Magali Jasiek

Alexandre Karras

Veronique Le Guern

Evguenia Krastinova

Rafik Mesbah

Stanislas Faguer

Noemie Jourde-Chiche

Anne-Laure Fauchais

Laurent Chiche

Emmanuelle Dernis

Ve ´ronique Le Guern

Noe ´mie Jourde-Chiche

Guillaume Moulis

Jean-Baptiste Fraison

Estibaliz Lazaro

Perrine Jullien

Eric Hachulla

Alain Le Quellec

Philippe Re ´my 17

Aure ´lie Hummel

Nathalie Costedoat-Chalumeau

Pierre Ronco

Philippe Vanhille

Vannary Meas-Yedid

Carole Cordonnier

Sophie Ferlicot

Laurent Daniel

Raphaele Seror

Xavier Mariette

Eric Thervet

He ´le `ne Franc ¸ois

Benjamin Terrier

He ´ Le ` Ne Franc

A multicentre study of 95 biopsy-proven cases of renal disease in primary Sjögren's syndrome

Keywords: primary Sjo ¨gren's syndrome, tubulointerstitial nephritis, cryoglobulinaemia

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

Introduction

Primary SS (pSS) is a chronic autoimmune disorder characterized by lymphocytic infiltration of exocrine glands, resulting in significant loss of secretory function, with oral and eye dryness. The course of the disease is also characterized by the occurrence of extraglandular features, with frequent multi-organ involvement, including the kidney. However, overt renal disease is a rare event during pSS, and its prevalence was found to be around 35% in most recent series [START_REF] Malladi | Primary Sjo ¨gren's syndrome as a systemic disease: a study of participants enrolled in an international Sjo ¨gren's syndrome registry[END_REF]. The main types of pSSassociated nephropathies previously described are tubulointerstitial nephritis (TIN) and membrano-proliferative GN (MPGN) related to cryoglobulinaemia. In addition, electrolyte disturbances such as renal tubular acidosis are frequent when appropriately screened [START_REF] Bossini | Clinical and morphological features of kidney involvement in primary Sjo ¨gren's syndrome[END_REF]. Therefore, as tubulointerstitial involvement is sometimes insidious and slowly progressive, renal disease is probably underdiagnosed, a fact that may account for the low reported prevalence in large pSS retrospective cohorts [START_REF] Goules | Clinically significant renal involvement in primary Sjo ¨gren's syndrome: clinical presentation and outcome[END_REF][START_REF] Goules | Clinically significant and biopsy-documented renal involvement in primary Sjo ¨gren syndrome[END_REF]. Very few reports of biopsy-proven renal disease in pSS have been published so far: the largest recent studies having reported 41 patients in China [START_REF] Ren | Renal involvement and followup of 130 patients with primary Sjo ¨gren's syndrome[END_REF], 33 in Europe [START_REF] Goules | Clinically significant renal involvement in primary Sjo ¨gren's syndrome: clinical presentation and outcome[END_REF] and 24 in the USA [START_REF] Maripuri | Renal involvement in primary Sjo ¨gren's syndrome: a clinicopathologic study[END_REF]. For this reason, treatment of pSS-related nephropathy is not codified, and data in the literature are lacking.

In the present study, we conducted a French nationwide analysis of 95 biopsy-proven pSS-related nephropathies. We describe here the clinical and biological presentation of pSS-associated renal disease, the renal pathology data, and the response to treatment.

Methods

Study population

This retrospective study, the REnal INvolvement in SS study, was conducted in French universities and general hospitals across departments of Internal Medicine, Nephrology and Rheumatology. The study was performed in accordance with the ethical standards of the Declaration of Helsinki and was approved by the institutional review board of Ho ˆpital Europe ´en Georges Pompidou. We used electronic newsletters and emails to contact practitioners registered on the Socie ´te Ńational Franc ¸aise de Me ´decine Interne, the Socie ´te ´de Ne ´phrologie-Socie ´te ´Francophone de Dialyse and the Club Rhumatismes et Inflammation (section of the French Society of Rheumatology) websites, and screened the French Assessment of Systemic Signs and Evolution in SS (ASSESS) database [START_REF] Gottenberg | Serum levels of beta2-microglobulin and free light chains of immunoglobulins are associated with systemic disease activity in primary Sjo ¨gren's syndrome. Data at enrollment in the prospective ASSESS cohort[END_REF]. Inclusion criteria were: pSS diagnosis, based on the AmericanEuropean Consensus Group (AECG) criteria [START_REF] Vitali | Classification criteria for Sjo ¨gren's syndrome: a revised version of the European criteria proposed by the American-European Consensus Group[END_REF] or the enlarged AECG criteria, defined by the presence of three or more of four AECG items among subjective oral dryness, subjective ocular dryness, anti-SSA/SSB positivity and postivie minor salivary gland biopsy results, as previously described [START_REF] Maldini | Epidemiology of primary Sjo ¨gren's syndrome in a French multiracial/multiethnic area[END_REF]; and presence of nephropathy, based on renal biopsy findings. Renal biopsy was performed for clinically relevant renal symptoms according to local practice. We excluded secondary SS associated with RA, SLE, scleroderma or PM. Positivity of anti-DNA antibodies, chronic hepatitis C, chronic HIV infection, sarcoidosis and cervical radiotherapy were considered as exclusion criteria.

Patients' characteristics

Practitioners in charge of the patients collected clinical and biological data at diagnosis of renal involvement and during follow-up. EULAR SS Disease Activity Index (ESSDAI) was recorded at the time of nephropathy diagnosis [START_REF] Seror | EULAR Sjogren's syndrome disease activity index: development of a consensus systemic disease activity index for primary Sjogren's syndrome[END_REF]. Data were retrospectively collected and centrally reviewed, including the renal pathology findings. Kidney function was assessed by the estimated glomerular filtration rate (eGFR) using the Modification of Diet in Renal Disease Study equation [START_REF] Levey | A more accurate method to estimate glomerular filtration rate from serum creatinine: a new prediction equation. Modification of Diet in Renal Disease Study Group[END_REF], and renal dysfunction was defined as an eGFR of < 60 ml/min/1.73 m 2 . Chronic kidney disease (CKD) was defined by the presence of a renal dysfunction for at least 3 months and acute kidney injury (AKI) as an eGFR decrease of > 25%, known or presumed to have occurred within the prior 3-month period.

Response to therapy

The assessment of renal response to therapy was based on the evolution of eGFR during follow-up. A renal response was defined by an improvement of eGFR >20%, compared with baseline values.

Interstitial fibrosis quantification and characterization of renal cellular infiltrate Kidney biopsy specimens were centrally collected when possible for additional analyses, including computerassisted quantification of cortical interstitial fibrosis, using segmentation method and colour image analysis, as previously described [START_REF] Servais | Quantification of interstitial fibrosis by image analysis on routine renal biopsy in patients receiving cyclosporine[END_REF]. Interstitial leucocyte infiltrate was quantified (in a double-blinded manner using semiquantitative analysis on Masson's trichrome slides) for 25 patients. Characterization of the interstitial cellular infiltrate was also performed for 20 patients by imunohistochemistry (using anti-CD3, anti-CD19 and anti-CD138 primary antibodies for T cell, B cell and plasma cell detection, respectively). These patients were randomly selected and did not differ from the patients of the entire cohort (supplementary Table S5, available at Rheumatology Online). Cellular infiltration was characterized as predominant when it represented the majority of the cells infiltrating the kidney slide.

Statistical analyses

Data are expressed as mean (S.D.) or median (range) for continuous variables, and as number (percentage) for non-continuous variables, respectively. Paired t-tests were used to analyse the evolution of paired values, and Fischer tests to compare non-paired values. Baseline characteristics of patients according to the improvement of eGFR 520% throughout follow-up were compared using the Chi-square test and the Wilcoxon rank-sum test for categorical and continuous variables, respectively. This 20% threshold corresponded to the median value of eGFR improvement in the cohort. We estimated the eGFR dynamics over time, accounting for the correlation among repeated measurements within each individual, through linear mixed models with random intercept and random slope. The best model (Akaike's criterion) was obtained with no slope change. We modelled eGFR dynamics for the first 60 months, during which the median number of eGFR measurements was 4 per subject (min = 2; max = 4). Slopes of eGFR were compared between the groups. Mean eGFR evolution was depicted by plotting the mixed model predictions. P < 0.05 was considered as statistically significant. Analyses were performed with STATA software (release 13; Stata Corp., College Station, TX, USA).

Results

Patient's characteristics

A total of 95 patients were included, including 86 women and 9 men, and the median agewas 49 years (range 17.987.3) at pSS diagnosis. pSS-related characteristics at diagnosis are summarized in Table 1. Eight-two patients (86.3%) met the AECG criteria, while 13 patients (13.7%) met the enlarged criteria. Another autoimmune disease was associated with pSS in 11 patients (11.6%): autoimmune hepatitis (n = 4), Hashimoto's thyroiditis (n = 3), primary biliary cirrhosis (n = 2), coeliac disease (n = 1) and myasthenia gravis (n = 1). All 22 patients (23.2%) with negative anti-SSA or SSB antibodies had positive minor salivary gland biopsy (focus score 51). RF was present in 53/82 patients (64.6%) and cryoglobulinaemia in 26/89 (29.2%). Hypergammaglobulinaemia, defined as total gammaglobulin level above 16 g/l, was found in 55/81 patients (67.9%). No measurement of serum IgG4 was available.

The median ESSDAI score at renal biopsy was 16 (941) (detailed in supplementary Table S1, available at Rheumatology Online). In association with renal involvement, the ESSDAI items reflecting 'biological activity' (78.9%) and 'articular involvement' (28.4%) were the most frequently present. Only 12 patients (12.6%) had isolated renal involvement.

Characteristics of initial renal presentation

The median age at diagnosis of renal involvement was 50.9 years (range 1787). The characteristics of nephropathy are summarized in Table 2. Diagnosis of nephropathy was concomitant with pSS diagnosis in most cases (57.9%), and preceded pSS diagnosis in 10.5%. Renal test abnormalities included renal dysfunction in 86.3% of all included patients (CKD in 54.7%, AKI in 31.6%), followed by proteinuria (26.3%), isolated electrolyte disturbances (17.9%) (acidosis and hypokalaemia), nephrolithiasis (9.5%) and nephrocalcinosis (5.3%). At diagnosis, severe renal dysfunction was rare, with only 32.6% of patients displaying an eGFR below 30 ml/min/1.73 m 2 and only one patient requiring dialysis.

Analysis of kidney biopsy specimens

Kidney biopsies were performed between 1977 and 2013, including 81/95 (85%) biopsies after 2000. The mean number of glomeruli per specimen was 19 (range 263), with a median number of sclerotic glomeruli of 4 (range 032). Renal biopsy showed TIN in 93/95 (97.9%) cases, which was isolated in 73/95 (76.8%) patients. Glomerular lesions were found in 22/95 (23.2%) patients and were associated with interstitial cellular infiltration in 20/22 cases. Glomerular disease included polyclonal cryoglobulinaemia-related MPGN in 8 patients, and other glomerulopathies (such as membranous nephropathy or focal segmental glomerulosclerosis) in 14 cases, (Fig. 1). No feature suggestive of IgG4-related disease was noted.

Light microscopy examination of kidney biopsy among patients with TIN revealed that the cellular infiltrate was mainly composed of lymphocytes, but contained plasma cells in 45/66 (68%) cases. We were able to perform a centralized semi-quantitative analysis of the interstitial cellular infiltrate in 25 cases with TIN, and a qualitative immunohistochemistry analysis in 20 patients, whereas other specimens were lacking (Table 3). The degree of cellular interstitial infiltration varied from <25% of the total interstitial area to > 75% in some cases. Interstitial cellular infiltrates were positive for T cell markers in 20/20 (100%) patients, B cell markers in 19/20 (95%) and plasma-cell markers in 15/20 (75%) patients. T cell infiltration was predominant in 13/20 (65%), B cell infiltration in 2/20 (10%) and plasma-cell infiltration in 5/20 (25%) of kidney biopsies.

A computerized quantification of the cortical interstitial fibrosis was performed in 37 kidney biopsy specimens. The median fibrotic surface area was 27.1% (14.373), with interstitial fibrosis involving >25% of the total renal cortical area in 25 (67.6%) of all studied cases (Table 3).

Finally, correlations of immunological features with histological findings showed that patients with isolated anti-SSA antibodies (71.4 vs 28.6%, P < 0.05) and those with both anti-SSA/SSB antibodies (88 vs 12%, P < 0.05) more frequently had isolated TIN rather than pSSassociated glomerular disease (supplementary Table S2, available at Rheumatology Online).

Treatment

Treatment used for pSS-related nephropathy is summarized in supplementary Table S3, available at Rheumatology Online. Eighty-one patients (85.3%) received immunosuppressive treatment. Sixty patients received CSs alone, and 21 patients received CSs in combination with other immunosuppressive agents, mostly rituximab (RTX) (n = 18). The median initial daily dose of prednisone was 55 mg (range 580), and the median dose of prednisone was 10 mg/day (range 060) at 6 months, 8 mg/day (040) at 12 months and 0 mg/day (range 060) at the end of follow-up. Patients received RTX as first-line therapy in 12 cases and as second-line in 6. RTX was given at the dose of 1 g at days 1 and 15 for 10 patients, and at the dose of 375 mg/m 2 /week during 4 weeks, for 8 patients. The mean ESSDAI was higher in patients treated (P = 0.0003) at the end of follow-up (Fig. 2A). Among the 64 patients with isolated TIN and at least two available eGFR values during follow-up, renal function significantly improved, with eGFR increasing from 35.0 (1.9) ml/min/ 1.73 m 2 to 43.0 (2.3) ml/min/1.73 m 2 at M6 and 42.5 (2.4) ml/min/1.73 m 2 at M12. Improvement of eGFR occurred regardless of the treatment used (Fig. 2B). In the eight patients with cryoglobulinaemia-related MPGN, mean eGFR also significantly improved at M12 and at last follow-up (P = 0.03) (Fig. 2C).

We next analysed the impact of the CSs/RTX combination, CSs/non-RTX combination and CSs alone in terms of achieving an eGFR gain of >20% compared with baseline, in the subgroup of patients with isolated TIN. In univariate analysis, renal improvement was more frequent among patients with severe initial renal dysfunction. Surprisingly, hypertension was associated with a good renal prognosis (P = 0.02). Positivity of anti-SSA or anti-SSB antibodies was associated with poor prognosis (P = 0.01 and 0.05, respectively). None of the pathology findings was associated with renal prognosis throughout follow-up (Table 4). There was no difference between the combinations of immunosuppressive agents plus CSs compared with CSs in univariate analysis, or in multivariate analysis, after adjustment for age at renal biopsy, hypertension and the presence of both anti-SSA or SSB antibodies (supplementary Table S4, available at Rheumatology Online). After modeling the eGFR dynamics over time, we found no slope difference between the CSs/ RTX combination, CSs/non-RTX combination and CSs alone (supplementary Fig. S1, available at Rheumatology Online). With a median follow-up of 60.4 months (range 12.1174.7), only five patients (5%) reached end-stage renal disease, including four patients with isolated TIN and one with focal segmental glomerulosclerosis associated with TIN. During this period, two cases of lymphoma were reported: one breast cancer and one lung cancer. Both cases involved patients with cryoglobulinaemia. Five patients receiving immunosuppressive agents (including RTX in three cases) presented with severe infection. Five patients died: one from septic shock, one from lung cancer and three from unknown causes.

Discussion

Renal involvement related to pSS, although only slowly progressive, can lead to chronic renal dysfunction that can impact long-term prognosis. The present study, which is the largest series of biopsy-proven pSS-related nephropathies published so far, focused on the clinical, biological and pathological presentation of this complication.

In our study, pSS diagnosis was based on the AECG criteria, which were satisfied in the vast majority of cases. However, as objective measurement of sicca symptoms was missing at pSS diagnosis in a minority of patients, we choose to use the enlarged AECG criteria, based on the presence of 53 out of the four items, including mandatory positive anti-SSA/SSB serology and/or positive minor salivary gland biopsy results [START_REF] Maldini | Epidemiology of primary Sjo ¨gren's syndrome in a French multiracial/multiethnic area[END_REF]. In addition, to avoid analysis bias, we excluded patients with secondary SS, associated with other systemic rheumatic diseases. Our study population was homogeneous, with clear female predominance and a median age of 49 years, as expected in pSS patients without renal disease [START_REF] Malladi | Primary Sjo ¨gren's syndrome as a systemic disease: a study of participants enrolled in an international Sjo ¨gren's syndrome registry[END_REF][START_REF] Baldini | Primary Sjogren's syndrome as a multi-organ disease: impact of the serological profile on the clinical presentation of the disease in a large cohort of Italian patients[END_REF]. Of note, a high proportion of patients presented with hypergammaglobulinaemia and/or cryoglobulinaemia, features consistent with an important B cell polyclonal activation.

This study provides important insights into the presentation of renal involvement during pSS. Among the most important findings, renal disease was diagnosed in the majority of cases either simultaneously or closely after pSS diagnosis, and presented as CKD, mild proteinuria and/or electrolyte disturbances. This presentation was in accordance with the tubulointerstitial lesions observed on kidney biopsies. Our results confirmed those of previous series, suggesting that the large majority of patients had TIN [START_REF] Goules | Clinically significant renal involvement in primary Sjo ¨gren's syndrome: clinical presentation and outcome[END_REF][START_REF] Ramos-Casals | Systemic involvement in primary Sjogren's syndrome evaluated by the EULAR-SS disease activity index: analysis of 921 Spanish patients (GEAS-SS Registry)[END_REF][START_REF] Maripuri | Renal involvement in primary Sjo ¨gren's syndrome: a clinicopathologic study[END_REF], a lesion that mimics minor salivary gland infiltration by both lymphocytes and plasma cells [START_REF] Guellec | Diagnostic value of labial minor salivary gland biopsy for Sjo ¨gren's syndrome: a systematic review[END_REF]. In contrast, glomerular disease was found in < 25% of cases, mainly related to cryoglobulinaemic MPGN. Although membranous nephropathy has previously been frequently associated with pSS [START_REF] Laraki | Membranous glomerulonephritis during primary Gougerot-Sjo ¨gren syndrome[END_REF], we observed only four such cases in our series.

Regarding correlations between immunological and histopathological features, we found that the presence of anti-SSA and particularly anti-SSB antibodies was very frequent in pSS interstitial nephropathy, suggesting that these pSS patients warrant careful renal function work-up. Of note, anti-SSA and anti-SSB antibodies were, respectively, found in 76.8 and 53.8% of our patients, whereas their prevalence was usually roughly 55% for anti-SSA and 33% for anti-SSB, in previous large epidemiological studies [START_REF] Gottenberg | Serum levels of beta2-microglobulin and free light chains of immunoglobulins are associated with systemic disease activity in primary Sjo ¨gren's syndrome. Data at enrollment in the prospective ASSESS cohort[END_REF]. Importantly, the presence of anti-SSA and/or anti-SSB is often considered as a marker of disease activity in pSS [START_REF] Candon | Quantitative assessment of antibodies to ribonucleoproteins in primary Sjo ¨gren syndrome: correlation with B-cell biomarkers and disease activity[END_REF][START_REF] Gottenberg | Correlation of serum B lymphocyte stimulator and b2 microglobulin with autoantibody secretion and systemic involvement in primary Sjo ¨gren's syndrome[END_REF].

Our study is one of the first to provide a detailed analysis of the cells infiltrating the kidney in pSS-related TIN. As reported in salivary glands [START_REF] Guellec | Diagnostic value of labial minor salivary gland biopsy for Sjo ¨gren's syndrome: a systematic review[END_REF], we found that T cell and B cell infiltration was a constant feature within the kidney. Evans et al. [START_REF] Evans | Tubulointerstitial nephritis in primary Sjo ¨gren syndrome: clinical manifestations and response to treatment[END_REF] recently showed in 12 patients with TIN that the interstitial infiltrate was predominantly a CD4 + T cell infiltrate. In contrast to the case in other causes of TIN, such as immune-allergic, infectious or sarcoidosis TIN, plasma cells were found in 75% of pSS-related TIN cases and were even predominant in 25% of biopsies. Of note, none of our patients had features consistent with IgG4-related renal disease, that is, presence of storiform fibrosis or pseudotumoral infiltration of the kidneys [START_REF] Saeki | IgG4-related kidney disease[END_REF]. Therefore, clinicians should be aware that the presence of plasma cells in the setting of TIN is very suggestive of pSS-related nephropathy. Finally, these findings suggest that specific therapies targeting B cells or plasma cells could be relevant, presuming that those cellular populations have a pathogenic role in this disease.

One of the main objectives of this study, despite its retrospective nature, was to evaluate the impact of therapeutic intervention on renal function. In contrast to previous findings by Goules et al. [START_REF] Goules | Clinically significant and biopsy-documented renal involvement in primary Sjo ¨gren syndrome[END_REF] that TIN was benign, four of our patients with isolated TIN reached end-stage renal disease. Our results are consistent with those of Maripuri et al. [START_REF] Maripuri | Renal involvement in primary Sjo ¨gren's syndrome: a clinicopathologic study[END_REF], who showed that treatment with glucocorticoids or other immunosuppressive agents can hamper the progression of renal disease over time. In our study, the mean GFR improved from 39.8 at baseline to 49.9 ml/min/1.73 m 2 at 1 year. The improvement in eGFR after treatment may be considered as modest, but this limitation is probably explained by the high level of renal fibrosis observed at presentation, reflecting the silent and insidious progression of TIN. It is important to underline that more than twothirds of the renal biopsies (analysed with computerized quantification of fibrosis) had >25% interstitial fibrosis, a threshold that is considered to be a strong prognostic marker, associated with poor renal prognosis in most nephropathies.

We observed that patients with more severe initial renal dysfunction were more prone to experience good renal response, especially patients with cryoglobulinaemia-related MPGN (Fig. 2C). This is most likely due to an earlier diagnosis for patients with proliferative glomerular disease, who will usually present with hypertension, high-range proteinuria and acute renal failure, in contrast with TIN patients who present with low-range proteinuria, no hypertension and slowly progressive renal failure. Along this line, the finding that hypertension was weakly associated with a good renal prognosis is probably related to associated confounders, such as better medical surveillance or the use of anti-hypertensive therapies. Of note, we were neither able to correlate the severity of interstitial fibrosis, nor the characteristics of the cellular infiltrate (i.e. B cell or plasma cells predominant in the infiltration) with the response to therapy, probably because of the small number of samples in which precise characterization was done. Interestingly, presence of anti-SSA or anti-SSA/SSB antibodies was associated with poor renal outcome.

A large majority of patients included in this study received CSs, and it is not possible to draw definitive conclusions concerning the efficacy of this treatment in pSS TIN when compared with supportive therapy, since the exact course of TIN patients without steroid treatment cannot be anticipated. Importantly, we found no additional benefit of immunosuppressive agents or RTX in comparison with CSs alone in patients with TIN. Among the limitations of our study, we must therefore underline the fact that immunosuppressive agents or RTX were administered according to the clinician's decision, in the absence of therapeutic guidelines and probably more frequently for patients with severe renal disease.

The efficacy of RTX in pSS is still controversial, as the TEARS trial failed to demonstrate any efficacy of this drug. However, patients included in this trial had low disease activity, and the primary outcome concerned fatigue, dryness and pain, but not the evolution of systemic involvement [START_REF] Devauchelle-Pensec | Treatment of primary Sjo ¨gren syndrome with rituximab: a randomized trial[END_REF]. In contrast, a retrospective registry showed good efficacy and safety of RTX on pSS systemic manifestations, including kidney involvement [START_REF] Gottenberg | Efficacy of rituximab in systemic manifestations of primary Sjogren's syndrome: results in 78 patients of the AutoImmune and Rituximab registry[END_REF].

Overall, these findings suggest that CSs alone seem to be an acceptable therapeutic option as first-line therapy in patients with TIN; given the prognosis, they are possibly better than supportive treatment alone. Immunosuppressive agents or RTX should probably be used in patients with refractory or relapsing disease, as well as in patients with cryoglobulinaemic GN [START_REF] Terrier | Management of noninfectious mixed cryoglobulinemia vasculitis: data from 242 cases included in the CryoVas survey[END_REF].

Moreover, since B-cell or plasma-cell infiltration is frequently found in pSS TIN, B-cell targeting agents such as RTX could be evaluated in CS sparing strategies.

In conclusion, our cohort may represent a subgroup of pSS patients, since renal involvement was severe enough to prompt a kidney biopsy. However, our study demonstrates that TIN represents the more frequent renal manifestation of pSS. Presence of either anti-SSA or anti-SSA/ SSB antibodies is predictive of both the risk of developing TIN, but also of experiencing poor renal outcome. Our data reveal that pSS-related TIN is characterized by frequent and sometimes predominant plasma-cell infiltration of the kidney, an observation that should be highly suggestive of pSS diagnosis. Finally, renal function significantly improved with treatment, although we found no clear benefit of immunosuppressive agents or RTX in comparison with CSs alone. Prospective and controlled trials are thus needed to define the place of immunosuppression and especially Bcell targeting agents in pSS-related TIN.

FIG. 1 6 )

 16 FIG.1Renal biopsy findings in the 95 patients with pSS

FIG. 2

 2 FIG. 2 Evolution of renal function during follow-up

TABLE 1

 1 Characteristics of patients with biopsy-proven pSS-related renal involvement

	Number of patients	95
	Demography
	Gender	86 F/9 M
	Age, median (IQR)	49.0 (17.987.3)
	pSS diagnosis, n (%)
	AECG criteria (n = 95)	82 (86.3)
	Enlarged criteria only (n = 95)	13 (13.7)
	Other autoimmune diseases	11 (11.6)
	associated with pSS (n = 95)
	Dry eyes (n = 93)	91 (97.8)
	Subjective (n = 93)	82 a (88.2)
	Objective (n = 65)	56 (86.2)
	Dry mouth (n = 94)	89 (93.7)
	Subjective (n = 94)	87 b (92.6)
	Objective (n = 42)	32 (76.2)
	ESSDAI, median (IQR)	16.0 (941)
	Comorbidities, n (%)
	Hypertension (n = 95)	27 (28.4)
	Diabetes (n = 95)	7 (7.4)
	Hypertension and diabetes (n = 95)	6 (6.3)
	Immunological features, n (%)
	ANA (+) (n = 95)	85 (89.5)
	Anti-SSA (+) (n = 95)	73 (76.8)
	Anti-SSB (+) (n = 93)	50 (53.8)
	Anti-SSA and SSB (À) (n = 95)	22 (23.2)
	Salivary gland biopsy (+),	82 (96.5)
	Chisholm score 53, (n = 85)
	RF (+) (n = 82)	53 (64.6)
	Cryoglobulinaemia (+) (n = 89)	26 (29.2)
	Low C3 (<800 mg/l) (n = 75)	13 (17.3)
	Low C4 (<150 mg/l) (n = 76)	27 (35.5)
	Hypergammaglobulinaemia	55 (67.9)
	(516 g/l) (n = 81)
	IQR: interquartile range; AECG: AmericanEuropean
	Consensus Group; ESSDAI: European SS Disease Activity
	Index. Missing data for a 'two subjective ocular symptoms'
	and

b 'one subjective dry mouth'.

TABLE 2

 2 Characteristics of renal involvement at presentation Fourteen patients (14.7%) received neither CSs nor immunosuppressive agents, mainly because of severe renal fibrosis considered to be irreversible in the majority of cases. Only one patient refused to receive CSs for the treatment of TIN.

	Number of patients	95
	Age, median (IQR), years	50.9 (787.3)
	Delay between renal symptoms	0 (À265.7, 352.1)
	and pSS, median (IQR), months
	Delay between renal symptoms/
	pSS, n (%)
	A least 1 year before	10 (10.5)
	Simultaneous (1 year)	55 (57.9)
	15 years after	20 (21.1)
	>5 years after	10 (10.5)
	Renal dysfunction (eGFR <60), n (%)	82 (86.3)
	Acute kidney injury	30 (31.6)
	CKD	52 (54.7)
	Proteinuria
	Presence, n (%)	25 (26.3)
	Dosage, mean (S.D.), g/day	1.4 (0.3)
	Isolated electrolyte disturbances, n (%)	17 (17.9)
	Lithiasis, n (%)	9 (9.5)
	Nephrocalcinosis, n (%)	5 (5.3)
	CKD stages, n (%)
	II (560 ml/min/1.73 m 2 )	13 (13.7)
	IIIa (4559 ml/min/1.73 m 2 )	16 (16.8)
	IIIb (3044 ml/min/1.73 m 2 )	34 (35.8)
	IV (1529 ml/min/1.73 m 2 )	23 (24.2)
	V (<15 ml/min/1.73 m 2 )	8 (8.4)
	Unknown	1 (1.1)

IQR: interquartile range; CKD: chronic kidney disease.

with RTX than in those not treated with RTX [22.6 (2.2) vs 17.3 (0.7); P = 0.03].

2

at M6 (P = 0.0001), 49.9 (3.0) ml/min/1.73 m 2 at M12 (P = 0.00009) and 47.5 (2.5) ml/min/1.73 m 2

TABLE 4

 4 Baseline characteristics according to the improvement or not of eGFR in patients with tubulointerstitial nephritis

		No eGFR gain or	eGFR gain
	Patient characteristics	a gain of < 20% (n = 32)	of 5 20% (n = 32)P-value a
	Demography and comorbidities
	Gender, female, n (%)	30 (94)	30 (94)	0.9
	Age at diagnostic, median (IQR), years	46 (3163)	54 (3466)	0.4
	Age at kidney biopsy, median (IQR), years	51 (3565)	58 (3366)	0.5
	Hypertension, n (%)	3 (9)	11 (34)	0.02 b
	Diabetes, n (%)	1 (3)	3 (9)	0.5
	pSS characteristics
	ESSDAI, median (IQR)	15 (1218)	16 (1220)	0.3
	AECG criteria, n (%)	28 (88)	26 (81)	0.5
	Enlarged criteria, n (%)	4 (13)	6 (19)	0.4
	(i) Anti-SSA, n (%)	31 (97)	23 (72)	0.01
	(i) Anti-SSA and -SSB, n (%)	25 (78)	17 (53)	0.05
	Chisholm score 53, n (%)	26 (82)	27 (84)	1
	RF positivity, n (%)	21 (66)	15 (47)	0.3
	Cryoglobulinaemia positivity, n (%)	9 (28)	7 (22)	0.5
	Serum gammaglobulin, median (IQR), g/l	23 (1732)	20 (1638)	0.8
	Hypergammaglobulinaemia (>16 vs <16 g/l), n (%)	26 (81)	25 (78)	0.8
	Characteristics of renal involvement
	Delay between renal disease/symptoms of pSS, year, median (IQR)	1.5 (0.55.4)	1.0 (0.14.3)	0.3
	(i) eGFR at inclusion, median (IQR), ml/min/1.73 m 2	40 (3352)	30 (2037)	0.002
	Isolated electrolyte disturbances, n (%)	6 (19)	10 (31)	0.2
	Lithiasis, n (%)	5 (16)	3 (9)	0.2 b
	Nephrocalcinosis, n (%)	4 (13)	1 (3)	0.2 b
	Proteinuria, median (IQR), g/24 h	0.5 (0.31.0)	0.5 (0.20.8)	0.5
	P/C, median (IQR), g/mmol	0.07 (0.030.1)	0.05 (0.020.1)	0.4
	Lymphocyte infiltration, n (%)	32 (100)	26 (81)	0.2
	Plasmocyte infiltration, n (%)	23 (72)	16 (50)	0.5
	Fibrosis >25% vs 425%, n (%)	8/13 (62)	13/18 (72)	0.9
	Degree of cellular infiltration<50% vs 550%, n (%)	9/21 (43)	12/21 (57)	0.08 b

a Baseline characteristics were compared using the Chi-square test and the Wilcoxon rank-sum test for dichotomous and continuous variables, respectively. b The Fisher exact test was used as appropriate. IQR: interquartile range; AECG: AmericanEuropean Consensus Group; ESSDAI: European SS Disease Activity Index; eGFR: estimated glomerular filtration rate; P/C: protein/creatinine ratio; pSS: primary SS.

Acknowledgements

The authors thank all the clinicians and pathologists who participated in the REINSS study: C. Agard, A. Audemard, S. Bally, J.J. Boffa, D. Le Thi Huong-Boutin, P. Cacoub, P. Cathebras, G. Choukroun, E. Daugas, J.G. Fuzibet, C. Garrouste, M. Hamidou, P.Y. Hatron, D. Joly, J.E. Kahn, B. Knebelmann, T. Kofman, B. Laurent-Pilonchery, C. Lavigne, G. Le Guenno, T. Lobbedez, J. Schmidt, C. Sordet, A. Barbier-Dupas, A. Guillaudeau, A. Moreau, B. Ambrosetti, B. Mougenot, B. Mac-Gregor, C. Deminiere, C. Guilbeau-Frugier, C. Mussini, D. Droz, D. Desvaux-Belghiti, F. Comoz, H. Perrochia, H. Beaufils, J. Verine, L.H. Noel, J.P. Duong Van Huyen, A. Modesto, L. Marcellin, M. Mignon-Conte, D. Nochy, N. Patey-Mariaud De Serre, P. Rouvier, S. Lepreux, V. Algagnac-Oskman, M.-C. Copin, B. Gosselin, B. Delisle, F. Paraf, I. Brocheriou, J.-P. Saint-Are, J.-L. Kermeny, P. Callard.

Author contributions: M.J., A.K., V.L.G., X.M., E.T., H.F. and B.T. designed the study. M.J., A.K., V.L.G., R.M., S.F., N.J.C., A.L.F., L.C., E.D., G.M., J.B.F., E.L., P.J., E.H., A.L.Q., P.R., A.H., N.C.C., P.R., P.V., V.M.Y., A.M.S., C.C., S.F., L.D., R.S., X.M., E.T., H.F. and B.T. collected data. M.J., E.K. and H.F. performed the statistical analysis. M.J., H.F., V.L.G., X.M., E.T., A.K. and B.T. interpreted the data. M.J., A.K., V.L.G., H.F. and B.T. wrote the manuscript.

Funding: No specific funding was received from any bodies in the public, commercial or not-for-profit sectors to carry out the work described in this manuscript.

Supplementary data

Supplementary data are available at Rheumatology Online.


Gaudin

O Peyrusse
email: olivier.peyrusse@univ-amu.fr

J.-M André

P Jonnard

J Gaudin

Modeling of the interaction of an XFEL with large finite samples

Keywords: numbers: 52.65.-y, 41.50.th, 42.55.Vc, 52.25.Os

Modeling of the interaction of an x-ray free-electron laser with large finite samples

I. INTRODUCTION

X-ray free-electron lasers (XFELs) provide unprecedented high brilliance, good coherence and very short pulse duration in the X-ray regime. Their specifications have opened new frontiers for experiments in physics and chemistry (see [1] and references therein). In this context, the use of focusing optics which enable strong enhancement of X-ray intensity contributes to the performance of these sources. In particular, reflective mirrors can achieve the highest focusing efficiency [2]. However, XFEL characteristics impose severe requirements on the optics used to guide and shape the x-ray pulses. Here the onset of damage is difficult to evaluate and can occur well before the material reaches its melting temperature. Furthermore, in the hard x-ray regime and at grazing incidences, the damage threshold needs to be evaluated by taking into consideration the energetic photoelectrons that can remove deposited energy from the interaction region. This particular example illustrates how XFELs involve a new regime of radiation-matter interaction which has to be investigated. The development of a theoretical model describing this interaction is difficult since the sample enters a regime refered as warm dense matter (WDM) regime, not very well understood.

A large effort has been put in the development of numerical tools to model the intense monochromatic X-ray irradiation of matter. These models use either particle or continuum approaches [3][4][5][6]. Because of the complex physics underlying the XFEL interaction with matter, the most convenient approach depends on the X-ray intensity, on the photon energy and on the size of the samples.

In this article, we present a model for the interaction of monochromatic X-rays with large finite samples which has resulted in a code called X-RIM. The model must deal with pulse duration shorter than the electron-ion relaxation times so that electrons and ions must be considered separately as two subsystems. Then, the model consists of a one-dimensional one-fluid, two-temperature model. It includes the elasto-plastic behavior of solids. In X-RIM, X-ray electric field amplitude in the material obeys the Helmholtz equation while the interaction is described with a complex index of refraction which depends on density and temperatures through a non-local thermodynamical equilibrium (NLTE) modeling of the X-ray opacity. Moreover, our model includes a deterministic treatment of the transport of energetic photoelectrons. It is well known that x-rays interact with the atomic inner-shells and that atomic processes such as photoionization, autoionization, photoexcitation as well as collisional secondary processes coupling different atomic configurations, are involved during and after the x-ray pulse. This specific aspect as well as the completness of the configuration list is discussed elsewhere [7,[START_REF] Peyrusse | Modern Methods in Collisional-Radiative Modeling of Plasmas[END_REF]. From the deposited energy, electronic temperature is obtained consistently with the population kinetics through a linearization procedure. The model depends on external physical constants such that shear modulus, yield strength, thermal conduction coefficient and electron-ion equilibration time. Throughout a simulation these parameters are needed for a broad range of temperature and density. This point can be a problem to address in independent studies. The computer tool described here allows one to imagine pump-probe experiments where the dynamics of an adequately pumped sample can be probed, then modeled and fitted as a function of rather unknown parameters in the WDM regime (electron-ion collision frequency, conductivity, shear modulus, etc). The values chosen or given to these parameters are a way of testing microscopic theories or abinitio derived thermophysical properties [START_REF] Wang | [END_REF][10][11]. Furthermore, the development of such a code could be of crucial help for optimizing the design of optics for x-ray radiation sources.

Most of the current beamlines use flat mirrors to steer the beam, and Kirkpatrick-Baez mirrors to achieve high focusing. X-ray mirrors are made of a substrate (most of the time Si, or SiO2) coated with a thin layer (between 50 to 100 nm) of a reflective element. Other solutions like Fresnel zone plates [12], or Berylium lens [13] are also used to focus the beam. These latter ones are used at normal incidence angle, while mirrors are set up at very low grazing incidence. To sum up, this article is of interest for both fundamental studies of X-ray interaction with matter and applied studies important for X-ray optics.

In Section II, we give a detailed description of the X-RIM model. Then, focusing on a rather low intensity regime of irradiation, we discuss the behavior of different samples of interest for X-ray optics (Section III). In particular, we investigate how absorption takes place and we present profiles of temperature deep inside irradiated materials (Section III).

In the last section (Section IV), we summarize and present the conclusion.

II. OVERVIEW AND MODELING OF THE PHYSICAL PROCESSES

We consider the phenomena arising when a short, intense X-ray pulse interacts with a solid material. the dominant interaction is photoionization of the atoms in the solid. If the photon energy range is typically between 100 eV and a few tens of keV, the primary absorption process is inner-shell photoionization. Within a few femtoseconds, shell holes decay mainly by autoionization, giving Auger electrons in addition to the primary photoelectrons. From these electrons, a collisional-radiative cascade takes place where secondary electrons resulting from collisional ionization are produced. At this step, one faces the problem of evaluating the incident X-ray electric field inside the material, along with the non-local thermodynamical equilibrium (NLTE) atomic physics problem where populations of inner-shell ionized atoms must be followed as a function of time. From these populations, one can built the opacity in the material. At each time, a complex refractive index can then be calculated which in turn allows one to compute the X-ray electric field. From the local NLTE opacity and from the local value of the electric field, it is possible to compute a local source of free electrons. As mentioned in the introduction, it seems suitable to consider two main classes of x-ray induced free electrons: those which deposit quasi-locally their energy (i.e. those thermalizing locally) and those (i.e. the most energetic) which can escape the interaction zone and deposit their energy elsewhere. After typically an hundred of femtoseconds, thermalized free electrons start to transfer their energy to the ions. Well after the end of the pulse, i.e. after about one picosecond, the heated matter experiences a macroscopic motion. In the continuum approximation, this behavior can be described by a set of hydrodynamics equations which must describe first the coupling electron-ion and then, the macroscopic ionic motion. As a consequence, there are two main classes of physical problems to present and to discuss here: (i) the internal X-ray electric field and energy deposition, with a special attention on the complex refractive index and on the transport of the energetic free electrons, (ii) the hydrodynamics and potential elasto-plastic behavior.

A. Internal X-ray electric field and energy deposition

The electric field in the material

A typical XFEL pulse (in the Self-Amplified Spontaneous Emission Mode) has a bandwith of 0.2% which gives about 20 eV for 10 keV photons. However, a typical pulse is rather made of many quasi-monochromatic spikes lying within this bandwidth. Consequently, we will consider the problem of a monochromatic (ω o ) plane-wave incident on a stack of homogeneous Lagrangian layers. Indeed, consistent with a 1D plane-parallel hydrodynamics description, we consider the material as a juxtaposition of cells of constant mass while the cell density and thickness may vary over time (Lagrangian description). Depending on the polarisation and on the angle of incidence, a part of the wave is reflected while the other propagates or is absorbed in the medium (see Fig. 1). From macroscopic Maxwell equations, it results that the electric field in the material obey a wave-equation. Writing each component of the electric field as E i = Ẽi exp iω o t where i stands for (x, y, z), the problem amounts to solving the Helmholtz equation written as

∆ Ẽi + ω 2 o c 2 ñ2 Ẽi = 0 (1)
inside the material, at every instant. Here ñ is the complex refractive index which must be calculated in each cell of the material. We note that, because of the generally small extinction lengths, we neglected here the time-dependence of the Ẽi even though there are some indications that small time-delay effects may exist in the context of short pulse X-ray irradiation [14,15].

The discretization of the medium is along x and the propagation is supposed in the plane (x, y). In S polarization, one follows only component Ẽz written as Ẽz = ẽz (x) exp (ik o y cos θ), while in P polarization, one follows components Ẽy and Ẽx written as Ẽy = ẽy (x) exp (ik o y cos θ) and Ẽx = ẽx (x) exp (ik o y cos θ), respectively.

One of the most convenient approach to calculate the electric field inside a stratified structure is the matrix approach. Indeed, it turns out that a stratified structure made of isotropic and homegeneous cells and plane-parallel interfaces can be described by 2×2 matrices using the fact that the tangential component of the electric field is continuous and that Eq. ( 1) is linear. Elegant and efficient methods exist [START_REF] Knittl | Optics of Thin Films[END_REF][START_REF] Leif | [END_REF], the only difference here is the use of cells of variable thickness corresponding to our Lagrangian mesh. Also, while our refractive index is here a scalar quantity, the matrix method can be extended to anisotropic media where one deals with dielectric tensors [START_REF] Yeh | Optical Waves in Layered Media[END_REF].

The complex refractive index

As mentioned above, the complex index of refraction ñ of each Lagrangian cell is needed in the modeling of the electric field in the medium. It depends on the material of the cell and on its microscopic state at each instant in a simulation. ñ is conveniently written as

[19] ñ = 1 -δ -iβ (2)
with δ = roλ 2 o 2π N f 1 and β = roλ 2 o 2π N f 2 .
N is the density of atoms and λ o the wavelength. Quantities f 1 and f 2 are the real and the imaginary part of the atomic scattering factor.

They are linked to the local opacity per atom κ(ω) through the relations

f 1 (ω o ) = Z * + b ω 2 κ(ω) ω 2 o -ω 2 dω, (3)
and,

f 2 (ω o ) = π 2 ω o bκ(ω o ) (4)
where b -1 = πhcr o , r o being the classical electron radius.

This set of relations are the so-called Kramers-Kronig relations. Z * is the atomic number corrected for relativistic effects [START_REF] Henke | [END_REF]. The opacity N κ(ω) is a sum of bound-free, boundbound and free-free processes and depends on the local populations of the atomic levels in the considered cell. In X-RIM, our atomic description is a Detailed Configuration Accounting (DCA) approach working with a predefined list of configuration depending on the material and on the photon energy [5,[START_REF] Peyrusse | Modern Methods in Collisional-Radiative Modeling of Plasmas[END_REF]. As a consequence, at each instant in the simulation, the opacity is built from the current configuration populations resulting from a collisionalradiative (CR) calculation. This point is extensively described in dedicated textbooks (see for instance [START_REF] Bauche | Atomic Properties in Hot Plasmas -From Levels to Superconfigurations[END_REF]). It is important to note that the opacity built in this way reflects the strong time-dependent aspect of the interaction with the XFEL as well as the more or less complex kinetics which may occur during the pulse. This out of equilibrium aspect of the complex refractive index is an important feature of the model. Evaluation of integral in f 1 requires some care since the pole makes the evaluation of the integral difficult (principal value of the integral). We chose the Maclaurin's formula in which all points but the pole are taken for the summation [START_REF] Ohta | [END_REF]. Another point is the frequency step and the frequency range which obviously depend on ω o (the energy of XFEL photons). From the lower bound zero, we chosen safely an upper bound of 15 times ω o with a step of 0.5 eV.

In Fig. 2, we display the opacity N κ of cold solid Ru and of solid warm Ru (in a restricted range of photon energy), as calculated in line by X-RIM. Note that for low photon energies (< 20 eV) the opacity has been artificially considered as a constant in order to avoid the low energy divergence of the classical Kramers free-free opacity used in this work. For photon energies below the first photoionization threshold, a proper treatment of the free-free opacity would be required (for a discussion on this point, see for instance [22]). However beyond the XUV range, for such an important Z and in this range of temperature, the contribution to the opacity comes from the bound electrons. In cold solid Ru, the contribution comes from ion Ru + given that one assumes an initial ionization of 1 (one valence electron in the solid). It is worth noting that ionization thresholds have been lowered by a quantity called ionization potential depression (IPD). The IPD model used here is a common formulation of the Stewart-Pyatt model [23]. From the Kramers-Kronig relations, the real part of the complex refractive index is deduced from κ through a calculation of the atomic scattering factor f 1 . The optical constants f 1 of Ru corresponding to Fig. 2 are displayed in Fig. 3.

These figures illustrate how the spectral features of the absorption spectrum (photoionization thresholds, lines) gives peculiar counterparties on the real part of the refractive index.

Energy deposition and transport of energetic photoelectrons

From the values of the electric field components, one obtains at each instant, the local power deposited in the material (per volume unit) through the Joule-Lenz law P = j. E in

which j = σ E = iω o o (ñ 2 -1). E.
The time average over one period of P reads finaly,

P abs =< P >= 1 2 σ 1 ( Ẽ * x Ẽx + Ẽ * y Ẽy + Ẽ * z Ẽz ) (5)
where

σ 1 = -2 (ñ) (ñ) o ω o . (6)
The strict use of Eq. ( 5) would imply that P abs is a total power deposited locally on the free electrons according to the local value or ñ and E.

As mentioned in the introduction, the way the XFEL beam deposits its energy cannot be described by a simple local source term such as Eq. ( 5). In the multi-keV X-ray range, absorption generates rather energetic photoelectrons that can escape from the region of interaction and deposit their energy in neighboring regions of the material, while sub-keV electrons deposit their energy rather locally in a nanoscale volume. For that reason, it is important to mitigate a purely local absorption (producing sub-keV photoelectrons) and to model the transport and the energy deposition of the energetic electrons. In order to relax the local character of the energy deposition given by Eq. ( 5), one first observes that it is possible to extract from the opacity calculation, the contribution of the processes giving electrons in a particular class, e.g. the photoelectrons in some kinetic energy range. Considering the keV or multi-keV electrons, one can define a power (per volume unit) deposited on these electrons which is

P keV = N κ keV (ω o ) ω o φ o (7)
where φ o is the local X-ray flux (deduced from the local value of the electric field, i.e. proportional to |E| 2 ) while N κ keV (ω o ) is the local opacity at ω o corresponding the photoionization processes giving keV and multi-keV electrons. Of course these energetic photoelectrons can be divided into groups of different energies according to the corresponding bound-free opacity and it is possible to define a local source of energetic electrons in each group. At this point, one may also consider the direction of the emitted photoelectrons with respect to direction and polarisation of incident photons. Except for one-electron atoms (see [24] and references therein), very few calculations of the corresponding differential photoionization cross-section have been done. We are led to consider real atoms as hydrogenic atoms and use the results obtained for the K-shell, L-shell, etc. For instance for K-shell, the differential cross-section has the form σ ∝ sin 2 θ (1 -v c cos θ ) -4 where θ is the angle between the inci-dent photon and ejected electron direction, v is the velocity of the ejected electron. While valid only for the photoionization of K-shell electrons, it seems that in many practical situations, the use of this distribution for other shells does not introduce appreciable errors in electron transport applications [25]. We used this distribution in the simulations presented in the next Section.

As a consequence of our distinction between low-energy electrons and energetic electrons, the power deposited locally by sub-keV electrons in the material becomes

D 1 = P abs -P keV (8)
while keV or multi-keV electrons must be transported before energy deposition. For that purpose, we implemented a deterministic transport model in 1D planar geometry.

Considering the energetic electrons only and at the step of the opacity calculation, one first calculates the local source of photoelectrons Q(E, x, t, µ). Q is a number of photoelectrons of energy E produced per interval of energy, per volume unit, per solid angle unit and per time unit, at x, at time t and of travelling direction µ = cos Ω). Note that E represents here a group of energies.

Denoting W (E, x, t, µ) as the angular fluence of energetic photoelectrons, W obeys the transport equation

1 v ∂W ∂t + µ ∂W ∂x = Q + C(W ) ( 9
)
where v is such that E = 1 2 mv 2 . C is the collision operator which can be written as

C(W ) = ∞ 0 dE dΩ σ s (E , E, Ω. Ω )W (E , x, t, Ω ) -Σ(E)W (E, x, t, µ) (10)
where

Σ(E) = 2π +1 -1 dµ o ∞ 0 σ s (E , E, µ o )dE . σ s (E , E, µ o )
is the whole differential scattering cross-section of the electrons in the material (it contains the contribution of elastic and inelastic scattering processes). The dot product Ω. Ω = cos µ o gives the scattering angle. An expansion of C around µ o = 1 (forward-peaked scattering) and around E = E (continuous slowing-down approximation) gives the Fokker-Planck approximation [26],

C = ξ 1 (x, E) 2 L µ W + ∂ ∂E [S(x, E)W ], (11)
where ξ 1 (x, E) is the first-order scattering transport coefficient,

ξ 1 (x, E) = 2πN (x) +1 -1 (1 -µ o )dµ o ∞ 0 σ s (E , E, µ o )dE (12)
in which N (x) is the local density of scattering atoms. Also in Eq. (11),

L µ = ∂ ∂µ (1 -µ 2 ) ∂ ∂µ
is the angular laplacian operator which can be discretized with appropriate Gauss-Legendre abcisses and weights [27]. S(x, E) is the stopping power formally written as

S(x, E) = 2πN (x) +1 -1 dµ o ∞ 0 E σ inel (E , E, µ o )dE ( 13
)
where σ inel is the cross-section for inelastic collisions (mostly ionizations). Note that many publications and databases provide this stopping power.

Finally, the energy deposited locally by the electrons described by the angular fluence W reads

D 2 (x, t) = 2π +1 -1 dµ ∞ 0 S(x, E)W (E, x, t, µ)dE. ( 14
)
This term enters in the source term S e of the set of Equations 15 (see below). Among the practical details concerning the implementation of this deterministic transport model, we shall mention a use of the Mott angular differential cross-section [START_REF] Mott | The Theory of Atomic Collisions[END_REF] (elastic scattering)

with the usual choice of the Wentzel screening parameter. We note that in collisions with atoms, elastic scattering is the prevailing contribution to angular deflections of electrons.

To include deflections coming from inelastic collisions, we use the current approximation where the square of the atomic number Z in the Mott angular differential cross-section is replaced by Z(Z + 1). This allows us to calculate the first order scattering factor ξ 1 (Eq. ( 12)) for each energy group. For the stopping power S, we chosen to use results existing in specialized databases (see below).

B. Hydrodynamics and elasto-plastic behavior

Well after the end of the pulse, i.e. after about one picosecond, the heated matter experiences an hydrodynamics movement driven by the stress tensor. In other words, the layer boundaries sketched in Fig. 1 start to move. Based on this picture, and in 1D planar geometry, the following set of Lagrangian equations characterizes evolution of both the temperatures (T e for the electrons and T i for the ions) and the density of the material:

             d dt ρ = -ρ ∂v ∂x d dt v = + 1 ρ ∂(σxx) ∂x d dt e = -Pe ρ ∂v ∂x + Se ρ d dt i = -P i ρ ∂v ∂x + S i ρ ( 15
)
where ρ is the mass density, v is the fluid velocity while e and i are the electron and ion internal energy (per mass unit), respectively. σ xx is the component xx of the stress tensor.

Finally, the two source terms are of the form S e = -n e 3me

m i ν ei k(T e -T i ) + κ e ∂ 2 Te ∂x 2 + D 1 + D 2 and S i = n e 3me m i ν ei k(T e -T i )
, respectively. The first term in S e and S i is the volumic energy transfer rate between the electrons (of mass m e ) and the ions (of mass m i ), in which ν ei is the electron collision frequency. In order to consider materials that undergo a transition between the solid state and the plasma state, an heuristic prescription for ν ei is discussed in various articles [5, [START_REF] Eidmann | [END_REF]30]. Otherwise, this parameter can be considered as a free and adjustable parameter of the model to be deduced from a comparison with experiments or obtained from a specific modeling. These remarks are also valid for the electron thermal conductivity coefficient κ e present in the thermal conduction term (second term of S e ). The last two terms D 1 and D 2 , are the volumic power deposited locally by the XFEL and, the volumic power deposited by the energetic electrons, respectively. They are given by Eq. [START_REF] Peyrusse | Modern Methods in Collisional-Radiative Modeling of Plasmas[END_REF] and Eq. ( 14) respectively and they depend on how the absorption takes place and on how the energetic free electrons deposit their energy. This point has been addressed in Section II-A-3.

In order to consider low flux interaction conditions and to address damage problem in X-ray optics, it is important to account for the elasto-plastic behavior of material. For that reason, unlike the Ref.

[5], we consider in the previous set of equations the component xx of the stress tensor (1D planar geometry). This component can be split into two parts

σ xx = -(P e + P i ) + s xx ( 16
)
where P e , P i are the thermodynamical electron and ion pressure, respectively and s xx is the deviatoric part describing the elasto-plastic behavior. s xx reads

s xx = 2G (x) ( 17
)
where G is the shear modulus and (x) is the deformation or strain obeying the equation (in Lagrangian form)

d dt = 2 3 dv dx . (18)
Beyond some deformation, i.e. beyond the elastic behavior, the material becomes plastic when s xx ≥ 2 3 Y (in the Von Mises sense) in such a way that s xx = min(s xx , 2 3 Y ), in which Y is the yield strength. The two mechanical parameters G and Y , generally known experimentally, depends on the thermodynamical properties. In particular, they increase with increasing pressure and they decrease with increasing temperature. To model this behavior, semiempirical models have been developed [31]. More simply, G and Y decrease when the material goes toward the liquid phase and vanishes at melting so that these two parameters can be multiplied by a simple decreasing function of the internal energy i . More precisely, such a function (varying between 1 and 0) depends on the difference i -( i ) o and must vanish for i = ( i ) m where is ( i ) o is the ion internal energy in the cold material and ( i ) m is the ion internal energy at melting. A simple polynomial of the third degree may satisfy this requirement. During a simulation, the state of a given Lagrangian cell can be tracked and labeled as elastic solid, plastic solid or fluid.

C. Description of the computational model

We have developed a computer program X-RIM to solve the equations presented in the previous Sections. This work relies on the 1D hydrocode already presented in Ref. [5]. To summarize, we consider the propagation of an X-ray pulse in a 1D material consisting in a juxtaposition of layers as shown in Fig 1 . Axes are oriented so that the axe of stratification x is normal to the surface (y, z). Propagation takes place in the plane (x, y). At each time step, the populations of the atomic populations are obtained in each cell from a configurationaverage NLTE atomic model (see Refs.

[5] and [START_REF] Peyrusse | Modern Methods in Collisional-Radiative Modeling of Plasmas[END_REF] for more details). Follow a calculation of the local opacity and of the local complex refractive index (Eqs. (2-4)) with the hypotheses discussed in section II-A-2. Then, at each time step during the pulse, the Helmoltz equation (Eq. ( 1)) is solved to get a map of the electric field into the material. This electric field is used to calculate the energy deposition according to the hypotheses discussed in section II-A-3 i.e. by separating the contribution of processes giving (relatively) low-energy electrons and the contribution of photoionization processes giving energetic electrons. These last electrons are supposed to obey a deterministic transport equation (Eq. ( 9)). During and after the XFEL pulse, the coupling of the free electrons with the ions (i.e. the heating of the material) and then the positions of the layer boundaries (in our Lagrangian description) are driven by a set of hydrodynamical equations (Eq. ( 15)).

The electron collision frequency ν ei used here is an effective frequency interpolating between an effective electron-phonon frequency and the plasma electron collision frequency.

Details on this interpolation procedure and the electron-phonon frequency effectively used are given in Ref. [5]. The same kind of interpolation is used for the electron-ion frequency involved in the electron thermal conductivity κ e .

For the ion Equation-of-State (EOS), i.e. the formulation of both i and P i , we use the QEOS model [32]. Electron EOS, i.e. ( e , P e ), is the sum of three contributions: a cold (or zero-temperature) part, a thermal contribution of the bound electrons and a contribution of the free electrons. The latter part is built from the quantum electron gas model. Again, see Ref.

[5] for more details.

Among other the computational details, it is important to note that for the stopping power, we used results of the ESTAR database of NIST [33]. We solve Eq. ( 9) with the Lax-Wendroff explicit finite differencing scheme for the advection part (left side of Eq. ( 9) including Q) while for the right part C (see Eq. ( 11)), we use an implicit differencing scheme in angle and in energy for the angular and energy derivatives, respectively. Calculations presented below has been performed using 40 groups of energetic photoelectrons and 20 Gauss-Legendre directions. Energetic electrons are typically considered between 2 keV and ω o -χ where χ is the smallest encountered ionization threshold. Typically, we used 50-100 spatial zone in the simulations.

For completness, we mention an omission in this modeling: the surface charging and the possibility of electron reflux into the material. Indeed, escaping energetic photoelectrons undergo a restoring force due to surface charging. However, it is unlikely that these electrons will return exactly from where they come from. Nevertheless, this place limits in the accuracy of our simulations in the sense that near the surface target, the possible cooling due to the electron escape is probably overestimated.

We simulated the behavior of samples of a few materials in 1D planar geometry. In particular, we focused on the behavior of a multilayer material irradiated by a low intensity pulse at grazing incidence. This is important to quantify the radiation damage of X-ray optics. In this context, the physical aspects discussed in this article (wave-like nature of the radiation-matter interaction, transport of the energetic photoelectrons, elasto-plastic behavior of materials) are crucial for a proper modeling of the problem. It is clear that for other applications such as the creation of warm dense matter [34] or of hot dense plasmas [35] involving high-intensities and not so grazing incidences, these physical aspects are less relevant (with the exception of the energetic electron transport). We note also that in these higher intensity conditions, atomic kinetics may be more complex. Concerning the possibility of damage, our conjecture is not only that the ion temperature must remain below the melting temperature but that the material must remains below the plastic threshold on a whole simulation. This is the condition for considering a material as undamaged after the interaction with an XFEL pulse.

A. Grazing incidence hard x-ray mirrors

We first illustrate the results of the computer model X-RIM with an example where the damage threshold has been measured in a complex high-Z material used for X-ray mirror coating: Ru. Its high density implies a high critical angle and allows it to operate at high grazing angles. For this kind of material, it is important to know its single shot damage threshold limit. Then, Ru coated flat Si mirror substrates have been exposed to hard XFEL radiation [36] : photon energies of 7 and 12 keV, pulse energies up to 100 µJ and durations of 20 fs. Because the sample densities and the thickness were measured using Mo K-α radiation (17.5 keV), the reflectivity of a Ru coated substrate of Si (50 nm of Ru on 1 µm thick Si) as calculated with X-RIM at this energy is shown in Fig. 4. In the case of XFEL irradiation and in order to obtain large reflectivities (i.e. greater than 90 %), grazing incidences of 4.5 mrad and 3.0 mrad have been used at 7 keV and 12 keV respectively. at 7 keV [36]. The laser comes from the right, i.e. the ruthenium side, at a grazing incidence of 4.5 mrad. In Fig. 5, the calculation considers a purely local absorption of XFEL photons (with a purely diffusive treatment of electron transport driven by the term κ e ∂ 2 Te ∂x 2 in the third equation of system (15)) while in Fig. 6, the transport of energetic photoelectrons (≥ 2000 eV) is taken into account. These calculations uses 40 velocity groups with the deterministic formalism described above. Here, we did not include the secondary electrons coming from the KLL-Auger relaxation process (they are considered as locally thermalized). Comparison of Fig. 5 and Fig. 6 illustrates the strong importance of the energetic electron transport in these conditions of irradiation as underlined elsewhere [36][START_REF] London | Proc. of SPIE[END_REF][START_REF] Kim | [END_REF]. Calculations that do not take into the electron transport of energetic electrons (cf. Fig. 5) give sharp profiles and lead to a strong overestimation of the electron temperature (and after equilibration, to an ion temperature well above the melting temperature of Ru) unlike the calculations presented in Fig. 6. In the latter case however, it is worth noting that, after about one picosecond, the Ru layer has not melted but reaches the plastic state if one uses the standard value of the yield strength of Ru i.e. 0.372 GPa (see the discussion of Section II-B).

In that sense, these calculations should indicate that significant damage has occured at this measured threshold fluence. Nevertheless, there are some indications that thin films deposited on subtstrates present significantly high values of the yield strength compared with bulk materials [39,40]. The reasons given are that grain boundaries and interfaces 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 The intensity is of 1.55 × 10 16 W/cm 2 , the (Gaussian) pulse duration is 20 fs (FWHM) and the photon energy is 7 keV. XFEL radiation comes from the right with a grazing incidence of 4.5 mrad.

The thick black curve represents the initial density profile of the sample.

between layers may act as obstacles against dislocation motion. Unfortunately, exact values for Y depend on the film material and thickness as well as on the substrate itself so that one can hardly get a precise estimation of this mechanical quantity for a film. Tentatively, we multiplied Y by a factor 5-6 (which is a rather important value but consistent with results reported for thin films [39,40]). In the irradiation conditions of Fig. 6, the target thus remains in its elastic state. Here, we observed that after 10 ps, the electronic temperature of Ru has decreased and reached a value of about 0.03 eV so that one cannot expect further deformation. Consequently, since the plastic state has not been reached with this significantly increased value of the yield strength, the material can be considered as undamaged.

One can remark here that these results may vary according to others parameters such as the electron-ion equilibration times (calculated in our model to be of about 10-20 ps in Si and 60 ps in Ru if T e is less than 0.5 eV) or the electron thermal conduction (not very important here compared with the effect of energetic electrons), however most of the uncertainty comes from the mechanical parameters. mrad and a fluence of 4700 J/cm 2 (the measured damage threshold fluence [36]). Consistently with measurements and a yield strength multiplier of 5, no cell has reached the plastic state during the whole simulation. On the other hand, a plastic state in the external layer of Ru is reached when the fluence is enhanced to 6000 J/cm 2 .

B. Hard x-ray normal incidence irradiation conditions

In the normal incidence conditions where hard x-rays have large penetration depths, the transport of energetic electrons has also some importance not only because it affects the energy deposition but also because a part of these electrons escape from the external layers. This is illustrated by a simulation of a 1 mm thick bulk Si sample irradiated at normal incidence by a 20 fs pulse of 10 keV photons with an intensity of 3.1 × 10 15 W/cm 2 . Figs. 8 and9 display temperature profiles at different instants of the ion temperature T i and of the electron temperature T e , respectively. As shown in Fig. 8, this intensity allows one just to obtain the melting temperature of Si (i.e. 0.145 eV) in the external layers (see the inset corresponding to the few external microns). Even though the corresponding fluence of 0.62 µJ/µm 2 is probably a very rough estimation of the ablation fluence, it remains close to a measurement of the ablation threshold of Si, namely 0.78 µJ/µm 2 in these conditions of irradiation [START_REF] Koyama | Proc. of SPIE[END_REF]. After 3 ps, T i equilibrates with the electron temperature T e whose history reflects the deposition of the XFEL photon energy and the transport of the energetic electrons (see Fig. 9). What is noticeable in Fig. 9 is a decrease of the electron temperature on the external edge as a result of the escaping energetic electrons carrying away an important part of the energy (see the inset corresponding to the few external microns of the target). In reality, this cooling is probably overestimated because of the charge build-up of the surface which is not considered here. Indeed, for large samples, escaping electrons are likely to return in the target because of this surface charging. However, exact trajectories of these electrons in the vacuum are difficult to predict as well as their expected return.

At higher intensity, this electron cooling effect (although probably overestimated) remains visible (Fig. 10). Furthermore, a significantly higher proportion of energetic electrons escape from the interaction zone which explains why the electron temperature does not follow linearly any increase of the XFEL intensity. Also, the electron cooling effect is expected to be more important for much thinner targets (i.e. of thickness less than a few microns).

IV. SUMMARY AND CONCLUSIONS

We have extended a model developed previously for considering the interaction of high intensity XFEL pulses with bulk matter, to take into account low fluence conditions as well as grazing incidences and multi-keV photon energies. This has been done by (i) including a wave-equation solver for the XFEL electric field in the material, (ii) modeling the transport of energetic photoelectrons, (iii) considering the elasto-plastic behavior of the material. Both points (i) and (ii) involves a proper determination of the XFEL absorption which, through a partitioning of the produced photoelectrons allows us to determine whether the energy is deposited locally or not locally.

We have analyzed the behavior of materials of interest for X-ray optics when they are irradiated at grazing or normal incidence by single, intense XFEL pulses. We have shown that our computer code X-RIM predicts fluence thresholds consistent with experiments. We plan to apply this computer tool to the important problem of the damage dynamics of a periodic stack of bilayers irradiated by a XFEL. In a high intensity regime and at normal incidence, this 1D model is also a useful tool if the thickness of the irradiated sample (or at least the longitudinal penetration depth) remains small or of the order of the focal spot size.

Of course, uncertainties remain concerning, (1) the treatment of solid density atomic physics that could affect the complex refractive index, (2) the equation-of-state, (3) the various thermophysical parameters poorly known in the WDM regime, (4) the mechanical parameters (G, Y) in the solid state regime. Better values for these quantities would improve the accuracy of the model predictions. We believe that improved values for these parameters can be determined by comparison with further experiments.

FIG. 1 :

 1 FIG. 1: (Color online) Sketch of the 1D material irradiated by a plane-wave in the X-ray range. The material is discretized into different layers, the thicknesses of which may vary over time. E i , E r and E t are the incident, reflected and transmitted electric fields, respectively. k i , k r and k t being the corresponding wavevectors. θ is the grazing incidence angle. Components of the electric field separate into 2 components: component S (along the z-axis) and component P (in the plane xy).

FIG. 2 :FIG. 3 :

 23 FIG.2:(Color online) Calculated absorption coefficient N κ vs photon energy for cold solid (ρ = 12.2 g.cm -3 ) Ru (black) and for warm solid Ru (red).

Fig. 5 FIG. 4 :

 54 Fig. 5 and Fig. 6 display calculated temperature profiles at different times, in the whole target (Si/Ru) for a XFEL beam fluence of 310 J/cm 2 i.e. the measured threshold for damage

  FIG. 5: (Color online) Calculated electron temperature profiles at different times in a Si/Ru target without transport of energetic photoelectrons (purely local absorption and diffusive transport).

Fig. 7 3 )

 73 Fig.7displays a simulation performed for 12 keV photons, a grazing incidence angle of

3 )

 3 FIG. 7: (Color online) Calculated electron temperature profiles at different times in a Si/Ru target with transport of energetic photoelectrons. The intensity is of 2.35 × 10 17 W/cm 2 , the pulse (Gaussian) duration is 20 fs (FWHM) and the photon energy is 12 keV. XFEL radiation comes from the right with a grazing incidence of 3 mrad.

FIG. 8 : 7 FIG. 9 :

 879 FIG. 8: (Color online) Calculated ion temperature profiles in a 1 mm thick bulk Si target with transport of energetic photoelectrons. The intensity is of 3.1 × 10 15 W/cm 2 , the pulse (Gaussian) duration is 20 fs (FWHM) and the photon energy is 10 keV. XFEL radiation comes from the right at normal incidence.

4 FIG. 10 :

 410 FIG.10:(Color online) Calculated electron temperature profiles in a 1 mm thick bulk Si target with transport of energetic photoelectrons. The intensity is of 9 × 10 15 W/cm 2 , the pulse (Gaussian) duration is 20 fs (FWHM) and the photon energy is 10 keV. XFEL radiation comes from the right at normal incidence.