Giovanni ; Storto

Eugenio Di Giorgio

Amalia De Renzo

Laura Micol

Giuseppe Cerciello

Anna Nardelli

Daniela Capacchione

Elena Castaldi

Giovanni Ortosecco

Leonardo Pace

Eugenio Di

Assessment of metabolic activity by PET-CT with F-18-FDG in patients with T-cell lymphoma

Keywords: T-cell lymphomas, FDG, metabolic activity, standard uptake value, positron emission tomography

INTRODUCTION

T-cell lymphomas (TL) comprise a heterogeneous group of uncommon diseases accounting for approximately 10-15% of all non-Hodgkin's lymphomas. Compared to B-cell lymphomas, TL more frequently involve extranodal sites and have a worse prognosis. The staging of patients with malignant lymphoma take account of physical examination, instrumental procedures and results from peripheral blood and bone marrow involvement, if any. Recently, 18

DESIGN AND METHODS

Patients

Twenty patients (7 women; 54.9 ± 16.5 y) [Table 1] with histologically proven T-cell lymphoma underwent baseline PET/CT scan 2-3 weeks prior to start a first line therapy. All patients underwent neck echography, chest, abdomen and pelvis CT and bone marrow biopsy. Diagnoses were made according to the WHO classification. Number and dimensions of involved nodal and extra-nodal sites (regions) were evaluated at PET/CT (as per patient tumoral burden). Results of PET/CT scans obtained from 20 patients (7 women; 53.6 ± 13.9 y) with indolent B-NHL and 20 patients (7 women; 54.1 ± 14.5 y) presenting extra-nodal aggressive B-NHL served as control. All patients signed informed consent in accordance with the Declaration of Helsinki.

Immunohistochemical staining

Ki-67-stained sections of tissue specimens from cases of T-cell lymphoma were available to determine the distribution of proliferating cells and proliferation rate of cells. A monoclonal mouse anti-human ki-67 antigen, clone MIB-1 (Dakocytomation, Danmark) was used. Ki-67 staining was assessed by subjectively evaluating the percentage of positive cells.

Imaging modalities

PET and CT were carried out with a commercial combined PET/CT scanner (Discovery LS; GE Milwaukee, WI, USA) combining an Advance NXi PET scanner and a Light Speed Plus four rows Raw PET data were reconstructed with and without attenuation correction into transverse, sagittal, and coronal images. Attenuation correction was based on CT attenuation coefficients, which were determined by iterative reconstruction. All images were reviewed at a workstation by using PET/CT fusion software (Volumetrix for PET, GE). Each PET/CT study was interpreted by three experienced observers in consensus; one of them was also a radiologist. The examiners first evaluated the CT images alone. Mass sizes (nodal or extra-nodal) were visually estimated and the most significant was measured at least in maximum diameter by using a vendor-provided software. A lesion in the baseline study was defined as an identifiable mass or lymph node larger than 1 cm in minimum diameter with soft-tissue/mediastinal window settings. Up to 6 hottest (representative) lesions per patient were considered. Then, maximum standardized uptake values, body weight corrected (SUVmax), were determined by using vendor-provided software on PET scans, according to the European Organization

Statistical analysis

Data are expressed as proportions or as means ± 1 standard deviations, as appropriate. One-way ANOVA was used for comparing means between groups. Post hoc analysis with Bonferroni correction was performed. Differences between mean values were assessed by Student's t test (twotailed probability) for unpaired data. Linear regression analysis was used to assess the relationship between mean SUVmax and Ki-67 values, on a patient basis. A probability (p) value < 0.05 was considered significant.

RESULTS

Patient characteristics of each group (study and controls) are summarized in Table 1

Tumor burden and Ki-67 in TL

Analyzing patient tumour burden, a mean number of 3.8 ± 2 and of 4.7 ± 2 sites were involved in the group with TL and with indolent B-NHL, respectively (p=ns) whereas extra-nodal aggressive B-NHL had 1.5 ± 0.4 sites involved (p<0.05 vs TL and vs indolent B-NHL). Tissue specimens were collected from 13 nodal locations, 4 cutaneous/subcutaneous lesions, 2 from bone marrow and one from gastric involvement. Mean Ki-67 was 64% ± 15% in patients with TL (range: 30%-80%). On a patient basis, No relationship between individual mean SUVmax and Ki-67 values was observed (r = 0.1; p = 0.6).

Semiquantitative assessment in TL patients and controls

Considering all detected representative lesions, the mean SUVmax in patients with TL was lower as compared to either indolent and extra-nodal high grade B-NHL (6.1 ± 4 vs 9.1 ± 6, and vs 8.9 ± 5, respectively; p< 0.01) (Figure 1) .

When the SUVmax was assessed only just in the extra-nodal locations, the TL group still presented lower values than the other groups (5.3 ± 3 vs 9.5 ± 6 and 8.9 ± 5, respectively; p< 0.05)

(Figure 2A) (Fig 1A).
Finally, the SUVmax in nodal locations of patients with TL was lower than in nodal indolent

DISCUSSION

In the present study we found that patients with TL had a significant but low metabolic activity at PET scan, regardless of a considerable tumour burden, an aggressive clinical behaviour and high values of Ki-67. These patients had a lower uptake index in the involved regions than patients presenting either indolent or aggressive B-NHL. Our setting showed a distinctive involvement of extra-nasal sites, mostly cutaneous, visceral or with bone marrow locations and some patients had nodal concern. All subjects had a significant tumour burden, an aggressive behaviour and a positive scan. Nevertheless, comparing the degree of tracer uptake between extra-nodal and nodal locations of TL patients was not different. (i.e. supposed disparities in blood support), we were not able to detect differences, being it significant but moderate in both districts.) Irrespective of locations involved, mechanisms of metabolic deregulation during the genesis of T-cell lymphomas and cell transformation might be more complex than merely meeting the cellular requirements of growth and proliferation.

Although clear evidence is lacking at this time, biologic differences among specific pathologic subtypes of lymphoma can be evocated in determining different degrees of FDG uptake. For instance, T-cell lymphomas histologically can show a broad cytologic spectrum even if apoptosis and necrosis are common (Cheung et al, 2003), both affecting FDG uptake. Higher activity of glucose-6phosphatase might determine a lower FDG uptake as in some histological subtypes of Hodgkin's disease (Reed-Sternberg and Hodgkin cells) and in hepatocellular carcinoma (Pangalis & Tsavaris,

1986; Torizuka et al, 1995).

There is the evidence that some TL cells mirror, histo-pathologically, the mononuclear cells or even the typical Sternberg-Reed cells of Hodgkin's disease [START_REF] Whitcomb | T-cell lymphoma-leukemia. Pathologic observations in three cases[END_REF]). Finally, it should be considered the expression of membrane glucose transporters, Glut-1 (Aloj et al, 1999) or the role of lectins, a group of specific membrane proteins that preferentially bind to carbohydrates inside and outside the cells, playing a critical function in cancer biology and in various SUV values obtained at extra-nodal level in TL were compared with those of sex-and age matched patients having extra-nodal aggressive lymphoma; once more the degree of FDG uptake in TL was significantly lower, confirming that reasons other than aggressiveness are implicated in cellular tracer accumulation.

Although there still may be considerable overlap between SUVs of benign tissue and those of malignant tissue, we quantified the intensity of FDG activity using a SUV cut-off of around 2.5 to 3.0 which has been indicated evocative of malignant involvement (Lucignani et al, 2004).

Our findings suggest a careful quantitative evaluation of FDG PET images in such disease as not always the higher the proliferation the higher the metabolic activity. The (As) accurate staging of TL is essential for providing the patient with appropriate prognostic information and for guiding the clinician in selecting the treatment option according to the tumour metabolic activity which seems not directly related to the proliferation. As a result, alternative drugs such as receptor-selective, immune-modulating or apoptosis-inducing agents rather than those disrupting cancer growth by direct cytotoxic effect could be preferred (see targeted therapy).

The analysis of additional patients will provide further information for FDG-PET impact on the management of such disease. In addition, the comprehension of the exact cellular mechanisms sustaining different metabolic profiles in various lymphomas will be of crucial value.

F

  F-fluorodeoxyglucose positron emission tomography-computed tomography (PET/CT) has been widely recognized as a suitable diagnostic tool in staging (Jerusalem et al, 2001a), follow-up and tumour response assessment of lymphoma patients (Mikhaeel et al, 2000; Naumann et al, 2004; Zinzani et al, 2004; Castellucci et al, 2005); its prognostic role has been evoked (Jerusalem et al, 1999; Mikhaeel et al, 2005; Haioun et al, 2005; Cheson et al, 2007). PET/CT offers the advantage of functional tissue characterization, which is mostly independent from morphologic criteria assessing the extent of the disease; it has higher sensitivity than conventional anatomic imaging modalities thus often redirecting patient management. Different 18 F-fluorodeoxyglucose (FDG) uptake among different subtypes of lymphoma has been reported (Schoder et al, 2005; Wong et al, 2005; Tsukamoto et al, 2007), which could be explained by the complex histological profile including the proliferating capacity of tumour cells, the proportion of viable tumour cells and the presence of reactive non-malignant cells (Kubota et al, 1993). The evidence that the degree of FDG accumulation correlates well with the percentage of proliferative cells in the biopsied samples rather than with the histologic grade has been reported mainly in B-cell non-Hodgkin's lymphomas (Laskinen-Kallio et al, 1991). Conversely, only few data have pointed about the amount of tracer uptake and the role of PET/CT in staging T-cell neoplasms which usually exhibit aggressive clinical behaviour (Kako et al, 2007; Karantanis et al, 2008; Khong et al, 2008; Tang et al, 2009). Accordingly, we evaluated the results from the semiquantitative assessment of a staging PET/CT in patients with histologically proven T-cell the findings to Ki-67 proliferation index and to those obtained in patients presenting indolent-or aggressive B-cell non-Hodgkin lymphoma. This imaging modality might have a role for tailoring therapy in such setting redirecting to alternative drugs or new combination treatment algorithms on a metabolic assessment basis.

  PET/CT imaging was acquired 75-90 minutes after intravenous administration of 12-15 mCi (444-555 MBq) of FDG. During the 4-6 hours prior to imaging patients were fasting. Blood glucose level was determined in all patients before FDG administration and a cut-off value of less than (160 mg/dL) 8.88 mmol/L was considered appropriate to perform examination. Helical CT (pitch 1.5; 120 mAs; 120 kVp) was performed without intravenous contrast medium. PET scan was performed with four minutes per bed position and six to eight bed positions per patient, depending on patient height. Raw computed tomography (CT) data were reconstructed into transverse images with a 4.25-mm section thickness. Sagittal and coronal CT images were generated by reconstruction of transverse data.

for

  Research and Treatment of Cancer's (EORTC) PET recommendations (Young et al, 1999) and to recent upgraded guidelines (Cheson et al, 2007; Juweid et al, 2007). Briefly, focal or diffuse FDG uptake above background in a location mismatched with normal anatomy or physiology was interpreted as abnormal and was considered to be (evocative) indicative of a lymphoma lesion. The /CT was considered positive if PET/CT detected a lymphoma lesion in at least one site detected by CT series or by physical examination.

  according to the prevalence of the histological subtypes. (Mean) The Number of extra-nodal and nodal sites per subtype (are) is also shown. PET/CT scan was positive in all patients of the study and control groups. CT scans performed as a part of PET/CT study allowed to identify extra-nodal and nodal sites in all groups of patients. ; (most representative lesions were measured at least in maximum diameter, those of TL patients are recorded in Table 2) (Classification of TL patients and controls According to the WHO classification the histological subtypes of TL patients were: peripheral T-cell lymphoma, unspecified (PTCL-U, n = 9), anaplastic large cell lymphoma (ALCL, n = 6), mycosis fungoides (MF, n = 3), extranodal NK/T-cell lymphoma, nasal type (ENKL, n = 1), angioimmunoblastic T-cell lymphoma (AILT, n = 1). ) Bone marrow biopsy was positive in 11/20 patients with TL. Fourteen patients with TL were classified as having a stage IV disease, three a stage a stage II. A total of 12 bone marrow lesions were detected, eight cutaneous/subcutaneous lesions were recorded in MF patients, one in a ALCL patient. (In the group of indolent NHLs, 18 patients were classified as having follicular lymphoma and two as extranodal marginal zone B-cell lymphoma (EMZL). Among patients with extra-nodal aggressive NHL, 17 had diffuse large B-cell lymphoma, two showed blastic EMZL and one patient had a intravascular large B-cell lymphoma.) In patients with indolent NHL 6/20 had both extra-nodal and nodal sites involved, 8/20 had nodal and 6/20 exclusive extranodal involvement.

  (6.4 ± 3 vs 8.8 ± 6; p< 0.05) (Figure2B) (Fig1B). Within the TL or indolent B-NHL groups there was no difference in SUVmax values between extra-nodal and nodal sites. In TL patients the mean SUVmax of bone marrow lesions was 6.0 ± 3, it was not different from that of the other extra-nodal lesions grouped (5.0 ± 3; p=ns).No differences were found between SUVmax values of indolent B-NHL and extra-nodal aggressive B-NHL.

(

  Although several studies have strengthened the value of FDG-PET in the assessment of malignant lymphoma at diagnosis [1,2,6,18] , its role in revealing different amount of FDG uptake among various histological subtype of lymphoid malignancies is still under evaluation. [10, 19-23] As a result, the histological subtype of malignancy should be carefully considered in assessing PET-FDG results.) Natural killer (NK)/T-cell lymphomas represent a rare type of lymphoma derived from either activated NK cell or, rarely, cytotoxic T cells. It usually originates in the nasal cavity/nasopharynx and invades the surrounding tissues; a delay in diagnosis could result in a fatal outcome. T-cell neoplasms frequently display aggressive clinical behaviour and a prognosis not as good as B-cell neoplasms, (Rudiger et al, 2002; Mounier et al, 2004 ) mostly because of the high frequency of advanced-stage disease and extranodal involvement, as a consequence of intrinsic histopathological features. The

  this tumour emphasizes the need for more effective and useful diagnostic tools in order to implement beneficial therapeutic strategies. There are only few published data substantiating the use of PET/CT in patients presenting Tcell lymphoma (Kako et al, 2007; Khong et al, 2008; Suh et al, 2008) being it alternatively allotted as a valuable or useless tool in such disease (Karantanis et al, 2008; Khong et al, 2008; Tang et al, 2009). Our data confirm the findings of others who demonstrated a high positive rate of pre-therapeutic PET/CT in most subtypes of T-cell neoplasms, comparable to what observed for B-cell neoplasms (Haioun et al, 2005). All patients enrolled in the present study showed significant amount of tracer uptake at PET-CT scan. Previous studies have demonstrated a substantial overlap in the intensity of FDG uptake between patients with indolent and aggressive B-cell lymphoma in the low SUV range (Rodriguez et al,1995; Lapela et al, 1995). However, Schoder et al. (2005) reported that the intensity of FDG uptake is generally lower in indolent as compared with aggressive non-Hodgkin's lymphoma and that the likelihood for aggressive disease increases in parallel with increases in SUV. This is not the case of TL, in which mechanisms other than those sustaining aggressiveness should be considered. In our setting, for instance, mean SUVmax of lesions was constantly low, though significant, either for nodal or extra-nodal involved districts while TL patients showed high values of Ki-67 which is an established marker of cell proliferation. (Uptake of FDG in lymphoma is influenced by many different biological factors, including molecular, such as oncogene expression as well as patho-physiological aspects such as tumor perfusion, apoptosis and viable cell fraction, effects of hypoxia, amount of inflammatory cell infiltration and lastly by substrate utilization. However,) There are some evidences (Kako et al, 2007; Karantanis et al, 2008) that PET/CT appears to be less sensitive for cutaneous and bone marrow involvement in TL and, in general, for the detection of extranasal sites. On the other hand, et al, 2008; Berk et al, 2008) have focused on the potential of PET/CT for improving the staging and the monitoring of response to therapy in TL. This was proven for cutaneous, nodal, and visceral disease in cutaneous T-cell lymphoma as well as for detection of subcutaneous nodules in extranodal nasal NK/T-cell lymphomas, highlighting the important role of FDG-PET in the staging of these diseases.

  Taking into account the above mentioned issues, we aimed to compare FDG avidity of extra-nodal and nodal locations of TL patients with that detected in both extra-nodal and nodal locations of sex-and age matched patients presenting indolent B-NHL. Patients with TL showed lower FDG uptake in both nodal and extra-nodal locations as compared to those of indolent B-NHL. Recent data in indolent lowgrade lymphomas (follicular and marginal zone lymphomas) (showed) have shown positive FDG uptake despite a low tumour cell proliferation (Hoffmann et al, 2003; Elstrom et al, 2003; Jerusalem et al, 2001b; Najjar et al, 2001), (detecting metabolic activity in both follicular and marginal zone lymphomas [42, 43] ). The fact that follicular lymphomas almost invariably showed high FDG uptake demonstrates that the histologic grading is not the most important predictor for FDG avidity (Elstrom et al, 2003).

(

  To our knowledge this is the first study comparing the degree of FDG uptake in TL with matched both indolent or aggressive B-NHL. Comparisons with other published studies are most data are derived from solid tumours and nodal-based lymphomas, and no previous studies have been published comparing SUVs in TL and B-NHL.)

Figure 1 .

 1 FIGURE LEGEND

Table 1 .

 1 Patients characteristics

	o
	r
	P
	e e r
	R e v i e w
	13

MF, mycosis fungoides; PTCL-U, peripheral T-cell lymphoma unspecified; ENKL, extra-nodal natural killer/T-cell lymphoma, nasal type; ALCL, anaplastic large cell lymphoma; AILT, angioimmunoblastic T-cell lymphoma; NHL, non-Hodgkin lymphoma; FL, follicular lymphoma; EMZL, extranodal marginal zone B-cell lymphoma; DLBCL, diffuse large B-cell lymphoma; LBCL, large B-cell lymphoma.

Table 2 .

 2 Characteristics of patients with TL at the time of FDG-PET scan PTCL-U, peripheral T-cell lymphoma unspecified; ENKL, extra-nodal natural killer/T-cell lymphoma, nasal type; ALCL, anaplastic large cell lymphoma; AILT, angioimmunoblastic T-cell lymphoma; *, most representative; BM, bone marrow; Pos., positive; Neg., negative

	Pts	Sex	Age	Histology	* Locations	* CT lesion	BM	Stage	PET results
	N°		(yrs)		extranodal/nodal	size (cm)	biopsy
	1	M	55	MF	subcutaneous in	2.1x2.5/	Neg.	IV	+
					chest/right axilla	2.0x2.6
	2	M	59	PTCL-U	left iliac region	2.2x2.4/	Pos.	IV	+
					(bone)/right inguinal	2.3x1.9
	3	F	51	ALCL	-/right iliac region	1.8x1.2	Pos.	IV	+
	4	F	20	PTCL-U	C4 (bone)/ -	2.8x2.5	Pos.	IV	+
	5	M	48	ALCL	-/ant-sup mediastinum	2.3x2.0	Pos.	IVA	+
	6	F	38	PTCL-U	mediastinal tissue/ -	1.9x2.9	Neg.	IIE	+
	7	M	21	ALCL	-/ant-sup mediastinum	1.6x2.7	Pos.	IV	+
	8	F	35	ENKL	posterior rino-farinx/right	1.3x1.4/	Neg.	IIE	+
					submandibular	1.2x2.7
	9	M	57	PTCL-U	right hepatic flexure	1.9x3.0	Pos.	IV	+
					(colon)/ -
	10	M	52	MF	left neck (skin)/bilateral	1.1x1.2/	Neg.	IIIE	+
					inguinal	1.2x1.4
	11	M	67	PTCL-U	posterior naso-farinx/right	1.2x2.0/	Pos.	IV	+
					iliac region	2.3x3.1
	12	M	79	MF	left neck (skin);	1.2x1.0/	Neg.	IV	+
					prostate/bilateral inguinal	1.5x2.7
	13	M	70	PTCL-U	-/ paratracheal	1.7x1.9	Neg.	IIB	+
	14	M	53	AITL	-/left iliac region	1.3x1.9	Pos.	IV	+
	15	F	46	ALCL	-/ant-sup mediastinum	1.7x2.6	Neg.	III	+
	16	F	56	PTCL-U	acromion (bone)/ ant-sup	1.1x1.5/	Pos.	IV	+
					mediastinum	1.5x1.5
	17	M	62	PTCL-U	gastric/paraortic	1.8x3.1/	Neg.	IV	+
						1.3x2.8
	18	M	76	PTCL-U	-/left neck	1.5x2.5	Neg.	III	+
	19	F	68	ALCL	subcutaneous in leg /left	1.6x2.2/	Pos.	IV	+
					iliac region	1.5x1.6
	20	M	63	ALCL	spleen/left neck	1.8x2.5/	Pos.	IV	+
						1.2x1.6
	MF, mycosis fungoides;

British Journal of Haematology

AUTHORSHIP AND DISCLOSURES

The authors have indicated they have no financial conflicts of interest.

All co-authors have reviewed this paper and approve of its publication. In particular, G.S. designed the research, cared for research subjects, collected data, analyzed data, conceived, wrote and reviewed the manuscript. E.D.G., A.D.R., L.M. and G.C. cared for research subjects and reviewed the manuscript. A.N., D.C., E.C., and G.O. collected, analyzed data and reviewed the manuscript. A.N. and E.C. performed radiochemistry procedures, analyzed data and reviewed the manuscript. B.R. and L.P. reviewed the manuscript.

RUNNING HEADS

Assessment of metabolic activity in T-cell lymphoma


François ; Maurier

Ste

Nadine Petitpain
email: n.petitpain@chu-nancy.fr

J-F Guichard

L Javot

P Tréchot

Olanzapine and pancytopenia with severe folate deficiency

To the editor : Olanzapine, a thienobenzodiazepine serotonin-dopamine antagonist, is a recent atypical antipsychotic agent, the structural and pharmacologic profile is similar to that of clozapine.

Although clozapine is effective in refractory schizophrenia, its use is limited by the occurrence of hematologic adverse effects, such as agranulocytosis which occurs in 0.5%-2% of treated patients [1]. Many studies have suggested a better hematologic safety profile for olanzapine, but impaired granulocytopoiesis and thrombocytopenia have recently been reported [2][3][4][5][6]. No case of pancytopenia with megaloblastic anemia has been described with the use of olanzapine. We report a case of pancytopenia with severe folate deficiency in a schizophrenic patient treated with olanzapine.

A 56-year-old man, who worked as a painter, was a heavy smoker, had bad dental health, and a history of chronic psychosis was hospitalized on 11 August 2008 with severe pancytopenia, suggested by purpura. He had been living with his brother and had been treated for many years with haloperidol decanoate (3 mg every 3 weeks), loxapine (100 mg/day), amisulpride (800 mg/day), tropatepine (10 mg/day), diazepam (10 mg/day), and clorazepate (50 mg/day).

Because of recent worsening of his psychotic symptoms, olanzapine was introduced (20 mg/day) in May 2008.

The laboratory data on admission and during the early period of hospitalization are shown in Table 1. The hemogram and reticulocyte count revealed pancytopenia with macrocytic normochromic aregenerative anemia. The folate plasmatic level was dramatically decreased, total protein and cobalamin were slightly decreased and renal and hepatic functions were normal. Shortly after admission, only olanzapine was discontinued. The patient received probabilist cobalamin supplementation with no immediate hematologic improvement. On 14 August, he began to receive intravenous folate supplementation (50 mg/day). Ten days later, a huge reticulocyte crisis was observed, whereas the white blood cell and platelet counts were significantly increased (Table 1). Meanwhile, the results of bone marrow aspiration confirmed the diagnosis of megaloblastic anaemia. The patient was supplemented with folic acid over 2 weeks. Because the medical staff decided against restarting olanzapine, the loxapine dosage was increased to 150 mg/day with a good response. Computerized tomography imaging showed hepatic abnormalities consistent with portal hypertension. On 4 September, the macrocytic anemia recovered and folate supplementation was discontinued. One month after admission, gastric endoscopy revealed grade I peptic esophagitis without esophageal varix. The biopsies were all negative, thus ruling out celiac disease. At this time, his laboratory data only showed a slight macrocytosis with no anemia. At the 3-month followup, olanzapine had not been restarted and the patient had a perfect clinical examination, despite discontinuation of folate supplementation.

Cases involving pancytopenia with olanzapine are rare [2], and no case of megaloblastic anemia has been reported with this drug. For our patient, we observed pancytopenia due to folate deficiency. The sequence of events strongly suggests that olanzapine was responsible for this hematologic disorder with i) a 3-month delay of onset, ii) the absence of recurrence after drug withdrawal, despite cessation of folate supplementation, and iii) delayed reticulocytosis after vitamin supplementation .

Folate deficiency is usually linked to decreased intake or decreased proximal small bowel absorption. The poor dental health of our patient is likely to have slowly favored decreased folate intake, but endoscopic examination did not reveal abnormalities consistent with impaired small bowel absorption. However, chronic inadequate folate intake is not sufficient to explain the association of pancytopenia that occurred 3 months after the beginning of olanzapine treatment. Other hypotheses, based on anticonvulsant and recent cough mixture cases of drug-induced folate deficiency imply methylenetetrahydrofolate reductase (MTHFR) polymorphism and hyperhomocysteinemia [7,8]. Unfortunately, neither MTHFR phenotype and homocysteine level were determined in our patient. Olanzapine, with less neutrophil toxicity than clozapine, is expected to be a safer alternative. However, olanzapine can induce impaired granulocytopoiesis and prolonged granulocyte depression in patients switched from clozapine to olanzapine [3]. The mechanism underlying olanzapine-induced neutrophil toxicity remains unclear. Toxicity of olanzapine on granulocyte-macrophage progenitors has been demonstrated in vitro [9], and like other neutropenic drugs, olanzapine generates a reactive metabolite that is able to bind to human neutrophils [1,10].

The link between olanzapine treatment and thrombocytopenia is less well documented [4][5][6].

One case of fatal thrombocytopenia 3 weeks after the start of olanzapine was reported in a 78year-old patient with a history of idiopathic thrombocytopenic purpura and mild renal insufficiency [4]. This patient had a plasma level of olanzapine 10 times higher than expected.

In our patient, renal function was normal and the plasma level of olanzapine was not determined on admission. It is noteworthy that our patient received other treatment when pancytopenia was diagnosed. Haloperidol, loxapine, and amisulpride are associated with hematologic adverse effects, like many other neuroleptics. As none of these medications were discontinued, it is unlikely that haloperidol, loxapine, or amisulpride were causal. Moreover, the loxapine dosage was increased after olanzapine withdrawal with no recurrence of any blood cell dyscrasia. This first case of pancytopenia with severe folate deficiency associated with olanzapine underlines the necessity of hematologic controls during the administration of this drug. To the editor :

Olanzapine, a thienobenzodiazepine serotonin-dopamine antagonist, is a recent atypical antipsychotic agent, the structural and pharmacologic profile is similar to that of clozapine.

Although clozapine is effective in refractory schizophrenia, its use is limited by the occurrence of hematologic adverse effects, such as agranulocytosis which occurs in 0.5%-2% of treated patients [1]. Many studies have suggested a better hematologic safety profile for olanzapine, but impaired granulocytopoiesis and thrombocytopenia have recently been reported [2][3][4][5][6]. No case of pancytopenia with megaloblastic anemia has been described with the use of olanzapine. We report a case of pancytopenia with severe folate deficiency in a schizophrenic patient treated with olanzapine.

A 56-year-old man, who worked as a painter, was a heavy smoker, had bad dental health, and a history of chronic psychosis was hospitalized on 11 August 2008 with severe pancytopenia, suggested by purpura. He had been living with his brother and had been treated for many years with haloperidol decanoate (3 mg every 3 weeks), loxapine (100 mg/day), amisulpride (800 mg/day), tropatepine (10 mg/day), diazepam (10 mg/day), and clorazepate (50 mg/day).

Because of recent worsening of his psychotic symptoms, olanzapine was introduced (20 mg/day) in May 2008.

The laboratory data on admission and during the early period of hospitalization are shown in Table 1. The hemogram and reticulocyte count revealed pancytopenia with macrocytic normochromic aregenerative anemia. The folate plasmatic level was dramatically decreased, total protein and cobalamin were slightly decreased and renal and hepatic functions were normal. Shortly after admission, only olanzapine was discontinued. The patient received probabilist cobalamin supplementation with no immediate hematologic improvement. On 14

August, he began to receive intravenous folate supplementation (50 mg/day). Ten days later, a huge reticulocyte crisis was observed, whereas the white blood cell and platelet counts were significantly increased (Table 1). Meanwhile, the results of bone marrow aspiration confirmed the diagnosis of megaloblastic anaemia. The patient was supplemented with folic acid over 2 weeks. Because the medical staff decided against restarting olanzapine, the loxapine dosage was increased to 150 mg/day with a good response. Computerized tomography imaging showed hepatic abnormalities consistent with portal hypertension. On 4 September, the macrocytic anemia recovered and folate supplementation was discontinued. One month after admission, gastric endoscopy revealed grade I peptic esophagitis without esophageal varix. The biopsies were all negative, thus ruling out celiac disease. At this time, his laboratory data only showed a slight macrocytosis with no anemia. At the 3-month followup, olanzapine had not been restarted and the patient had a perfect clinical examination, despite discontinuation of folate supplementation.

Cases involving pancytopenia with olanzapine are rare [2], and no case of megaloblastic anemia has been reported with this drug. For our patient, we observed pancytopenia due to folate deficiency. The sequence of events strongly suggests that olanzapine was responsible for this hematologic disorder with i) a 3-month delay of onset, ii) the absence of recurrence after drug withdrawal, despite cessation of folate supplementation, and iii) delayed reticulocytosis after vitamin supplementation .

Folate deficiency is usually linked to decreased intake or decreased proximal small bowel absorption. The poor dental health of our patient is likely to have slowly favored decreased folate intake, but endoscopic examination did not reveal abnormalities consistent with impaired small bowel absorption. However, chronic inadequate folate intake is not sufficient to explain the association of pancytopenia that occurred 3 months after the beginning of olanzapine treatment. Other hypotheses, based on anticonvulsant and recent cough mixture cases of drug-induced folate deficiency imply methylenetetrahydrofolate reductase (MTHFR) polymorphism and hyperhomocysteinemia [7,8]. Unfortunately, neither MTHFR phenotype and homocysteine level were determined in our patient. Olanzapine, with less neutrophil toxicity than clozapine, is expected to be a safer alternative. However, olanzapine can induce impaired granulocytopoiesis and prolonged granulocyte depression in patients switched from clozapine to olanzapine [3]. The mechanism underlying olanzapine-induced neutrophil toxicity remains unclear. Toxicity of olanzapine on granulocyte-macrophage progenitors has been demonstrated in vitro [9], and like other neutropenic drugs, olanzapine generates a reactive metabolite that is able to bind to human neutrophils [1,10].

The link between olanzapine treatment and thrombocytopenia is less well documented [4][5][6].

One case of fatal thrombocytopenia 3 weeks after the start of olanzapine was reported in a 78year-old patient with a history of idiopathic thrombocytopenic purpura and mild renal insufficiency [4]. This patient had a plasma level of olanzapine 10 times higher than expected.

In our patient, renal function was normal and the plasma level of olanzapine was not determined on admission. It is noteworthy that our patient received other treatment when pancytopenia was diagnosed. Haloperidol, loxapine, and amisulpride are associated with hematologic adverse effects, like many other neuroleptics. As none of these medications were discontinued, it is unlikely that haloperidol, loxapine, or amisulpride were causal. Moreover, the loxapine dosage was increased after olanzapine withdrawal with no recurrence of any blood cell dyscrasia.

This first case of pancytopenia with severe folate deficiency associated with olanzapine underlines the necessity of hematologic controls during the administration of this drug.

Table 1 :

 1 Summary of laboratory data

		Units/	11/8 13/8 14/8 18/8 20/8 25/8 27/8	2/9	18/9
		[normal
		range]
	Hemoglobin	g/l	82	91	85	75	85	71	83	97	129
		[130-180]
	MGV	Fl	124.6 111.1 104.2 104 104.8 118.2 116.3 112.9 102
		[78-94]
	White blood	10 9 /l	1.1	1.7	1.58		3.89 12.12 17.77 5040 6100
	cell count	[4-11]
	Platelets	10 9 /l	39	28	27	16	52	174	323	507	121
		[150-400]
	Reticulocytes	10 9 /l*			15.2			368	158 235.6
	Cobalamin	ng/l	141
		[>300]
	Folate	nmol/l	0							>45.4
	(serum)	[4.5-21]
	Folate	nmol/l								>3100
	(erythrocytes)	[400-1300]
	Protein	g/l	65				61
		[65-80]
	Creatinine	mg/l	10.3				8				10
		[5-12]
	*: reticulocytes>80.10 9 /l are consistent with regenerative anemia
	The authors declare that they have no conflict of interest.

Table 1 :

 1 Summary of laboratory data : reticulocytes>80.10 9 /l are consistent with regenerative anemia The authors declare that they have no conflict of interest.

		Units/	11/8 13/8 14/8 18/8 20/8 25/8 27/8	2/9	18/9
		[normal
		range]
	Hemoglobin	g/l	82	91	85	75	85	71	83	97	129
		[130-180]
	MGV	Fl	124.6 111.1 104.2 104 104.8 118.2 116.3 112.9 102
		[78-94]
	White blood	10 9 /l	1.1	1.7	1.58		3.89 12.12 17.77 5040 6100
	cell count	[4-11]
	Platelets	10 9 /l	39	28	27	16	52	174	323	507	121
		[150-400]
	Reticulocytes	10 9 /l*			15.2			368	158 235.6
	Cobalamin	ng/l	141
		[>300]
	Folate	nmol/l	0							>45.4
	(serum)	[4.5-21]
	Folate	nmol/l								>3100
	(erythrocytes)	[400-1300]
	Protein	g/l	65				61
		[65-80]
	Creatinine	mg/l	10.3				8				10
		[5-12]

*

European Journal of Clinical Pharmacology


Orazio Caffo
email: orazio.caffo@apss.tn.it

Stefania Fallani

Elena Marangon

Stefania Nobili

Maria Iris Cassetta

Viviana Murgia

Federica Sala

Andrea Novelli

Enrico Mini

Massimo Zucchetti

Enzo Galligioni

Á S Nobili

Á M I Cassetta

Á A Novelli

Á E Mini

Á F Sala

Á M Zucchetti

Pharmacokinetic study of gemcitabine, given as prolonged infusion at fixed dose

Keywords: Non-small-cell lung cancer, Chemotherapy, Cisplatin, Gemcitabine, Prolonged infusion, Pharmacokinetics

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Gemcitabine, 2 0 ,2 0 -difluoro-2 0 -deoxycytidine (dFdC) is a pyrimidine antimetabolite, with a broad spectrum of antitumor activity [START_REF] Mini | Cellular pharmacology of gemcitabine[END_REF], representing one of the reference drugs in combination chemotherapy of non-small-cell lung cancer (NSCLC) [START_REF] Sandler | Phase III trial of gemcitabine plus cisplatin versus cisplatin alone in patients with locally advanced or metastatic non-small-cell lung cancer[END_REF].

Gemcitabine is a pro-drug that enters the cell by means of nucleoside transporters and becomes active through an intracellular transformation catalyzed by deoxycytidine kinase to the final triphosphate form (dFdCTP) in a ratelimiting reaction [START_REF] Mini | Cellular pharmacology of gemcitabine[END_REF][START_REF] Abbruzzese | A phase I clinical, plasma, and cellular pharmacology study of gemcitabine[END_REF][START_REF] Grunewald | Saturation of 2 0 , 2 0 -difluorodeoxycytidine 5 0 -triphosphate accumulation by mononuclear cells during a phase I trial of gemcitabine[END_REF]. Gemcitabine pharmacokinetics in humans are characterized by rapid elimination and extensive deamination to its inactive 2 0 ,2 0 -difluorodeoxyuridine (dFdU) that is the main metabolite present in plasma [START_REF] Abbruzzese | A phase I clinical, plasma, and cellular pharmacology study of gemcitabine[END_REF][START_REF] Grunewald | Saturation of 2 0 , 2 0 -difluorodeoxycytidine 5 0 -triphosphate accumulation by mononuclear cells during a phase I trial of gemcitabine[END_REF].

Gemcitabine is usually administered intravenously as a 30-min infusion at the dose of 1,000-1,200 mg/m 2 , but previous studies suggest that under these conditions, the formation of dFdCTP in normal peripheral blood mononuclear cells (PBMC) is saturated when gemcitabine concentrations of 15-20 lM are achieved in plasma [START_REF] Abbruzzese | A phase I clinical, plasma, and cellular pharmacology study of gemcitabine[END_REF][START_REF] Grunewald | Saturation of 2 0 , 2 0 -difluorodeoxycytidine 5 0 -triphosphate accumulation by mononuclear cells during a phase I trial of gemcitabine[END_REF]. Gemcitabine delivered as a fixed dose rate (FDR) infusion of 10 mg/m 2 /min has been suggested to be more effective than when administered as the standard 30-min infusion [START_REF] Abbruzzese | A phase I clinical, plasma, and cellular pharmacology study of gemcitabine[END_REF][START_REF] Grunewald | Saturation of 2 0 , 2 0 -difluorodeoxycytidine 5 0 -triphosphate accumulation by mononuclear cells during a phase I trial of gemcitabine[END_REF][START_REF] Tempero | Randomized phase II comparison of dose-intense gemcitabine: 30-min infusion and fixed dose rate infusion in patients with pancreatic adenocarcinoma[END_REF]. This schedule seems adequate for maintenance of the plasma target level for a longer period of time [START_REF] Tempero | Randomized phase II comparison of dose-intense gemcitabine: 30-min infusion and fixed dose rate infusion in patients with pancreatic adenocarcinoma[END_REF][START_REF] Santini | A phase I trial of fixed dose rate gemcitabine plus capecitabine in metastatic cancer patients[END_REF], but the available pharmacokinetic data are still too limited to draw definitive conclusions.

In terms of clinical outcomes, the picture is unclear. For example, in pancreatic cancer, despite the promising results from a randomized phase II study showing a survival advantage for patients treated with FDR infusion [START_REF] Tempero | Randomized phase II comparison of dose-intense gemcitabine: 30-min infusion and fixed dose rate infusion in patients with pancreatic adenocarcinoma[END_REF], a more recent report failed to confirm the same advantage for FDR infusion that led an increased toxicity [START_REF] Poplin | Phase III, randomized study of gemcitabine and oxaliplatin versus gemcitabine (fixed-dose rate infusion) compared with gemcitabine (30-min infusion) in patients with pancreatic carcinoma E6201: a trial of the eastern cooperative oncology group[END_REF].

We have therefore conducted a phase II study in advanced NSCLC patients, aimed to investigate the pharmacokinetics of gemcitabine and dFdU administered as a FDR of 10 mg/m 2 /min at doses of 600 and 1,200 mg/m 2 in combination with 75 mg/m 2 of cisplatin.

Methods

Patients, study design and objectives

On the basis of results from our previous dose-escalating study with cisplatin and gemcitabine administered at 7 different dose levels (from 600 to 1,200 mg/m 2 ) [START_REF] Caffo | Prolonged infusion (PI) of gemcitabine (G) in combination with cisplatin (C) in patients with advanced non-small cell lung cancer (NSCLC): preliminary results of a dose-finding and pharmacokinetic (PK) study[END_REF] as prolonged FDR in advanced NSCLC, we planned to explore both the plasmatic and intracellular pharmacokinetics of gemcitabine at the lower (600 mg/m 2 ) and higher (1,200 mg/m 2 ) levels of dose.

From July 2005 to February 2006, we enrolled 8 patients affected by advanced NSCLC and treated at the Department of Oncology of Santa Chiara Hospital, Trento, Italy. The treatment schedule was planned to evaluate both intraand inter-patient pharmacokinetic variability of gemcitabine given as a prolonged FDR infusion in combination with cisplatin.

According to the study plan, the patients were divided into two groups receiving different initial doses of the drug: 4 patients treated with 600 mg/m 2 gemcitabine 60-min i.v. infusion (Group A), and 4 patients treated with 1,200 mg/m 2 gemcitabine 120-min i.v. infusion (Group B), both at FDR of 10 mg/m 2 /min on days 1 and 8 of a 21-day cycle (at first cycle). At the second cycle, all patients received 1,200 mg/m 2 gemcitabine 120-min i.v. infusion (FDR of 10 mg/m 2 /min) on days 1 and 8 of a 21-day cycle.

At each cycle, gemcitabine was administered alone on day one and in combination with 75 mg/m 2 of cisplatin on day 8. Cisplatin was infused for 30 min, immediately before the administration of gemcitabine.

All patients, in absence of progressive disease, received further 4 courses of gemcitabine 1,200 mg/m 2 on days 1 and 8 plus cisplatin 75 mg/m 2 on day 8 for a total number of 6 cycles: in these courses, no pharmacokinetic analyses were planned.

All of the patients gave their written informed consent to participate in the study, which was approved by the local Ethics Committee.

Pharmacokinetic study

Blood samples were collected at the following times: preinfusion, 15 and 30 min during the infusion, at the end of the infusion, 30 min, 2, 4, and 24 h after the end of the infusion.

Blood was withdrawn by venipuncture, immediately placed into heparinized tubes containing 50 lL of the cytidine deaminase inhibitor tetrahydrouridine (10 mg/mL, in bi-distilled water) (Calbiochem, La Jolla, CA, USA) to prevent the spontaneous deamination of gemcitabine to dFdU and placed in an ice bath until following separation of plasma and PBMC. After centrifugation at 4°C for 10 min at 2,500 rpm, the plasma fraction was separated and stored frozen at -20°C until analysis of gemcitabine and dFdU. Then the cellular fraction, kept at 4°C, was suspended (1:1) with phosphate buffered saline (PBS) and layered over 5 ml Ficoll-Hypaque (Pharmacia, Stockholm, Sweden) (specific gravity 1.077 g/mL) (2:1 v/v) and centrifuged at 1,000 g for 25 min at 4°C. The buffy coat containing PBMC was collected, washed twice with 20 mL ice-cold PBS (4°C) and centrifuged at 400 g for 8 min at 4°C. Finally, the cell pellet was resuspended in 110 lL PBS. An aliquot of 10 lL of the suspension was used to determine the cell number by automatic coulter counter system. The rest of the suspension was immediately stored at -40°C until shipping and analysis.

The determination of the plasma concentrations of gemcitabine and dFdU was performed by using the recently published method based on HPLC coupled to mass spectrometry [START_REF] Marangon | Simultaneous determination of gemcitabine and its main metabolite, dFdU, in plasma of patients with advanced non-small-cell lung cancer by high-performance liquid chromatography-tandem mass spectrometry[END_REF].

The concentration of dFdCTP in PBMC cells was determined by HPLC (lowest sensitivity limit 0.25 mg/L) as described by Kirstein et al. [START_REF] Kirstein | High-performance liquid chromatographic method for the determination of gemcitabine and 2 0 , 2 0 -difluorodeoxyuridine in plasma and tissue culture media[END_REF] and Huang et al. [START_REF] Huang | Analysis of intracellular nucleoside triphosphate levels in normal and tumor cell lines by high-performance liquid chromatography[END_REF] for chromatographic conditions and final sample preparation following deproteinization, respectively.

Pharmacokinetic analysis

Gemcitabine and dFdU pharmacokinetic parameters were calculated with WinNonLin Pro Node 4.1 pharmacokinetic software (Pharsight Co., Mountain View, CA, USA) by using a non-compartmental approach.

Statistical analysis

Two-tailed t test was used to determine the significance of differences between measurements of dFdCTP values.

Results

Patients' characteristics

The patients' (6 men and 2 women) median age was 61.5 years (range 34-69). All tumors were adenocarcinoma. Three patients achieved a partial response, while 5 patients experienced progressive disease. All patients relapsed with a median progression-free survival of three months (range 1-7 months); the median survival was 4.5 months (range 2-34). Grade 3-4 haematological toxicities were observed only in two patients of group A during the second course of therapy.

Pharmacokinetic data

Plasma pharmacokinetics

Gemcitabine C max values were in the range 10-20 lM, approximately at the end of the constant rate infusion of 60 or 120 min in all the patients, in all courses monitored. After the end of infusion, gemcitabine concentrations rapidly declined from plasma with a T 1/2 of less than 30 min. Gemcitabine was detectable up to 4 h post infusion at concentrations close to the limit of quantitation (Table 1).

Table 1 reports the means and standard deviations of the main pharmacokinetic parameters of gemcitabine and dFdU.

Cellular pharmacokinetics

A wide interpatient variability in dFdCTP peak concentrations was observed (data not shown) (Table 2).

In both the groups, the mean peak concentrations of dFdCTP in PBMC were higher on day 8 of the first cycle of treatment when compared to day 1 of the same cycle of treatment. In both cases, the difference was not statistically significant.

In group A, the mean peak concentrations of dFdCTP after the infusion of 1,200 mg/m 2 gemcitabine during a 120-min period were similar at day 1 and day 8 of the cycle 2. Within this group of patients, a slight increase in the peak levels of dFdCTP was observed at day 1 of cycle 2 when compared to day 1 of cycle 1.

In Group B, the mean dFdCTP peak concentration was similarly increased, albeit not significantly, at day 8 when compared to day 1 of the second cycle. An increase in mean peak dFdCTP concentrations was also noted at day 1 and 8 of cycle 2 when compared to the same days of cycle 1 in group B.

Intergroup variations were also observed. In group B patients, the intracellular concentrations of dFdCTP were significantly or nearly significantly higher than corresponding values in group A patients.

Peak dFdCTP concentrations occurred within a wide interval of time ranging from 15 min after the start of gemcitabine infusion and 4 h after the end of it, independently of its duration (60 or 120 min). No significant variations in T max among cycles and days of treatment were observed within each group of patients. A trend to a later occurrence of the dFdCTP peak concentration was observed in patients of group B when compared to those of group A.

Planned comparisons

The comparison of pharmacokinetic parameters between different patients receiving two different dose levels in the first course showed a significant difference in terms of gemcitabine C max (23.8 vs. 18.7; P = 0.01) and AUC exp (7.7 vs. 10.5; P = 0.005) on day 1 and in terms of intracellular peak concentration on day 8 (173.0 vs. 316.9; P = 0.009). The same comparison within patients treated with two different dose levels in the first two chemotherapy courses showed that both gemcitabine AUC exp and dFdU pharmacokinetic parameters were significantly higher when the patients received the higher dose in the second course (data not shown). No differences were observed assessing the differences in patients who received the same dose in the first two courses of gemcitabine.

Discussion

Gemcitabine is one of the most widely used anticancer drugs and is usually administered at the dose of 1,000-1,200 mg/m 2 over 30-min infusion. However, in recent years, several studies have tested both different infusion durations and drug doses to optimize the relationship between the drug delivery and anticancer activity. Since the intracellular metabolic conversion of gemcitabine is limited by saturation of the deoxycytidine kinase activity occurring at low-dose level, the 10 mg/m 2 /min FDR infusion could enhance the drug antitumor activity [START_REF] Tempero | Randomized phase II comparison of dose-intense gemcitabine: 30-min infusion and fixed dose rate infusion in patients with pancreatic adenocarcinoma[END_REF][START_REF] Ceribelli | Prolonged gemcitabine infusion in advanced non-small cell lung carcinoma: a randomized phase II study of two different schedules in combination with cisplatin[END_REF].

In our previous study we observed an apparent dosedependent gemcitabine pharmacokinetics [START_REF] Caffo | Prolonged infusion (PI) of gemcitabine (G) in combination with cisplatin (C) in patients with advanced non-small cell lung cancer (NSCLC): preliminary results of a dose-finding and pharmacokinetic (PK) study[END_REF]. Between 600 and 1,200 mg/m 2 , the drug plasma levels appeared to be not related to the doses. The critical level of 15 lM, necessary to exert the pharmacological activity [START_REF] Abbruzzese | A phase I clinical, plasma, and cellular pharmacology study of gemcitabine[END_REF][START_REF] Grunewald | Saturation of 2 0 , 2 0 -difluorodeoxycytidine 5 0 -triphosphate accumulation by mononuclear cells during a phase I trial of gemcitabine[END_REF], was achieved and maintained successfully in the majority of patients treated at the low or middle doses, but surprisingly, not in patients treated at the high doses. To clarify these surprising findings, we evaluated the intra-and interpatient differences in plasmatic and intracellular gemcitabine pharmacokinetics when the drug was administered at two different doses as FDR.

As a result, the target plasma concentration of 15 lM was successfully achieved in all patients treated with 600 mg/m 2 and in 14/16 of those treated with 1,200 mg/m 2 . The pharmacokinetic parameters were clearly different when the patients received a higher gemcitabine dose in the second cycle compared to the lower dose of the first course, confirming that, in this case, the intra-patient variability was mainly due to the dose level; in the same time, the intracellular drug levels were not modified, as expected by the FDR administration system. Comparing the pharmacokinetic parameters of different patients treated at different dose levels, the results appeared to be quite similar: in this case, the inter-patient variability may have a role in masking the effect of the different dose levels.

Various studies have evaluated intracellular dFdCTP concentrations in PBMC after gemcitabine treatment in patients with solid tumors [START_REF] Tempero | Randomized phase II comparison of dose-intense gemcitabine: 30-min infusion and fixed dose rate infusion in patients with pancreatic adenocarcinoma[END_REF][START_REF] Van Moorsel | Pharmacokinetic schedule finding study of the combination of gemcitabine and cisplatin in patients with solid tumors[END_REF][START_REF] Patel | Phase II clinical investigation of gemcitabine in advanced soft tissue sarcomas and window evaluation of dose rate on gemcitabine triphosphate accumulation[END_REF][START_REF] De Lange | No evidence of gemcitabine accumulation during weekly administration[END_REF][START_REF] Soo | A multicentre randomised phase II study of carboplatin in combination with gemcitabine at standard rate or fixed dose rate infusion in patients with advanced stage non-small-cell lung cancer[END_REF]. However, it is difficult to compare our results with those of the literature because the published studies were performed administering gemcitabine at variable doses and with variable infusion times. When a standard 30-min infusion was employed, gemcitabine doses ranged from 800 to 2,200 mg/m 2 [START_REF] Tempero | Randomized phase II comparison of dose-intense gemcitabine: 30-min infusion and fixed dose rate infusion in patients with pancreatic adenocarcinoma[END_REF][START_REF] Van Moorsel | Pharmacokinetic schedule finding study of the combination of gemcitabine and cisplatin in patients with solid tumors[END_REF][START_REF] Patel | Phase II clinical investigation of gemcitabine in advanced soft tissue sarcomas and window evaluation of dose rate on gemcitabine triphosphate accumulation[END_REF][START_REF] De Lange | No evidence of gemcitabine accumulation during weekly administration[END_REF][START_REF] Soo | A multicentre randomised phase II study of carboplatin in combination with gemcitabine at standard rate or fixed dose rate infusion in patients with advanced stage non-small-cell lung cancer[END_REF]. When gemcitabine was administered as a FDR infusion of 10 mg/m 2 /min, infusion times ranged from 75 to 150 min [START_REF] Tempero | Randomized phase II comparison of dose-intense gemcitabine: 30-min infusion and fixed dose rate infusion in patients with pancreatic adenocarcinoma[END_REF][START_REF] Soo | A multicentre randomised phase II study of carboplatin in combination with gemcitabine at standard rate or fixed dose rate infusion in patients with advanced stage non-small-cell lung cancer[END_REF]. Although in our study FDR was the same, the infusion times were different, i.e. 60 and 120 min. The increase in the accumulation of dFdCTP in PBMC that occurred at day 8 when compared to day 1 in almost all treatment conditions may have been due to enzyme induction, particularly of deoxycytidine kinase necessary for gemcitabine phosphorylation, that is an activation reaction allowing the drug to be incorporated into DNA and exert its therapeutic action in cells [START_REF] Csapo | Similar changes were induced by Cladribine and by gemcitabine, in the deoxypyrimidine salvage, during short-term treatments[END_REF]. The administration of cisplatin before that of gemcitabine at day 8 of each cycle might have also played a relevant role in the induction of deoxycytidine kinase. The ability of cisplatin to induce nucleotide excision repair processes may up-regulate several enzymes, including deoxycytidine kinase, needed to increase deoxyribonucleotides levels (e.g. dFdCTP) [START_REF] Van Moorsel | Pharmacokinetic schedule finding study of the combination of gemcitabine and cisplatin in patients with solid tumors[END_REF]. They also reported a similar trend to increased dFdCTP concentrations in white blood cells in patients with solid tumors treated with a gemcitabine-cisplatin combination although with a longer interval (24 h) between platinum and gemcitabine.

In our study, only a slight increase in PBMC dFdCTP concentrations occurred at cycle 2 when compared to cycle 1 in both patient groups and both days of treatment with the exception of day 8 in group A patients. These variations were, however, not statistically significant. No evidence of dFdCTP accumulation was observed by De Lange et al. [START_REF] De Lange | No evidence of gemcitabine accumulation during weekly administration[END_REF] as a function of courses and days of treatment.

In group B patients undergoing a longer time of FDR drug infusion (120 min), the intracellular concentrations of dFdCTP were substantially higher when compared to group A patients receiving the same fixed dose rate for a shorter infusion time (60 min).

Similar results were also reported by Tempero et al. [START_REF] Tempero | Randomized phase II comparison of dose-intense gemcitabine: 30-min infusion and fixed dose rate infusion in patients with pancreatic adenocarcinoma[END_REF] for an identical FDR infusion of gemcitabine lasting 150 min versus a standard 30-min infusion of a higher gemcitabine dose (2,200 mg/m 2 ). In a cohort of patients receiving gemcitabine at different dose rates (1,000 mg/m 2 over a standard 30-min infusion on week 1 and over 150min infusion on week 2) Patel et al. [START_REF] Patel | Phase II clinical investigation of gemcitabine in advanced soft tissue sarcomas and window evaluation of dose rate on gemcitabine triphosphate accumulation[END_REF] reported a C max of dFdCTP at the end of infusion (150 min), which was higher than the plateau levels achieved after the 30-min infusion. An investigation by Soo et al. [START_REF] Soo | A multicentre randomised phase II study of carboplatin in combination with gemcitabine at standard rate or fixed dose rate infusion in patients with advanced stage non-small-cell lung cancer[END_REF] comparing a combination of carboplatin and gemcitabine at standard rate or fixed dose rate infusion in patients with advanced stage NSCLC reported similar AUC of intracellular dFdCTP in both treatment arms even though the dose of gemcitabine was higher in the standard rate arm (1,000 mg/ m 2 over 30 min vs. 750 mg/m 2 over 75 min).

Although the activation mechanisms of gemcitabine are well-known in terms of metabolic pathways, some aspects are relevant for future developments. Several studies have underlined that gemcitabine activity could be related to specific polymorphisms in the deoxycytidine kinase, cytidine deaminase, and/or gemcitabine transporter genes [START_REF] Javle | Polymorphisms of genes involved in gemcitabine metabolism correlate with prognosis in patients receiving neoadjuvant therapy for pancreatic cancer[END_REF]. Moreover, gemcitabine may have anticancer activity also when it is administered at very low doses and with an infusion rate much lower than the well-known level of 10 mg/m 2 /min: the drug delivered at 250 mg/m 2 in 360-min infusion has in fact been demonstrated to be active in NSCLC patients [START_REF] Xiong | Phase II trial of lowdose gemcitabine in prolonged infusion and cisplatin for advanced non-small cell lung cancer[END_REF][START_REF] Zwitter | Gemcitabine in brief versus prolonged low-dose infusion, both combined with cisplatin, for advanced non-small cell lung cancer: a randomized phase II clinical trial[END_REF] and in bladder cancer patients [START_REF] Khaled | Primary chemotherapy with lowdose prolonged infusion gemcitabine and cisplatin in patients with bladder cancer: a Phase II trial[END_REF].

In conclusion, several pharmacokinetic and clinical studies suggested that gemcitabine has a complex pharmacological profile, where the transporter proteins play a central role able to influence the drug activity. In this view, data from the present study, although obtained from a very limited sample of patients, may be of value for future researches covering a wider range of clinically useful dose rates of gemcitabine with a better exploitation of its activating pharmacokinetic and metabolic mechanisms.

Table 1

 1 Main pharmacokinetic parameters of gemcitabine and dFdU in NSCLC patients

			Day 1				Day 8
			C max	AUC exp	AUC inf	T 1/2	C max	AUC exp	AUC inf	T 1/2
			(lM)	(lg*h/mL)	(lg*h/mL)	(min)	(lM)	(lg*h/mL)	(lg*h/mL)	(min)
	Gemcitabine
	Group A (n = 4)
	Course 1 (600 mg/m 2 )	Mean	23.8	7.7	7.7	19	22.5	6.4	-	-
		SD	2.5	1	1	6.7	4.1	1.46	-	-
		CV (%)	10.5	13	13	35.2	18.2	22.9	-	-
	Course 2 (1,200 mg/m 2 )	Mean	34.5	17.1	17.1	27.8	27.5	14.4	14.4	19
		SD	20	5.8	5.8	6.6	5	2.3	2.3	6.6
		CV (%)	58.8	33.9	33.9	23.7	18.2	15.7	15.8	33.9
	Group B (n = 4)
	Course 1 (1,200 mg/m 2 )	Mean	18.7	10.5	10.5	19.9	21.2	10.4	11.5	21.2
		SD	1.9	0.8	0.8	6.9	7.1	3.2	3	6.1
		CV (%)	10	7.8	7.8	34.8	33.4	31.2	26.1	28.6
	Course 2 (1,200 mg/m 2 )	Mean	17.8	9.2	9.3	23.7	21.6	1.1	-	-
		SD	4.1	2.7	3.4	13.3	12.5	5	-	-
		CV (%)	22.6	30.1	36.4	56.1	57.7	45.5	-	-
	dFdU
	Group A (n = 4)
	Course 1 (600 mg/m 2 )	Mean	90.7	215.2	269.1	10.7	89.2	204	281.3	13.4
		SD	10.3	31.8	57.9	2.4	17.2	33.7	75.7	5.8
		CV (%)	11.4	14.8	21.5	22.7	19.3	16.5	26.9	43.1
	Course 2 (1,200 mg/m 2 )	Mean	138.5	346.7	436.3	10.2	162.8	329.6	371.1	8.3
		SD	31.6	84.6	177.9	4.7	55.2	96.2	105.7	1
		CV (%)	22.9	24.4	40.8	45.8	33.9	29.2	28.5	12.2
	Group B (n = 4)
	Course 1 (1,200 mg/m 2 )	Mean	94.8	220.3	253.6	7.1	99.6	214.5	255	10.3
		SD	32.2	136.8	154.7	3.2	35.5	81	92.6	2.9
		CV (%)	34	62.1	61	45.6	35.7	37.8	36.3	28.1
	Course 2 (1,200 mg/m 2 )	Mean	94.1	194.8	232.9	10	88.8	215.2	271.9	11.5
		SD	19.4	43.9	58	2.3	15.5	37.5	47.1	3.3
		CV (%)	20.6	22.6	24.9	23	17.5	17.4	17.3	28.6

Table 2

 2 Mean intracellular peak concentrations and T max of dFdCTP in peripheral blood mononuclear cells

	Day 1		Day 8 (with cisplatin)
	Peak concentrations (pmol 9 10 6 cells)	T max (min)	Peak concentrations (pmol 9 10 6 cells)	T max (min)

Cancer Chemother Pharmacol (2010) 65:1197-1202


X W Gao

Ch Zhang

J Sladek

V Sladek

Fracture Analysis of Functionally Graded Materials by a BEM

Keywords: A. Functional composites, B. Fracture, B. Mechanical properties, C. Crack, C. Computational mechanics

In this paper, crack analysis in two-dimensional (2-D), continuously nonhomogeneous, isotropic and linear elastic functionally graded materials (FGMs) is presented. For this purpose, a boundary element method (BEM) based on a boundarydomain integral equation formulation is developed. An exponential variation with spatial variables is assumed for Young's modulus of the FGMs, while a constant Poisson's ratio is considered. Fundamental solutions for homogeneous, isotropic and linear elastic solids are applied in the formulation. To avoid displacement gradients in the domain-integral, normalized displacements are introduced. By using the radial integration method, the domain-integral is transformed into boundary integrals over the global boundary. The normalized displacements in the domain-integral are approximated by a combination of radial basis functions and polynomials in terms of global coordinates, which leads to a meshless scheme. Special attention of the analysis is devoted to the computation of the most important crack-tip characterizing parameters of cracked FGMs, namely the stress intensity factors. To show the effects of the material gradation on the stress intensity factors, numerical examples are presented and discussed.

1.

x -direction parallel to the crack is described by the an exponential law

Introduction

In recent years, a new class of composite materials, the so-called functionally graded materials (FGMs), attracted many research interests in materials and engineering sciences [START_REF] Suresh | Fundamentals of functionally graded materials[END_REF][START_REF] Miyamoto | Functionally graded materials[END_REF]. FGMs are advantageous over classical homogeneous materials with only one material constituent, because FGMs consist of more material constituents and they combine the desirable properties of each constituent. As a representative example for FGMs, we just mention the metal/ceramic FGMs, which are compositionally graded from a ceramic phase to a metal phase. Metal/ceramic FGMs can incorporate advantageous properties of both ceramics and metals such as the excellent heat, wear, and corrosion resistances of ceramics and the high strength, high toughness, good machinability and bonding capability of metals without severe internal thermal stresses.

However, ceramics have a brittle nature, and microcracks or crack-like defects are often induced in the fabrication process or under the in-service loading conditions. Thus, fracture and fatigue analysis of FGMs is an important research issue to the design, optimization, and novel engineering applications of FGMs. For cracked FGMs with general geometry and loading conditions, advanced numerical methods have to be applied, because of the high mathematical complexity of the corresponding governing partial differential equations with variable coefficients, and because the most available analytical methods can be successfully applied to cracked FGMs only with very simple geometry and loading conditions. In this context, we just mention the singular integral equation method [START_REF] Delale | The crack problem for a nonhomogeneous plane[END_REF][START_REF] Erdogan | The crack problem for bonded nonhomogeneous materials under antiplane shear loading[END_REF][START_REF] Erdogan | Fracture mechanics of functionally graded materials[END_REF][START_REF] Konda | The mixed mode crack problem in a nonhomogeneous elastic medium[END_REF][START_REF] Chan | The crack problem for nonhomogeneous materials under antiplane shear loading -A displacement based formulation[END_REF], the classical finite element method (FEM) [START_REF] Eischen | Fracture of nonhomogeneous materials[END_REF][START_REF] Gu | Cracks in functionally graded materials[END_REF][START_REF] Gu | Crack deflection in functionally graded materials[END_REF][START_REF] Gu | A simplified method for calculating the crack tip field of functionally graded materials using the domain integral[END_REF][START_REF] Anlas | Numerical calculation of stress intensity factors in functionally graded materials[END_REF][START_REF] Marur | Numerical analysis of crack-tip fields in functionally graded materials with a crack normal to the elastic gradient[END_REF][START_REF] Li | Experimental investigation of the quasi-static fracture of functionally graded materials[END_REF][START_REF] Zou | On the multiple isoparametric finite element method and computation of stress intensity factor for cracks in FGMs[END_REF], the graded finite element method [START_REF] Santare | Use of a graded finite element to model the behavior of nonhomogeneous materials[END_REF][START_REF] Kim | Isoparameteric graded finite elements for nonhomogeneous isotropic and orthotropic materials[END_REF][START_REF] Kim | Finite element evaluation of mixed-mode stress intensity factors in functionally graded materials[END_REF][START_REF] Kim | T-stress, mixed-mode stress intensity factors, and crack initiation angles in functionally graded materials: a unified approach using the interaction integral method[END_REF], the extended finite element method (XFEM) [START_REF] Dolbow | On the computation of mixed-mode stress intensity factors in functionally graded meterials[END_REF], the element-free Galerkin method (EFG) [START_REF] Chen | Element free Galerkin methods for fracture of functionally graded materials[END_REF][START_REF] Rao | Mesh-free analysis of cracks in isotropic functionally graded materials[END_REF], the boundary integral equation method (BIEM) or boundary element method (BEM) [START_REF] Yue | Boundary element analysis of crack problems in functionally graded materials[END_REF][START_REF] Ch | Numerical analysis of cracked functionally graded materials[END_REF][START_REF] Ch | Effects of material gradients on transient dynamic mode III SIFs in a FGM[END_REF][START_REF] Ch | Antiplane crack analysis of a functionally graded material by a BIEM[END_REF][START_REF] Ch | Transient dynamic analysis of cracked functionally graded materials[END_REF], and the meshless Petrov-Galerkin method (MLPG) [START_REF] Sladek | An advanced numerical method for computing elastodynamic fracture parameters in functionally graded materials[END_REF][START_REF] Sladek | A meshless local boundary integral equation method for dynamic anti-plane shear crack problem in functionally graded materials[END_REF][START_REF] Sladek | The MLPG method for crack analysis in anisotropic functionally graded materials[END_REF][START_REF] Sladek | Evaluation of fracture parameters for crack problems in FGM by a meshless method[END_REF].

Although the BEM has been successfully applied to homogeneous, isotropic and linear elastic solids for many years, its application to FGMs is yet very limited due the fact that the corresponding fundamental solutions or Green' s functions for general FGMs are either not available or mathematically too complex [START_REF] Martin | On Green' s function for a threedimensional exponentially-graded elastic solid[END_REF][START_REF] Chan | Green' s function for a two-dimensional exponentially-graded elastic medium[END_REF]. The nonhomogeneous nature of FGMs prohibits an easy construction and implementation of fundamental solutions for general FGMs.

In this paper, crack analysis in 2-D, continuously nonhomogeneous, isotropic and linear elastic FGMs is presented. For this purpose, a boundary-domain integral equation formulation is applied. For simplicity, an exponential variation of Young's modulus and constant Poisson's ratio are assumed. Fundamental solutions for homogeneous, isotropic and linear elastic solids are applied in the present formulation, which results in a boundary-domain integral equation formulation due to the materials nonhomogeneity.

To avoid displacement gradients in the domain integral, normalized displacements are introduced. The radial integration method of Gao [START_REF] Gao | The radial integration method for evaluation of domain integrals with boundary-only discretization[END_REF][START_REF] Gao | A boundary element method without internal cells for two-dimensional and three-dimensional elastoplastic problems[END_REF] is applied to convert the arising domain-integral into boundary-integrals over the global boundary of the cracked solids. Basis functions consisting of a combination of radial basis functions and polynomials in terms of global coordinates are used to approximate the normalized displacements in the domain-integral. In this manner, a meshless scheme is obtained, which requires only conventional boundary discretization and additional interior nodes instead of cells or meshes. An advantage of the present BEM is that it is easy to implement and can be easily incorporated into an existing BEM code for homogeneous, isotropic and linear elastic solids. Special attention of the analysis is devoted to the investigation of the material gradation on the stress intensity factors. Numerical examples for cracks parallel and perpendicular to the material gradation are presented and discussed.

Boundary-Domain Integral Equations

We consider 2-D, continuously nonhomogeneous, isotropic and linear elastic FGMs. In the absence of body forces, the equilibrium equations are given by , 0

ij j σ = , (1)
where ij σ represents the stress tensor, a comma after a quantity represents spatial derivatives and repeated indexes denote summation. It is assumed that the Young's modulus ( ) E x of the FGMs depends on Cartesian coordinates while Poisson's ratio ν is constant. In this case, the elasticity tensor ( ) ijkl C x can be written as 0 ( ) ( )

ijkl ijkl C C µ = x x , (2)
where

( ) ( ) 2 (1 )
E µ ν = + x x , 0 2 1 2 ijkl ij kl ik jl il jk C ν δ δ δ δ δ δ ν = + + - . (3)
In Eqs.

ij ijkl k l ijkl k l C u C u σ µ = = x . ( 4
)
The traction vector i t on the boundary of the considered domain is related to the stress components by

i ij j t n σ = , ( 5
)
where j n is the outward unit normal vector to the boundary Γ of the domain Ω .

The weak-form of the equilibrium equations ( 1) can be written as , 0

jk k ij U d σ Ω ⋅ Ω = , (6)
where ( , ) ij U x y is the weight or test function. Substitution of Eq. ( 4) into Eq. ( 6) and application of Gauss's divergence theorem yield

0 0 , , , 0 ij j ij j rsjl ir sl j rsjl ir s l j U t d T u d C U u d C U u d µ µ µ Γ Γ Ω Ω Γ - Γ + Ω + Ω = , (7)
where

ij ijl l T n = Σ , ( 8
) 0 , , , , 2 1 2 ijl rsjl ir s ik k jl ij l il j C U U U U ν δ ν Σ = = + + - . ( 9
)
For the weight function ( , ) ij U x y , we choose the displacement fundamental solutions for homogeneous, isotropic and linear elastic solids, which satisfy the following partial differential equations 0 , ( )

rsjl ir sl ij C U δ δ = - - x y , ( 10
)
where ( ) δ -

x y is the Dirac delta function. The solution ( , ) ij U x y of Eq. ( 10) is given by the Kelvin's displacement fundamental solutions for homogeneous, isotropic and linear elastic solids with µ =1, which can be written as [36] , , 1 (3 4 ) ln( ) 8 ( 1)

ij ij i j U r r r ν δ π ν = - - - - , (11)
where r = x y . Substitution of Eq. ( 10) into Eq. ( 7) and application of the sifting property of the Dirac delta function lead to

( ) ( , ) ( ) ( , ) ( ) ( , ) ( ) i ij j ij j ij j u U t d T u d V u d Γ Γ Ω = Γ - Γ + Ω x y x y x x y x x y , ( 12
)
where the normalized displacements ( ) i u x and the normalized shear modulus ( ) µ x are defined by

( ) ( ) ( ) i i u u µ = x x x , (13)
( ) log ( )

µ µ = x x . ( 14
)
The fundamental solutions ( , ) ij T x y and ( , ) ij V x y arising in Eq. ( 12) can be expressed as [START_REF] Gao | Boundary element programming in mechanics[END_REF] ( )(

)

, , , , , , 1 1 2 2 4 (1 ) ijl il j ij l jl i i j l r r r r r r r ν δ δ δ π ν Σ = - - + - + - , (15) ( )( ) ( ) , , , , , 1 1 2
(1 2 ) 2 4 (1 ) ij ijl l i j j i ij i j l l T n n r n r r r r n r ν ν δ π ν = Σ = - - - + - + - , (16)
( )( ) ( ) , , , , , , , , , 1 1 2
(1 2 ) 2 4 (1 ) ij ijl l i j j i ij i j l l V r r r r r r µ ν µ µ ν δ µ π ν = Σ = - - - + - + - . ( 17
)
Boundary-domain integral equations for boundary points can be obtained by letting → Γ y in Eq. ( 12). It should be remarked here that the boundary-domain integral equations [START_REF] Anlas | Numerical calculation of stress intensity factors in functionally graded materials[END_REF] are formulated in terms of the boundary tractions j t and the normalized displacements j u in contrast to most previous boundary-domain integral equation formulations for nonhomogeoneous, isotropic and linear elastic solids, where displacement gradients arise in the domain-integral. This feature not only facilitates the numerical implementation, but also leads to a high accuracy of the computational method. In the case of an exponential law for Young's modulus or shear modulus such as that used in this analysis, it can be seen from Eq. ( 14) that ,i µ is constant and ( , ) ij V x y thus becomes very simple for integration.

Numerical Solution Procedure

The boundary discretization of the boundary-domain integral equations ( 12) can be done by following the usual procedure applied in the conventional BEM. A key issue to the numerical solution procedure is how to compute the domain integral in [START_REF] Anlas | Numerical calculation of stress intensity factors in functionally graded materials[END_REF]. Compared to the classical mesh-or cell-integration method, a transform of the domain-integral in [START_REF] Anlas | Numerical calculation of stress intensity factors in functionally graded materials[END_REF] into boundary integrals over the global boundary is advantageous. In this paper, the radial integration method (RIM) of Gao [START_REF] Gao | The radial integration method for evaluation of domain integrals with boundary-only discretization[END_REF][START_REF] Gao | A boundary element method without internal cells for two-dimensional and three-dimensional elastoplastic problems[END_REF] is used for this purpose. In the RIM, the normalized displacements in the domain-integral of Eq. ( 12) are approximated by a series of prescribed basis functions as used in the dual reciprocity method [START_REF] Nardini | A new approach for free vibration analysis using boundary elements[END_REF]. Previous including our own investigations have demonstrated that a combination of the radial basis functions and the polynomials in terms of global coordinates is very promising (see e.g. [START_REF] Golberg | Some recent results and proposals for the use of radial basis functions in the BEM[END_REF] for more details). For this reason, the normalized displacements ( ) i u x are approximated by 0 ( ) ( )

A A k i i i k i A u R a x a α φ = + + x , ( 18
) 0 A i A α = , 0 A A i j A a x = , (19)

0, ,

A A A A A A R R R R d R d d d R d φ - + - ≤ ≤ = ≥ , ( 20
)
where A d is the support size for the application point A.

The unknown coefficients A i α and k i a in Eq. ( 18) can be determined by applying the application point A in Eqs. ( 18) and ( 19) to every node. Then, a set of linear algebraic equations can be obtained as

= ⋅ u φ α φ α φ α φ α , ( 21
)
where α α α α is a vector containing the coefficients A i α for all application points and k i a . If two nodes do not coincide, the matrix φ φ φ φ is invertible and therefore

1 - = ⋅ u α φ α φ α φ α φ . ( 22
)
Substitution of Eq. ( 18) into the domain-integral of Eq. ( 12) results in

0 A A k ij j j ij j ij k j ij V u d V d a V x d a V d α φ Ω Ω Ω Ω Ω = Ω + Ω + Ω . ( 23
)
The domain-integrals in Eq. ( 23) can be transformed into boundary integrals by using RIM [START_REF] Gao | The radial integration method for evaluation of domain integrals with boundary-only discretization[END_REF][START_REF] Gao | A boundary element method without internal cells for two-dimensional and three-dimensional elastoplastic problems[END_REF] as ( )

1 0 0 1 1 ,k A A k k ij j j ij j ij j k j ij r r r r V u d F d a F d a y a F d , r n r n r n Γ α Ω Γ Γ ∂ ∂ ∂ Ω = Γ + Γ + + Γ ∂ ∂ ∂ ( 24
)
where

0 r A A ij ij F rV dr φ = , 1 2 0 r ij ij F r V dr = , 0 0 r ij ij F rV dr = . ( 25
)
It should be noted here that in the radial integrals of Eq. ( 24), the term ,i r is constant and the relation

i i ,i x y r r = + ( 26
)
is used for the transformation from x to r. The last two radial integrals in Eq. ( 25) can be evaluated analytically. The first radial integral in Eq. ( 25) is regular and can be computed numerically by using standard Gaussian quadrature formula. To this end, the function ( ) A R φ should be expressed as a function of r , which can be done by using the following relation (see Fig. 1)

2 2 R r sr R , = + + (27)
where

2 ,i i s r R = , ( 28
)
A i i R R R = - = ⋅ y x , ( 29
)
A i i i R y x = -. ( 30
)
After numerical integrations and substitution of Eq. ( 22) into Eq. ( 24 Equations ( 31) and ( 32) can be written in a more compact form as

b b b b i i i i - = - 0 A V x y A I V u y , ( 33
)
where I is the identity matrix. From Eq. ( 33), both the boundary unknowns b x and the normalized internal displacements i u can be obtained numerically. Then, the true displacements can be obtained by using Eq. ( 13). Subsequently, the stresses at internal points can be computed by a usual data post-processing.

Computation of Stress Intensity Factors

Since the asymptotic crack-tip field for continuously nonhomogenous, isotropic and linear elastic solids has the same structure as that for homogeneous, isotropic and linear elastic solids [START_REF] Eischen | Fracture of nonhomogeneous materials[END_REF], the stress intensity factors can be computed by using the following

relations 2 1 ( , ) 2 2 ( , ) 1 tip I II K u r K u r r π µ π π κ = + , ( 34
)
3 4 , plane strain, (3 ) /(1 ), plane stress,

ν κ ν ν - = - + ( 35
)
where r is the radial coordinate with the origin at the crack-tip, 1 ( , )

u
I II r u r K K r r u r α α α α α α α α ⋅ = ⋅ , ( 36
)
where [START_REF] Nardini | A new approach for free vibration analysis using boundary elements[END_REF] in which the displacements at the discrete nodes r α from the crack-tip are used.

1 1 2 2 ˆ( ) ( , ) 2 2 ˆ( ) ( , ) 1 tip u r u r u r u r α α α α π µ π π κ = + ,

Numerical Results

In the first numerical example, we consider an edge crack parallel to the material gradation in a rectangular FGM plate, which is subjected to a uniform tensile loading as depicted in Fig. 2. The geometry of the cracked plate is described by: plate width ( )

x E x E e α = , 0 1 ln w E b E α = , (38)
with 0 E and w E being the Young's modulus at the left and the right side of the plate.

Poisson's ratio is taken as ν =0.25 and plane strain condition is assumed in the numerical calculations. Due to symmetry of the problem, only one half of the plate is considered. 180 boundary nodes and 251 internal nodes are used in the discretization.

On the global boundary of the plate, 3-noded quadratic boundary elements are applied.

To consider the stress concentration near the crack-tip, dense boundary and internal nodes are used as shown in Fig. 3.

To test the accuracy of the implemented BEM, the computed displacement component 2 u on the crack-face is compared with the FEM result in Fig. 4, which shows an excellent agreement between both results. Here, the Young's moduli are taken as 0 10000 E = and 50000 w E = . In the FEM calculation, 900 cubic four-noded graded elements are applied, where a linear variation of Young's modulus within each element is assumed.

To investigate the sensitivity of the implemented BEM to the used number and distribution of internal nodes, numerical calculations with different node numbers and distributions have been also carried out. The number of boundary nodes (i.e., 180) has been taken as constant, while the number and the distribution of internal nodes have been varied. In particular, non-uniform node distribution with 112, 57, 17 and 0 internal nodes, and uniform node distribution with 112 and 55 internal nodes have been applied.

Our numerical results have shown that some internal nodes are required to achieve a sufficient accuracy of the numerical results [40]. Otherwise the implemented BEM is quite robust and insensitive with respect to the used number and distribution of internal nodes. Generally speaking, an increase in the number of internal nodes improves the accuracy of the present numerical method, but it may increase the required computing time drastically.

The normalized mode-I stress intensity factor Here, the mode- II stress intensity factor is identically zero due to the symmetry of the problem with respect to the crack-line.

In the second numerical example, we consider an edge crack perpendicular to the material gradation in a rectangular functionally graded plate as depicted in Fig. 6. The cracked plate has the same geometry and loading condition as in the first numerical example. The material gradation in the 2 x -direction perpendicular to the crack is described by an exponential law as 2 2 0 ( )

x E x E e β = , 0 1 ln 2 H E h E β = , (39)
with 0 E and H E being the Young's modulus at the bottom and the top side of the cracked plate. Also here, Poisson's ratio is taken as ν =0.25 and plane strain condition is assumed in the numerical calculations. Young's modulus at the bottom side of the plate is fixed as 0 10000 E = , while Young's modulus at the bottom side of the plate is varied.

For the discretization of the cracked plate, 139 boundary elements with 254 boundary nodes and 502 internal nodes are used. The node distribution is shown in Fig. 7. β . The normalized mode-II stress intensity factor is zero in the case of a homogeneous plate (i.e., 0 h β = ), and its amplitude increases with increasing amplitude of the gradient parameter h β . This implies that the material gradation may affect the crack propagation direction, since the mode-II stress intensity factor has direct relevance to the direction of the crack propagation.

Conclusions

In this paper, fracture analysis of cracked FGMs by a BEM is presented. By using fundamental solutions for homogeneous, isotropic and linear elastic solids, a boundarydomain integral equation formulation is obtained. Normalized displacements are introduced to avoid displacement gradients in the domain-integral. A numerical solution procedure is developed to solve the boundary-domain integral equations. The domainintegral is transformed into boundary integrals over the global boundary of the considered domain by using the radial integration method of Gao [START_REF] Gao | The radial integration method for evaluation of domain integrals with boundary-only discretization[END_REF][START_REF] Gao | A boundary element method without internal cells for two-dimensional and three-dimensional elastoplastic problems[END_REF]. The present method is a meshless method, which is suitable for crack analysis of 2-D continuously nonhomogeneous, isotropic and linear elastic FGMs. Numerical results show that the materials gradient may have significant influences on the stress intensity factors.

  (2) and (3), ( ) µ x denotes the shear modulus and ij δ represents the Kronecker delta.The stress tensor ij σ and the displacement gradients ,

  from the application point A to the filed point x, A i α and k i a are unknown coefficients to be determined, and A j x denotes the coordinates at the application point A which consists of all boundary nodes and some internal points. In this analysis, the following 4-th order spline-type radial basis function is applied

  ), an expression for the domain-integral in terms of the normalized displacement vector u at all nodes can be obtained. If the BEM model consists of N b boundary nodes and N i internal nodes and after invoking the boundary conditions, the boundary-domain integral equations (12) lead to the following system of linear algebraic equations (31) and (32), the sizes of the matrices b A and i A are 2N b ×2N b and 2N i ×2N b , while b V and i V are 2N b ×2N t and 2N i ×2N t with N t = N b +N i , respectively. The vector b x with a size of 2N b ×1 contains the unknown normalized boundary displacements or the unknown boundary tractions, while the vector u ~ with a size of 2N t ×1 consists of the unknown normalized boundary displacements and all normalized internal displacements. It should be noted here that after invoking the boundary conditions, the columns of the matrices b V and i V corresponding to the known boundary displacement nodes should be taken as zero.

  r π and 2 ( , ) u r π are the crack-face displacements, tip µ is the shear modulus at the crack-tip, and I K and II K are the mode-I and mode-II stress intensity factors. In this analysis, a least-squares technique based on the minimization of the quadratic deviations of the displacements on the crack-faces is used for computing the stress intensity factors. This technique leads to the following simple formula[START_REF] Aliabadi | The boundary element method[END_REF]

b α is presented in Fig. 5 .

 5 The normalized mode-I stress intensity factor decreases with increasing dimensionless gradient parameter b α .

Figures 8 and 9

 9 Figures 8 and 9 show the influences of the dimensionless gradient parameter h β on

Fig. 1 :Fig. 2 :Fig. 3 :Fig. 5 :Fig. 6 :Fig. 7 :Fig. 8 : 25 -βhFig. 9 :

 1235678259 Fig. 1: Relationship between distances

Acknowledgements

Support by the German Research Foundation (DFG) under the project number ZH 15/10-1 is gratefully acknowledged.