Annarita Rita Giardina

Angelo Ferrante

Francesco Ciccia

Rosalia Impastato

Maria Concetta Miceli

Alfonso Principato

Professor Giovanni Triolo
email: g.triolo@unipa.it

A 2-year comparative open label randomized study of efficacy and safety of etanercept and infliximab in patients with ankylosing spondylitis

Keywords: ankylosing spondylitis, Key Word 2: etanercept, Key Word 3: infliximab

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Ankylosing Spondylitis (AS) is a debilitating disease predominantly affecting the spine characterised by axial skeletal ankylosis [START_REF] Khan | Update on spondyloarthropathies[END_REF]. Therapeutic options for AS, such as non-steroidal antiinflammatory drugs (NSAIDs), offer temporary pain relief with little if any clinical benefit on spinal mobility, and disease modifying antirheumatic drugs (DMARDs) do not appear to affect the spinal involvement of AS [START_REF] Zochling | Current evidence for the management of ankylosing spondylitis: systematic literature review for the ASAS/EULAR management recommendations in ankylosing spondylitis[END_REF] Tumour necrosis factor is a proinflammatory cytokine that appears to have a key role in the pathogenesis of AS inflammation [START_REF] Rudwaleit | Cytokine gene polymorphisms relevant for the spondyloarthropathies[END_REF]. Etanercept is a fully human recombinant protein, comprising two molecules of soluble TNF receptor p75 and the crystallisable fragment component of immunoglobulin G1, which specifically binds to and neutralises TNF-alpha. Several clinical study have shown that etanercept reduces disease activity in patients with spondyloarthropathies, including reactive arthritis and AS (4-8). Similar results have been reported with infliximab, a chimeric monoclonal antibody against TNF [START_REF] Braun | Treatment of active ankylosing spondylitis with infliximab: a randomised controlled multicentre trial[END_REF][START_REF] Braun | Two year maintenance of efficacy and safety of infliximab in the treatment of ankylosing spondylitis[END_REF][START_REF] Van Der Heijde | Efficacy and Safety of Infliximab in Patients With Ankylosing Spondylitis[END_REF][START_REF] Marzo-Ortega | Infliximab in combination with methotrexate in active ankylosing spondylitis: a clinical and imaging study[END_REF][START_REF] Braun | Long-Term efficacy and safety of infliximab in the treatment of ankylosing spondylitis[END_REF][START_REF] Braun | Major reduction in spinal inflammation in patients with ankylosing spondylitis after treatment with infliximab. Results of a multicenter, randomized, double-blind, placebo-controlled magnetic resonance imaging study[END_REF][START_REF] Hochberg | The benefit/risk profile of TNF-blocking agents: findings of a consensus panel[END_REF]. The aim of the present work was to compare the efficacy and safety of both etanercept and infliximab in patients with ankylosing spondylitis in a two-year open label randomised study.

Patients and Methods

Fifty consecutive patients that fulfilled the modified New York criteria for the diagnosis of AS [START_REF] Van Der Linden | Evaluation of diagnostic criteria for ankylosing spondylitis: a proposal for modification of the New York criteria[END_REF] were enrolled in the study. Patients had to be non responder to oral non steroidal anti inflammatory drugs and naïve for DMARDs or other TNF blocking agents. Patients with complete ankylosis (fusion) of the spine were excluded. Approval from an independent ethics committee was obtained, and all patients provided written informed consent to participate. Patients were randomised to receive alternatively etanercept or infliximab with a ratio of 1:1. Efficacy and safety evaluations were performed at weeks 2, 4, 8, and 12 and then every 3 months until 2 years. Criteria for inclusion were: active disease for at least three months, a BASDAI >4 and a VAS for spinal pain score >4. Patients received Etanercept at a 50 mg dose delivered subcutaneously weekly or Infliximab at 5 mg/Kg at week 0, 2, 6 and every 6 weeks for a period of 102 weeks. The clinical response to etanercept or infliximab was evaluated on the basis of response criteria recommended by the ASAS Working Group [START_REF] Anderson | Ankylosing Spondylitis Assessment Group preliminary definition of short-term improvement in ankylosing spondylitis[END_REF]. An ASAS 20 and 40 responder was defined as a patient who showed at least 20% or 40% improvement from baseline and had an absolute improvement from baseline of at least 1 unit (on a scale of 0-10) in at least 3 of the following 4 assessment domains: patient's global assessment, spinal pain, function according to the Bath Ankylosing Spondylitis Functional Index (BASFI), and morning stiffness (the average of the last 2 questions of the BASDAI) Disease activity was assessed bay the Bath Ankylosing Spondylitis Activity Index (BASDAI). Functional impairment was assessed by the 10 item Bath Ankylosing Spondylitis Functional Index (BASFI). The BASMI is an aggregate score (ranging from 0 to 10) of patient mobility assessments, including tragus-to-wall lumbar flexion (Schober test), cervical rotation, lumbar side flexion, and intermalleolar distance [START_REF] Jenkinson | Defining spinal mobility in ankylosing spondylitis (AS): the Bath AS Metrology Index[END_REF][START_REF] Fernández-Sueiro | Evaluation of ankylosing spondylitis spinal mobility measurements in the assessment of spinal involvement in psoriatic arthritis[END_REF]. Chest expansion is the difference between the circumference of the chest in maximal inspiration and that in maximal expiration. Patients were monitored for adverse events and abnormal laboratory test results over the course of the study. Vital signs were monitored, and standard haematology, serum chemistry, and urine analysis tests were evaluated.

Statistics

To compare mean differences between time points (week 0 vs week 12, week 54 and week 104), a unpaired t-test was applied. In the case of skewed distributions [CRP and erythrocyte sedimentation rate (ESR)] the corresponding non-parametric test (Wilcoxon test) was used. McNemar test was applied in the case of proportions

Results

A total of 50 patients were enrolled in the study; 25 were assigned to receive etanercept and 25 were assigned to infliximab. The average age of patients was 32.2 ± 8 years. 39 patients were male (79%) and the duration of symptoms was 15.6 +/-8.7 years. The treatment groups had similar baseline disease activity scores and demographic characteristics (Table I). No patients discontinued therapies. Although the difference was not statistically significant, more Infliximab patients than Etanercept patients responded at the ASAS 20 level as early as week 2 but sustained differences were not evident up to week 12. After 12 weeks, 19 of 25 patients (75 %) in the infliximab group were ASAS 20 responders compared with 15 of 25 patients (60%) in the etanercept group (Fig. 1). At the 12th week 55 % of patient treated with infliximab and 43 % of patients treated with etanercept were at the ASAS 40 level (fig. 1). At week 12 more infliximab than etanercept treated patients achieved a significantly reduction of BASDAI (4.8 vs 5.9; p < 0.005 and 3.5 vs 5.6; p< 0.005) (fig 2), and of BASFI (3.5 vs 5; p < 0.005 ) (fig 3). Acute phase reactants, BASMI, SP and HAQ significantly decreased in both group of patients from baseline during the observation period without differences between the two groups (not shown). Treatments were generally well tolerated and adverse events were mostly mild to moderate (Table 2). There were no discontinuations for safety reasons. In particular there was no cases of opportunistic infections, tuberculosis, congestive heart disease, demyelinating disorders, lupus-like syndromes and malignancy.

Discussion

TNF blocking agents are internationally considered to represent a major progress in the treatment of AS. Comparative studies with etanercept and infliximab treatments were not yet published, however. A 50 mg dose of etanercept delivered subcutaneously weekly and 6 weeks infusion of infliximab (5 mg/kg) produced rapid, significant, and sustained improvement in multiple clinical and laboratory measures of AS. Although no significant differences were observed at the end of the study, our results are consistent with a significant more rapid clinical improvement in the infliximab treated group. At week 12 in fact more infliximab than etanercept treated patients achieved a significantly reduction of BASDAI and BASFI, and there were also more responders in the infliximab group at the ASAS 20 and ASAS 40 level.

In general, improvements with both therapies were observed at two weeks and were sustained up to the end of observation. The potential interest of our study was also that patients enrolled in our study were outpatients that have to be considered at community levels.

CRP levels and ESR values significantly decreased and spinal mobility, as measured by Schober's test, significantly improved in patients undergoing both treatments.

The results of this randomised study suggest that treatment with both etanercept and infliximab are effective, safe, and well tolerated in patients with AS. Patients with AS, in fact, were treated continuously and we did not observed any reduction in efficacy with both treatments.

Fig. 1

 1 Fig. 1

Fig. 2

 2 Fig. 2

Fig. 3 to figures Figure 1 ASAS

 31 Fig. 3

Figure 2 BASDAIFigure 3 BASFI

 23 Figure 2BASDAI lebvels in patients trated with infliximab or etanercept at 12 w a significant decrease of BASDAI was obtained in patients treated with infliximab (p<0.005)

Fig. 2

 2 Fig.2

Fig. 3

 3 Fig.3

Table 1

 1 Baseline characteristics of patients

	Baseline characteristics	Treatment groups	p
		Infliximab (n=25)	Etanercept (n=25)
	Age (mean ± SD)	31.9 ± 9.2	32.6 ± 6.8	NS
	Sex, M/F	19/6	20/5	NS
	Duration of AS, years (mean ± SD)	15.4 ± 10.6	15.7 ± 6.5	NS
	HLA-B27 positive (%)	23 (92)	24 (96)	NS
	BASDAI score, 0-10 (mean ± SD)	6.5 ± 1.2	6.6 ± 1.1	NS
	BASFI score, 0-10 (mean ± SD)	6.1 ± 0.9	6.5 ± 1.1	NS
	BASMI score, 0-10 (mean ± SD)	3.7 ± 1.6	3.9 ± 1.7	NS
	Chest expansion, cm (mean ± SD)	2.9 ± 0.8	3.1 ± 0.9	NS
	Spinal Pain, 0-10 (mean ± SD)	6.0 ± 1.5	6.3 ± 1.2	NS
	Patient's global assessment, 0-10 VAS	6.4 ± 1.4	6.7 ± 1.4	NS
	CRP level, mg/l (mean ± SD)	25 ± 12.1	22.9 ± 10.5	NS
	ESR mm/1h (mean ± SD)	32.1 ± 14.6	29.6 ± 13.7	NS
	HAQ (mean ± SD)	1.5 ± 0.5	1.4 ± 0.5	NS

Table 2

 2 No (%) of Adverse Events (AE) through week 104

	AE	Infliximab	Etanercept	p
		n (%)	n (%)
	Injection site reactions	1 (4)	5 (25)	< 0.005
	Infusion reactions
	Headache	8 (32)	7 (28)	NS
	Diarrhea	2 (8)	1 (4)	NS
	Tachycardia	12 (48)	8 (32)	NS
	Hypertension	4 (16)	2 (8)	NS
	Abdominal pain	1 (4)	1 (4)	NS
	Uveitis	1 (4)	2 (8)	NS
	Optic neuritis	1 (4)	1 (4)	NS
	Arthralgia	4 (16)	3 (12)	NS
	Vertigo	1 (4)	2 (8)	NS
	Severe infections	2 (8)	1 (4)	NS

Table 1

 1 Baseline characteristics of patients

	Baseline characteristics	Treatment groups	p
		Infliximab (n=25)	Etanercept (n=25)
	Age (mean ± SD)	31.9 ± 9.2	32.6 ± 6.8	NS
	Sex, M/F	19/6	20/5	NS
	Duration of AS, years (mean ± SD)	15.4 ± 10.6	15.7 ± 6.5	NS
	HLA-B27 positive (%)	23 (92)	24 (96)	NS
	BASDAI score, 0-10 (mean ± SD)	6.5 ± 1.2	6.6 ± 1.1	NS
	BASFI score, 0-10 (mean ± SD)	6.1 ± 0.9	6.5 ± 1.1	NS
	BASMI score, 0-10 (mean ± SD)	3.7 ± 1.6	3.9 ± 1.7	NS
	Chest expansion, cm (mean ± SD)	2.9 ± 0.8	3.1 ± 0.9	NS
	Spinal Pain, 0-10 (mean ± SD)	6.0 ± 1.5	6.3 ± 1.2	NS
	Patient's global assessment, 0-10 VAS	6.4 ± 1.4	6.7 ± 1.4	NS
	CRP level, mg/l (mean ± SD)	25 ± 12.1	22.9 ± 10.5	NS
	ESR mm/1h (mean ± SD)	32.1 ± 14.6	29.6 ± 13.7	NS
	HAQ (mean ± SD)	1.5 ± 0.5	1.4 ± 0.5	NS


PhD Tuominen Risto
email: risto.tuominen@utu.fi

MD Möttönen Timo

MD Suominen Carita

MSc Vahlberg Tero

MStud Tuominen Sini

Risto Tuominen

Realtive importance of functions

Keywords: Perception, function, disability, rheumatoid arthritis

Objectives: To evaluate the relative importance of the twenty functions assessed by patients to produce the Health Assessment Questionnaire's Disability Index (HAQ-DI).

Methods: A systematic sample of 242 rheumatoid arthritis patients was drawn. Altogether 179 subjects returned acceptably filled questionnaire including perceived rheumatoid arthritis status in the 20 functions comprising the HAQ-DI. A separate complementary questionnaire was sent to 80 randomly chosen subjects. They were asked to estimate how important each of the same twenty items were in their everyday life by using the VAS method. Completely answered complementary questionnaires were returned by 66 (88%) of the subsample.

Results: The relative importance of the functions assessed varied considerably. Men assessed the highest importance to the ability to walk outdoors on flat ground and women to dressing themselves, getting in and out of bed and washing and drying their body. For both genders the least important function was the ability to take a tub bath. The importance of all other functions except taking a tub bath were positively highly significantly correlated with the importance of each of the other functions. Factor analyses with one factor model showed high loadings on most functions, with the exception of 'Take a tub bath'. The two factor model formed one factor with loadings concentrating slightly more on 'lower extremity' and another on 'upper extremity'.

Conclusions: Assessment of RA patients' functional status would benefit from further development of measures by giving relative weights from patients' perspective to those functional abilities which are used for index computations.

Introduction

Severity of rheumatoid arthritis is commonly studied by using numbers of painful or swollen joints [START_REF] Pinals | Preliminary criteria for clinical remission in rheumatoid arthritis[END_REF], severity of pain [START_REF] Huskisson | Measurement of pain[END_REF], global health indices [START_REF] Smedstad | Correlates of patients' global assessment of arthritis impact. A 2-year study of 216 patients with RA[END_REF], and by assessing clinical disease activity [START_REF] Van Der Heijde | Judging disease activity in clinical practice in rheumatoid arthritis: first step in the development of a disease activity score[END_REF][START_REF] Van Der Heijde | Development of a disease activity score based on judgment in clinical practice by rheumatologists[END_REF][START_REF] Prevoo | Validity and reliability of joint indices. A longitudinal study in patients with recent onset rheumatoid arthritis[END_REF] or functional ability [START_REF] Fries | Measurement of patient outcome in arthritis[END_REF][START_REF] Bruce | The Health Assessment Questionnaire (HAQ)[END_REF].

The most commonly used index to measure the functional ability and its limitations among RA patients has been Health Assessment Questionnaire Disability Index (HAQ-DI) [START_REF] Fries | Measurement of patient outcome in arthritis[END_REF][START_REF] Bruce | The Health Assessment Questionnaire (HAQ)[END_REF]. It comprises 20 functions which form eight domains [START_REF] Fries | Measurement of patient outcome in arthritis[END_REF]. A modified form of HAQ (MHAQ) was published by Pincus et al. in 1983 [9], where the number of functions was limited to eight, one question from each of the eight domains. However, both in the original HAQ-DI and in the MHAQ the selection of functions and their descriptions were based on professionals' intuitive choice [START_REF] Fries | Measurement of patient outcome in arthritis[END_REF][START_REF] Pincus | Assessment of patient satisfaction in activities of daily living using a modified Stanford health assessment questionnaire[END_REF][START_REF] Wolfe | Which HAQ is best? A comparison of the HAQ, MHAQ and RA-HAQ, a difficult 8 item HAQ (DHAQ), and a rescored 20 item HAQ (HAQ20): Analyses in 2491 rheumatoid arthritis patients following Leflunomide initaiation[END_REF]. Later studies have shown that disability items spontaneously generated by patients are different from those chosen by rheumatology health professionals [START_REF] Tugwell | The MACTAR patient preference functional priority approach for assessing improvement in physical disability in clinical trials in rheumatoid arthritis[END_REF][START_REF] Hewlett | Values for function in rheumatoid arthritis: patients, professionals, and public[END_REF][START_REF] Carr | Rheumatology outcomes: The patient's perspective[END_REF][START_REF] Verhoeven | Validity of the MACTAR questionnaire as a functional index in a rheumatoid arthritis clinical trial[END_REF][START_REF] Tuominen | Perceived functional disabilities among rheumatoid arthritis patients[END_REF]. The use of patient preferences has been suggested to supplement professionally chosen indicators [START_REF] Tuominen | Perceived functional disabilities among rheumatoid arthritis patients[END_REF][START_REF] Kvien | Patient perspective in outcome assesmentsperceptions or something else[END_REF].

The many functions measured to produce the overall index figures representing the patients' functional status are given equal weight in most index computations [START_REF] Fries | Measurement of patient outcome in arthritis[END_REF][START_REF] Bruce | The Health Assessment Questionnaire (HAQ)[END_REF][START_REF] Pincus | Assessment of patient satisfaction in activities of daily living using a modified Stanford health assessment questionnaire[END_REF][START_REF] Wolfe | Which HAQ is best? A comparison of the HAQ, MHAQ and RA-HAQ, a difficult 8 item HAQ (DHAQ), and a rescored 20 item HAQ (HAQ20): Analyses in 2491 rheumatoid arthritis patients following Leflunomide initaiation[END_REF][START_REF] Tugwell | The MACTAR patient preference functional priority approach for assessing improvement in physical disability in clinical trials in rheumatoid arthritis[END_REF][START_REF] Hewlett | Values for function in rheumatoid arthritis: patients, professionals, and public[END_REF][START_REF] Kvien | Patient perspective in outcome assesmentsperceptions or something else[END_REF]. Although the use of equal weight makes the index figure computation easier, there is no scientific evidence to support the underlying idea that all the functions are equally important when determining a patient's functional status. This holds both for clinical assessment by health professionals and for patient perception.

The Aim

The aim of this study was to evaluate the relative importance of the twenty functions assessed by patients to produce the Health Assessment Questionnaire's Disability Index. The ethical committee of the Hospital District of Southwest Finland had approved the study, and the sampled subjects received a written description of the sampling, study purposes as well as the planned use and storage of the information they were to provide. This was followed by a description of the subject's rights according to the Helsinki declaration. The participating subjects were asked to give their written informed consent and written permission to collect data from their patient registers.

Material and methods

A

In the first phase all 242 subjects were sent a questionnaire form, including the above mentioned descriptions and permissions, together with a prepaid return envelope. Before the actual study questions, example questions with answering models were presented to make the answering task easier. The first part of the study questions covered socio-economic background. For this study the following socio-economic variables were formed: age in years, gender, number of persons in the same household analysed as living alone or with someone, level of income calculated as household disposable monthly income per person in the household, and retirement status.

After these general well being and experience of pain in general during the last week were asked by using 10 centimetre long double anchored Visual Analog Scales (VAS), with endpoints from worst possible (0) to best possible (100). The patient's assessments of numbers of swollen and painful joints were solicited.

Following questions dealt with the twenty items included to calculate the HAQ disability index. The subject was asked to estimate her/his ability to perform the given function during the last week using the 10 centimetre long double anchored VAS, with endpoints from 'could not at all' (0) to 'could without any problems' (100). The Finnish version of HAQ Disability Index has some expressions which differ from the American version. Instead of asking the ability to shampoo ones hair, the Finnish version asks the ability to wash ones hair. Opening a new milk cartoon is replaced by asking the ability to prepare ones own food, and doing chores such as vacuuming or yardwork is replaced by simply asking the ability to vacuum.

Altogether 179 subjects returned acceptably filled questionnaires (73.9%). Detailed descriptions of sampling and the final sample characteristics have been reported earlier [START_REF] Tuominen | Perceived functional disabilities among rheumatoid arthritis patients[END_REF]. A separate systematic sample of 80 subjects was drawn from the respondents and a separate complementary questionnaire was sent to them. In this questionnaire the subjects were asked to estimate how important each of the same twenty items were in their everyday life. The estimates were solicited by using the 10 centimetre long double anchored VAS with endpoints from 'extremely unimportant' (0) to 'extremely important' (100). Completely answered complementary questionnaires were returned by 66 (88%) of the subsample.

The distributions of the perceived importance scores were skewed to the left, with the exception of taking a tub bath which was skewed to the right. The Mann-Whitney U-test was used to compare the difference in perceived importance of the functions between genders. The Wilcoxon Signed Rank test was used to estimate the differences between importance scores. The associations between continuous variables were ascertained with Spearman's rank correlation coefficients.

Exploratory factor analyses were performed for perceived importance functions with the maximum likelihood estimation method and oblimin-rotation. The correlation matrix used in factor analysis was computed using Spearman correlations coefficients due to skewed distributions. Statistical analyses were done using SPSS version 16.0 (SPSS Inc., Chicago, Illinois) and SAS Systems for Windows, release 9.1 (SAS Institute Inc., Cary, NC). P-values less than 0.05 were considered statistically significant.

Results

Overall, the relative importance of the functions assessed varied considerably. The subjects' socioeconomic background, general well-being and experience of pain or joint swelling did not affect the perceived importance. Men assessed the highest importance to the ability to walk outdoors on flat ground (97.0) and women to dressing themselves, getting in and out of bed and washing and drying their body (95.0). The least important function was the ability to take a tub bath both among men (6.5) and women (4.0). The differences between sexes were small (NS). (Table 1)

In seven out of the eight domains used for computing the traditional HAQ, the most important function was the one which had been selected for MHAQ to represent that domain. However, in the Hygiene domain the difference in importance between the most important function 'Getting on and off the toilet' and the one chosen for MHAQ 'Wash and dry your body' was unsubstantial (Table 1).

Perceived importance varied more between subjects than within subjects. Those who expressed higher importance in one function, usually considered also other functions important. The importance of all other functions except taking a tub bath were positively highly significantly (p<0.01) correlated (r= 0.324 -0.878) with the importance of each of the other functions. Also subjects' perceived functional status varied considerably and a significant (p<0.05) positive correlation between perceived importance and functional ability of the same functions were observed only in cutting ones own meat (r= 0.257), vacuuming (r= 0.276) and getting in and out of a car (r= 0.271).

Factor analyses with one factor model showed high loadings on most functions, with the exception of 'Take a tub bath', which had a very low loading. The two factor model formed one factor with loadings concentrating slightly more on 'lower extremity' and another on 'upper extremity'.

However, the two-factor solution did not provide a unique solution, because five functions loaded (loadings > 0.30) on both factors. Forcing more than two factors into the models did not provide any additional information. When evaluating the factor loadings within the eight domains forming the HAQ-DI score, practically all eight domains included more than one function with high loadings. However, the highest loadings within the domains were constantly observed for those eight functions which represent each domain in MHAQ computations except the difference in the Hygiene domain (Table 2).

Discussion

The Hospital District of Southwest Finland provides health care services for approximately 460 000 people residing in the area. It is assumed that the sampled patients are not selected, but the sample includes patients with varying lengths of disease history and in varying stages of RA at same proportions as in the RA patient population treated by rheumatologists [START_REF] Tuominen | Perceived functional disabilities among rheumatoid arthritis patients[END_REF]. However, one should be cautious when generalising the findings of this study to all RA patients. Although the participation rate can be considered good, our final sample size is small and patients with the mildest disease course were underrepresented. If the sample had been restricted to either more or less severe cases or those who had suffered from RA shorter or longer periods of time, the findings may have been different. Activities or functional abilities which are considered important in the early stages of disease may become less relevant as the pattern of disease changes [START_REF] Carr | Rheumatology outcomes: The patient's perspective[END_REF].

The observed variation in importance scores between the twenty functions included in HAD -DI implies that the functions have an inconsistent role in the RA patient's everyday life. If an overall measure of patients' functional ability is calculated by using equal weights for all functions, as is the case in e.g. HAQ and MHAQ [START_REF] Fries | Measurement of patient outcome in arthritis[END_REF][START_REF] Pincus | Assessment of patient satisfaction in activities of daily living using a modified Stanford health assessment questionnaire[END_REF], the index figures do not necessarily represent the burden RA causes to the patients. With a small sample of 25 inpatients with relatively severe disease status Hewlett et al. showed that HAQ items do not have universal value [START_REF] Tugwell | The MACTAR patient preference functional priority approach for assessing improvement in physical disability in clinical trials in rheumatoid arthritis[END_REF], which was corroborated by the present study findings. However, the present study finding that 'Taking a tub bath' was considered significantly less important than all other functions, was different from the earlier study, where none of the twenty functions were deemed unimportant [START_REF] Tugwell | The MACTAR patient preference functional priority approach for assessing improvement in physical disability in clinical trials in rheumatoid arthritis[END_REF]. One reason for this may be the difference in patients' disease status, or the fact that during the last decades tub baths have become more and more rare in Finland.

For MHAQ, Pincus et al. chose intuitively one function from each HAQ-DI domain, instead of the worst ability out of two or three functions used in the original index [START_REF] Pincus | Assessment of patient satisfaction in activities of daily living using a modified Stanford health assessment questionnaire[END_REF]. Present study findings support the choice they made intuitively earlier. Using 'Wash and dry your body' instead of 'Get on and off the toilet' probably does not limit the coverage of functions in MHAQ as movements included in sitting are included in 'Get in and out of bed' and 'Get in and out of car'. In an earlier study, RA patients with more severe disease status ranked practically the same functions as the most important ones within each of the eight domains [START_REF] Tugwell | The MACTAR patient preference functional priority approach for assessing improvement in physical disability in clinical trials in rheumatoid arthritis[END_REF], further validating the selection of Pincus et al. [START_REF] Pincus | Assessment of patient satisfaction in activities of daily living using a modified Stanford health assessment questionnaire[END_REF].

Factor analyses formed one distinct factor with high loadings in very different functions. These loadings did not accumulate to any particular HAQ domain, but all domains included functions with high loadings. One interpretation for this could be that most of the functions included in the original HAQ and all those in MHAQ are important components of RA patients' well-being. All eight functions intuitively chosen for MHAQ from the original list of twenty functions for HAQ-DI had high factor loadings. This may be considered as a further indication of rheumatology health professionals' success in their selection process [START_REF] Pincus | Assessment of patient satisfaction in activities of daily living using a modified Stanford health assessment questionnaire[END_REF]. On the other hand, development of the Recent-Onset Arthritis Disability (ROAD) questionnaire showed that soliciting the important functional items from health professionals with varying clinical backgrounds and variety of patients also produced an item pool of much wider variety [START_REF] Salaffi | Development of a functional disability measurement tool to asses early arthritis: the Recent-Onset Arthritis Disability (Road9 questionnaire[END_REF].

Functions related to movements of lower extremities have been suggested to play an important role in RA patients' disability [START_REF] Pincus | Further development of a physical function scale on a multidimensional health assessment questionnaire for standard care of patients with rheumatoid arthritis[END_REF][START_REF] Mason | The rapid assessment of disease activity in rheumatology (radar) questionnaire. Validity and[END_REF]. However, the ROAD questionnaire development suggested that upper extremity functions explain most of the variance. One reason for their contradictory finding may be the final selection of the items they used [START_REF] Salaffi | Development of a functional disability measurement tool to asses early arthritis: the Recent-Onset Arthritis Disability (Road9 questionnaire[END_REF]20]. In the present study, the roles of upper and lower extremities were distinguished as entities, as movements related to these activities had high loadings in factor analysis. Overall, the factor loadings may suggest that reduced locomotor activities of the extremities and large body movements are more important than fine movements of the upper extremity.

Perceived importance of the functions was only weakly related to the corresponding perceived functional ability. If the level of disability and perceived importance were strongly correlated, then the level of importance of a function could be considered as a proxy for the level of disability. The variations in these relationships indicate that subjects who experienced higher disability in some function were not necessarily those who considered that function important. Perceived importance was not directed by level of disability, which supports the idea that the assessments quantify independently the relative importance of the functions. However, this is somewhat in contradiction with the earlier suggestion that patient preferences for improvement in health may be an alternative for the assessment of important patient outcomes [START_REF] Kvien | Patient perspective in outcome assesmentsperceptions or something else[END_REF]. The patient's contribution to assessment of health status and treatment outcomes, as well as the magnitude of their role requires further experimentation [21].

Present study findings corroborate the view that assessment of RA patients' functional status would benefit from further development of measures by giving relative weights from patients' perspective to those functional abilities which are used for index computations. sensitivity to change of a patient self-reported measure of joint count and clinical status.

  systematic sampling from the patient registers of the Hospital District of Southwest Finland was done. From ten consecutive calendar months (March-December 2006) the first twenty patients with diagnose M05.8 (Arthritis Rheumatoides seropositiva) and ten patients who had been diagnosed as having M06.0 (Arthritis Rheumatoides seronegativa) formed the basic sample. The proportions of seropositive and seronegative patients was known to be approximately 2:1. The patient registers include RA patients treated by rheumatologists of Turku University Hospital. The patients with the mildest disease course are mostly treated in primary health care and are thus underrepresented in our sample. Anticipated number of patients with these diagnoses were not available for each calendar month, and the final sample comprised 242 patients, 59.3% with M05.8 diagnosis.

  Arthritis & Rheum 35:156-62. 20. Salaffi F, Stancati A, Neri R, Grassi W, Bombardieri S (2005) Measuring functional disability in early rheumatoid arthritis: the validity, reliability and resposiveness of the Recent-Onset Arthritis Disability (ROAD) index. Clin Exp Rheumatol 23(Suppl. 39):S31-S42. 21. Kirwan J, Heiberg T, Hewlett S, Hughes R, Kvien T, Ahlmen M, et al. (2003) Outcomes from the patient perspective workshop at OMERACT 6. J Rheumatol 30:868-72.

Table 1 .Table

 1 Median values with Interquartile ranges (IQR) for the perceived importance of the functions frequently asked from RA patients to estimate their functional ability by sex 93.0 75.5-97.8 93.0 74.5-98.0 93.0 76.3-98.0 Turn faucets on and off 92.0 84.8-99.0 93.0 66.5-98.0 93.0 79.3-98.0 P H

Table 1

 1 The function representing the domain in MHAQ by Pincus et al.(9) H = The function ranked highest in the domain by Hewlett et al.[START_REF] Tugwell | The MACTAR patient preference functional priority approach for assessing improvement in physical disability in clinical trials in rheumatoid arthritis[END_REF]

	. continued
	ACTIVITIES
	Run errands and shop	91.5 64.5-98.8	87.0 58.5-96.5	88.5 62.0-97.8	H
	Get in and out of a car	94.5 73.8-98.8	90.0 60.5-97.5	93.0 67.8-98.0 P
	Vacuum	85.5 54.0-97.5	65.0 45.0-96.0	70.5 45.0-95.8
	P =

Table 2 .

 2 Factor loadings of the relative importance scores based on Spearman's rank correlations

	One factor model	Two factor model
		Factor 1	Factor 2

Table 2 .

 2 continued P = The function representing the domain in MHAQ by Pincus et al.(9) H = The function ranked highest in the domain by Hewlett et al.[START_REF] Tugwell | The MACTAR patient preference functional priority approach for assessing improvement in physical disability in clinical trials in rheumatoid arthritis[END_REF]

Acknowledgements

This study has been supported by grants from the research funds of The Hospital District of Southwest Finland and from Wyeth AB.


Dr Satish M Rachapalli

R Malaiya

Tamt Mohd

R A Hughes

Successful treatment of Candida Discitis with 5-Flucytosine and Fluconazole

Case report:

Intervertebral discitis is a rare but recognised complication of systemic candidaemia. Once diagnosed, treatment often consists of a prolonged course of amphotericin B and/or Fluconazole. We report a case of Candida discitis, successfully treated with Fluconazole and 5-Flucytosine.

A 48 year old Caucasian lady underwent a computerised tomography (CT) scan of chest to investigate chronic cough. This showed an incidental finding of inflammation of the inter-vertebral disc between sixth and seventh thoracic vertebrae. Three months prior to this admission she underwent a laparotomy for division of adhesions. Her postoperative period was complicated by severe sepsis requiring admission to the intensive care unit. Blood cultures at that time grew Staphylococcus aureus and Candida and she was treated with Vancomycin, Amphotericin B and Fluconazole.

During this admission, she complained of mid thoracic pain and did not have any features of systemic infection. She was on a combination of four different analgesics for the management of the pain. On examination, she had minimal tenderness over the mid thoracic region without neurological signs. Investigations showed a C-reactive protein of 11 mg/l, normal white cell count and bone profile. Blood cultures were sterile and Candida antigen was not detected. Protein electrophoresis was normal and urine was negative for Bence-Jones proteins. An isotope bone scan showed a hot spot at 6 th and 7 th thoracic vertebrae disc (T6/7) and a MRI scan revealed a high signal in the same area with surrounding oedema and soft tissue swelling, suggestive of discitis (Figure). A CT guided biopsy showed inflammatory cells comprising of plasma cells and periodic acid Schiff (PAS) staining confirmed presence of Candida hyphae.

Fluconazole 400 mg once a day orally was commenced but she continued to complain of pain even after 2 weeks of treatment. Treatment was switched to intravenous Fluconazole and IV 5-Flucytosine for 2 weeks which resulted in symptomatic improvement and she was continued on oral Fluconazole 400mg for a further four weeks.

Candida discitis is a rare complication of systemic Candidaemia and is thought to spread haematogenously. [START_REF] Munk | Candida osteomyelitis and disc space infection of the lumbar spine[END_REF] To date, less than 60 such cases are reported. Invasive Candida infections have increased significantly over the past two decades [START_REF] Pappas | Guidelines for treatment of candidiasis[END_REF] and are commonly seen in immunosuppressed patients and intravenous drug users. The lack of typical inflammatory response and insidious course of the disease can make the diagnosis difficult.

The ideal duration for treatment of fungal discitis has not yet been established, and current practice is to continue therapy until clinical symptoms and inflammatory markers have normalised and/or imaging shows resolution of inflammation. Other treatment options include surgical intervention especially in cases of neurological compromise. [START_REF] Ugarriza | Candida albicans spondylodiscitis[END_REF][START_REF] Miller | Vertebral osteomyelitis due to Candida species: case report and literature review[END_REF] Increasing resistance of Candida species has also been noted. Candida discitits is increasingly being noticed in severely ill patients with candidaemia previously exposed to azoles. [START_REF] Pappas | Guidelines for treatment of candidiasis[END_REF] Of note, no relationship between the duration of the initial treatment of candidaemia and the subsequent occurrence of late complications has been established. [START_REF] Pemán | Spondylodiscitis caused by Candida krusei: case report and susceptibility patterns[END_REF] Evidence points towards Amphotericin IV being the first line agent of choice. [START_REF] Pemán | Spondylodiscitis caused by Candida krusei: case report and susceptibility patterns[END_REF] However as Amphotericin is nephrotoxic, some authors prefer Fluconazole due to its favourable side effect profile. [START_REF] Hennequin | Spondylodiskitis due to Candida albicans: report of two patients who were successfully treated with fluconazole and review of the literature[END_REF] Currently Flucytosine is recommended for use in combination therapy in resistant cases of chronic disseminated candidiasis. [START_REF] Pappas | Guidelines for treatment of candidiasis[END_REF] This case shows that it is important to consider fungal discitis in patients with back pain and a history of candidaemia. 5-Flucytosine might be a useful agent in treating candida discitis in an era where resistance to other anti-fungal agents is increasing.

Conflict of interest: None


Marcin Milchert

Katarzyna Fisher

Jacek Fliciński

Lidia Ostanek

Marek Brzosko

High prevalence of anti-beta-2-glycoprotein-I and anti-prothrombin antibodies in adult onset Still's disease. Comment on: "Portal vein thrombosis in adult onset Still's disease: a case

Dear Sir, We have followed with great interest case report by Morita et al. [START_REF] Morita | Portal vein thrombosis in adultonset Still's disease: a case report and literature review[END_REF] . In our group of 8 patients diagnosed with Adult-onset Still's disease (AOSD) according to Yamaguchi criteria we reviewed presence of antifospholipid antibodies. They were detected in 6 of 8 patients: the most common were anti-prothrombine (anti-PT) followed by anti-beta2 glycoprotein I antibodies (anti-β2GPI) (Table 1). D-dimer level was elevated in all patients, however, no related thrombosis was diagnosed. Only one patient presented with deep vein thrombosis at the time of AOSD relapse, and one woman had a history of miscarriage, however not fulfilling current antifospholipid syndrome classification criteria.

Our first reflection was how difficult differential diagnosis of liver disease in AOSD is, as raised serum aminotransferase (AT) level may be associated with high disease activity and preceded by a raise of acute phase reactants (Fig. 1). In a case described by Morita et al. [START_REF] Morita | Portal vein thrombosis in adultonset Still's disease: a case report and literature review[END_REF] , portal vein thrombosis occured in a patient with raised AT level and low general disease activity. Our patient with deep vein thrombosis at the time of AOSD relapse (Table 1, no 6) had also concurrent raise of serum AT level, and high concentration of anti-β2GPI and anti-PT. Raised concentration of antifospholipid antibodies might help to establish early diagnosis of thrombosis in patients with AOSD.

Coagulation abnormalities are frequently reported in AOSD: there have been reports of various clinical manifestations of thrombotic microangiopa thic haemolitic anaemia (TMHA) [START_REF] Sayarlioglu | Thrombotic thrombocytopenic purpura-hemolytic uremic syndrome and adult onset Still's disease: case report and review of the literature[END_REF] , disseminated intravascular coagulation (DIC) [START_REF] Shinohara | Calcinosis cutis and intestinal pseudoobstruction in a patient with adult onset Still's disease associated with recurrent relapses of disordered coagulopathy[END_REF] , and occasionally, antiphospholipid syndrome [START_REF] Jinnouchi | A case of brain ischemia presenting spectacular shrinking deficit in adult-onset Still's disease associated with antiphospholipid antibody syndrome[END_REF] . However, the knowledge on etiopathogenesis of hypercoagulative state in AOSD is limited. The presence of anticardiolipin antibodies (aCL) may contribute to the developement of TMHA [START_REF] Espinosa | Thrombotic microangiopathic haemolytic anaemia and antiphospholipid antibodies[END_REF] .

Only one of our patients was positive for aCL, in contrast with high prevalence of anti-β2GPI and anti-PT. These antibodies have not been routinely assessed in previous studies [START_REF] Morita | Portal vein thrombosis in adultonset Still's disease: a case report and literature review[END_REF][START_REF] Sayarlioglu | Thrombotic thrombocytopenic purpura-hemolytic uremic syndrome and adult onset Still's disease: case report and review of the literature[END_REF][START_REF] Shinohara | Calcinosis cutis and intestinal pseudoobstruction in a patient with adult onset Still's disease associated with recurrent relapses of disordered coagulopathy[END_REF] .

The presence of LAC in our patients is not easy to interpret also because Russel's Viper venom test revealed moderate probability of LAC. This might be also explained by deficiency of coagulation factors or presence of their unknown inhibitors, especially because activated partial thromboplastin time (APTT) was short or normal.

Prevalence of anti-β2GPI and anti-PT in AOSD should be further studied to reveal their role in coagulopathies.

Table 1 Characteristics of patients with AOSD: fibrinogen (normal 150-400), D -dimer (normal <500), alanine-aminotransferase (ALT, normal <30), ferritin (normal 30-400), anti-β2GPI (normal <20), aCL IgM (normal <12), anti-PT (normal <30). All tests were performed in patients with "active" disease, with exception o f control antibodies levels (written after slash) assessed at minimum 6 weeks interval. Lupus anticoagulant was detected using dRVVT. All other antibodies were detected by enzyme-linked immunosorbent assay (ELISA).

No

Fig. 1

 1 Fig. 1 Change of ESR and alanine-aminotransferase (ALT) values in one of the patients.


Prof Theodore Simos

D F Papadopoulos

Z A Anastassi

Theodore E Simos

A modified phase-fitted and

Keywords: Runge-Kutta-Nystr\"{o}m methods, Phase-fitted, Amplification-fitted, Schr\"{o}dinger, Phase-lag infinity Runge-Kutta-Nyström methods, Phase-fitted, Amplification-fitted, Schrödinger, Phase-lag infinity PACS: 0.260, 95.10

A new Runge-Kutta-Nystr\"{o}m method, with phase-lag and amplification error of order infinity, for the numerical solution of the Schr\"{o}dinger equation is developed in this paper. The new method is based on the Runge-Kutta-Nystr\"{o}m method with fourth algebraic order, developed by Dormand, El-Mikkawy and Prince. Numerical illustrations indicate that the new method is much more efficient than other methods derived for the same purpose.

Response to Reviewers: Reviewer #1: The authors modified the fourth order Runge-Kutta-Nystrom method. The numerical approach to me is OK. However, I think that efficiency evaluation of the method is too brief and more measurement is needed for a complete comparison with other methods. I suggest the authors use other ways to evaluate the performance of the new method. Moreover, it is of interest to know the efficiency at different orders.

We have read the reviewer's comments and we made the following changes:

•

We added the bound states problem using eight eigenenergies. We present the figures with the efficiency of the methods for these eigenenergies and some remarks on the results. There are no other RKN methods of the same order that are used for the solution of the Schrodinger equation so that we can compare with.

We have also changed Theorem 1, so that it is more clear and precise.

Introduction

The one-dimensional or radial Schrödinger equation has the form

y ′′ (x) + E - l(l + 1) x 2 -V (x) y(x) = 0, where 0 ≤ x < ∞ (1)
We call the term l(l+1)/x 2 the centrifugal potential, and the function V (x) the electric potential. In [START_REF] Dormand | Families of Runge-Kutta-Nyström formulae[END_REF], E is a real number denoting the energy, and l is a quantum number. The function W (x) = l(l + 1)/x 2 + V (x) denotes the effective potential, where lim x→∞ V (x) = 0 and so lim x→∞ W (x) = 0. The boundary condition are y(0) = 0 together with a second boundary condition, for large values of x, determined by the physical considerations.

This type of equations appears in many scientific areas including, theoretical physics, nuclear physics, molecular physics, physical chemistry, quantum chemistry. For the numerical solution of equation [START_REF] Dormand | Families of Runge-Kutta-Nyström formulae[END_REF] there is a high demand. The last three decades a lot of researchers have developed new methods to solve numerically the Schrödinger equation [3 -14].

The purpose of this paper is to construct an explicit Runge-Kutta-Nyström with fourth algebraic order, phase lag of order infinity and amplification error of order infinity. The new method is based on the coefficients of the well-known fourth algebraic order Dormand, El-Mikkawy and Prince Runge-Kutta-Nyström method [START_REF] Dormand | Families of Runge-Kutta-Nyström formulae[END_REF]. It is also based on the FSAL (first stage as last) technique, which means that the number of stages is four but it effectively uses only three stages at each step.

Modified Runge-Kutta-Nyström method

The general m-stage method for the equation

d 2 u(t) dt 2 = f (t, u(t)) (2)
is of the form

u n = u n-1 + hû n-1 + h 2 m i=1 b i f (t n-1 + c i h, f i ), ûn = ûn-1 + h m i=1 bi f (t n-1 + c i h, f i ), (3)
where

f i = u n-1 + hg i c i ûn-1 + h 2 i-1 j=1 α ij f (t n-1 + c j h, f j ), i = 1, . . . , m (4
) especially for the explicit Runge-Kutta-Nyström methods c 1 = 0 and more specifically for an FSAL explicit RKN method c m = 1 and c m,j = b i for j = i. The above expressions are presented in Table I.

As we can see in Table I, the Butcher table has an extra column as compared to a classical m-stage Runge-Kutta-Nyström method. For the classical method g i = 1, i = 2(1)m. In the present paper and based on the requirement of phase and amplification fitting, values g i , i = 2(1)m are variable and depend on z (which is the product of the frequency v and the step-size h). In section (4) we will present a development of a phase and amplification fitted four-stage Runge-Kutta-Nyström method of algebraic order four, based on the well-known DEP Runge-Kutta-Nyström [START_REF] Dormand | Families of Runge-Kutta-Nyström formulae[END_REF].

Phase lag and amplification error analysis for Runge-Kutta-Nyström methods

To develop the new method we use the test equation,

d 2 u(t) dt 2 = (iv) 2 u(t) =⇒ u ′′ (t) = -v 2 u(t), v ∈ R (5)
By applying the general method (3) to the test equation (5) we obtain the numerical solution

u n hû n = D n u 0 hû 0 , D = A(z 2 ) B(z 2 ) Ȧ(z 2 ) Ḃ(z 2 ) , z = vh, (6)
where A, B, Ȧ, Ḃ are polynomials in z 2 , completely determined by the parameters of the method (3). The exact solution of ( 5) is given by

u(t n ) = σ 1 [exp(iz)] n + σ 2 [exp(-iz)] n , (7)
where Substituting in [START_REF] Simos | A modified Runge-Kutta method with phaselag of order infinity for the numerical solution of the Schrödinger equation and related problems[END_REF], we have

σ 1,2 = 1 2 [u 0 ± (iû 0 ) v ] or σ 1,2 = |σ|exp(±iχ).
u(t n ) = 2|σ|cos(χ + nz). (8)
Furthermore we assume that the eigenvalues of D are ̺ 1 , ̺ 2 , and the consequent eigenvectors are [1,

v 1 ] T , [1, v 2 ] T , where v i = Ȧ/(ρ i -Ḃ), i = 1, 2.
The numerical solution of ( 5) is

u n = c 1 ρ n 1 + c 2 ρ n 2 , (9)
where

c 1 = - v 2 u 0 -hû 0 v 1 -v 2 , c 2 = v 1 u 0 -hû 0 v 1 -v 2 .
If ρ 1 , ρ 2 are complex conjugate, then c 1,2 = |c|exp(±iw) and ρ 1,2 = |ρ|exp(±ip). By substituting in (9), we have

u n = 2|c||ρ| n cos(w + np). ( 10
)
The following definition is originally formulated by van der Houwen and Sommeijer [2]. DEFINITION 1. (Phase-lag). Apply the RKN method (3) to the general method [START_REF] Van De Vyver | Modified explicit Runge-Kutta methods for the numerical solution of the Schrödinger equation[END_REF]. Then we define the phase-lag

Φ(z) = z -arccos(tr(D)/2 det(D))
. If Φ(z) = O(z q+1 ), then the RKN method is said to have phaselag order q. In addition, the quantity a(z) = 1 -det(D) is called amplification error.

Let us denote

R(z 2 ) = tr(D) = A(z 2 ) + Ḃ(z 2 ) Q(z 2 ) = det(D) = A(z 2 ) Ḃ(z 2 ) -Ȧ(z 2 )B(z 2 ) ( 11
)
where z = vh. From definition 1 it follows that

Φ(z) = z -arccos R(z 2 ) 2 Q(z 2 ) , a(z) = 1 -Q(z 2 ). ( 12
)
If at a point z, a(z) = 0, then the Runge-Kutta-Nyström method has zero dissipation at this point.

We can also put forward an alternative definition for the case of infinite order of phase lag.

DEFINITION 2. (Phase-lag of order infinity). To obtain phase-lag of order infinity the relation

Φ(z) = z -arccos R(z 2 ) 2 √ Q(z 2 ) = 0 must be hold.
From definition 2 we have the following theorem. THEOREM 1. If we have phase-lag of order infinity and at a point z, α(z) = 0 then,

z -arccos R(z 2 ) 2 √ Q(z 2 ) = 0 1 -Q(z 2 ) = 0 ⇒ R(z 2 ) = 2cos(z) Q(z 2 ) = 1
Proof.

Φ(z) = z -arccos R(z 2 ) 2 Q(z 2 ) = 0 ⇔ arccos R(z 2 ) 2 Q(z 2 ) = z ⇔ cos arccos R(z 2 ) 2 Q(z 2 ) = cos(z) ⇔ R(z 2 ) 2 Q(z 2 ) = cos(z) ⇔ R(z 2 ) Q(z 2 ) = 2cos(z). ( 13
)
Now for

a(z) = 0 ⇔ 1 -Q(z 2 ) = 0 ⇔ Q(z 2 ) = 1. ( 14
)
From the above relation and the expression (13) we have

R(z 2 ) = 2cos(z). ( 15
)
LEMMA 1. For the construction of a method with phase lag of order infinity and amplification error of order infinity, we must sutisfy the conditions R(z 2 ) = 2cos(z) and Q(z 2 ) = 1.

Derivation of the new Runge-Kutta-Nyström method

In this section we construct a four-stage explicit Runge-Kutta-Nyström method (presented in table I), based on R(z 2 ) = 2cos(z) and Q(z 2 ) = 1. Using the coefficients of DEP [START_REF] Dormand | Families of Runge-Kutta-Nyström formulae[END_REF] and for g 4 = 1, the method can be seen in table II. Now let us rewrite R(z 2 ) and Q(z 2 ) in the following form

R(z 2 ) = 2 -r 1 z 2 + r 2 z 4 -r 3 z 6 + . . . + r i z 2i = 0 Q(z 2 ) = 1 -q 1 z 2 + q 2 z 4 -q 3 z 6 + . . . + q i z 2i = 0 (16)
By computing the polynomials A, Ȧ, B, Ḃ in terms of RKN parameters we obtain the expressions of R(z 2 ) and Q(z 2 ). Then by applying them to the equations ( 14) and (15) we lead to the following formulas for the coefficients g 2 and g 3 .

Numerical illustrations

In this section we will apply our method to the radial Schrödinger equation. For the purpose of our numerical illustration we take the domain of integration as x ∈ [0, 15], using the Woods-Saxon potential:

V (x) = u 0 1 + q + u 1 q (1 + q) 2 , q = exp x -x 0 α , where (17)
u 0 = -50, α = 0.6, x 0 = 7, u 1 = - u 0 α 5.1. Resonanse problem
In the case of positive energies (E = k 2 ), the potential (V (x)) dies away faster than the centrifugal potential (l(l + 1)/x 2 ), so for a large number for x, Schrödinger equation effectively reduces to

y ′′ (x) + (k 2 - l(l + 1) x 2 )y(x) = 0 (18)
The above equation (18) has two linearly independent solutions, kxj l (kx) and kxn l (kx), where j l and n l are the spherical Bessel and Neumann functions, respectively. When x → ∞, the solution of equation (1) take the following asymptotic form

y(x) ≃ Akxj l (kx) -Bkxn l (kx) ≃ D sin kx - lπ 2 + tan(δ l )cos kx - lπ 2 , ( 19
)
where δ l is the scattering phase shift that may be calculated from the bellow formula

D. F. Papadopoulos, Z. A. Anastassi, T. E. Simos tan(δ l ) = y(x i )S(x i+1 ) -y(x i+1 )S(x i ) y(x i+1 )C(x i ) -y(x i )C(x i+1 ) , (20)
where S(x) = kxj l (kx), C(x) = kxn l (kx) and x i < x i+1 both exist in the asymptotic region.

For positive energies and for l = 0, we calculate the phase shift (δ l ) and then we compare it with the accurate value which is π/2. The boundary conditions for this eigenvalue problem are y(0) = 0 and y(x) = cos( √ Ex) for large x. We use the following eigenenergies E 1 = 53.588872 E 2 = 163.215341 E 3 = 341.495874 E 4 = 989.701916

Bound-states problem

In the case of negative energies (E < 0), we consider the eigenvalue problem with boundary conditions y(0) = 0 and y(x) = exp(-√ -Ex) for large x. In order to solve this problem numerically, by a chosen eigenvalue, we integrate forward from the point x = 0, backward from the point x = 15 and matching up the solution at some internal point in the range of integration.

For the Bound-states problem we use the following eigenenergies

E 1 = -49.457788728 E 2 = -48.148430420 E 3 = -41.232607772 E 4 = -38.122785096 E 5 = -26.873448915 E 6 = -22.588602257 E 7 = -8.676081670 E 8 = -3.908232481
We are going to compare our results to those derived by using the high order method of embedded Runge-Kutta-Nyström 4(3)4 method, of DEP (see [START_REF] Dormand | Families of Runge-Kutta-Nyström formulae[END_REF]), as well as to other methods derived for the numerical solution of the Schrödinger equation. The methods used in the comparison have been denoted by: • PAFRKN4: The new fourth-order RKN method with four stages (three effective stases with FSAL property), phase lag and amplification error of order infinity derived in Section 4.

• RKN4: The high order method of pair RKN 4(3)4 method of Dormand, El-Mikkawy and Prince [START_REF] Dormand | Families of Runge-Kutta-Nyström formulae[END_REF].

• MODRK4A: The modified exponentially fitted fourth order RK method with four stages of H. Van de Vyver [START_REF] Van De Vyver | Modified explicit Runge-Kutta methods for the numerical solution of the Schrödinger equation[END_REF] (Section 2.1).

• MODRK4B: The modified exponentially fitted fourth order RK method with four stages of H. Van de Vyver [START_REF] Van De Vyver | Modified explicit Runge-Kutta methods for the numerical solution of the Schrödinger equation[END_REF] (Section 2.2).

One way to measure the efficiency of the method is to compute the accuracy in the decimal digits, that is -log 10 (error at the end point) when comparing the phase shift to the actual value π/2 versus the computational effort measured by the log 10 (number of function evaluations required). The frequency is given by the suggestion of Ixaru and Rizea [START_REF] Gr | A numerov-like scheme for the numerical solution of the Schrödinger equation in the deep continuum spectrum of energies[END_REF]

v = √ E + 50, x ∈ [0, 6.5] √ E, x ∈ [6.5, 15]
In the figures we display the efficiency curves, that is the accuracy versus the computational cost measured by the number of function evaluations required by each method. Numerical results indicate that the new method derived in section 4, is very efficient for solving numerically the Schrödinger equation. More specifically we observe that, in the case of resonance problem our method is more accurate than MODRKB4 and MODRKA4 by two and three decimals respectively. Moreover the new method is more accurate by two decimals than RKN4 for the eigenvalues E = {53.588872, 163.215341, 341.495874} and by three decimals for the eigenvalue E = 989.701916. In the case of bound-states problem, PAFRKN4 is more accurate than MODRKB4 and MODRKA4 by three decimals. Also our method maintained more accurate than the classical RKN4 method, by two decimals for the eigenvalues E = {-49.457788728, -48.148430420} and by three decimals for the rest six eigenvalues.

Conclusion

A new Runge-Kutta-Nyström method of fourth algebraic order with four stages (three effective stages with FSAL property), is developed in the present paper. The new method is based on the very well known classical Dormand, El-Mikkawy and Prince fourth algebraic order Runge-Kutta-Nystöm method and it has phase lag of order infinity and ampli-D. F. Papadopoulos, Z. A. Anastassi, T. E. Simos fication error of order infintiy. The numerical results show that the new method is more efficient for the numerical solution of the Schrödinger equation than other methods derived for the same purpose, including the corresponding classical method of DEP.

D

  . F. Papadopoulos, Z. A. Anastassi, T. E. Simos

g 2 = 6 -

 26 -1/20 (-933120 + 459 z 6 -19440 z 4 + 276480 z 2 -18 (-597196800 + 597196800 cos(z) -576020 z 8 + 12913920 z 6 -98858880 z 4 + 349401600 z 2 + 8551 z 10 + 7102080 z 4 cos (z) -50803200 z 2 cos (z)) / (-186624 + 31104 z 2 -1332 z 4 + 17 z 6 )) + 648 ((-597196800 + 597196800 cos (z) -576020 z 8 + 12913920 z 6 -98858880 z 4 + 349401600 z 2 + 8551 z 10 + 7102080 z 4 cos (z) -50803200 z 2 cos (z))/ (z 2 -186624 + 31104 z 2 -1332 z 4 + 17 z 6 ))+ 933120 cos(z)z -2 17 z 4 -772 z 2 + 2304 + 597196800 cos (z) -576020 z 8 + 12913920 z 98858880 z 4 + 349401600 z 2 + 8551 z 10 + 7102080 z 4 cos (z) papdopoulos-anastassi-simos_rev.tex; 13/11/2009; 8:53; p.6

Figure 1 .

 1 Figure 1. Efficiency for the Schrödinger equation using E=53.588872

Figure 2 .Figure 3 .

 23 Figure 2. Efficiency for the Schrödinger equation using E=163.215341

Figure 4 .

 4 Figure 4. Efficiency for the Schrödinger equation using E=989.701916

Figure 12 .

 12 Figure 12. Efficiency for the Schrödinger equation using E=-3.908232481

Table I

 I

		. m-stage Runge-Kutta-Nystöm method
	c1	1	0
	c2	g2	α21
	c3	g3	α31	α32
	. . .	. . .	. . .	. . .
	cm	gm	αm,1 αm,2 . . .	αm,m-1
			b1	b2	. . .	bm-1	bm
			b1	b2	. . .	bm-1	bm

  Efficiency for the Schrödinger equation using E=-48.148430420 Efficiency for the Schrödinger equation using E=-38.122785096 Efficiency for the Schrödinger equation using E=-22.588602257

	14		A Modified Runge-Kutta-Nyström Method D. F. Papadopoulos, Z. A. Anastassi, T. E. Simos		13
		10 11 8 9 7 7 8 6	PAFRKN4 RKN4 MORK4A MODRK4B PAFRKN4 RKN4 MORK4A MODRK4B PAFRKN4 PAFRKN4 RKN4 RKN4 MORK4A MORK4A MODRK4B MODRK4B	WOODS-SAXON POTENTIAL WITH E=-48.148430420 WOODS-SAXON POTENTIAL WITH E=-38.122785096 WOODS-SAXON POTENTIAL WITH E=-3.908232481 WOODS-SAXON POTENTIAL WITH E=-22.588602257
		9 7 6 5
	-LOG(ERRMAX) -LOG(ERRMAX) -LOG(ERRMAX) -LOG(ERRMAX)	7 8 5 6 4 5 3 4
		6 4 3
		2
		5 3 2
		1
		4 2 1
		2.2 3 2.2 1 2.2 0 2.2 0	2.4 2.4 2.4 2.4	2.6 2.6 2.6 2.6	2.8 LOG(FUNCTION EVALUATIONS) 3 3.2 2.8 3 3.2 LOG(FUNCTION EVALUATIONS) 2.8 3 3.2 2.8 3 3.2 LOG(FUNCTION EVALUATIONS) LOG(FUNCTION EVALUATIONS)	3.4 3.4 3.4 3.4	3.6 3.6 3.6 3.6	3.8 3.8 3.8 3.8
	2.2 3 4 5 6 7 8 9 10 11 Figure 6. 2.2 -LOG(ERRMAX) 1 2 3 4 5 6 7 8 9 -LOG(ERRMAX) Figure 8. 2.2 0 1 2 3 4 5 6 7 8 -LOG(ERRMAX) 0 1 2 3 4 5 7 -LOG(ERRMAX) Figure 10. 2.2 6	2.4 PAFRKN4 RKN4 MORK4A MODRK4B 2.4 PAFRKN4 RKN4 MORK4A MODRK4B 2.4 PAFRKN4 RKN4 MORK4A MODRK4B 2.4 PAFRKN4 RKN4 MORK4A MODRK4B	2.6 WOODS-SAXON POTENTIAL WITH E=-49.457788728 2.8 3 3.2 2.6 2.8 3 3.2 3.4 WOODS-SAXON POTENTIAL WITH E=-41.232607772 2.6 2.8 3 3.2 3.4 WOODS-SAXON POTENTIAL WITH E=-26.873448915 2.6 2.8 3 3.2 3.4 WOODS-SAXON POTENTIAL WITH E=-8.676081670	3.4 3.6 3.6 3.6	3.6 3.8 3.8 3.8
					LOG(FUNCTION EVALUATIONS) LOG(FUNCTION EVALUATIONS) LOG(FUNCTION EVALUATIONS) LOG(FUNCTION EVALUATIONS)
	Figure 5. Efficiency for the Schrödinger equation using E=-49.457788728 Figure 7. Efficiency for the Schrödinger equation using E=-41.232607772 Figure 9. Efficiency for the Schrödinger equation using E=-26.873448915 Figure 11. Efficiency for the Schrödinger equation using E=-8.676081670
			papdopoulos-anastassi-simos_rev.tex; 13/11/2009; 8:53; p.11 papdopoulos-anastassi-simos_rev.tex; 13/11/2009; 8:53; p.12 papdopoulos-anastassi-simos_rev.tex; 13/11/2009; 8:53; p.13 papdopoulos-anastassi-simos_rev.tex; 13/11/2009; 8:53; p.14

papdopoulos-anastassi-simos_rev.tex; 13/11/2009; 8:53; p.3

papdopoulos-anastassi-simos_rev.tex; 13/11/2009; 8:53; p.5

papdopoulos-anastassi-simos_rev.tex; 13/11/2009; 8:53; p.7

papdopoulos-anastassi-simos_rev.tex; 13/11/2009; 8:53; p.9

papdopoulos-anastassi-simos_rev.tex; 13/11/2009; 8:53; p.15

Acknowledgements

We want to thank the anonymous reviewers for their careful reading of the manuscript and their fruitful comments and suggestions.