, and approval of the final version. CR: Writing the article and approval of the final version. SEK: Writing and editing the article and approval of the final version. AJF: Conception/design, assembly of data, writing and editing, and approval of the final version.</p>", "<title>Funding</title>", "<p>Open access funding provided by SCELC, Statewide California Electronic Library Consortium. Carol Ochoa was supported by the National Cancer Institute to conduct this study (K00CA264294-02).</p>", "<title>Data availability</title>", "<p>Data from the All of Us Research Program can only be accessed through the Researcher Workbench (<ext-link ext-link-type=\"uri\" xlink:href=\"https://workbench.researchallofus.org/login\">https://workbench.researchallofus.org/login</ext-link>) as per the informed consent of program participants. The investigators are prohibited to share raw-level data of participants in accordance with the user agreement established by this program. Therefore, it is not possible to provide a de-identified dataset for this manuscript.</p>", "<title>Declarations</title>", "<title>Competing interests</title>", "<p id=\"Par32\">The authors declare no competing interests.</p>", "<title>Ethical approval</title>", "<p id=\"Par34\">The All of Us Research Program Institutional Review Board (IRB) established that registered tier data available on the All of Us’ Workbench meets criteria for non-Human Subject Research. This project used registered tier data and required no IRB review. <ext-link ext-link-type=\"uri\" xlink:href=\"https://www.researchallofus.org/faq/do-i-need-institutional-review-board-irb-approval-from-my-own-institution-in-order-to-access-this-data-through-the-researcher-workbench/\">https://www.researchallofus.org/faq/do-i-need-institutional-review-board-irb-approval-from-my-own-institution-in-order-to-access-this-data-through-the-researcher-workbench/</ext-link>.</p>" ]

[ "<fig id=\"Fig1\"><label>Fig. 1</label><caption><p>Theoretical framework of the impact nativity has on healthcare delays</p></caption></fig>" ]

[ "<table-wrap id=\"Tab1\"><label>Table 1</label><caption><p>Descriptive characteristics of the sample and their association with SES barriers (<italic>n</italic> = 10,020)</p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th align=\"left\" rowspan=\"2\">Variables</th><th align=\"left\" colspan=\"4\">Number of socioeconomic barriers<break/><italic>n</italic> (%)</th><th align=\"left\" rowspan=\"2\"><italic>p</italic> value</th></tr><tr><th align=\"left\">No barriers<break/>(<italic>n</italic> = 3,112)</th><th align=\"left\">1<break/>(<italic>n</italic> = 4,314)</th><th align=\"left\">2<break/>(<italic>n</italic> = 1,514)</th><th align=\"left\">3 or more<break/>(<italic>n</italic> = 1,080)</th></tr></thead><tbody><tr><td align=\"left\" colspan=\"6\">Sex</td></tr><tr><td align=\"left\"> Female</td><td align=\"left\"> &gt; 2100 (&gt; 30%)</td><td align=\"left\">2688 (40.6%)</td><td align=\"left\"> &gt; 1000 (&gt; 15%)</td><td align=\"left\"> &gt; 500 (&gt; 10%)</td><td align=\"left\"/></tr><tr><td align=\"left\"> Male</td><td align=\"left\">991 (29.7%)</td><td align=\"left\">1596 (47.9%)</td><td align=\"left\">450 (13.5%)</td><td align=\"left\">295 (8.9%)</td><td char=\".\" align=\"char\"> &lt; 0.001</td></tr><tr><td align=\"left\"> Other^a</td><td align=\"left\"> ≤ 20 (&lt; 25%)</td><td align=\"left\">31 (47.7%)</td><td align=\"left\"> ≤ 20 (&lt; 20%)</td><td align=\"left\"> ≤ 20 (&lt; 20%)</td><td align=\"left\"/></tr><tr><td align=\"left\" colspan=\"6\">Age</td></tr><tr><td align=\"left\"> Median [Q1, Q3]</td><td align=\"left\">59.3 [51.3, 65.8]</td><td align=\"left\">68.1 [60.6, 73.2]</td><td align=\"left\">64.7 [52.9, 72.0]</td><td align=\"left\">58.9 [48.0, 67.3]</td><td char=\".\" align=\"char\"> &lt; 0.001</td></tr><tr><td align=\"left\" colspan=\"6\">Race/ethnicity</td></tr><tr><td align=\"left\"> Asian</td><td align=\"left\">70 (35.7%)</td><td align=\"left\"> &gt; 50 (&gt; 40%)</td><td align=\"left\"> &gt; 20 (&gt; 10%)</td><td align=\"left\"> ≤ 20 (&lt; 20%)</td><td align=\"left\"/></tr><tr><td align=\"left\"> Black</td><td align=\"left\">113 (17.6%)</td><td align=\"left\">178 (27.7%)</td><td align=\"left\">150 (23.4%)</td><td align=\"left\">201 (31.3%)</td><td align=\"left\"/></tr><tr><td align=\"left\"> Hispanic</td><td align=\"left\">143 (23.4%)</td><td align=\"left\">164 (26.7%)</td><td align=\"left\">137 (22.3%)</td><td align=\"left\">171 (27.8%)</td><td align=\"left\"/></tr><tr><td align=\"left\"> White</td><td align=\"left\">2687 (32.6%)</td><td align=\"left\">3759 (45.6%)</td><td align=\"left\">1154 (14.0%)</td><td align=\"left\">643 (7.8%)</td><td char=\".\" align=\"char\"> &lt; 0.001</td></tr><tr><td align=\"left\"> More than one pop</td><td align=\"left\">52 (34.7%)</td><td align=\"left\"> &gt; 50 (&gt; 40%)</td><td align=\"left\"> ≤ 20 (&lt; 20%)</td><td align=\"left\"> ≤ 20 (&lt; 20%)</td><td align=\"left\"/></tr><tr><td align=\"left\"> Other^b</td><td align=\"left\">47 (26.9%)</td><td align=\"left\">62 (35.4%)</td><td align=\"left\">32 (18.3%)</td><td align=\"left\">34 (19.4%)</td><td align=\"left\"/></tr><tr><td align=\"left\" colspan=\"6\">Annual income^e</td></tr><tr><td align=\"left\"> &lt; 35 K</td><td align=\"left\"> ≤ 20 (&lt; 20%)</td><td align=\"left\">140 (7.7%)</td><td align=\"left\">703 (38.4%)</td><td align=\"left\"> &gt; 900 (&gt; 40%)</td><td align=\"left\"/></tr><tr><td align=\"left\"> ≥ 35 K</td><td align=\"left\">2864 (39.6%)</td><td align=\"left\">3687 (51.0%)</td><td align=\"left\">641 (8.9%)</td><td align=\"left\">39 (0.5%)</td><td char=\".\" align=\"char\"> &lt; 0.001</td></tr><tr><td align=\"left\"> Other^c</td><td align=\"left\"> &gt; 200 (&gt; 20%)</td><td align=\"left\">488 (50.8%)</td><td align=\"left\">170 (17.7%)</td><td align=\"left\"> ≤ 60 (&lt; 10%)</td><td align=\"left\"/></tr><tr><td align=\"left\" colspan=\"6\">Marital status</td></tr><tr><td align=\"left\"> Single</td><td align=\"left\"> &gt; 600 (&gt; 15%)</td><td align=\"left\">1215 (35.1%)</td><td align=\"left\">778 (22.5%)</td><td align=\"left\"> &gt; 700 (&gt; 20%)</td><td align=\"left\"/></tr><tr><td align=\"left\"> Married/living with a partner</td><td align=\"left\">2405 (37.2%)</td><td align=\"left\">3071 (47.5%)</td><td align=\"left\">709 (11.0%)</td><td align=\"left\">282 (4.4%)</td><td char=\".\" align=\"char\"> &lt; 0.001</td></tr><tr><td align=\"left\"> Other^c</td><td align=\"left\"> ≤ 20 (&lt; 30%)</td><td align=\"left\">29 (30.9%)</td><td align=\"left\">27 (28.7%)</td><td align=\"left\"> ≤ 20 (&lt; 10.0%)</td><td align=\"left\"/></tr><tr><td align=\"left\" colspan=\"6\">Employed</td></tr><tr><td align=\"left\"> Yes</td><td align=\"left\"> &gt; 2700 (&gt; 50%)</td><td align=\"left\"> &gt; 900 (&gt; 20%)</td><td align=\"left\"> &gt; 250 (&gt; 5%)</td><td align=\"left\"> &lt; 100 (&lt; 5%)</td><td align=\"left\"/></tr><tr><td align=\"left\"> No</td><td align=\"left\"> ≤ 20 (&lt; 20%)</td><td align=\"left\">3273 (60.4%)</td><td align=\"left\">1161 (21.4%)</td><td align=\"left\">981 (18.1%)</td><td char=\".\" align=\"char\"> &lt; 0.001</td></tr><tr><td align=\"left\"> Other^c</td><td align=\"left\">48 (55.2%)</td><td align=\"left\"> ≤ 20 (&lt; 20%)</td><td align=\"left\"> ≤ 20 (&lt; 20%)</td><td align=\"left\"> ≤ 20 (&lt; 20%)</td><td align=\"left\"/></tr><tr><td align=\"left\" colspan=\"6\">Educational level</td></tr><tr><td align=\"left\"> College or more</td><td align=\"left\">3097 (34.5%)</td><td align=\"left\">4133 (46.0%)</td><td align=\"left\"> &gt; 1000 (&gt; 10%)</td><td align=\"left\"> &gt; 500 (&gt; 5%)</td><td align=\"left\"/></tr><tr><td align=\"left\"> ≤ High school or equivalent</td><td align=\"left\"> ≤ 20 (&lt; 20%)</td><td align=\"left\">160 (16.2%)</td><td align=\"left\"> &gt; 300 (&gt; 30%)</td><td align=\"left\">484 (49.1%)</td><td char=\".\" align=\"char\"> &lt; 0.001</td></tr><tr><td align=\"left\"> Other^c</td><td align=\"left\"> ≤ 20 (&lt; 30%)</td><td align=\"left\">22 (40.0%)</td><td align=\"left\"> ≤ 20 (&lt; 20%)</td><td align=\"left\"> ≤ 20 (&lt; 20%)</td><td align=\"left\"/></tr><tr><td align=\"left\" colspan=\"6\">Insured</td></tr><tr><td align=\"left\"> No</td><td align=\"left\"> ≤ 20 (&lt; 30%)</td><td align=\"left\"> ≤ 20 (&lt; 30%)</td><td align=\"left\">29 (22.1%)</td><td align=\"left\"> &gt; 50 (&gt; 50%)</td><td align=\"left\"/></tr><tr><td align=\"left\"> Yes</td><td align=\"left\"> &gt; 2500 (&gt; 30%)</td><td align=\"left\"> &gt; 4000 (&gt; 40%)</td><td align=\"left\">1463 (14.9%)</td><td align=\"left\">975 (9.9%)</td><td char=\".\" align=\"char\"> &lt; 0.001</td></tr><tr><td align=\"left\"> Other^d</td><td align=\"left\">20 (25.3%)</td><td align=\"left\"> &gt; 20 (30%)</td><td align=\"left\">22 (27.8%)</td><td align=\"left\"> ≤ 20 (&lt; 30%)</td><td align=\"left\"/></tr><tr><td align=\"left\" colspan=\"6\">Treatment</td></tr><tr><td align=\"left\"> No</td><td align=\"left\"> &gt; 2300 (&gt; 25%)</td><td align=\"left\"> &gt; 3300 (&gt; 40%)</td><td align=\"left\"> &gt; 1100 (&gt; 10%)</td><td align=\"left\"> &gt; 800 (&gt; 10%)</td><td char=\".\" align=\"char\"> &lt; 0.01</td></tr><tr><td align=\"left\"> Yes</td><td align=\"left\">796 (34.1%)</td><td align=\"left\">953 (40.8%)</td><td align=\"left\">356 (15.2%)</td><td align=\"left\">233 (10.0%)</td><td align=\"left\"/></tr><tr><td align=\"left\"> Missing</td><td align=\"left\"> ≤ 20 (&lt; 30%)</td><td align=\"left\"> ≤ 20 (&lt; 50%)</td><td align=\"left\"> ≤ 20 (&lt; 25%)</td><td align=\"left\"> ≤ 20 (&lt; 10%)</td><td align=\"left\"/></tr><tr><td align=\"left\" colspan=\"6\">Nativity</td></tr><tr><td align=\"left\"> US-Born</td><td align=\"left\">2869 (31.2%)</td><td align=\"left\">4003 (43.5%)</td><td align=\"left\">1379 (15.0%)</td><td align=\"left\">955 (10.4%)</td><td char=\".\" align=\"char\"> &lt; 0.001</td></tr><tr><td align=\"left\"> Foreign-Born</td><td align=\"left\">243 (29.8%)</td><td align=\"left\">312 (38.3%)</td><td align=\"left\">135 (16.6%)</td><td align=\"left\">125 (15.3%)</td><td align=\"left\"/></tr><tr><td align=\"left\" colspan=\"6\">Housing status</td></tr><tr><td align=\"left\"> Own</td><td align=\"left\"> &gt; 3000 (&gt; 40%)</td><td align=\"left\">3534 (47.6%)</td><td align=\"left\"> &lt; 800 (&lt; 10%)</td><td align=\"left\">125 (1.7%)</td><td align=\"left\"/></tr><tr><td align=\"left\"> Rent/other arrangements</td><td align=\"left\"> ≤ 20 (&lt; 30%)</td><td align=\"left\">734 (29.6%)</td><td align=\"left\">793 (32.0%)</td><td align=\"left\"> &gt; 800 (&gt; 35%)</td><td char=\".\" align=\"char\"> &lt; 0.001</td></tr><tr><td align=\"left\"> Other^c</td><td align=\"left\">50 (40.0%)</td><td align=\"left\">47 (37.6%)</td><td align=\"left\"> &gt; 20 (&gt; 17%)</td><td align=\"left\"> ≤ 20 (&lt; 30%)</td><td align=\"left\"/></tr><tr><td align=\"left\" colspan=\"6\">Cancer type</td></tr><tr><td align=\"left\"> Bladder</td><td align=\"left\">65 (26.5%)</td><td align=\"left\">105 (42.9%)</td><td align=\"left\">53 (21.6%)</td><td align=\"left\">22 (9.0%)</td><td char=\".\" align=\"char\"> &lt; 0.001</td></tr><tr><td align=\"left\"> Blood</td><td align=\"left\">238 (31.3%)</td><td align=\"left\">350 (46.1%)</td><td align=\"left\">111 (14.6%)</td><td align=\"left\">61 (8.0%)</td><td align=\"left\"/></tr><tr><td align=\"left\"> Bone</td><td align=\"left\"> &gt; 25 (&gt; 25%)</td><td align=\"left\">39 (36.5%)</td><td align=\"left\"> ≤ 20 (&lt; 30%)</td><td align=\"left\"> ≤ 20 (&lt; 30%)</td><td align=\"left\"/></tr><tr><td align=\"left\"> Brain</td><td align=\"left\">35 (27.8%)</td><td align=\"left\">49 (38.9%)</td><td align=\"left\">22 (17.5%)</td><td align=\"left\">20 (15.9%)</td><td align=\"left\"/></tr><tr><td align=\"left\"> Breast</td><td align=\"left\">1031 (34.8%)</td><td align=\"left\">1283 (43.3%)</td><td align=\"left\">408 (13.8%)</td><td align=\"left\">240 (8.1%)</td><td align=\"left\"/></tr><tr><td align=\"left\"> Cervical</td><td align=\"left\">188 (27.3%)</td><td align=\"left\">202 (29.4%)</td><td align=\"left\">126 (18.3%)</td><td align=\"left\">172 (25.0%)</td><td align=\"left\"/></tr><tr><td align=\"left\"> Colon rectal</td><td align=\"left\">131 (30.6%)</td><td align=\"left\">168 (39.3%)</td><td align=\"left\">73 (17.1%)</td><td align=\"left\">56 (13.1%)</td><td align=\"left\"/></tr><tr><td align=\"left\"> Endocrine</td><td align=\"left\"> ≤ 20 (&lt; 30%)</td><td align=\"left\">31 (46.3%)</td><td align=\"left\"> ≤ 20 (&lt; 30%)</td><td align=\"left\"> ≤ 20 (&lt; 30%)</td><td align=\"left\"/></tr><tr><td align=\"left\"> Endometrial</td><td align=\"left\">70 (25.6%)</td><td align=\"left\">128 (46.7%)</td><td align=\"left\">50 (18.3%)</td><td align=\"left\">26 (9.5%)</td><td align=\"left\"/></tr><tr><td align=\"left\"> Head neck</td><td align=\"left\"> &gt; 50 (&gt; 30%)</td><td align=\"left\">79 (42.9%)</td><td align=\"left\">25 (13.6%)</td><td align=\"left\"> ≤ 20 (&lt; 10%)</td><td align=\"left\"/></tr><tr><td align=\"left\"> Kidney</td><td align=\"left\">89 (30.4%)</td><td align=\"left\">122 (41.6%)</td><td align=\"left\">48 (16.4%)</td><td align=\"left\">34 (11.6%)</td><td align=\"left\"/></tr><tr><td align=\"left\"> Lung</td><td align=\"left\">39 (16.1%)</td><td align=\"left\">109 (45.0%)</td><td align=\"left\">55 (21.7%)</td><td align=\"left\">39 (16.1%)</td><td align=\"left\"/></tr><tr><td align=\"left\"> Other</td><td align=\"left\">427 (29.8%)</td><td align=\"left\">611 (42.6%)</td><td align=\"left\">211 (14.7%)</td><td align=\"left\">186 (13.0%)</td><td align=\"left\"/></tr><tr><td align=\"left\"> Ovarian</td><td align=\"left\">60 (27.3%)</td><td align=\"left\">89 (41.5%)</td><td align=\"left\">29 (13.2%)</td><td align=\"left\">42 (19.1%)</td><td align=\"left\"/></tr><tr><td align=\"left\"> Prostate</td><td align=\"left\">382 (28.3%)</td><td align=\"left\">727 (53.9%)</td><td align=\"left\">165 (12.2%)</td><td align=\"left\">74 (5.5%)</td><td align=\"left\"/></tr><tr><td align=\"left\"> Thyroid</td><td align=\"left\">245 (38.2%)</td><td align=\"left\">223 (34.7%)</td><td align=\"left\">104 (16.2%)</td><td align=\"left\">70 (10.9%)</td><td align=\"left\"/></tr><tr><td align=\"left\" colspan=\"6\">Healthcare delay</td></tr><tr><td align=\"left\"> No</td><td align=\"left\">2178 (31.6%)</td><td align=\"left\">3286 (47.7%)</td><td align=\"left\">914 (13.3%)</td><td align=\"left\">514 (7.5%)</td><td char=\".\" align=\"char\"> &lt; 0.001</td></tr><tr><td align=\"left\"> Yes</td><td align=\"left\">934 (29.9%)</td><td align=\"left\">1028 (32.9%)</td><td align=\"left\">600 (19.2%)</td><td align=\"left\">566 (18.1%)</td><td align=\"left\"/></tr><tr><td align=\"left\" colspan=\"6\">Health literacy</td></tr><tr><td align=\"left\"> ≤ 9</td><td align=\"left\"> ≤ 20 (&lt; 10%)</td><td align=\"left\">52 (25.1%)</td><td align=\"left\"> &gt; 20 (10%)</td><td align=\"left\">126 (44.9%)</td><td char=\".\" align=\"char\"> &lt; 0.001</td></tr><tr><td align=\"left\"> &gt; 9</td><td align=\"left\">3048 (31.8%)</td><td align=\"left\">4188 (43.7%)</td><td align=\"left\">1435 (14.9%)</td><td align=\"left\">922 (9.6%)</td><td align=\"left\"/></tr></tbody></table></table-wrap>", "<table-wrap id=\"Tab2\"><label>Table 2</label><caption><p>Descriptive characteristics of the sample and their association with healthcare delays (<italic>n</italic> = 10,020)</p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th align=\"left\">Variables</th><th align=\"left\">No healthcare delay<break/>(<italic>n</italic> = 6,892)</th><th align=\"left\">Any healthcare delay (<italic>n</italic> = 3,128)</th><th align=\"left\"><italic>p</italic> value</th></tr></thead><tbody><tr><td align=\"left\" colspan=\"4\">Sex</td></tr><tr><td align=\"left\"> Female</td><td align=\"left\">4239 (64.0%)</td><td align=\"left\">2384 (36.0%)</td><td align=\"left\"/></tr><tr><td align=\"left\"> Male</td><td align=\"left\">2608 (78.3%)</td><td align=\"left\">724 (21.7%)</td><td char=\".\" align=\"char\"> &lt; 0.001</td></tr><tr><td align=\"left\"> Other^a</td><td align=\"left\">45 (69.2%)</td><td align=\"left\">20 (30.8%)</td><td align=\"left\"/></tr><tr><td align=\"left\" colspan=\"4\">Age</td></tr><tr><td align=\"left\"> Median [Q1, Q3]</td><td align=\"left\">66.4 [579, 72.3]</td><td align=\"left\">57.8 [46.7, 65.7]</td><td char=\".\" align=\"char\"> &lt; 0.001</td></tr><tr><td align=\"left\" colspan=\"4\">Race/ethnicity</td></tr><tr><td align=\"left\"> Asian</td><td align=\"left\">132 (67.4%)</td><td align=\"left\">64 (32.7%)</td><td align=\"left\"/></tr><tr><td align=\"left\"> Black</td><td align=\"left\">374 (58.3%)</td><td align=\"left\">267 (41.7%)</td><td align=\"left\"/></tr><tr><td align=\"left\"> Hispanic</td><td align=\"left\">345 (56.1%)</td><td align=\"left\">270 (43.9%)</td><td align=\"left\"/></tr><tr><td align=\"left\"> White</td><td align=\"left\">5856 (71.0%)</td><td align=\"left\">2387 (29.0%)</td><td char=\".\" align=\"char\"> &lt; 0.001</td></tr><tr><td align=\"left\"> More than one pop</td><td align=\"left\">89 (59.3%)</td><td align=\"left\">61 (40.7%)</td><td align=\"left\"/></tr><tr><td align=\"left\"> Other^b</td><td align=\"left\">96 (54.9%)</td><td align=\"left\">79 (45.1%)</td><td align=\"left\"/></tr><tr><td align=\"left\" colspan=\"4\">Annual income^e</td></tr><tr><td align=\"left\"> &lt; 35 K</td><td align=\"left\">914 (50.0%)</td><td align=\"left\">916 (50.0%)</td><td align=\"left\"/></tr><tr><td align=\"left\"> ≥ 35 K</td><td align=\"left\">5300 (73.3%)</td><td align=\"left\">1930 (26.7%)</td><td char=\".\" align=\"char\"> &lt; 0.001</td></tr><tr><td align=\"left\"> Other^c</td><td align=\"left\">678 (70.6%)</td><td align=\"left\">282 (29.4%)</td><td align=\"left\"/></tr><tr><td align=\"left\" colspan=\"4\">Marital status</td></tr><tr><td align=\"left\"> Single</td><td align=\"left\">2103 (60.8%)</td><td align=\"left\">1357 (39.2%)</td><td align=\"left\"/></tr><tr><td align=\"left\"> Married/living with a partner</td><td align=\"left\">4734 (73.2%)</td><td align=\"left\">1732 (26.8%)</td><td char=\".\" align=\"char\"> &lt; 0.001</td></tr><tr><td align=\"left\"> Other^c</td><td align=\"left\">55 (58.5%)</td><td align=\"left\">39 (41.5%)</td><td align=\"left\"/></tr><tr><td align=\"left\" colspan=\"4\">Employed</td></tr><tr><td align=\"left\"> Yes</td><td align=\"left\">2911 (64.4%)</td><td align=\"left\">1608 (35.6%)</td><td align=\"left\"/></tr><tr><td align=\"left\"> No</td><td align=\"left\">3923 (72.5%)</td><td align=\"left\">1491 (27.6%)</td><td char=\".\" align=\"char\"> &lt; 0.001</td></tr><tr><td align=\"left\"> Other^c</td><td align=\"left\">58 (66.7%)</td><td align=\"left\">29 (33.3%)</td><td align=\"left\"/></tr><tr><td align=\"left\" colspan=\"4\">Educational level</td></tr><tr><td align=\"left\"> College or more</td><td align=\"left\">6273 (69.9%)</td><td align=\"left\">2707 (30.1%)</td><td align=\"left\"/></tr><tr><td align=\"left\"> ≤ High school or equivalent</td><td align=\"left\">585 (59.3%)</td><td align=\"left\">401 (40.7%)</td><td char=\".\" align=\"char\"> &lt; 0.001</td></tr><tr><td align=\"left\"> Other^c</td><td align=\"left\">34 (61.8%)</td><td align=\"left\">21 (38.2%)</td><td align=\"left\"/></tr><tr><td align=\"left\" colspan=\"4\">Insured</td></tr><tr><td align=\"left\"> No</td><td align=\"left\">50 (38.2%)</td><td align=\"left\">81 (61.8%)</td><td align=\"left\"/></tr><tr><td align=\"left\"> Yes</td><td align=\"left\">6798 (69.3%)</td><td align=\"left\">3012 (30.7%)</td><td char=\".\" align=\"char\"> &lt; 0.001</td></tr><tr><td align=\"left\"> Other^d</td><td align=\"left\">44 (55.7%)</td><td align=\"left\">35 (44.3%)</td><td char=\".\" align=\"char\"/></tr><tr><td align=\"left\" colspan=\"4\">Nativity</td></tr><tr><td align=\"left\"> US-Born</td><td align=\"left\">6364 (69.1%)</td><td align=\"left\">2841 (30.9%)</td><td char=\".\" align=\"char\">0.01</td></tr><tr><td align=\"left\"> Foreign-Born</td><td align=\"left\">528 (64.8%)</td><td align=\"left\">287 (35.2%)</td><td align=\"left\"/></tr><tr><td align=\"left\" colspan=\"4\">Housing status</td></tr><tr><td align=\"left\"> Own</td><td align=\"left\">5521 (74.4%)</td><td align=\"left\">1898 (25.6%)</td><td align=\"left\"/></tr><tr><td align=\"left\"> Rent/other arrangement</td><td align=\"left\">1296 (52.3%)</td><td align=\"left\">1180 (47.7%)</td><td char=\".\" align=\"char\"> &lt; 0.001</td></tr><tr><td align=\"left\"> Other^c</td><td align=\"left\">75 (60.0%)</td><td align=\"left\">50 (40.0%)</td><td align=\"left\"/></tr><tr><td align=\"left\" colspan=\"4\">Treatment</td></tr><tr><td align=\"left\"> No</td><td align=\"left\"> &gt; 5200 (&gt; 65%)</td><td align=\"left\"> &gt; 2300 (&gt; 30%)</td><td char=\".\" align=\"char\">0.96</td></tr><tr><td align=\"left\"> Yes</td><td align=\"left\">1610 (68.9%)</td><td align=\"left\">728 (31.1%)</td><td align=\"left\"/></tr><tr><td align=\"left\"> Missing</td><td align=\"left\"> ≤ 13 (&lt; 65%)</td><td align=\"left\"> ≤ 20 (&lt; 40%)</td><td align=\"left\"/></tr><tr><td align=\"left\" colspan=\"4\">Cancer type</td></tr><tr><td align=\"left\"> Bladder</td><td align=\"left\">193 (78.8%)</td><td align=\"left\">52 (21.2%)</td><td char=\".\" align=\"char\"> &lt; 0.001</td></tr><tr><td align=\"left\"> Blood</td><td align=\"left\">549 (72.2%)</td><td align=\"left\">210 (27.8%)</td><td align=\"left\"/></tr><tr><td align=\"left\"> Bone</td><td align=\"left\">67 (62.6%)</td><td align=\"left\">40 (37.4%)</td><td align=\"left\"/></tr><tr><td align=\"left\"> Brain</td><td align=\"left\">72 (57.1%)</td><td align=\"left\">54 (42.9%)</td><td align=\"left\"/></tr><tr><td align=\"left\"> Breast</td><td align=\"left\">2076 (70.1%)</td><td align=\"left\">886 (42.9%)</td><td align=\"left\"/></tr><tr><td align=\"left\"> Cervical</td><td align=\"left\">316 (45.9%)</td><td align=\"left\">372 (54.1%)</td><td align=\"left\"/></tr><tr><td align=\"left\"> Colon rectal</td><td align=\"left\">295 (68.9%)</td><td align=\"left\">133 (31.1%)</td><td align=\"left\"/></tr><tr><td align=\"left\"> Endocrine</td><td align=\"left\">46 (68.7%)</td><td align=\"left\">21 (31.3%)</td><td align=\"left\"/></tr><tr><td align=\"left\"> Endometrial</td><td align=\"left\">172 (62.8%)</td><td align=\"left\">102 (37.2%)</td><td align=\"left\"/></tr><tr><td align=\"left\"> Head neck</td><td align=\"left\">129 (70.1%)</td><td align=\"left\">55 (29.9%)</td><td align=\"left\"/></tr><tr><td align=\"left\"> Kidney</td><td align=\"left\">198 (67.6%)</td><td align=\"left\">95 (32.4%)</td><td align=\"left\"/></tr><tr><td align=\"left\"> Lung</td><td align=\"left\">176 (72.7%)</td><td align=\"left\">66 (27.3%)</td><td align=\"left\"/></tr><tr><td align=\"left\"> Other</td><td align=\"left\">979 (68.2%)</td><td align=\"left\">456 (31.8%)</td><td align=\"left\"/></tr><tr><td align=\"left\"> Ovarian</td><td align=\"left\">134 (60.9%)</td><td align=\"left\">86 (39.1%)</td><td align=\"left\"/></tr><tr><td align=\"left\"> Prostate</td><td align=\"left\">1120 (83.1%)</td><td align=\"left\">228 (16.9%)</td><td align=\"left\"/></tr><tr><td align=\"left\"> Thyroid</td><td align=\"left\">370 (57.6%)</td><td align=\"left\">272 (42.4%)</td><td align=\"left\"/></tr><tr><td align=\"left\" colspan=\"4\">Health literacy</td></tr><tr><td align=\"left\"> ≤ 9</td><td align=\"left\">178 (55.1%)</td><td align=\"left\">145 (44.9%)</td><td char=\".\" align=\"char\"> &lt; 0.001</td></tr><tr><td align=\"left\"> &gt; 9</td><td align=\"left\">6714 (69.2%)</td><td align=\"left\">2983 (30.8%)</td><td align=\"left\"/></tr><tr><td align=\"left\" colspan=\"4\">SES barrier</td></tr><tr><td align=\"left\"> 0</td><td align=\"left\">2178 (70.0%)</td><td align=\"left\">934 (30.0%)</td><td align=\"left\"/></tr><tr><td align=\"left\"> 1</td><td align=\"left\">3286 (76.2%)</td><td align=\"left\">1028 (23.9%)</td><td align=\"left\"/></tr><tr><td align=\"left\"> 2</td><td align=\"left\">914 (60.4%)</td><td align=\"left\">600 (39.6%)</td><td align=\"left\"/></tr><tr><td align=\"left\"> 3+</td><td align=\"left\">514 (47.6%)</td><td align=\"left\">566 (52.4%)</td><td align=\"left\"/></tr></tbody></table></table-wrap>", "<table-wrap id=\"Tab3\"><label>Table 3</label><caption><p>Results from the multivariable regression analysis of risk factors for health care delay and by nativity status among cancer survivors from the All of Us Research Program</p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th align=\"left\">Variables</th><th align=\"left\">Adjusted odds ratios<break/><italic>n</italic> = 9,820<break/>OR (95%CI)</th><th align=\"left\">US-Born<break/><italic>n</italic> = 9,022<break/>OR (95%CI)</th><th align=\"left\">Foreign-Born<break/><italic>n</italic> = 798<break/>OR (95%CI)</th></tr></thead><tbody><tr><td align=\"left\" colspan=\"4\">Nativity</td></tr><tr><td align=\"left\"> US-Born</td><td align=\"left\">Ref</td><td align=\"left\">–</td><td align=\"left\">–</td></tr><tr><td align=\"left\"> Foreign-Born</td><td align=\"left\">1.04 (0.87–1.25)</td><td align=\"left\">–</td><td align=\"left\">–</td></tr><tr><td align=\"left\" colspan=\"4\">Health literacy</td></tr><tr><td align=\"left\"> p-trend</td><td align=\"left\">0.92 (0.89–0.95)<italic>*</italic>*<italic>*</italic></td><td align=\"left\">0.91(0.88–0.94)***</td><td align=\"left\">0.98(0.90–1.07)</td></tr><tr><td align=\"left\"> ≤ 9</td><td align=\"left\">1.20 (0.89–1.59)</td><td align=\"left\">1.41 (1.02–1.97)*</td><td align=\"left\">0.66 (0.34–1.25)</td></tr><tr><td align=\"left\"> &gt; 9</td><td align=\"left\">Ref</td><td align=\"left\">Ref</td><td align=\"left\">Ref</td></tr><tr><td align=\"left\" colspan=\"4\">SES barrier factors index</td></tr><tr><td align=\"left\"> p-trend</td><td align=\"left\">1.29 (1.23–1.36)***</td><td align=\"left\">1.27 (1.21–1.34)***</td><td align=\"left\">1.72 (1.43–2.08)***</td></tr><tr><td align=\"left\"> 0</td><td align=\"left\">Ref</td><td align=\"left\">Ref</td><td align=\"left\">Ref</td></tr><tr><td align=\"left\"> 1</td><td align=\"left\">0.98 (0.88–1.10)</td><td align=\"left\">0.97 (0.86–1.09)</td><td align=\"left\">1.15 (0.76–1.74)</td></tr><tr><td align=\"left\"> 2</td><td align=\"left\">1.65 (1.43–1.90)***</td><td align=\"left\">1.53 (1.32–1.78)***</td><td align=\"left\">4.35 (2.61–7.34)***</td></tr><tr><td align=\"left\"> 3+</td><td align=\"left\">2.18 (1.84–2.58)***</td><td align=\"left\">2.10 (1.76–2.50)***</td><td align=\"left\">3.83 (2.14–6.98)***</td></tr></tbody></table></table-wrap>" ]

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[ "<table-wrap-foot><p>Single = includes those who reported: divorced, widowed, separated, and never married. p-values were obtained from chi-square and Kruskal–Wallis tests. Per “All of Us” data use agreement policy, groups &lt; 20 participants are shown as ≤ 20 (%) with a corresponding &gt; (%) category to prevent deriving counts &lt; 20 from other values. No all percentages equal to 100</p><p>Q1 = Quartile 1(25%), Q3 = Quartile 3(75%)</p><p><italic>SES</italic> socioeconomic include (education, income, insurance, housing, and employment status)</p><p>Data values included in these categories: ^aOther and missing</p><p>^bNone of this, another population, and prefer not to answer</p><p>^cPrefer not to answer and missing</p><p>^dMissing, do not know, and prefer not to answer, NA = Missing</p><p>^eIncome reported in US dollar, Cancer type “other” also includes esophageal, eye, pancreatic, and stomach cancers</p></table-wrap-foot>", "<table-wrap-foot><p>Single = includes those who reported: divorced, widowed, separated, and never married. <italic>p</italic> values were obtained from chi-square and Mann–Whitney tests. Any healthcare delay includes delays due to transportation, living in a rural area where healthcare providers are too far, nervousness about seeing a healthcare provider, could not get time off work, could not get childcare, cannot leave adult unattended due to being a caretaker, could not afford copays, deductible was too high or could not afford it or had to pay out of pocket for some or all procedures. Per “All of Us” data use agreement policy, groups &lt; 20 participants are shown as ≤ 20 (%) with a corresponding &gt; (%) category to prevent deriving counts &lt; 20 from other values. No all percentages equal to 100</p><p>Q1 = Quartile 1(25%), Q3 = Quartile 3(75%)</p><p><italic>SES</italic> socioeconomic</p><p>Data values included in these categories:^aOther and missing</p><p>^bNone of this, another population, and prefer not to answer</p><p>^cPrefer not to answer and missing</p><p>^dMissing, do not know, and prefer not to answer, NA = Missing</p><p>^eIncome reported in US dollar, Cancer type “other”: includes esophageal, eye, pancreatic, and stomach cancers</p></table-wrap-foot>", "<table-wrap-foot><p>p-trends were obtained by assessing SES barriers and Health Literacy as continuous measures</p><p>Adjusted odds ratios (OR) for: sex, race/ethnicity, age, marital status, active treatment, and cancer type</p><p><italic>SES</italic> socioeconomic, <italic>Ref</italic> reference group, <italic>CI</italic> confidence interval</p><p>Significant <italic>p</italic> values ***&lt; 0.001, **&lt; 0.01, *&lt; 0.05</p></table-wrap-foot>", "<fn-group><fn><p><bold>Publisher's Note</bold></p><p>Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.</p></fn></fn-group>" ]

[ "<graphic xlink:href=\"10552_2023_1782_Fig1_HTML\" id=\"MO1\"/>" ]

[ "<media xlink:href=\"10552_2023_1782_MOESM1_ESM.docx\"><caption><p>Supplementary file1 (DOCX 19 KB)</p></caption></media>" ]

{ "acronym": [], "definition": [] }

[ "<title>Introduction</title>", "<p id=\"Par5\">As a major discovery in the twentieth century, antibiotics play an irreplaceable role in human health, animal epidemic prevention, and disease treatment (Coleman et al. ##REF##26196923##2015##). However, due to the overuse of antibiotics, a variety of bacteria have developed drug resistance, including the intensively studied <italic>Staphylococcus aureus</italic> (<italic>S. aureus</italic>) through a variety of escape routes to avoid drug access, which can cause various animal diseases such as mastitis, endometritis, sepsis, and other systemic symptoms, seriously endangering the production of livestock and poultry and the development of animal husbandry (Coleman et al. ##REF##26196923##2015##; Kim et al. ##REF##31358625##2019##). Drug resistance was transmitted through the environment, humans, and livestock (Davies and Davies ##REF##20805405##2010##), causing huge economic losses (El-Sayed Ahmed et al. ##UREF##1##2020##). Vancomycin is considered the last line of defense against infection by drug-resistant Gram-positive bacteria (G⁺), but resistance has quickly developed with the increase in drug use in clinical practice (Rao ##REF##7772805##1995##). Therefore, there is an urgent need for an antibiotic substitution to address this issue.</p>", "<p id=\"Par6\">Antimicrobial peptides (AMPs), as an important part of the natural immune barrier, exhibit antibacterial characteristics, immune regulation, and multiple other functions (Shinohara et al. ##REF##7638204##1995##; Huttner and Bevins ##REF##10367766##1999##; Peters et al. ##REF##21060861##2010##; Wang et al. ##REF##34090965##2021a##, ##REF##34009979##b##). The DBAASP database contains 20,523 antibacterial peptides and derivatives (Pirtskhalava et al. ##REF##33151284##2021##),  being considered a rich resource for AMPs screening (Pinto et al. ##REF##31127270##2019##). AMPs generally have the characteristics of cationic, hydrophobic, short sequence length (&lt; 50aa), and unique three-dimensional (3D) structure including α-helix, β-sheet, and β-turn or coil; these play a vital role in the functional activity of AMPs for better understanding of the structure-activity relationship (Cao et al. ##REF##25261129##2015##; Wang et al. ##REF##29523806##2018##; Wang et al. ##REF##34090965##2021a##, ##REF##34009979##b##; Wu et al. ##REF##35725522##2022##). The AMPs, consisting of only amino acid in term of narrow definition, have multiple antibacterial mechanisms including destroying the cell membrane or wall or causing the contents to leak (Miao et al. ##UREF##8##2016##; Glukhov et al. ##REF##18186149##2008##; Jhong et al. ##REF##30380085##2019##; Lee et al. ##REF##20553419##2011##; Ma et al. ##REF##28885829##2017##); this is different from traditional heterocyclic peptide antibiotics, for which it is not easy to produce drug resistance. Therefore, they are becoming a research hotspot for the substitution of antibiotics (Gao et al. ##REF##34673132##2021##; Pinto et al. ##REF##31127270##2019##; Wu et al. ##UREF##13##2021##). Plectasin is a fungal defensin, isolated from <italic>Pseudoplectania nigrella</italic>, which can inhibit the synthesis of the bacterial cell wall by binding to the cell wall precursor Lipid II. It has a strong killing effect on G⁺ (Mygind et al. ##REF##16222292##2005##; Schneider et al. ##REF##20508130##2010##) and the ability to treat peritonitis and pneumonia caused by <italic>Streptococcus pneumoniae</italic> <italic>and</italic>\n<italic>S.aureus</italic> (Mygind et al. ##REF##16222292##2005##). However, its high cost and low druggability limit its clinical applications. In order to solve the above problems, 12 kinds of plectasin-derived peptides were screened, analyzed, and verified in the DBAASP database (Zhang et al. ##REF##23624708##2014##, ##REF##33534018##2021##; Othman et al. ##UREF##9##2018##; Wang et al. ##REF##34090965##2021a##, ##REF##34009979##b##; Huang et al. ##REF##35524777##2022##), AP138 (DBAASPS_12115) created by Lociuro was predicted to be the best one among plectasin-derived peptides with high antimicrobial activity and a long postantibiotic effect (PAE) against G⁺ bacteria including <italic>S. aureus</italic> and methicillin-resistant <italic>Staphylococcus aureus</italic> (MRSA) (Lociuro et al. ##UREF##6##2015##; Groo et al. ##REF##30532539##2018##). Compared with plectasin, there are three main differences in AP138: (i) Five amino acids, 9D, 13 M, 14Q, 17N, and 26 K, are mutated to 9S, 13L, 14R, 17R, and 26R, respectively; (ii) the positive charge increased from + 1 to + 4.5; (iii) the second structure α-helix (30.75 to 20.8%) and β-sheet (20.8 to 12.5%), β-turn (95.8 to 120.8%) changed obviously (Table ##TAB##0##1## and ##SUPPL##0##S1##). These studies and bioinformatics analysis all indicated that AP138 has research and development value in clinic therapeutics. However, AP138 is chemically synthesized due to D-type Arg at 26th site in sequence, so suffer from the heterologous expression difficulty and high cost (Koo and Seo ##UREF##5##2019##). Heterologous expressions may solve the problem of high cost, but the amino acid composition will be natural L-type amino acids (Lee et al. ##REF##20553419##2011##; Lin et al. ##REF##29675221##2018##). Therefore, the potential new different properties and mechanism of L-type AP138 (defined as AP138L-arg26 (DBAASPS_12115)) should be characterized as one of the major goals of this work (Groo et al. ##REF##30532539##2018##; Pinto et al. ##REF##31127270##2019##; Patel et al. ##REF##19805558##2009##; Umerska et al. ##REF##27113868##2016##; McEwen and Collignon ##UREF##7##2018##).\n</p>", "<p id=\"Par7\">In this study, plectasin-derived peptide AP138L-arg26 was expressed in <italic>Pichia pastoris</italic> (<italic>P. pastoris</italic>) cells X-33 with a 5-L fermenter, and in vitro analysis of structure, antibacterial activity, stability, drug resistance, toxicity, and safety was performed. Finally, its bactericidal mechanism against <italic>S. aureus</italic> was revealed.</p>" ]

[ "<title>Materials and methods</title>", "<title>Strains and cell lines</title>", "<p id=\"Par8\">Gram-positive bacteria: <italic>S. aureus</italic> (ATCC 25923, 43300), <italic>Staphylococcus epidermidis</italic> (<italic>S. epidermidis</italic>) (ATCC 12228, 35984), and <italic>Streptococcus agalactiae</italic> (<italic>S. agalactiae</italic>) ATCC 13813 were purchased from American Type Culture Collection (ATCC). <italic>S. aureus</italic> CVCC 546 and <italic>Streptococcus dysgalactiae</italic> (<italic>S. dysgalactiae</italic>) CVCC 3938 were purchased from the China Veterinary Culture Collection Center (CVCC). <italic>S. aureus</italic> E48 was donated by Northwest Agriculture and Forestry University<italic>. S. agalactiae</italic> CAU-FRI-2022-01 and <italic>S. agalactiae</italic> CAU-FRI-2022-02 were donated by China Agricultural University. Gram-negative bacteria: <italic>Escherichia coli</italic> (<italic>E. coli</italic>) ATCC 25922 were purchased from ATCC. <italic>Salmonella enteritidis</italic> (<italic>S. enteritidis</italic>) CVCC 3377 and <italic>Salmonella pullorum</italic> (<italic>S. pullorum</italic>) CVCC1789 were purchased from CVCC. <italic>Shigella flexneri</italic> (<italic>S. flexneri</italic>) (CMCC 3926, 51571) were purchased from National Center for Medical Culture Collections (CMCC). <italic>E. coli</italic> O157 (CICC 21530), <italic>Pseudomonas aeruginosa</italic> (<italic>P. aeruginosa</italic>) (CICC 21625, CICC21630) were purchased from China Center of Industrial Culture Collection (CICC). <italic>S. aureus</italic> CAAS-FRI-2023-01 <italic>and</italic> CAAS-FRI-2023-02 were separated from Tianjin Aoxin Animal Husbandry Sheep Farm and Huanxian Sheep Farm, respectively. <italic>E. coli</italic> DH5α, <italic>P. pastoris</italic> X-33, and pPICZαA were purchased from Invitrogen (Beijing, China). RAW 264.7 mice macrophages were obtained from Peking Union Medical College. Bovine endometrial epithelial cell line BNCC35923 was purchased from by BeNa Culture Collection (Beijing, China).</p>", "<title>Reagents</title>", "<p id=\"Par9\">The recombinant plasmid pPICZαA-AP138L-arg26 was synthesized by Sangon Biotech Co., Ltd. (Shanghai, China). Plasmid extraction kits, antibiotics (vancomycin and ceftiofur sodium), Dulbecco’s modified Eagle medium (DMEM), and fetal bovine serum (FBS) were purchased from Tiangen Co., Ltd, China Meilungel and Gibco (China), respectively. Other reagents were analytical grade.</p>", "<title>Model animal</title>", "<p id=\"Par10\">The female ICR mice (SPF, 6–8 weeks, 20–25 g/mouse) were purchased from the Vital River Laboratories (VRL, Beijing, China). Mice acclimated to the environment for 1 week before the experiment. Animal experiments strictly complied with the requirements for animal handling and welfare of the Laboratory Animal Ethical Committee and its Inspection of the Feed Research Institute of Chinese Academy of Agricultural Sciences (CAAS) (AEC-CAAS-20090609).</p>" ]

[ "<title>Results</title>", "<title>Analysis of AP138 and designation of AP138L-arg26</title>", "<p id=\"Par34\">Plectasin was searched for as a keyword in the DBAASP database, and 12 sequences were retrieved; these derived peptides, AP138, NZ2114, NZX, MP1102, and MP1106, and other reported peptides DLP4, ID3, and P2 were also collected. Their physicochemical properties are shown in Table ##TAB##0##1##. AP138L-arg26 (GFGCNGPWSEDDLRCHRHCKSIKGYR_L26G GYCAKGGFVCKCY) was predicted to have the highest AMP possibility (0.992) and charge (+ 4.5) (Table ##TAB##0##1##) and so was considered the plectasin-derived peptide. Through secondary and 3D structure analysis, we found that AP138L-arg26 conformation changed greatly with reduced α-helix/β-sheet and increased β-turn compared with the parent peptide plectasin (Table ##TAB##0##1##). Meanwhile, the electron cloud of amino acids at the 9th, 13th, 14th, 17th, and 26th positions changed significantly with the mutant amino acids; especially at the 13th, 14th, and 17th positions, more than 2 electron clouds changed at α-Helix, which may affect the function of peptides (Table ##TAB##0##1## and Fig. ##FIG##0##1##).</p>", "<title>Design, expression, and purification of AP138L-arg26</title>", "<p id=\"Par35\">The recombinant plasmid pPICZαA-AP138L-arg26 was linearized and transferred into <italic>P. pastoris</italic> X-33 competent cells. The recombinant plasmid sequence length was 3000–5000 bp (Fig. ##FIG##1##2##a). The positive transformants were screened using the inhibition zone test, and transformants AP138L-arg26-8, AP138L-arg26-28, AP138L-arg26-68, and AP138L-arg26-94 had a good antibacterial effect with a large and clear inhibition zone (Fig. ##FIG##1##2##b). Tricine-SDS-PAGE analysis of AP138L-arg26 expression in shaking flask level showed that the protein band was around 4.6 kDa, which was consistent with the predicted value of 4.46 kDa (Fig. ##FIG##1##2##c). The positive transformants AP138L-arg26 with the highest effect (AP138L-arg26-8) was chosen and expressed with a 5-L fermenter, which displayed high production compared with the shaking flask (Fig. ##FIG##1##2##d, e). The total protein of fermentation supernatant was 3.1 mg/mL and the biomass was 0.36 g/mL after induction in a 5-L fermenter at 120 h (Fig. ##FIG##1##2##f). The peptide was purified using the cation-exchange column (AKTAxpress system), and only a target peptide was detected at around 4.6 kDa through Tricine-SDS-PAGE band and a single peak was detected in matrix-assisted laser desorption/ionization time-of-flight mass spectrometry (MALDI-TOF/TOF MS) of 4464.21 Da (Fig. ##FIG##1##2##g, h).</p>", "<title>MICs and MBCs</title>", "<p id=\"Par36\">The results of MICs and MBCs showed that AP138L-arg26 had potent antimicrobial activity against G⁺ bacteria such as <italic>S. aureus</italic> (2–16 µg/mL), <italic>Streptococcus</italic> (4 µg/mL), and <italic>S. epidermidi</italic><italic>s</italic> (4 or 8 µg/mL) with MIC values of 4–16 µg/mL, and the MBCs were around fourfold higher than MICs (Table ##TAB##1##2##). Antimicrobial activity was slightly lower than AP138 (Groo et al ##REF##30532539##2018##) and vancomycin, but AP138L-arg26 retained good activity. These results show that AP138L-arg26 has potential for drug development against G⁺.\n</p>", "<title>Time killing curves and PAE</title>", "<p id=\"Par37\">As shown in Fig. ##FIG##2##3##a, all selected concentrations of AP138L-arg26 (1 × , 2 × , 4 × MIC) could kill 99.99% <italic>S. aureus</italic> ATCC 43300 within 1.5 h, while the antibiotic vancomycin needed at least 6 h. Meanwhile, AP138L-arg26 had longer PAE with 0.9 h, 1.25 h, and 1.91 h at 1 × , 2 × , 4 × MIC than vancomycin (0.27, 0.45, 1.18) (Fig. ##FIG##2##3##b). These results suggest that AP138L-arg26 had a faster and longer bactericidal effect than vancomycin, even within a low concentration (1 × MIC), which may be related to the different bactericidal mechanisms of AP138L-arg26 and vancomycin.</p>", "<title>Intracellular activity of AP138L-arg26 against <italic>S. aureus</italic></title>", "<p id=\"Par38\">As shown in Fig. ##FIG##2##3##c, d, AP138L-arg26 could kill intracellular <italic>S. aureus</italic> CVCC 546 in mouse macrophages RAW264.7 and bovine endometrial epithelial cells BNCC359233. The peptide of AP138L-arg26 could kill more than 70% of bacteria at a concentration of 5 × MIC, and the bactericidal rate was up to 85% at 50 × MIC. Therefore, AP138L-arg26 exhibited potent intracellular bactericidal ability, laying the foundation for in vivo applications.\n</p>", "<title>The synergism assay of AP138L-arg26 with antibiotic</title>", "<p id=\"Par39\">As shown in Table ##TAB##2##3##, the FICI values of peptides AP138L-arg26 and vancomycin, ceftiofur sodium, ampicillin, or streptomycin sulfate ranged from 0.375 to 0.75, showing a synergistic effect. AP138L-arg26 had the lowest FICI value (0.375) with ceftiofur sodium. The bacterial growth curve results showed that AP138L-arg26 (8µg/mL, 2 × MIC) and sublethal levels of ceftiofur sodium alone (0.25 µg/mL, 1/8 MIC) had little effect on the growth of <italic>S. aureus</italic> CVCC 546 (Fig. ##FIG##2##3##e), but their combination sharply inhibited the growth of <italic>S. aureus</italic> CVCC 546. Therefore, the combination of AP138L-arg26 and ceftiofur sodium had the potential to be used to cure infection by <italic>S. aureus</italic>.\n</p>", "<title>Stability in vitro assay</title>", "<p id=\"Par40\">These results show that AP138L-arg26 maintained its original MIC value (8 µg/mL) after incubation at different temperatures (20 °C, 40 °C, 60 °C, 80 °C), but the antibacterial activity was lost (&gt; 64 µg/mL) under 100 °C. The antimicrobial activity of AP138L-arg26 was unaffected by gastric juice and different pH values (2–10). AP138L-arg26 was sensitive to trypsin, losting activity within 30 min after incubation (&gt; 64 µg/mL) (Table ##TAB##3##4##), this would be the key limitation factor at least special for its oral administration.\n</p>", "<title>Structure analysis</title>", "<p id=\"Par41\">The secondary structures of AP138L-arg26 were measured in a different environment of 20 mM SDS and 50% TFE solution, which was used to simulate the hydrophobic environment of eukaryotic cells and the bacterial cell membrane environment, respectively. The results showed a positive peak at 196 nm and two negative peaks at 208 nm and 228 in 50% TFE solution, indicating an α/β spatial structure in these environments (Figure ##SUPPL##0##S2##). However, the second structure of AP138L-arg26 had obviously changed (α-Helix, 4.72 to 6.06%; β-sheet, 47.78 to 42.82%; β-turn, 15.6 to 17.4%) in 20 mM SDS, especially for the increased α-Helix (1.34% increase), which could enhance membrane interactions between bacteria and AP138L-arg26 (Table ##SUPPL##0##S1##).</p>", "<title>High safety of AP138L-arg26 in vitro and vivo</title>", "<p id=\"Par42\">As shown in Fig. ##FIG##3##4##a, b, there was only 3% hemolysis and over 75% cell viability at a high concentration of AP138L-arg26 (256 µg/mL) in vitro, which indicated that AP138L-arg26 had low hemolysis and cytotoxicity. In vivo, mice (ICR, 6-8 weeks) were intraperitoneally injected with AP138L-arg26 at a concentration of 10 mg/kg (Fig. ##FIG##3##4##c–g). Compared with the control group, there were no significant differences in the whole blood detection indexes and biochemical indexes of the AP138L-arg26 treatment group. Moreover, the tissues of the AP138L-arg26 treatment group such as the heart, kidney, spleen, lung, and liver had no hyperemia, bleeding, hyperplasia, or other lesions in the full field, being almost same as those of the control group (Fig. ##FIG##3##4##d). These data indicate that the AP138L-arg26 peptide has excellent safety and good potential as clinical therapeutic drugs.</p>", "<title>The mechanism of AP138L-arg26 against <italic>S. aureus</italic> CVCC 546</title>", "<title>SEM</title>", "<p id=\"Par43\">The morphology changes of bacteria after treatment with AP138L-arg26 were observed using SEM. As shown in Fig. ##FIG##4##5##a, the surface of <italic>S. aureus</italic> CVCC 546 (nearly 100%) was smooth and spherical without peptide treatment. However, 60–80% of cell structures changed significantly with surface roughness, granular secretion, perforation, and deformation after treatment with 2 × MIC peptide AP138 at 0.5, 1, and 2 h. More than 80% of <italic>S. aureus</italic> CVCC 546 had significantly changed structure at 1 h. The results showed that AP138L-arg26 could destroy the structure of <italic>S. aureus</italic>.</p>", "<title>The effect of AP138L-arg26 on the cell membrane</title>", "<p id=\"Par44\">According to the SEM results demonstrated that AP138L-arg26 caused disruption to the cell membrane or wall. Thus, a PI/SYTO9 assay was conducted to prove the destructive effect of AP138L-arg26 on <italic>S. aureus</italic> CVCC 546. The untreated group was almost stained green by SYTO9 (over 99%), demonstrating that they were predominantly living cells. The AP138L-arg26 treatment group could disrupt the cell membrane integrity with more than 50% cells stained red by PI (Fig. ##FIG##4##5##b). K⁺ leakage further illustrates the disruptive effect of AP138L-arg26 on bacterial membrane, the results showed that the AP138L-arg26 treated group significantly increased the extracellular K⁺ leakage level (0.05 mg/L) compared to the untreated bacteria (0.02 mg/L), similar to the positive control (0.1% TritonX-100) (Fig. ##FIG##4##5##d). It was also found that AP138L-arg26 could affect the membrane potential, as measured using DiSC₃(5) fluorescence staining, the relative fluorescence units (RFU) values of the untreated and 1 × MIC AP138L-arg26-treated groups were around 2000, and the 4 × MIC AP138L-arg26 and nisin (positive control) groups were above 3000 (a 1000 increase) (Fig. ##FIG##4##5##c), which was mainly due to the depolarization caused by the destructive effect of the AP138L-arg26 on the bacterial membrane when the special compounds insert into lipid bilayers, they can change the membrane fluidity and disrupt the normal plasma membrane fluidity homeostasis, which further leads to leakage of cellular components and bacterial death. Therefore, a dye, Laurdan was used to measure the <italic>S. aureus</italic> membrane fluidity and quantified it using the Laurdan GP index. The results showed that the Laurdan GP of <italic>S. aureus</italic> decreased from 0.22 to 0.18 after treatment with 2 × MIC AP138L-arg26, which indicated an increase in the fluidity of the bacterial membrane due to the interaction of the AP138L-arg26 with the membrane (Fig. ##FIG##4##5##e). These results suggested that AP138L-arg26 could destroy the cell membrane of <italic>S. aureus</italic>.</p>", "<title>The effect of AP138L-arg26 on bacterial metabolism</title>", "<p id=\"Par45\">In order to further study whether AP138L-arg26 could affect the metabolism to kill bacteria, bacterial respiration was detected using an ATP kit. The results showed that AP138L-arg26 could increase intracellular ATP: the RLU values of 1 × , 2 × , 4 × MIC AP138L-arg26-treated groups (10,700, 11,500, and 11,900, respectively) were 7.6, 8.2, and 8.5 times higher than that of the untreated group (1400), which indicated a concentration-dependent pattern (Fig. ##FIG##5##6##a). The increase in ROS causes a series of cascading reactions such as oxidation of lipids, proteins, and damage to DNA, inducing bacteria death. After treatment with 1 × , 2 × , and 4 × MIC AP138L-arg26, the fluorescence values increased from 600 (blank control, CK) to 930, 1000, and 1400 with concentration-dependent (Fig. ##FIG##5##6##b). LDH levels decreased after antimicrobial peptide incubation. The LDH activity % of CK, 1 × , 2 × , and 4 × MIC AP138L-arg26 were 100%, 24%, 16%, and 16% respectively, which indicated a concentration-dependent pattern (Fig. ##FIG##5##6##c). The results showed that AP138 could induce the ROS generation of <italic>S. aureus</italic> CVCC 546, ultimately promoting cell death and reducing the probability of resistance generation.</p>" ]

[ "<title>Discussion</title>", "<p id=\"Par46\"><italic>S</italic>. <italic>aureus</italic> is a pathogenic bacterium that greatly threatens human health and the production efficiency of livestock and poultry. Many bacteria have developed drug resistance, particularly MRSA (Pinto et al. ##REF##31127270##2019##). The rise of drug resistance has triggered a public health crisis; therefore, there is an urgent need to develop new antibacterial agents to alleviate bacterial resistance (Peters et al. ##REF##21060861##2010##; Li et al. ##REF##32582062##2020##). The unique bactericidal mechanism and multiple functions of AMPs have been extensively studied as antibiotic substitutes (Rao ##REF##7772805##1995##; Peters et al. ##REF##21060861##2010##). Among them, plectasin is the first fungal defensin extracted from <italic>Pseudoplectania nigrella (Saprophytic ascomycetes)</italic>; it kills G⁺ bacteria but has low activity and some cytotoxicity. Therefore, researchers have conducted in-depth studies on the peptides derived from plectasin (Mygind et al. ##REF##16222292##2005##; Schneider et al. ##REF##20508130##2010##). AP138 is a derived peptide with potential for clinical development, obtained by chemical synthesis due to the D type Arg at the 26th position of the sequence (Stecher et al. ##REF##24413669##2014##; Groo et al. ##REF##30532539##2018##). In this work, natural amino acid sequence (L-type amino acid) was obtained using heterologous expression to reduce the high cost of chemical synthesis, and the activity and bactericidal mechanism of the L-type AP138 were analyzed.</p>", "<p id=\"Par47\">As can be seen in Table ##TAB##0##1##, it was predicted that AP138L-arg26 would have a high charge (+4.5) and antimicrobial peptide potential. Compared with plectasin, the substitution of five amino acids (D9S, M13L, Q14R, N17R, and K26R) significantly changed the physicochemical properties of AP138L-arg26 including an increased positive charge (+1 to  +4.5) and hydrophobicity (32 to 33%) (Table ##TAB##0##1##), making it a good AMP possibility. Although D-type modification of the Arg in the sequence (D-type AP138) could effectively improve the tolerance of the antimicrobial peptide to trypsin which this property cannot be maintained in our AP138L-arg26, its chemical synthesis is difficult on a large scale and at low cost, increasing the challenges and difficulties of clinical development (Stecher et al. ##REF##24413669##2014##; Umerska et al. ##REF##27113868##2016##; Groo et al. ##REF##30532539##2018##; Koo and Seo ##UREF##5##2019##. Heterologous expression can achieve high yields, thus reducing the cost of developing the target protein. The exogenous expression technology of proteins in a y<italic>east</italic> expression system or <italic>E. coli</italic> expression system is relatively mature, but the expression of small molecule peptides (&lt; 50 aa) is relatively difficult execpt few successful cases from fungal defensins and others (Zhang et al. ##REF##21558006##2011##; Zhang et al. ##REF##23624708##2014##; Cao et al. ##REF##25261129##2015##; Yang et al. ##REF##31025073##2019##; Shen et al. ##REF##33815324##2021##; Gries ##REF##36232802##2022##; Zeng et al. ##REF##34990807##2022##). In addition, the AMPs Retrocyclin-101, and Protegrin-1, and Abaecin was also expressed in a prokaryotic expression system with a low yield (Lee et al. ##REF##20553419##2011##). More details in our previous works, many antimicrobial peptides were successfully expressed in <italic>P. pastoris</italic> with high yield at level of  1.0-3.0 g/L (supernatant) and even higher level via high-density fermentation such as NZ2114, NZX, ID3 and NZL, with continous optimisation including enhancing the microbial biomass, controlling the fermentation and induction conditions, the increasing production capacity of equipment per unit volume and modificating expression system (Zhang et al. ##REF##23624708##2014##. Liu et al. ##REF##31900561##2020##; Feng et al. ##REF##22189867##2012##; Li et al ##REF##32582062##2020##; Shen et al. ##REF##33815324##2021##; Hao et al. ##UREF##3##2023##; Jin et al. ##UREF##4##2023##). In this study, the recombinant vector AP138L-arg26 was successfully constructed and expressed for the first time at high levels in <italic>P. pastoris</italic>. The total protein concentrations of the fermentation supernatant and microbial biomass were up to 3.1 mg/mL (95% purity) and 0.36 g/mL, respectively, after 120 h high density fermentation in a 5-L fermenter (Fig. ##FIG##1##2##), higher than those of NZ2114 (2.390 mg/mL, 94.8% purity) (Zhang et al. ##REF##23624708##2014##), and the production cost of AP138L-arg26 was significantly lower than that of chemical synthesis (solid phase).</p>", "<p id=\"Par48\">The MIC is one of the key indicators for the early screening of active AMPs, for which values of MICs under 16 µg/mL may be clinically relevant (Rao ##REF##7772805##1995##; Patel et al. ##REF##19805558##2009##; Oh et al. ##REF##30936103##2019##). The MICs of AP138L-arg26 were 2–16 µg/mL (0.45–3.6 µM) against selected standard and clinical <italic>S. aureus</italic>, <italic>S. epidermidis</italic>, <italic>S. dysgalactiae</italic>, and <italic>S. agalactiae</italic> (Table ##TAB##1##2##). Although the MIC values were lower than those of D-type AP138 (0.125–4 µg/mL), it is still superior to lincomycin, which is mainly used clinically against G⁺ bacteria. The structure and function of AMPs are closely related, and in-depth investigation of the structure-activity relationship is an essential step in studying AMPs (Rost and Sander ##REF##7892171##1994##; Rao ##REF##7772805##1995##). It was analyzed that AP138L-arg26 has a special CSαβ structure with  +4.5 net charge and 33% hydrophobic ratio in a normal environment with ddH₂O. In the simulated bacterial membrane structure (20 mM SDS), AP138L-arg26 showed significant secondary structure changes, increased α-helix structure (4.72 to 6.06%), which may be related to the bactericidal function of the peptide, while it does not change significantly in the simulated eukaryotic cell membrane structure (50% TFE buffer), suggesting no damage to cells (Figure ##SUPPL##0##S2##). The killing curves showed that 1 × , 2 × , 4 × MIC of AP138L-arg26 could kill all <italic>S. aureus</italic> ATCC 43300 within 1.5 h, which was shorter than that of vancomycin (6 h) (Fig. ##FIG##2##3##a). These results showed that the AMP AP138L-arg26 could rapidly killed bacteria. The PAE is an important guide for the rational use of clinical drugs, and the re-evaluation of adverse effects of antibiotics and combination drugs. In this study, the PAE of AP138L-arg26 was longer than that of antibiotics (Fig. ##FIG##2##3##b), indicating that low doses and long dosing intervals are feasible, which could reduce the amount of drugs used and alleviate the problem of drug resistance caused by the misuse of antibiotics (Li et al. ##REF##32582062##2020##). <italic>Staphylococcus</italic> pathgens can evade antibiotics by entering cells (Wang et al. ##REF##29523806##2018##). AP138L-arg26 had the ability to enter mammalian cells and can kill infected bacteria within cells such as <italic>S. aureus</italic> CVCC 546 in mouse macrophages RAW264.7 and bovine uterine epithelial cells in this study (Fig. ##FIG##3##4##c, d). Stability and safety are the factors that must be controlled when drugs enter the clinic, as high toxicity and low stability will hinder AMPs’ application (Koo and Seo ##UREF##5##2019##). AP138L-arg26 retained its antimicrobial activity in different concentrations of salt ions, pH, temperature, and pepsin, but it was sensitive to trypsin and lost antimicrobial activity for 30 min. These results showed that it remained the antimicrobial activity (MIC against <italic>S. aureus</italic> ATCC 43300, 8 µg/mL) and has the potential for topical drug development (Table ##TAB##1##2##), whereas it may need to be encapsulated for oral administration (Umerska et al. ##REF##28848347##2017##; Groo et al. ##REF##30532539##2018##). AP138L-arg26 had a better safety rating in vitro and in vivo, as reflected in the high cell survival (over 75% for mouse macrophages RAW264.7) and the low hemolysis (less than 3%) at a high concentration of 256 µg/mL of AP138L-arg26 in vitro (Fig. ##FIG##3##4##a, b). After intraperitoneal injection of 10 mg/kg AP138L-arg26 for 1 week, it was found that there were no obvious differences in whole blood detection indexes (&lt; 4.5%), biochemical index (&lt; 4%). The histological sections of the heart, liver, spleen, lung, and kidney were not damaged and retained a normal structure (no bleeding, hyperemia, or hyperplasia) indicating the safety of AP138L-arg26 in vivo (Fig. ##FIG##3##4##c–g). These results indicate that AP138L-arg26 has a high safety profile as a drug in <italic>vivo</italic> and in <italic>vitro</italic>.</p>", "<p id=\"Par49\">In general, the researchers suggest that the unique positive charge and hydrophobic properties of most AMPs could interact with bacterial cell membranes and exert bactericidal functions (Gao et al. ##REF##34673132##2021##; Pinto et al. ##REF##31127270##2019##; Zheng et al. ##REF##35722282##2022##). Changes in bacterial morphology were first observed using SEM after treatment with 2 × MIC AP138L-arg26 for 1 h, and most bacteria had rough surfaces, depressions, and granular secretions (Fig. ##FIG##4##5##a). It was further shown that AP138L-arg26 had a directly destructive effect on the cell membrane through PI staining, K⁺ leakage, and membrane fluidity assays (Fig. ##FIG##4##5##a–e). AP138L-arg26 might mainly be considered to interact with the negatively charged components on the surface of the bacterial membrane through itself charges, and then insert into the cell membrane, interfering with its orderly arrangement, causing further destruction (She et al. ##UREF##10##2022##; Wang et al. ##REF##34090965##2021a##, ##REF##34009979##b##). This was similar to the case with the antimicrobial peptide 5j and L007-0069 (She et al. ##UREF##10##2022##). AP138L-arg26 also affect the bacterial metabolism: (i) After the incubation of AP138L-arg26 with bacteria, intracellular ATP levels were elevated (maximum: 8.5-fold), which may cause bacteria to switch from a dormant state to an active state that is more conducive to them being killed (Shi et al. ##REF##36071151##2022##). (ii) Intracellular ROS were increased twofold with 2 × MIC of AP138L-arg26, which suggests that bacteria could be damaged through bacterial auto-oxidation (Shi et al. ##REF##36071151##2022##), this is an important way in which drugs could damage bacteria. (iii) At the same time, the amount of LDH decreased (lowest, 16%), which may limit the level of respiratory metabolism due to LDH being an essential enzyme in the respiratory chain (Wang et al. ##REF##34090965##2021a##, ##REF##34009979##b##) (Fig. ##FIG##5##6##).  All in all, AP138L-arg26 may kill bacteria by damaging cell membranes and influencing bacterial metabolism (ATP, ROS, LDH).</p>", "<p id=\"Par50\">Finally, we changed the D-type AP138D to L-type AP138L-arg26, and it was successfully expressed in <italic>P. pastoris</italic> with high production. It was demonstrated that AP138L-arg26 had a faster and longer bactericidal effect compared with conventional antibiotics, high stability, and excellent safety in vivo and in vitro. It was revealed that AP138L-arg26 has multiple bactericidal mechanisms including membrane rupture and metabolism imbalance, which is also an important reason that antimicrobial peptides do not tend to develop resistance. The better derived AMPs of plectasin will be worth of developing in the future so that the excellent property of high resistance to trypsin hydrolysis in AP138 construct could be merged and maintained in new AMP derivative with a high expression level.</p>" ]

[]

[ "<title>Abstract</title>", "<p id=\"Par1\">The low activity and yield of antimicrobial peptides (AMPs) are pressing problems. The improvement of activity and yield through modification and heterologous expression, a potential way to solve the problem, is a research hot-pot. In this work, a new plectasin-derived variant L-type AP138 (AP138L-arg26) was constructed for the study of recombination expression and druggablity. As a result, the total protein concentration of AP138L-arg26 was 3.1 mg/mL in <italic>Pichia pastoris</italic> X-33 supernatant after 5 days of induction expression in a 5-L fermenter. The recombinant peptide AP138L-arg26 has potential antibacterial activity against selected standard and clinical Gram-positive bacteria (G⁺, minimum inhibitory concentration (MIC) 2–16 µg/mL) and high stability under different conditions (temperature, pH, ion concentration) and 2 × MIC of AP138L-arg26 could rapidly kill <italic>Staphylococcus aureus</italic> (<italic>S. aureus</italic>) (&gt; 99.99%) within 1.5 h. It showed a high safety in vivo and in vivo and a long post-antibiotic effect (PAE, 1.91 h) compared with vancomycin (1.2 h). Furthermore, the bactericidal mechanism was revealed from two dimensions related to its disruption of the cell membrane resulting in intracellular potassium leakage (2.5-fold higher than control), and an increase in intracellular adenosine triphosphate (ATP), and reactive oxygen species (ROS), the decrease of lactate dehydrogenase (LDH) and further intervening metabolism in <italic>S. aureus</italic>. These results indicate that AP138L-arg26 as a new peptide candidate could be used for more in-depth development in the future.</p>", "<title>Key points</title>", "<p id=\"Par2\">\n<italic>• The AP138L-arg26 was expressed in the P. pastoris expression system with high yield</italic>\n</p>", "<p id=\"Par3\">\n<italic>• The AP138 L-arg26 showed high stability and safety in vitro and in vivo</italic>\n</p>", "<p id=\"Par4\">\n<italic>• The AP138L-arg26 killed S. aureus by affecting cell membranes and metabolism</italic>\n</p>", "<title>Supplementary Information</title>", "<p>The online version contains supplementary material available at 10.1007/s00253-023-12947-w.</p>", "<title>Keywords</title>" ]

[ "<title>Biological information analysis of AP138L-arg26</title>", "<p id=\"Par11\">AP138L-arg26 and plectasin derived AMP sequences were obtained from DBAASP database (<ext-link ext-link-type=\"uri\" xlink:href=\"https://www.dbaasp.org/home\">https://www.dbaasp.org/home</ext-link>) (Pirtskhalava et al. ##REF##33151284##2021##), their physical and chemical properties, 3D structure, and antimicrobial peptide possibility were predicted by APD database (<ext-link ext-link-type=\"uri\" xlink:href=\"https://aps.unmc.edu/AP/\">https://aps.unmc.edu/AP/</ext-link>) (Wang et al. ##UREF##12##2016##), I-TASSER (<ext-link ext-link-type=\"uri\" xlink:href=\"https://zhanggroup.org/I-TASSER/\">https://zhanggroup.org/I-TASSER/</ext-link>) (Yang and Zhang ##REF##25883148##2015##), and CAMP database (<ext-link ext-link-type=\"uri\" xlink:href=\"http://www.camp.bicnirrh.res.in/\">http://www.camp.bicnirrh.res.in/</ext-link>), respectively (Waghu et al. ##UREF##11##2014##).</p>", "<title>Expression, purification, and identification of AP138L-arg26</title>", "<p id=\"Par12\">The nucleic acid sequence of AP138L-arg26 (GenBank ID: 2736833) was subjected to codon preference using the Reverse Translate Tool (<ext-link ext-link-type=\"uri\" xlink:href=\"http://www.bioinformati\">http://www.bioinformati</ext-link> cs.org/sms2/rev_trans.html), then a Kex2 signaling peptide cleavage site was added at its N-terminus (Figure ##SUPPL##0##S1##) and inserted into the eukaryotic expression vector pPICZαA between the double enzyme (<italic>Xho</italic>I and <italic>Xba</italic>I) cleavage site, to construct the plasmid pPICZαA-AP138L-arg26<italic>.</italic> Additionally, the pPICZαA-AP138L-arg26 was digested using <italic>Pme</italic>I and transformed into competent <italic>P. pastoris</italic> X-33 cells via electroporation. Peptide purification was carried out with the AKTAxpress system. Expression of AMP AP138L-arg26 was identified using Tricine-SDS-PAGE firstly, and the purified AP138L-arg26 was identified using MALDI-TOF/TOF MS (Ultraflextreme, Bruker, Germany), finally, the concentration was detected using Bradford assay kits.</p>", "<title>Antimicrobial activity and pharmacodynamic analysis in vitro</title>", "<title>Minimum inhibitory concentration (MIC) and minimum bactericidal concentration (MBC)</title>", "<p id=\"Par13\">To determine the MIC values of AP138L-arg26, we used the broth microdilution technique as previously described (Andrews. ##REF##11420333##2001##; Wiegand et al. ##REF##18274517##2008##). In brief, AP138L-arg26 peptide were diluted from 1280 to 2.5 µg/mL and added to a 96-cells plate with 10 µL per well. The bacteria in the logarithmic stage were diluted to 10⁵ CFU/mL with 90 µL per well. All plates were cultured in a 37 °C constant-temperature incubator; no visible growth of bacteria was the MIC after 18 h incubation. The MBC method was also based on the CLSI 2021 guidelines. In brief, after the bacteria were incubated at 1 × , 2 × , 4 × , and 8 × MIC of AP138L-arg26, the cultures were coated with Mueller-Hinton Agar (MHA) plates, the concentrations of AP138L-arg26 with 99.99% the bacteria killed was defined as MBC.</p>", "<title>Time-killing curve assay</title>", "<p id=\"Par14\">The time-killing curve of AP138L-arg26 against MRSA ATCC 43300 was used to analyze the pharmacodynamics. The method was based on previous laboratory experiments (Flamm et al. ##UREF##2##2019##; Yang et al. ##REF##31025073##2019##). Briefly, exponential-phase bacteria were diluted to 1 × 10⁵ CFU/mL, and peptides underwent a twofold gradient dilution setting of 4 × , 2 × , and 1 × MIC. There was incubation in a 37 °C thermostatic shaker at 200 rpm for 0–24 h to collect samples, which were plated on MHA plates. Colony numbers were recorded at all collected time points.</p>", "<title>PAE assay</title>", "<p id=\"Par15\">The method of PAE of AP138L-arg26 against MRSA ATCC 43300 or <italic>S. aureus</italic> CVCC 546 was described previously (Wang et al. ##REF##29523806##2018##). The formula is PAE = <italic>T</italic> − <italic>C</italic> (<italic>T</italic> is the time required for the number of colonies in the sample treatment group to increase by tenfold, and <italic>C</italic> is the time required for the untreated group).</p>", "<title>Intracellular antibacterial activity</title>", "<p id=\"Par16\">The method of intracellular bactericidal effect was described previously (Wang et al. ##REF##29523806##2018##). Only the number of cells and bacteria changed from 2.5 × 10⁵ to 2 × 10⁵ cells/mL.</p>", "<title>Synergism with antibiotics</title>", "<title>Fractional inhibitory concentration index (FICI)</title>", "<p id=\"Par17\">The synergistic effects of AP138L-arg26 with different antibiotics were evaluated using a checkerboard assay. The synergistic effect was calculated using the FICI as follows: FICI = FIC of AP138L-arg26 + FIC of antibiotic; FIC = MIC_c/MIC_a, where MIC_c is the MIC of the peptide and antibiotic in combination, and MIC_a is the MIC of the peptide/antibiotic alone (Blier et al. ##REF##20008946##2010##). Three parallel experiments were performed for each group. The efficacy of combination therapy was defined as FICI ≤ 0.5, 0.5 &lt; FICI ≤ 1, 1 &lt; FICI ≤ 4, and FICI &gt; 4 indicating synergy, addition, no difference, and antagonism, respectively.</p>", "<title>Growth curve of <italic>S. aureus</italic></title>", "<p id=\"Par18\">Samples were taken during the logarithmic growth of <italic>S. aureus</italic> and diluted to 1 × 10⁵ CFU/mL (Delpech et al. ##UREF##0##2019##). Different concentrations of AP138L-arg26, ceftiofur sodium, or a combination of them with an equal volume of bacteria were added to a 96-well microplate. The growth curves were recorded using a fully automatic growth curve recording instrument.</p>", "<title>Stability analysis</title>", "<title>Artificial gastric and intestinal juice stability</title>", "<p id=\"Par19\">AP138L-arg26 was incubated in artificial gastric and intestinal juice (Beijing Coolaber Technology Co., Ltd., Beijing, China) for 0.083, 0.167, 0.25, 0.75, and 1 h. The antimicrobial activity of AP138L-arg26 against MRSA ATCC 43300 was tested through the MIC assay.</p>", "<title>Thermal, pH, and salt stability</title>", "<p id=\"Par20\">The thermal, pH, and salt stability of AMPs were studied previously (Zhang et al. ##REF##21558006##2011##). Briefly, the AP138L-arg26 was incubated at different temperatures (4/20/40/60/80/100 °C), pH values (2/4/6/8/10), and kinds of salt ions (50/100/150/200/300 mM NaCl, 1.25/2.5/5 mM KCl, MgCl₂, CaCl₂) for 1 h at 37 °C, respectively. The antimicrobial activity of AP138L-arg26 against MRSA ATCC 43300 was tested using MIC assay.</p>", "<title>Circular dichroism (CD) spectrum assay</title>", "<p id=\"Par21\">The secondary structures of AP138L-arg26 were detected using CD (Bio-Logic MOS450 spectropolarimeter, France) in a simulated eukaryotic cell environment (50% trifluoroethanol, TFE) and bacterial membrane environment (20 mM sodium dodecyl sulfate, SDS) (Yao et al. ##REF##28870607##2018##).</p>", "<title>Safety evaluation in vitro and in vivo</title>", "<title>Hemolysis activity</title>", "<p id=\"Par22\">The hemolysis of AP138L-arg26 to fresh ICR mouse erythrocytes was described previously (Zheng et al. ##REF##34491399##2021##). Briefly, 8% of mouse erythrocytes were incubated with different concentrations of antimicrobial peptide AP138L-arg26 (256, 128, 64, 32, 16, 8, 4, 2, 1, 0.5 µg/mL) in equal volumes. The blank and positive control was phosphate buffered saline (PBS) and 0.1% Triton X-100, respectively.</p>", "<title>Cytotoxicity</title>", "<p id=\"Par23\">The cytotoxicity of AP138L-arg26 to mouse macrophages RAW264.7 was detected by the thiazolyl blue tetrazolium bromide (MTT) method as described previously (Shen et al. ##REF##33815324##2021##).</p>", "<title>Acute toxicity in mice</title>", "<p id=\"Par24\">AP138L-arg26 was administrated by intraperitoneal injection (<italic>n</italic> = 4 per group, 10 mg/kg, body weight 25 g) every day for 1 week (Shi et al. ##REF##36071151##2022##). After 7 days, the mice were euthanized to collect anticoagulant whole-blood and tissues (liver, spleen, kidney, and lung), which were used for whole blood and biochemical detection, or tissue sections (hematoxylin-eosin (HE) staining), respectively.</p>", "<title>Antimicrobial mechanism of peptides</title>", "<title>Scanning electron microscopy (SEM) assay</title>", "<p id=\"Par25\">The effect of peptides on changes in bacterial morphology was observed by scanning electron microscopy. Briefly the exponential phase <italic>S. aureus</italic> CVCC 546 cells (1 × 10⁹ CFU/mL) were incubated with 2 × MIC peptides at 37 °C for 0.5 h, 1 h, and 2 h. The untreated bacteria were the negative control. The processing methods and steps of the samples were described in detail in our previous study (Li et al. ##REF##32582062##2020##). Briefly, there were two important steps: (1) Sample preparation: incubation, cleaning, fixation, drying, gold spraying. (2) Observation using microscope. Only one concentration was used to deal with bacteria at different times (0.5, 1, and 2 h).</p>", "<title>SYTO9/propidium iodide (PI) assay</title>", "<p id=\"Par26\">To explore the disruption of bacterial cell membranes by the AP138L-arg26, a SYTO9/PI kit was used to detect the integrity of cell membranes (L7007 LIVE/DEAD^R BacLight™ Bacerial Viability Kits). The unique feature of this kit is that SYTO9 alone can pass through integral cell membranes, while PI can only penetrate damaged cell membranes. Briefly, the <italic>S. aureus</italic> CVCC 546 cells (1 × 10⁹ CFU/mL) were incubated with AP138L-arg26 (2 × MIC, 37 °C for 1 h), and a volume of 3 µL SYTO9/PI (1.5 µL: 1.5 µL) was added to 1 mL of bacterial suspension. Finally, the results were observed using fluorescence microscopy.</p>", "<title>Membrane fluidity assay</title>", "<p id=\"Par27\">The final concentration of 10 µM Laurdan was used to detect the effect of AP138L-arg26 on bacterial cell membrane fluidity (Shi et al. ##REF##36071151##2022##). Briefly, (1) co-incubation of bacteria with Laurdan; (2) the stained bacteria are co-incubated with the peptide; (3) spectrophotometer (Tecan, Männedorf, Switzerland) detection. The calculation formula: generalized polarization (GP) = (I435 − I490)/(I435 + I490).</p>", "<title>Membrane depolarization assay</title>", "<p id=\"Par28\">To further explore the effect of AP138L-arg26 on the bacterial membrane, the membrane probe DiSC₃(5) was selected to detect the change in membrane potential (Wang et al. ##REF##31974490##2020##). Briefly, <italic>S. aureus</italic> was washed and resuspended in PBS to 1 × 10⁸ CFU/mL and incubated with 0.5 mM membrane dyes DiSC₃(5) at 37 °C for 1 h in the dark. Then, 90 µL of stained bacterial suspension and 10 µL of AP138 L-arg26 (1 × , 2 × , 4 × MIC) were mixed and added to black and clean 96-cell plates and the fluorescence intensity was determined using a spectrophotometer (excitation wavelength 622/emission wavelength 670 nm, Infinite M200).</p>", "<title>Potassium ion (K⁺) leakage</title>", "<p id=\"Par29\">The integrity effect of antimicrobial peptide AP138L-arg26 on bacterial cell membranes was further verified, and the K⁺ leakage assay was previously described (Li et al. ##REF##32582062##2020##). Briefly, the main procedures were as follows: Firstly, prepare the <italic>S. aureus</italic> CVCC 546 bacteria suspension at a concentration of 1 × 10⁸ CFU/mL. Secondly, incubate the bacterial suspension with 2 × MIC AP138L-arg26 at different times (15/30/60/90/120 min) at 37 °C, with untreated cells and nisin used as negative and positive controls. Finally, the supernatants were detected using inductively coupled plasma mass spectrometry (ICP-MS) (SantaClara, CA, USA).</p>", "<title>Intracellular adenosine triphosphate (ATP) determination</title>", "<p id=\"Par30\">As ATP is the most direct source of energy for living organisms, the effect of antimicrobial peptides on ATP was explored with an ATP assay kit (Beyotime, Shanghai, China). Briefly, exponential-stage <italic>S. aureus</italic> CVCC 546 (1 × 10⁸ CFU/mL) were incubated with different concentrations of AP138L-arg26 (1 × , 2 × , 4 × MIC) for 1 h, and the pellets were collected, lysed, and centrifuged to harvest the intracellular supernatant. The luminescence was detected using an Infinite M200 Microplate reader (Tecan, Luminescence signals).</p>", "<title>Lactic dehydrogenase (LDH) activity</title>", "<p id=\"Par31\">As disruption of the cell membrane structure leads to the release of LDH from the cytoplasm into the culture medium, the detection of LDH in bacteria can further reveal the antibacterial mechanism (Shi et al. ##REF##36071151##2022##). Briefly, <italic>S. aureus</italic> cells were co-incubated with AP138L-arg26 for 6 h. Later, the pellets were collected and sonicated (3 s/10 s, 30 times). The intracellular LDH activity was detected using an Infinite M200 Microplate reader (Tecan, Luminescence signals). The calculated results of LDH% = treated cells/the control.</p>", "<title>Reactive oxygen species (ROS) measurements</title>", "<p id=\"Par32\">The probe 2′,7′-Dichlorodihydrofluorescein diacetate (DCFH-DA) was used to measure the level of ROS in bacteria after incubated with AP138L-arg26 (Wang et al. ##REF##34090965##2021a##, ##REF##34009979##b##; Shi et al. ##REF##36071151##2022##). Briefly, exponential stage bacteria were incubated with DCFH-DA (10 µM, 37 °C, 0.5 h), and the stained bacteria were treated with AP138L-arg26 (1 × , 2 × , 4 × MIC) for 1 h. The ROS were detected at excitation wavelength (488 nm) and emission wavelength (525 nm) using a microplate reader (Tecan, Männedorf, Switzerland).</p>", "<title>Statistical analysis</title>", "<p id=\"Par33\">The software GraphPad Prism (version 8, USA) was used to analyze all data, and ANOVA was the method to determine the statistical significance. The results are presented as means ± standard deviation (SD). A <italic>P</italic> value of &lt; 0.05 was considered statistically significant.</p>", "<title>Supplementary Information</title>", "<p>Below is the link to the electronic supplementary material.</p>" ]

[ "<title>Acknowledgements</title>", "<p>We acknowledge Chunli Li from the Core Facility at the Institute of Microbiology at the Chinese Academy of Sciences (CAS) for the technical support with SEM and Tong Zhao for her technical support with FACS analysis.</p>", "<title>Author contribution</title>", "<p>KZ and JW conceived and designed the research. DT, RM, NY, and YH conducted experiments. KZ, NY, and JW evaluated data. KZ, NY, and JW wrote and revised the manuscript. All authors read and approved the manuscript.</p>", "<title>Funding</title>", "<p>This work was supported by the National Natural Science Foundation of China (Grant No. 31872393), National Key Research and Development Plan—High Expression of Thiopeptides and their Analogs (Grant No. 2022YFC2105000-03, 2022–2026), and the Innovation Program of Agricultural Science and Technology (ASTIP) in CAAS (Grant No. CAAS-ASTIP-2017-FRI-02) and its key projects (Grant No. CAAS-ZDRW202111 and Grant No. CAAS-ZDXT 201808).</p>", "<title>Data availability</title>", "<p>All data generated or analyzed during this study are included in this published article (and its supplementary information files).</p>", "<title>Declarations</title>", "<title>Ethical approval</title>", "<p id=\"Par51\">The mouse experiment was performed according to the Animal Care and Use Committee of the Feed Research Institute of the Chinese Academy of Agricultural Sciences (CAAS) and approved by the Laboratory Animal Ethical Committee and its Inspection of the Feed Research Institute of CAAS (AEC-CAAS-20090609).</p>", "<title>Competing interests</title>", "<p id=\"Par52\">The authors declare no competing interests.</p>" ]

[ "<fig id=\"Fig1\"><label>Fig. 1</label><caption><p>The 3D structure analysis of plectasin (left) and AP138L-arg26 (right). <bold>a</bold> Molecular modeling of plectasin and AP138L-arg26. <bold>b</bold> Electrostatic surface of plectasin (left) and AP138L-arg26 (right). Blue and red represent positive and negative charge, respectively</p></caption></fig>", "<fig id=\"Fig2\"><label>Fig. 2</label><caption><p>The construction of the pPICZαA-AP138L-arg26 plasmid and the expression of AP138L-arg26 in <italic>P. pastoris</italic> X-33 at the shaking flask and fermenter level. <bold>a</bold> Recombinant plasmids pPICZαA-AP138L-arg26 and linearized gel electrophoresis, lanes 1–2 were pPICZαA-AP138L-arg26 (left) and linearized plasmid. <bold>b</bold> Screening better AP138L-arg26 positive transformants AP138L-arg26-8, AP138L-arg26-28, AP138L-arg26-68, and AP138L-arg26-94 by inhibition zone against <italic>S. aureus</italic> ATCC 43300. <bold>c</bold> Tricine-SDS-PAGE analysis of AP138L-arg26 expression in shaking flask level, lanes 1–4 were fermentation supernatants of induction at 0 h, 24 h, 48 h, and 96 h, respectively. <bold>d</bold> Antimicrobial activity against <italic>S. aureus</italic> ATCC 43300 of AP138L-arg26 in fermenter level was analyzed by inhibition zone assay at 0 h, 24 h, 48 h, 96 h, and 120 h, respectively. <bold>e</bold> Tricine-SDS-PAGE analysis of AP138L-arg26 expression in fermenter level, lanes 1–6 were fermentation supernatants of induction at 0 h, 24 h, 48 h, 96 h, and 120 h, respectively. <bold>f</bold> Time curves of the total protein levels and cell wet weights at 0 h, 24 h, 48 h, 96 h, and 120 h, respectively. <bold>g</bold> Tricine-SDS-PAGE analysis of AP138L-arg26 purification. <bold>h</bold> MALDI-TOF MS analysis of the purified AP138L-arg26</p></caption></fig>", "<fig id=\"Fig3\"><label>Fig. 3</label><caption><p>The pharmacodynamic evaluation of antimicrobial peptide AP138L-arg26 in extracellular and intracellular activity. <bold>a</bold> Time killing curve of AP138L-arg26 against <italic>S. aureus</italic> ATCC 43300, vancomycin (Van) as positive control, untreated group (CK) as blank control. <bold>b</bold> Post-antibiotic effect of AP138L-arg26 against standard <italic>S. aureus</italic> ATCC 43300, Van as control. Intracellular activity of AP138L-arg26 against <italic>S. aureus</italic> CVCC 546 in <bold>c</bold> mouse macrophage RAW 264.7 and <bold>d</bold> Bovine uterine epithelial cells (BEND). Asterisk (*) indicates the significance between control and treatment groups. ***<italic>P</italic> &lt; 0.001. Results are expressed as the means from three biological replicates ± SD (<italic>n</italic> = 3). <bold>e</bold> Growth curve of <italic>S. aureus</italic> CVCC 546 after treatment with different combination of AP138L-arg26 and Ceftiofur Sodium (Cef)</p></caption></fig>", "<fig id=\"Fig4\"><label>Fig. 4</label><caption><p>The safety of AP138L-arg26 in vitro and in vivo. <bold>a</bold> Hemolytic activity of AP138L-arg26 against fresh mouse red blood cells. <bold>b</bold> Cytotoxicity of the AP138L-arg26 against RAW264.7 cells. ICR female mice (<italic>n</italic> = 4) were intraperitoneally administered with AP138L-arg26 (10 mg/kg) daily for a week. <bold>c</bold>–<bold>g</bold> Histology images (H&amp;E) stained (<bold>d</bold>), whole-blood cell profiles (<bold>e</bold>, <bold>f</bold>), serum biochemical 7 index (g) of mice after treatment with AP138L-arg26 for a week. CK, untreated group; scale bar, 100 µm</p></caption></fig>", "<fig id=\"Fig5\"><label>Fig. 5</label><caption><p>Study on the effect of AP138L-arg26 on cell membranes. <bold>a</bold> The morphological changes of <italic>S. aureus</italic> CVCC 546 were observed by SEM after AP138L-arg26 treatment. <bold>b</bold> The destruction of cell membranes by antimicrobial peptide AP138L-arg26 was observed by SYTO9/PI. <bold>c</bold> Detection of cell membrane potential after incubation of AP138L-arg26 with <italic>S. aureus</italic> CVCC 546. <bold>d</bold> The K⁺ leakage after incubation of AP138L-arg26 with <italic>S. aureus</italic> CVCC 546. <bold>e</bold> Membrane fluidity. Asterisk (*) indicates the significance between control and treatment groups, <italic>P</italic> &lt; 0.05; (ns) indicates the no significance between control and treatment groups. CK, untreated group</p></caption></fig>", "<fig id=\"Fig6\"><label>Fig. 6</label><caption><p>Effects of AP138L-arg26 on cell metabolism. <bold>a</bold> the level of Intracellular ATP, <bold>b</bold> the level of ROS, <bold>c</bold> the level of LDH. CK, untreated group. Asterisk (*) indicates the significance between control and treatment groups. ***<italic>P</italic> &lt; 0.001. Results are expressed as the means from three biological replicates ± SD (<italic>n</italic> = 3)</p></caption></fig>" ]

[ "<table-wrap id=\"Tab1\"><label>Table 1</label><caption><p>Physicochemical properties and likelihood prediction of antimicrobial peptide</p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th align=\"left\">Peptide</th><th align=\"left\">Sequence</th><th align=\"left\">AMP probability</th><th align=\"left\">Hydrophobic ratio</th><th align=\"left\">Net charge + </th><th align=\"left\">GRAVY − </th><th align=\"left\">Hydrophobicity</th><th align=\"left\">Amphiphilicity index</th><th align=\"left\">Protein-binding potential</th><th align=\"left\">Molecular weight</th></tr></thead><tbody><tr><td align=\"left\">Plectasin</td><td align=\"left\">GFGCNGPWDEDDMQCHNHCKSIKGYKGGYCAKGGFVCKCY</td><td char=\".\" align=\"char\">0.9495</td><td align=\"left\">32%</td><td align=\"left\">1</td><td char=\".\" align=\"char\">0.695</td><td char=\".\" align=\"char\">—</td><td char=\".\" align=\"char\">1.15</td><td char=\".\" align=\"char\">1.4</td><td char=\".\" align=\"char\">4407.99</td></tr><tr><td align=\"left\">AP138L-arg26</td><td align=\"left\">GFGCNGPWSEDDLRCHRHCKSIKGYRGGYCAKGGFVCKCY</td><td char=\".\" align=\"char\">0.992</td><td align=\"left\">33%</td><td align=\"left\">4.5</td><td char=\".\" align=\"char\">0.674</td><td char=\".\" align=\"char\">3.42</td><td char=\".\" align=\"char\">1.25</td><td char=\".\" align=\"char\">1.88</td><td char=\".\" align=\"char\">4460.125</td></tr><tr><td align=\"left\">NZ2114</td><td align=\"left\">GFGCNGPWNEDDLRCHNHCKSIKGYKGGYCAKGGFVCKCY</td><td char=\".\" align=\"char\">0.9895</td><td align=\"left\">33%</td><td align=\"left\">3.5</td><td char=\".\" align=\"char\">0.672</td><td char=\".\" align=\"char\">3.5</td><td char=\".\" align=\"char\">1.18</td><td char=\".\" align=\"char\">1.52</td><td char=\".\" align=\"char\">4417.049</td></tr><tr><td align=\"left\">NZX</td><td align=\"left\">GFGCNGPWSEDDIQCHNHCKSIKGYKGGYCARGGFVCKCY</td><td char=\".\" align=\"char\">0.9355</td><td align=\"left\">33%</td><td align=\"left\">2.5</td><td char=\".\" align=\"char\">0.5775</td><td char=\".\" align=\"char\">3.05</td><td char=\".\" align=\"char\">1.12</td><td char=\".\" align=\"char\">1.43</td><td char=\".\" align=\"char\">4389.981</td></tr><tr><td align=\"left\">MP1102</td><td align=\"left\">GFGCNGPWQEDDVKCHNHCKSIKGYKGGYCAKGGFVCKCY</td><td char=\".\" align=\"char\">0.88</td><td align=\"left\">33%</td><td align=\"left\">3.5</td><td char=\".\" align=\"char\">0.6475</td><td char=\".\" align=\"char\">4.47</td><td char=\".\" align=\"char\">1.24</td><td char=\".\" align=\"char\">1.28</td><td char=\".\" align=\"char\">4460.125</td></tr><tr><td align=\"left\">DLP4</td><td align=\"left\">ATCDLLSPFKVGHAACAAHCIARGKRGGWCDKRAVCNCRK</td><td char=\".\" align=\"char\">0.9455</td><td align=\"left\">50%</td><td align=\"left\">6.0</td><td char=\".\" align=\"char\">0.13</td><td char=\".\" align=\"char\">—</td><td char=\".\" align=\"char\">0.86</td><td char=\".\" align=\"char\">1.72</td><td char=\".\" align=\"char\">4275.26</td></tr><tr><td align=\"left\">ID13</td><td align=\"left\">ATCDLLSPFKVGHAACAAHCIARGKRGGWCDGRAVCNCRK</td><td char=\".\" align=\"char\">0.96</td><td align=\"left\">50%</td><td align=\"left\">5.5</td><td char=\".\" align=\"char\">0.0425</td><td char=\".\" align=\"char\">5.68</td><td char=\".\" align=\"char\">0.77</td><td char=\".\" align=\"char\">1.56</td><td char=\".\" align=\"char\">4203.96</td></tr></tbody></table></table-wrap>", "<table-wrap id=\"Tab2\"><label>Table 2</label><caption><p>The MIC and MBC of AP138L-arg26 and vancomycin</p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th align=\"left\" rowspan=\"2\">Strain</th><th align=\"left\" colspan=\"2\">MIC (µg/mL)</th><th align=\"left\" colspan=\"2\">MBC (µg/mL)</th></tr><tr><th align=\"left\">AP138L-arg26</th><th align=\"left\">Van</th><th align=\"left\">AP138L-arg26</th><th align=\"left\">Van</th></tr></thead><tbody><tr><td align=\"left\"><italic>Staphylococcus aureus</italic> ATCC 25923</td><td align=\"left\">16</td><td align=\"left\">1</td><td align=\"left\">32</td><td align=\"left\">2</td></tr><tr><td align=\"left\"><italic>Staphylococcus aureus</italic> ATCC 43300</td><td align=\"left\">8</td><td align=\"left\">1</td><td align=\"left\">8</td><td align=\"left\">1</td></tr><tr><td align=\"left\"><italic>Staphylococcus aureus</italic> E48</td><td align=\"left\">2</td><td align=\"left\">1</td><td align=\"left\">8</td><td align=\"left\">8</td></tr><tr><td align=\"left\"><italic>Staphylococcus aureus</italic> CVCC 546</td><td align=\"left\">4</td><td align=\"left\">1</td><td align=\"left\">32</td><td align=\"left\">1</td></tr><tr><td align=\"left\"><italic>Staphylococcus aureus</italic> CAAS-FRI-2023-01</td><td align=\"left\">4</td><td align=\"left\">1</td><td align=\"left\">16</td><td align=\"left\">2</td></tr><tr><td align=\"left\"><italic>Staphylococcus aureus</italic> CAAS-FRI-2023-02</td><td align=\"left\">8</td><td align=\"left\">2</td><td align=\"left\">32</td><td align=\"left\">8</td></tr><tr><td align=\"left\"><italic>Staphylococcus epidermidis</italic> ATCC 12228</td><td align=\"left\">4</td><td align=\"left\">2</td><td align=\"left\">16</td><td align=\"left\">4</td></tr><tr><td align=\"left\"><italic>Staphylococcus epidermidis</italic> ATCC 35984</td><td align=\"left\">8</td><td align=\"left\">2</td><td align=\"left\">16</td><td align=\"left\">4</td></tr><tr><td align=\"left\"><italic>Streptococcus dysgalactiae</italic> CVCC 3938</td><td align=\"left\">4</td><td align=\"left\">2</td><td align=\"left\">32</td><td align=\"left\">2</td></tr><tr><td align=\"left\"><italic>Streptococcus agalactiae</italic> ATCC 13813</td><td align=\"left\">4</td><td align=\"left\">2</td><td align=\"left\">8</td><td align=\"left\">4</td></tr><tr><td align=\"left\"><italic>Streptococcus agalactiae</italic> CAU-FRI-2022-01</td><td align=\"left\">4</td><td align=\"left\">2</td><td align=\"left\">32</td><td align=\"left\">8</td></tr><tr><td align=\"left\"><italic>Streptococcus agalactiae</italic> CAU-FRI-2022-02</td><td align=\"left\">4</td><td align=\"left\">2</td><td align=\"left\">32</td><td align=\"left\">4</td></tr></tbody></table></table-wrap>", "<table-wrap id=\"Tab3\"><label>Table 3</label><caption><p>Synergism of peptide AP138L-arg26 with antibiotics</p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th align=\"left\" rowspan=\"2\">Combination</th><th align=\"left\" rowspan=\"2\">Variety</th><th align=\"left\" colspan=\"4\"><italic>S. aureus</italic> CVCC 546</th></tr><tr><th align=\"left\">MIC_a (µg/mL)</th><th align=\"left\">MIC_c (µg/mL)</th><th align=\"left\">FIC</th><th align=\"left\">FICI index</th></tr></thead><tbody><tr><td align=\"left\" rowspan=\"2\">AP138L-arg26-vancomycin</td><td align=\"left\">AP138L-arg26</td><td align=\"left\">4</td><td align=\"left\">1</td><td align=\"left\">0.25</td><td align=\"left\">0.5</td></tr><tr><td align=\"left\">Vancomycin</td><td align=\"left\">1</td><td align=\"left\">0.25</td><td align=\"left\">0.25</td><td align=\"left\"/></tr><tr><td align=\"left\" rowspan=\"2\">AP138L-arg26-Streptomycin sulfate</td><td align=\"left\">AP138L-arg26</td><td align=\"left\">4</td><td align=\"left\">2</td><td align=\"left\">0.5</td><td align=\"left\">1</td></tr><tr><td align=\"left\">Streptomycin sulfate</td><td align=\"left\">1</td><td align=\"left\">0.5</td><td align=\"left\">0.5</td><td align=\"left\"/></tr><tr><td align=\"left\" rowspan=\"2\">AP138L-arg26-ceftiofur sodium</td><td align=\"left\">AP138L-arg26</td><td align=\"left\">4</td><td align=\"left\">0.25</td><td align=\"left\">0.0625</td><td align=\"left\">0.3125</td></tr><tr><td align=\"left\">Ceftiofur sodium</td><td align=\"left\">1</td><td align=\"left\">0.25</td><td align=\"left\">0.25</td><td align=\"left\"/></tr><tr><td align=\"left\" rowspan=\"2\">AP138L-arg26-ampicillin</td><td align=\"left\">AP138L-arg26</td><td align=\"left\">4</td><td align=\"left\">4</td><td align=\"left\">1</td><td align=\"left\">1.0625</td></tr><tr><td align=\"left\">Ampicillin</td><td align=\"left\">0.125</td><td align=\"left\">0.0078</td><td align=\"left\">0.0625</td><td align=\"left\"/></tr></tbody></table></table-wrap>", "<table-wrap id=\"Tab4\"><label>Table 4</label><caption><p>The stability of AMPs in different environments</p></caption><table frame=\"hsides\" rules=\"groups\"><thead><tr><th align=\"left\" colspan=\"6\">AP138L-arg26 against <italic>S. aureus</italic> CVCC 546</th></tr><tr><th align=\"left\">Condition</th><th align=\"left\">MIC</th><th align=\"left\">Condition</th><th align=\"left\">MIC</th><th align=\"left\">Condition</th><th align=\"left\">MIC</th></tr></thead><tbody><tr><td align=\"left\">Heat (20 °C)</td><td align=\"left\">8</td><td align=\"left\">KCl (1.25 mM)</td><td align=\"left\">8</td><td align=\"left\">50% serum (0.5 h)</td><td align=\"left\">8</td></tr><tr><td align=\"left\">Heat (40 °C)</td><td align=\"left\">8</td><td align=\"left\">KCl (2.5 mM)</td><td align=\"left\">8</td><td align=\"left\">50% serum (1 h)</td><td align=\"left\">8</td></tr><tr><td align=\"left\">Heat (60 °C)</td><td align=\"left\">8</td><td align=\"left\">KCl (5 mM)</td><td align=\"left\">8</td><td align=\"left\">50% serum (2 h)</td><td align=\"left\">8</td></tr><tr><td align=\"left\">Heat (80 °C)</td><td align=\"left\">8</td><td align=\"left\">CaCl₂ (1.25 mM)</td><td align=\"left\">8</td><td align=\"left\">50% serum (4 h)</td><td align=\"left\">8</td></tr><tr><td align=\"left\">Heat (100 °C)</td><td align=\"left\"> &gt; 64</td><td align=\"left\">CaCl₂ (2.5 mM)</td><td align=\"left\">8</td><td align=\"left\">50% serum (6 h)</td><td align=\"left\">8</td></tr><tr><td align=\"left\">pH 2</td><td align=\"left\">8</td><td align=\"left\">CaCl₂ (5 mM)</td><td align=\"left\">16</td><td align=\"left\">50% serum (8 h)</td><td align=\"left\">8</td></tr><tr><td align=\"left\">pH 4</td><td align=\"left\">16</td><td align=\"left\">MgCl₂ (1.25 mM)</td><td align=\"left\">8</td><td align=\"left\">Gastric juice (0.25 h)</td><td align=\"left\">4</td></tr><tr><td align=\"left\">pH 6</td><td align=\"left\">16</td><td align=\"left\">MgCl₂ (2.5 mM)</td><td align=\"left\">16</td><td align=\"left\">Gastric juice (0.5 h)</td><td align=\"left\">4</td></tr><tr><td align=\"left\">pH 8</td><td align=\"left\">8</td><td align=\"left\">MgCl₂ (5 mM)</td><td align=\"left\">16</td><td align=\"left\">Gastric juice (1 h)</td><td align=\"left\">4</td></tr><tr><td align=\"left\">pH 10</td><td align=\"left\">16</td><td align=\"left\">25% serum (0.5 h)</td><td align=\"left\">8</td><td align=\"left\">Gastric juice (2 h)</td><td align=\"left\">8</td></tr><tr><td align=\"left\">NaCl (50 mM)</td><td align=\"left\">8</td><td align=\"left\">25% serum (1 h)</td><td align=\"left\">16</td><td align=\"left\">Intestinal juice (0.25 h)</td><td align=\"left\" rowspan=\"4\"> &gt; 64</td></tr><tr><td align=\"left\">NaCl (100 mM)</td><td align=\"left\">8</td><td align=\"left\">25% serum (2 h)</td><td align=\"left\">16</td><td align=\"left\">Intestinal juice (0.5 h)</td></tr><tr><td align=\"left\">NaCl (150 mM)</td><td align=\"left\">8</td><td align=\"left\">25% serum (4 h)</td><td align=\"left\">8</td><td align=\"left\">Intestinal juice (1 h)</td></tr><tr><td align=\"left\">NaCl (200 mM)</td><td align=\"left\">16</td><td align=\"left\">25% serum (6 h)</td><td align=\"left\">16</td><td align=\"left\">Intestinal juice (2 h)</td></tr><tr><td align=\"left\">NaCl (300 mM)</td><td align=\"left\">16</td><td align=\"left\">25% serum (8 h)</td><td align=\"left\">8</td><td align=\"left\">Control (without treatment)</td><td align=\"left\">8</td></tr></tbody></table></table-wrap>" ]

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