S Mathieu
email: smathieu@chu-clermontferrand.fr

• B Pereira

• A Saraux

• C Richez

• B Combe

• M Soubrier

Disease-modifying drug retention rate according to patient age in patients with early rheumatoid arthritis: analysis of the ESPOIR cohort

Keywords: Rheumatoid arthritis, ESPOIR cohort, Elderly, Drug retention

Physicians are sometimes hesitant to use disease-modifying antirheumatic drugs (DMARDs) in elderly patients with rheumatoid arthritis (RA), as they are deemed too fragile, although there are no sufficient scientific evidence. We aimed to compare DMARD treatment retention in early RA patients from the ESPOIR cohort, according to age upon inclusion. Overall, treatment retention was evaluated as the percentage of patients whose DMARDs were not stopped, with stratification by age group: < 50, 50-64, and > 65 years. Survival curves were measured using the Kaplan-Meier method. Of the entire ESPOIR cohort (n = 813), 7% were > 65 years old. Methotrexate (MTX) was used by 521 patients, and was the sole DMARD for 198 patients. MTX treatment retention appeared better in patients > 65 years old compared to < 50 years old [HR 0.45 (0.25; 0.81); p = 0.008, n = 195/198] with adjustment on sex, smoking, positive anti-cyclic citrullinated peptide antibodies, positive rheumatoid factor, body mass index, changes in DAS28 and corticosteroid treatment. The proportion of patients using etanercept (n = 111), and this drug's retention rate, did not differ according to patient age. The proportion of patients treated with adalimumab (n = 104) was significantly higher in patients < 50 years old (p = 0.003), and treatment retention was marginally better among younger patients [HR 1.68 (0.88; 3.22), p = 0.12]. Within the ESPOIR cohort, DMARD retention did not appear to differ according to age-except for better retention of MTX treatment in patients 50-64 years old, and of adalimumab in patients < 50 years old.

Introduction

Rheumatoid arthritis (RA) is an inflammatory joint disease that causes structural damage, potentially leading to disability and significant quality-of-life reduction. Such outcomes can be prevented by early initiation of appropriate treatment, with close "tight control" follow-up [START_REF] Goekoop-Ruiterman | DAS-driven therapy versus routine care in patients with recent-onset active rheumatoid arthritis[END_REF]. Prospects for remission are rising due to the emergence of biotherapies, such as tumor necrosis factor α (TNF-alpha) inhibitors, tocilizumab, abatacept, and rituximab [START_REF] Felson | American College of Rheumatology/European league against rheumatism provisional definition of remission in rheumatoid arthritis for clinical trials[END_REF]. Corticosteroids can help manage inflammation and disease activity, including among elderly patients [START_REF] Bijlsma | Glucocorticoids in the treatment of rheumatoid arthritis: still used after 65 years[END_REF]. However, oral corticosteroids are associated with increased risks of infections and cardiovascular diseases, even at low daily doses [START_REF] Avina-Zubieta | Immediate and past cumulative effects of oral glucocorticoids on the risk of acute myocardial infarction in rheumatoid arthritis: a population-based study[END_REF]. Thus, disease-modifying antirheumatic drug (DMARD) treatments must be carefully optimized to minimize risk while achieving clinical remission, specifically involving the absence of structural degradation [START_REF] Grigor | Effect of a treatment strategy of tight control for rheumatoid arthritis (the TIC-ORA study): a single-blind randomised controlled trial[END_REF].

As the general population becomes increasingly older, the population of RA patients is also aging. The number of patients over 75 years of age with RA is growing, and most still lead active lives and want to be able to continue enjoying their hobbies. Most of these patients want their disease to be well managed; therefore, current recommendations advise treating elderly RA patients using the same methods employed in younger sufferers, and setting the same remission objectives [START_REF] Lahaye | Effectiveness and safety of abatacept in elderly patients with rheumatoid arthritis enrolled in the French Society of Rheumatology's ORA registry[END_REF]. However, it seems that there is less prescription of DMARD and bDMARD in the elderly. This impression is based more on discussion with colleagues or on the experience of rheumatologists than on scientific evidence. The literature is lacking in this subject of area and somewhat contradictory regarding advice for the management or RA in elderly patients [START_REF] Lahaye | Management of inflammatory rheumatic conditions in the elderly[END_REF]. Huscher et al. concluded to a potential treatment deficit in older RA patients in 2013 although Sugihara et al. reported that low disease activity is a realistic goal in elderly RA patients [START_REF] Huscher | Clinical presentation, burden of disease and treatment in young-onset and late-onset rheumatoid arthritis: a matched-pairs analysis taking age and disease duration into account[END_REF]. In their study population of 151 patients, Sugihara et al. applied the "treat-to-target" strategy in 83% of cases at 6 months, and in 75% of cases at 1 year. In 50% of patients, no evolution of structural damage was observed after 1 year [START_REF] Sugihara | Structural and functional outcomes of a therapeutic strategy targeting low disease activity in patients with elderly-onset rheumatoid arthritis: a prospective cohort study (CRANE)[END_REF]. Moreover, several studies have reported that treatment duration and biotherapy efficacy do not differ between elderly and young RA patients [START_REF] Cho | Drug retention and safety of TNF inhibitors in elderly patients with rheumatoid arthritis[END_REF][START_REF] Harigai | Safety and effectiveness of abatacept in Japanese non-elderly and elderly patients with rheumatoid arthritis in an all-cases post-marketing surveillance[END_REF][START_REF] Lurati | Safety of etanercept in elderly subjects with rheumatoid arthritis[END_REF]. However, these findings are not unanimously supported. Payet et al. assessed 1,709 patients from the AutoImmunity and Rituximab (AIR) registry, comparing rituximab efficacy and tolerance across different age groups. The study cohort included 417 patients aged 64-75 years, and 191 patients over 75 years of age. These elderly and very elderly patients received fewer lines of TNF-alpha inhibitor compared to younger patients. Treatment retention duration did not differ across the groups, but the infection rate was higher in the elderly subjects [START_REF] Payet | Efficacy and safety of rituximab in elderly patients with rheumatoid arthritis enrolled in a French Society of Rheumatology registry[END_REF]. Similarly, Pers et al. evaluated the effects of tocilizumab in RA patients over and under 65 years old, and found that tolerance and treatment retention were identical in both groups, but clinical efficacy was poorer among patients over 65 years of age [START_REF] Pers | Efficacy and safety of tocilizumab in elderly patients with rheumatoid arthritis[END_REF]. Several studies of the effects of TNF-alpha inhibitors in elderly patients have also concluded that efficacy is poorer in this population [START_REF] Soubrier | Rheumatoid arthritis in the elderly in the era of tight control[END_REF], while tolerance remains similar to in other age groups [START_REF] Radovits | Tumour necrosis factoralpha antagonists in the management of rheumatoid arthritis in the elderly: a review of their efficacy and safety[END_REF][START_REF] Fleischmann | Risk: benefit profile of etanercept in elderly patients with rheumatoid arthritis, ankylosing spondylitis or psoriatic arthritis[END_REF][START_REF] Genevay | Tolerance and effectiveness of anti-tumor necrosis factor alpha therapies in elderly patients with rheumatoid arthritis: a population-based cohort study[END_REF][START_REF] Bathon | Safety and efficacy of etanercept treatment in elderly subjects with rheumatoid arthritis[END_REF]. Notably, these elderly patients (> 75 years) have different profiles than younger patients, exhibiting more comorbidities (potential renal failure, cardiovascular diseases, etc.) and increased polypharmacy. Thus, physicians are more hesitant to prescribe DMARDs in these elderly patients due to fear of greater risk of adverse effects [START_REF] Ogasawara | Observational cross-sectional study revealing less aggressive treatment in Japanese elderly than nonelderly patients with rheumatoid arthritis[END_REF][START_REF] Iwanaga | Risk factors of adverse events during treatment in elderly patients with rheumatoid arthritis: an observational study[END_REF][START_REF] Tutuncu | Do patients with olderonset rheumatoid arthritis receive less aggressive treatment?[END_REF][START_REF] Setoguchi | Tumor necrosis factor-alpha antagonist use and heart failure in elderly patients with rheumatoid arthritis[END_REF][START_REF] Köller | Response of elderly patients with rheumatoid arthritis to methotrexate or TNF inhibitors compared with younger patients[END_REF][START_REF] Schiff | Long-term experience with etanercept in the treatment of rheumatoid arthritis in elderly and younger patients: patient-reported outcomes from multiple controlled and open-label extension studies[END_REF]. In such cases, good tolerance to treatment is the priority, and DMARDs are not necessarily optimized for achieving remission [START_REF] Matsubara | Longterm retention rate and risk factor for discontinuation due to insufficient efficacy and adverse events in Japanese patients with rheumatoid arthritis receiving etanercept therapy[END_REF][START_REF] Alivernini | Leflunomide treatment in elderly patients with rheumatoid or psoriatic arthritis: retrospective analysis of safety and adherence to treatment[END_REF].

The ESPOIR cohort study, a large national multicenter longitudinal and prospective cohort initiated by the French Society of Rheumatology included patients with undifferentiated arthritis or rheumatoid arthritis, of less than 6 months disease duration. In the present study, our main objective was to compare DMARD treatment retention over the follow-up in the ESPOIR cohort, according to patient age at inclusion and disease activity.

Methods

Study population

We analyzed all 813 patients of the French cohort "Etude et Suivi des Polyarthrites Indifférenciées Récentes (ESPOIR)" [START_REF] Combe | The ESPOIR cohort: a ten-year follow-up of early arthritis in France: methodology and baseline characteristics of the 813 included patients[END_REF] and retained only the 698 patients that fulfilled diagnosis ACR/EULAR criteria of rheumatoid arthritis. ESPOIR cohort study initiated by The French Society of Rheumatology initiated is a large national multicenter, longitudinal and prospective cohort, to set up databases to allow various investigations on diagnosis, prognostic markers, epidemiology, pathogenesis and medico-economic factors in the field of early arthritis and rheumatoid arthritis.

Objective and primary endpoint

In this study, our primary objective was to evaluate treatment retention of DMARDs (methotrexate [MTX], hydroxychloroquine, TNF-alpha inhibitors, tocilizumab, abatacept, and rituximab) in the ESPOIR cohort over 5 years of follow-up, with comparisons between different age groups (> 65 years, 50-64 years, and < 50 years). The primary endpoint was treatment retention of DMARD, evaluated at every patient consultation during the 5-year follow-up of the ESPOIR cohort. We decided to analyze only the five first years of follow-up of the 10-year ESPOIR cohort because the number of patients followed after 5 years was too low to draw strong conclusions. We recorded the date of initiation of each DMARD, and then analyzed the DMARDs that were still being received at each follow-up visit.

Retention was defined as the same treatment still being received at the same dosage or at an altered dosage. A temporary suspension of treatment for less than 6 months was not considered as a treatment cessation. On the other hand, beyond 6 months, the treatment was considered stopped and if represcribed, we noted a new start date. Discontinued DMARD was defined as when a treatment was changed, including cessation of the previous treatment. When a treatment was still being administered but in combination with another DMARD (bi-or tritherapy), this was defined as discontinued monotherapy.

Statistical analysis

Statistical analyses for a two-sided type I error at 5% were performed using Stata software (Version 13; StataCorp, College Station, USA). Continuous data were described as mean ± SD or median [interquartile range (IQR)], according to statistical distribution. We evaluated the assumption of normality using the Shapiro-Wilk test. Patient age was first considered as a quantitative criterion, then by category according to clinical relevance: > 65 years, 50-64 years, and < 50 years. These age groups were compared at baseline using ANOVA or Kruskal-Wallis tests if the assumptions of ANOVA were not met for continuous parameters, and with the Chi-squared or Fisher's exact tests for categorical variables. Where appropriate (omnibus p value of < 0.05), post hoc tests were used to account for multiple comparisons: Tukey-Kramer or Dunn after ANOVA, and Kruskal-Wallis and Marascuillo after Chi-squared or Fisher's exact tests. Treatment retention during the 5 years of follow-up of the ESPOIR cohort was defined as the percentage of patients whose treatment was not discontinued or modified, as assessed at each follow-up visit. In analyses, comparing the overall retention of DMARD treatment among the different patient age groups during the 5-year follow-up in the ESPOIR cohort, any treatment discontinuation was considered as right-censored data.

Estimates were calculated using the Kaplan-Meier method. Next, between-group comparisons were performed using the log-rank test for univariate analyses, and Cox proportional-hazards regression for multivariate analyses. The following covariates were fixed according to univariate results and clinical relevance: sex, smoking, positive anticyclic citrullinated peptide antibodies (ACPA), rheumatoid factor, body mass index (BMI), changes in disease activity score (DAS28), and corticosteroid treatment. We studied the proportional-hazard hypothesis using the Schoenfeld test and plotting residuals, and tested the interactions between possible prognostic factors. The results were expressed as hazard ratios (HRs) and 95% confidence intervals. A sensitivity analysis was conducted for the same statistical analysis plan with age considered a continuous variable.

ESPOIR was registered in ClinicalTrials under No. NCT03666091.

Results

Our present analyses included the 698 RA patients in the ESPOIR cohort. The mean age upon inclusion was 48.6 ± 12.1 years. Around half of the population were < 50 years old (335/698, 48%), and 7% (n = 50) were over 65 years of age. Table 1 summarizes the comparison of patient characteristics upon inclusion according to age.

Patients over 65 years of age were primarily male, nonsmokers, had a higher BMI, and presented with a higher mean DAS28 score (indicating more significant biological inflammatory syndrome).

Information regarding corticosteroid therapy was available at baseline only for 86 patients. We did not have data at baseline for the other patients and we did not know if information was missing or if the patient did not use corticosteroid. Among the 86 patients on corticosteroid treatment at baseline, 13 (15.1%) were used dose higher or equal to 0.5 mg/kg/day. The mean corticosteroid dose did not statistically differ according to age at baseline (p = 0.96): 19.4 ± 15.6 mg/day in patients < 50 years (n = 40), 18.7 ± 13.7 mg/day in 50-64 years (n = 39) and 17.1 ± 13.2 mg/day in patients > 65 years (n = 7). The mean dose of corticosteroid after 1 year of follow-up in the ESPOIR cohort was already not different in the 3 age groups (p = 0.81) and was lower compared with baseline: 7.8 ± 6.6 mg/day in patients < 50 years (n = 133), 7.6 ± 5.2 mg/day in 50-64 years (n = 147) and 8.3 ± 10.1 mg/ day in patients > 65 years (n = 20).

Over the 5-year follow-up, 83 patients did not receive DMARDs. In this group, the mean age upon inclusion was 49.6 ± 13.0 years, 18/83 (22%) were men, 39/83 (47%) were nonsmokers, and the mean DAS28 upon admission was 4.81 ± 1.41. Among these 83 patients, 39 (47%) had erythrocyte sedimentation rates (ESR) of ≥ 20 mm/h, 15 (18%) tested positive for anti-CCP antibodies, 23 (28%) presented rheumatoid factors with titers of > 10, and 36 (43%) had median Sharp scores of > 1 upon inclusion. In this patients group (n = 83), 21 patients were lost to follow-up from the start, and 9 patients were lost after one or two follow-up inhibitors' retention rate did not differ according to patient age (p = 0.38) (Fig. 3). For etanercept (n = 111), the proportion of RA patients using this treatment was not different among inclusion age (p = 0.61), and treatment retention for etanercept did not differ according to patient age upon inclusion [multivariable analysis: 0.60 (0.26; 1.40), p = 0.24 for 50-64 years compared to those < 50 years, n = 73/111]. A total of 104 patients were treated with adalimumab, and the rate of adalimumab use was significantly higher among patients < 50 years old (22%, 66//298) than patients of 50-64 years old (12%, 34/275) and patients > 65 years old (10%, 4/42) (p = 0.003). Adalimumab treatment retention appeared to be marginally worse among the oldest patients [univariate analysis: HR 1.72 (0.99; 3.02), p = 0.055], and this difference remained marginally significant in multivariate analysis [HR 1.68 (0.88; 3.22), p = 0.12; n = 93/104]. The Sensitivity analysis with age considered as continuous variable found the same results for treatment retention for MTX, etanercept and adalimumab. The conclusions were similarly supported by these results (data not shown).

Discussion

Our present results showed that DMARD treatment retention did not appear to fundamentally differ according to patient age upon inclusion in the ESPOIR cohort-except that MTX retention was better among patients 50-64 years old, and adalimumab retention was better among patients < 50 years old. The ESPOIR cohort included few elderly individuals, with only 59 patients (7%) over 65 years of age. Although those who were ≥ 65 years old upon inclusion were ≥ 75 years of age by the end of the 10-year follow-up. This reflects common practice for elderly patients, yet poses challenges for analysis. As this study examined real-life follow-up and prescription practices, the number of treatment sequences is important, thus, rendering it difficult to draw solid conclusions.

Our analyses of the use and retention of biologics treatments primarily concerned etanercept and adalimumab. We were unable to analyze data for other individual TNFalpha inhibitors or biotherapies due to the small numbers of samples. This can largely be explained by the fact that some biotherapies (e.g., certolizumab, golimumab, and tocilizumab) had only recently received market approval, and were thus less common 10 years ago compared to other TNF-alpha inhibitors. Moreover, the use of infliximab by only 21 patients was likely because the included patients were mostly younger people, whose lifestyles and professional activities are better served by subcutaneously administered biotherapies. In our sample, nearly identical numbers of patients received etanercept or adalimumab. In the Swedish registry, collated between 2003 and 2011, etanercept was the most commonly prescribed TNF-alpha inhibitor [START_REF] Neovius | Drug survival on TNF inhibitors in patients with rheumatoid arthritis comparison of adalimumab, etanercept and infliximab[END_REF], while worldwide prescription rates show adalimumab at the top of the list. It is unfortunate that we were unable to study a larger sample of patients receiving abatacept (n = 18), since this treatment is often prioritized in elderly subjects, largely due to its reduced infection risk [START_REF] Chen | Risk of hospitalized infection and initiation of abatacept versus TNF inhibitors among patients with rheumatoid arthritis: a propensity score-matched cohort study[END_REF] and good efficacy [START_REF] Lahaye | Effectiveness and safety of abatacept in elderly patients with rheumatoid arthritis enrolled in the French Society of Rheumatology's ORA registry[END_REF][START_REF] Harigai | Safety and effectiveness of abatacept in Japanese non-elderly and elderly patients with rheumatoid arthritis in an all-cases post-marketing surveillance[END_REF]. Ebina et al. recently described the tolerance of biotherapies and reasons for discontinuation among RA patients > 65 years old, and reported that abatacept presented the best treatment retention in this population based on observed adverse effects and efficacy [START_REF] Ebina | Drug tolerability and reasons for discontinuation of seven biologics in elderly patients with rheumatoid arthritis-the ANSWER cohort study[END_REF].

We cannot exclude the possibility that MTX treatment retention was better among the elderly population because TNF-alpha inhibitors were less commonly used in this demographic. The results of our present multivariate analysis were inconclusive. The impact of corticosteroid therapy is also as it is possible that MTX treatment was maintained for a longer duration among elderly patients because they received larger corticosteroid doses, enabling better control of disease activity. However, our analysis showed that the mean corticosteroid dose did not differ according to patient age. Additionally, the percentage of patients whose corticosteroid dosage was halved over time did not differ between the age groups (< 50 years, 50-64 years, and > 65 years). In the Corpus study, RA management was compared between patients over and under 75 years of age, revealing that the elderly patients exhibited less recourse to TNF-alpha inhibitors and received higher doses of corticosteroids [START_REF] Oishi | Treatment of active rheumatoid arthritis: comparison of patients younger vs older than 75 years (CORPUS cohort)[END_REF]. The Corpus study included patients from 2007 to 2009, a time period during which physicians may have been more hesitant to intensify treatment in elderly subjects. This phenomenon may explain why MTX retention was better among elderly patients, especially MTX monotherapy, since TNF-alpha inhibitors were less frequently prescribed. We obtained similar results in the ESPOIR cohort, albeit with more frequent prescription of TNF-alpha inhibitors. The most commonly prescribed was etanercept, which is recognized as offering better tolerance and lower infection risk.

This study had several limitations. One was the upper age limit (< 70 years) at inclusion in the ESPOIR cohort, which did not allow us to analyze patients of > 75 years old. It would have been interesting to observe very elderly patients, as it may be this age group that differs the most from younger populations with regards to DMARD prescriptions. On the other hand, the 10-year follow-up period of the ESPOIR cohort was an advantage, as it enabled us to assess whether patients aged > 65 years upon admission had different follow-ups once they reached > 75 years of age compared to younger patients. Our analysis revealed no apparent difference in follow-up or prescriptions. Within the entire study cohort of 813 patients with early-stage RA, 138 received no treatment. This number may seem surprising, and raise questions regarding the reliability of the cohort. However, these 138 patients included 39 for whom an RA diagnosis was excluded, and 51 who were lost to follow-up, and the rest of this group had RA that went into remission without requiring treatment. This demonstrates that the cohort results were similar to in common practice and real-life populations, where there is doubt surrounding diagnoses, patients do not show up for consultation, some stop their treatment, and some no longer need drugs because they are in remission.

The presently analyzed data from the ESPOIR cohort appears to validate the recommendation that elderly patients with RA should be treated and followed-up the same as younger patients, with recourse to biotherapies to obtain remission.

  visits. In 20 patients, rapid DAS remission was achieved, with no structural progression. Thirty-three patients exhibited Sharp score increases of over 0.5 point, of whom 27 exhibited low disease activity or DAS remission, and the other 6 exhibited moderate or strong disease activity.Among the patients who received DMARDs, 267 received one DMARD, 173 received two, 103 received three, 36 received four, and 36 received five or more. MTX was taken by 521 patients, and was used as monotherapy in 198. The proportion of patients receiving MTX monotherapy did not significantly differ among the three age groups: 72.7% of patients < 50 years, 76.4% of patients 50-64 years, and 68.4% of patients > 65 years (p = 0.67). By the same way, the percentage of patients started on MTX as first DMARD in each group of age was not different (60.5% in patients < 50 years (n = 296), 63.3% in 50-64 years (n = 275) and 59.5% in patients > 65 years (n = 42); p = 0.76). Seventy-four percent of the 698 RA patients are still present in the cohort at 5 years. The median time of stopping MTX is 15.5 months [IQR: 5.6-29.5]. MTX treatment retention appeared to be better among the oldest subjects (Fig. 1) (p = 0.01). This result was confirmed by multivariable analysis adjusted on sex, smoking, positive ACPA, FR, BMI, changes in DAS28 and corticosteroid treatment. We found no impact on the corticosteroid use on the results of retention rate of MTX. Indeed, we also observed significantly better MTX retention in patients of 50-64 years compared to those < 50 years old [HR 0.45 (0.25; 0.81); p = 0.008, n = 195/198]. MTX treatment retention decreased with the number of treatments received by the patients (Fig. 2), but remained better among the oldest subjects irrespective of the number of DMARD treatment lines received: [HR = 0.66 (0.50; 0.87), p = 0.003, n = 491/521] Hydroxychloroquine was taken by 148 patients, and was the only DMARD used by 39 patients. Treatment retention among these patients did not differ according to age [multivariable analysis: HR 1.15 (0.71; 1.84), p = 0.57 for 50-64 years compared to those < 50 years]. The percentage of patients treated with biologics (TNF-alpha inhibitor, rituximab, tocilizumab, and abatacept) was greater among the youngest subjects: 38% of patients < 50 years old (n = 113/298), 28% of patients 50-64 years old (n = 76/275), and 24% of patients > 65 years old (n = 10/42). In our analyses, we focused on the rates of TNF-alpha inhibitor use because the numbers of patients using the other biotherapies were too low: rituximab, n = 35; tocilizumab, n = 20; and abatacept, n = 18. The proportion of patients using TNFalpha inhibitors was higher among patients < 50 years old (110/298, 37%) than patients 50-64 years old (70/275, 25%) and patients > 65 years old (9/42, 21%) (p = 0.005).

Fig. 1

 1 Fig. 1 Methotrexate monotherapy retention according to age

Fig. 2

 2 Fig. 2 Methotrexate monotherapy retention according to number of disease-modifying antirheumatic drug lines received

Fig. 3

 3 Fig. 3 TNF-alpha inhibitors retention according to age

Table 1

 1 Comparison of three age categories according to characteristics upon inclusion Data are presented as n, % or as mean ± standard deviation y years old, DAS28 disease activity score, HAQ Health Assessment Questionnaire, BMI body mass index, ESR erythrocyte sedimentation rate, CRP C-reactive protein, RF rheumatoid factor, ACR American College of Rheumatology

		Under 50 y	50-64 y	Over 65 y	p value
	Female	274, 81.8%	229, 73.2%	33, 66.0%	0.006
	Smoker	175, 52.2%	151, 48.2%	15, 30.0%	0.013
	DAS28	5.12 ± 1.24	5.34 ± 1.21	5.68 ± 1.47	0.005
	HAQ	0.97 ± 0.67	1.06 ± 0.67	1.17 ± 0.81	0.096
	BMI	24.08 ± 4.65 25.97 ± 4.72 26.04 ± 3.20 < 0.001
	ESR	20 [11; 34]	24 [12; 45]	33 [22; 68]	< 0.001
	CRP	9 [5; 23]	10 [4; 27]	17 [6; 45]	0.010
	Positive RF	236, 70.5%	190, 60.7%	30, 60.0%	0.02
	Positive ACPA 174, 51.9%	122, 39.0%	19, 38.0%	0.002
	Sharp	1.0 [0.0; 3.3] 2.5 [1.0; 5.0] 4.0 [1.0; 9.5] < 0.001
	Comorbidities 199, 59.8%	221, 71.1%	44, 88.0%	< 0.001

Acknowledgements

We also wish to thank Nathalie Rincheval (CHU Montpellier and EA 2415) who did expert monitoring and data management and all the investigators who recruited and followed the patients (F. Berenbaum, Paris-Saint Antoine, MC. Boissier, Paris-Bobigny, A. Cantagrel, Toulouse, B. Combe, Montpellier, M. Dougados, Paris-Cochin, P. Fardellone et P. Boumier Amiens, B. Fautrel, Paris-La Pitié, RM. Flipo, Lille, Ph. Goupille, Tours, F. Liote, Paris-Lariboisière, O. Vittecoq, Rouen, X. Mariette, Paris Bicetre, P. Dieude, Paris Bichat, A. Saraux, Brest, T. Schaeverbeke, Bordeaux, J. Sibilia, Strasbourg). Funding An unrestricted grant from Merck Sharp and Dohme (MSD) was allocated for the first 5 years. Two additional grants from INSERM were obtained to support part of the biological database. The French Society of Rheumatology, Pfizer, Abbvie, Lilly, and Sanofi also supported the ESPOIR cohort study.


Damien Contou
email: damien.contou@ch-argenteuil.fr

Gaëtan Béduneau

Charlotte Rabault

Romain Sonneville

Antoine Marchalot

Rémi Coudroy

Damien Roux

Martin Cour

Julien Massol

Sébastien Préau

Damien Contou1

Gaetan Beduneau2

Charlotte Rabault3

Romain Sonneville3

Antoine Marchalot5

Remi Coudroy6

Damien Roux7

Martin Cour8

Julien Massol9

Sebastien Preau10

Nicolas De Prost11

The Hopeful

Study Group

Skin biopsy in adult patients with meningococcal purpura fulminans: a multicenter retrospective cohort study

Keywords: Purpura fulminans, Skin biopsy, ICU, Sepsis, Diagnosis, Meningitis, Infectious diseases, Neisseria meningitidis Abbreviations PF Purpura fulminans, ICU Intensive care unit, IQR Interquartile range, PCR Polymerase chain reaction

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.

Background

Purpura fulminans (PF) is a rare infectious disease carrying a high mortality and morbidity with 41% of the patients dying in the ICU and 28% of the survivors requiring limb amputations with a median number of 3 limbs amputated [START_REF] Contou | Clinical spectrum and short-term outcome of adult patients with purpura fulminans: a French multicenter retrospective cohort study[END_REF][START_REF] Contou | Long-term quality of life in adult patients surviving Purpura Fulminans an exposed-unexposed multicenter cohort study[END_REF][START_REF] Contou | Understanding purpura fulminans in adult patients[END_REF]. Neisseria meningitidis is the leading responsible bacterium accounting for two thirds of PF [START_REF] Contou | Clinical spectrum and short-term outcome of adult patients with purpura fulminans: a French multicenter retrospective cohort study[END_REF]. Obtaining a microbiological documentation of PF is crucial for confirming the diagnosis, as well as for adjusting the antibiotic therapy. It is also of paramount importance for public health interventions and postexposure chemoprophylaxis with antibiotic therapy and vaccination of persons having close contacts with a patient with meningococcal PF.

Given the high susceptibility of Neisseria meningitidis to β-lactam antibiotics, together with the high proportion of patients empirically treated before ICU admission [START_REF] Contou | Clinical spectrum and short-term outcome of adult patients with purpura fulminans: a French multicenter retrospective cohort study[END_REF], blood cultures may be sterile in half of the patients with meningococcal PF [START_REF] Contou | Clinical phenotype and outcomes of pneumococcal versus meningococcal purpura fulminans: a multicenter retrospective cohort study[END_REF]. Moreover, lumbar puncture has been shown to be of limited diagnostic value in this context [START_REF] Contou | Diagnostic yield of lumbar puncture in adult patients with purpura fulminans[END_REF], if not contra-indicated because of severe thrombocytopenia and coagulation disorders, which are almost constant in patients with PF [START_REF] Contou | Clinical spectrum and short-term outcome of adult patients with purpura fulminans: a French multicenter retrospective cohort study[END_REF][START_REF] Contou | Clinical phenotype and outcomes of pneumococcal versus meningococcal purpura fulminans: a multicenter retrospective cohort study[END_REF]. Given the tropism of Neisseria meningitidis for skin endothelium [START_REF] Contou | Understanding purpura fulminans in adult patients[END_REF][START_REF] Lerolle | Assessment of the interplay between blood and skin vascular abnormalities in adult purpura fulminans[END_REF], the microbiological examination of skin biopsy was previously suggested to be an interesting diagnostic tool in children with PF [START_REF] Staquet | Detection of Neisseria meningitidis DNA from skin lesion biopsy using real-time PCR: usefulness in the aetiological diagnosis of purpura fulminans[END_REF][START_REF] Arend | Prospective controlled study of the diagnostic value of skin biopsy in patients with presumed meningococcal disease[END_REF][START_REF] Periappuram | Rapid detection of meningococci from petechiae in acute meningococcal infection[END_REF][START_REF] Van Deuren | Rapid diagnosis of acute meningococcal infections by needle aspiration or biopsy of skin lesions[END_REF]. Skin biopsy is a simple and minimally invasive exam allowing to perform skin culture and polymerase chain reaction (PCR) to detect Neisseria meningitidis, even several days after the initiation of antibiotic therapy [START_REF] Staquet | Detection of Neisseria meningitidis DNA from skin lesion biopsy using real-time PCR: usefulness in the aetiological diagnosis of purpura fulminans[END_REF][START_REF] De Risi-Pugliese | Skin biopsy polymerase chain reaction for rapid microbiological diagnosis in patients with purpura fulminans[END_REF]. Only a few studies have assessed the diagnostic yield of skin biopsy in patients with a suspected meningococcal infection [START_REF] Staquet | Detection of Neisseria meningitidis DNA from skin lesion biopsy using real-time PCR: usefulness in the aetiological diagnosis of purpura fulminans[END_REF][START_REF] Arend | Prospective controlled study of the diagnostic value of skin biopsy in patients with presumed meningococcal disease[END_REF][START_REF] Periappuram | Rapid detection of meningococci from petechiae in acute meningococcal infection[END_REF][START_REF] Van Deuren | Rapid diagnosis of acute meningococcal infections by needle aspiration or biopsy of skin lesions[END_REF]. Most of these studies were performed in children and only one assessed the rentability of meningococcal PCR on skin biopsy [START_REF] Staquet | Detection of Neisseria meningitidis DNA from skin lesion biopsy using real-time PCR: usefulness in the aetiological diagnosis of purpura fulminans[END_REF]. Our aim was to evaluate the sensitivity of skin biopsy in adult patients with meningococcal PF.

Methods

Study design

We conducted a 17-year (2000-2016) multicenter retrospective cohort study including adult patients admitted to 43 intensive care units (ICU) in France (see the participating centers in the acknowledgement section) for a meningococcal PF. This observational study followed the Strengthening the Reporting of Observational Studies in Epidemiology (STROBE) reporting guideline. Meningococcal PF was defined by the association of a sudden and extensive purpura together with an acute circulatory failure needing vasopressor support [START_REF] Contou | Long-term quality of life in adult patients surviving Purpura Fulminans an exposed-unexposed multicenter cohort study[END_REF][START_REF] Lerolle | Assessment of the interplay between blood and skin vascular abnormalities in adult purpura fulminans[END_REF] and one or more microbiological sample positive (conventional culture or PCR) for Neisseria meningitidis in the blood, the cerebrospinal fluid or in skin biopsy. Patients with a nonmeningococcal PF, a noninfectious purpura and those with purpura in a context of infectious endocarditis were not included in the present study. Skin biopsy was performed at the discretion of the intensivist on a purpuric lesion by using a punch biopsy device after local anesthesia. The primary study endpoint was the rate of contributive skin biopsy. Skin biopsy was considered as contributive when culture grew Neisseria meningitidis and/or when PCR was positive for Neisseria meningitidis. All patients had blood cultures drawn upon ICU admission, and lumbar puncture was performed at the discretion of the intensivist.

Data collection

The investigator of each participating center was responsible for the identification of the patients, either from the hospital medical reports, using the function "research the files in which the word" purpura fulminans occurs of Microsoft Windows®, or through a search using the following International Classification of Diseases (Tenth Revision) codes: D65 (Disseminated intravascular coagulation), A39 (Meningococcal infection), and D65 (Disseminated intravascular coagulation). The hospital discharge reports of all identified patients were anonymized and then electronically or conventionally mailed to the main investigator (DC). Clinical charts were reviewed in order to check the inclusion criteria. Upon ICU admission and during ICU stay, data pertaining to demographics, comorbidities, clinical examinations, laboratory findings, microbiological investigations and therapeutic management were collected. Missing data were retrieved by queries to the investigators. Of note, two patients included in one of the participating centers have already been described in a previous cases series [START_REF] De Risi-Pugliese | Skin biopsy polymerase chain reaction for rapid microbiological diagnosis in patients with purpura fulminans[END_REF].

Ethics approval

This observational, non-interventional analysis of medical records was approved by the Institutional Review Board of the French Intensive Care Society in March 2016 (SRLF16-01).

Statistical analysis

Categorical variables were presented as number (percentage), and quantitative variables as mean ± standard deviation (SD) or median [interquartile range (IQR)], as appropriate. Characteristics of patients who had a skin biopsy performed or not was compared using Chisquare tests or Fisher's exact tests, as appropriate, for categorical variables and Student t tests or Mann-Whitney tests, as appropriate, for quantitative variables. All significance tests were two-sided, and the statistical significance level was set to 5%. Missing values were not imputed. All analyses were performed with R software (version 2.4.3, The R project for Statistical Computing, Vienna, Austria).

Results

Study population

Among the 306 patients admitted for PF, 195 had a meningococcal PF (64%) with a skin biopsy being performed in 68 (35%) of them (Fig. 1). Comparison between patients with and without a skin biopsy is detailed in Table 1. Meningococcemia was more frequent in patients without a skin biopsy (72% vs. 47%, p = 0.001) and patients without a skin biopsy performed had more frequent lumbar puncture performed than those who underwent a skin biopsy (76% vs. 40%, p < 0.001). Culture of cerebrospinal fluid was more frequently positive in patients without a skin biopsy performed than in others (73% vs. 44%, p = 0.001) (Table 1).

Results of skin biopsy culture and PCR

Skin biopsy was performed in median 1 [0-1] day after ICU admission and 1 [0-1] day after the initiation of antibiotic therapy. Standard culture of skin biopsy was performed in 61/68 (90%) patients and grew Neisseria meningitidis in 28 (46%) of them (Fig. 1). Neisseria meningitidis PCR on skin biopsy was performed in 51/68 (75%) patients and was positive in 50 (98%) of them (Fig. 1). Among these 50 positive meningococcal PCR, five were performed 3 days or more after initiation of antibiotic therapy. Finally, skin biopsy was considered as contributive in 60/68 (88%) patients, knowing that meningococcal PCR was not performed in the 8 patients with a non-contributive skin biopsy (Table 1). Identification of the meningococcal serogroup was obtained with skin biopsy (by conventional culture or/and PCR) in 48/68 (71%) patients (serogroup B: n = 29; serogroup C: n = 15; serogroup w135: n = 2 and serogroup Y: n = 2). Skin biopsy was the only positive microbiological exam (i.e., both blood and cerebrospinal fluid cultures, when performed, were sterile) in 29/195 (15%) of the patients with meningococcal PF. No significant bleeding was reported in any of the patients who underwent skin biopsy.

Discussion

Our study indicates that only one third of the patients with meningococcal PF had a skin biopsy performed. Skin biopsy seems to be contributive in most of the patients with meningococcal PF, especially when a meningococcal PCR is performed (up to 3 days after antibiotic therapy initiation).

Our 46% high rate of positive conventional skin culture compares well with those reported in previous studies, which ranged from 56 to 64% [START_REF] Arend | Prospective controlled study of the diagnostic value of skin biopsy in patients with presumed meningococcal disease[END_REF][START_REF] Periappuram | Rapid detection of meningococci from petechiae in acute meningococcal infection[END_REF][START_REF] Van Deuren | Rapid diagnosis of acute meningococcal infections by needle aspiration or biopsy of skin lesions[END_REF] but, as opposed to our study, these studies combined Gram examination and cultures. This 46% figure is higher than the 14% rate reported by Staquet and colleagues who did not consider Gram examination. We reported on a 98% high rate of positive meningococcal PCR on skin biopsy, which is inline with the 100% high rate previously reported by Staquet and colleagues in a smaller retrospective pediatric single-center study [START_REF] Staquet | Detection of Neisseria meningitidis DNA from skin lesion biopsy using real-time PCR: usefulness in the aetiological diagnosis of purpura fulminans[END_REF]. Overall, as previously reported [START_REF] Staquet | Detection of Neisseria meningitidis DNA from skin lesion biopsy using real-time PCR: usefulness in the aetiological diagnosis of purpura fulminans[END_REF], meningococcal PCR seems much more sensitive than conventional bacterial cultures and should be preferred in case of limited skin sample.

Given the higher rate of meningococcemia in patients without a skin biopsy obtained, one can speculate that skin biopsy was performed at day 1 because blood cultures remained sterile. Moreover, the higher rate of lumbar punctures performed in the group of patients without a skin biopsy may reflect an entrenched strategy of performing a lumbar puncture rather than a skin biopsy since coagulation disorder and Glasgow Coma Score did not differ between patients with and without a skin biopsy.

The main limitation of the study is inherent to its retrospective design. We acknowledge that a standardized protocol with a systematic realization of skin biopsy combining conventional culture and meningococcal PCR might have increased the proportion of patients with a contributive skin biopsy. The available data also did not allow us to comprehensively compute the diagnostic performances of skin biopsy. Indeed, having the total number of positive tests (standard culture and PCR) of PF patients, we could compute the sensitivity of skin biopsy, an informative parameter in this setting, but not the specificity, negative and positive predictive values, and likelihood ratios as we would have needed skin biopsy data in patients not having a meningococcal PF. Such data are currently not available, which is a limitation to our study.

Skin biopsy with conventional culture and meningococcal PCR has a global sensitivity of 88%. Given the high rentability of PCR as compared to conventional culture, meningococcal PCR on skin biopsy should be systematically considered in patients with suspected meningococcal PF in order to increase the diagnostic work-up, even several days after the initiation of antibiotic therapy.

Fig. 1

 1 Fig. 1 Flowchart of patients with meningococcal purpura fulminans. A skin biopsy was performed in 35% of patients (n = 68/195). The sensitivity of standard bacterial culture and meningococcal polymerase chain reaction (PCR) is displayed. *a meningococcal PCR was not performed in all eight patients with a non-contributive skin biopsy

Acknowledgements

The authors thanks all the members of the Hopeful Study group For the HOPEFUL Study Group: Martin Cour (Lyon), Romain Sonneville and Charlotte Rabault (Bichat, AP-HP), Gaetan Beduneau (Rouen), Sebastien Preau (Lille), Remi Coudroy (Poitiers), Damien Roux (Louis Mourier, AP-HP), Frederic Pene and Julien Massol (Cochin, AP-HP), Gwenhael Colin (La Roche-sur-Yon), Francois Barbier and Kamel Toufik (Orleans), Antoine Marchalot (Dieppe), Quentin Quelven (Rennes), Guillaume Schnell (Le Havre), Stephan Ehrmann and Helene Messet (Tours), Antoine Kimmoun (Nancy 1), Pascal Beuret (Roanne), Cedric Bruel (Saint-Joseph), Delphine Colling (Roubaix), Alexandre Conia (Chartres), Damien Contou (Argenteuil), Luis Ensenyat Martin and Danielle Reuter (Corbeil-Essonnes), Vincent Das and Marion Challier (Montreuil), Mathieu Jozwiak (Nice), Arnaud Galbois (Quincy sous Senart), Megan Fraisse (Ambroise Pare -AP-HP), Samir Jaber (Montpellier), Sebastien Jochmans (Melun), Pierre Kalfon (Chartres), Marie Conrad (Nancy 2), Alexandre Lautrette (Clermont Ferrand), Cedric Darreau (Le Mans), Nicolas Lerolle (Angers), Hugues Georges (Tourcoing), Bruno Megarbane (Lariboisiere, AP-HP), Claire Pichereau (Poissy), Tomas Urbina and Eric Maury (Saint-Antoine, AP-HP), Nicolas de Prost (Henri Mondor, AP-HP), Jean-Pierre Quenot (Dijon), Bertrand Sauneuf (Cherbourg), Matthieu Schmidt (Pitie Salpetriere, AP-HP), Xavier Valette (Caen), Lara Zafrani (Saint-Louis, AP-HP), Daniel Da Silva (Saint-Denis), Guillaume Rigault (Grenoble), Paul-Louis Woerther (Hopital Henri Mondor, AP-HP)

Funding

None.

Availability of data and materials

The dataset used and analyzed for the current study is available from the corresponding author on reasonable request.

Author contributions

DC and NDP are responsible for the conception and design. All the authors were responsible for analysis and interpretation of data. All authors read, critically reviewed, and approved the final manuscript. DC takes responsibility for the paper as a whole. All authors read and approved the final manuscript.

Declarations Ethics approval and consent to participate

This observational, non-interventional analysis of medical records was approved by the Institutional Review Board of the French Intensive Care Society in March 2016 (SRLF16-01).

Consent for publication

Not applicable.

Competing interests

On behalf of all authors, the corresponding author states that there is no conflict of interest.


Romain Biard
email: romain.biard@univ-lyon1.fr

Claude Lefèvre
email: clefevre@ulb.ac.be

Stéphane Loisel
email: stephane.loisel@univ-lyon1.fr

Haikady N Nagaraja

Asymptotic Finite-Time Ruin Probabilities for a Class of Path-Dependent Heavy-Tailed Claim Amounts Using Poisson Spacings

In the compound Poisson risk model, several strong hypotheses may be found too restrictive to describe accurately the evolution of the reserves of an insurance company. This is especially true for a company that faces natural disaster risks like earthquake or flooding. For such risks, claim amounts are often inter-dependent and they may also depend on the history of the natural phenomenon. The present paper is concerned with a situation of this kind where each claim amount depends on the previous interclaim arrival time, or on past interclaim arrival times in a more complex way. Our main purpose is to evaluate, for large initial reserves, the asymptotic finite-time ruin probabilities of the company when the claim sizes have a heavy-tailed distribution. The approach is based more particularly on the analysis of spacings in a conditioned Poisson process.

Introduction

The compound Poisson risk model is the central model proposed in insurance theory. An abundant literature has been devoted to its analysis and applications. The reader is referred e.g. to the comprehensive books by Rolski et al. (1999), Asmussen (2000) and Kaas et al. (2001). It is well recognized, however, that the simplifying hypotheses at the basis of the model may be too restrictive for certain insurance coverages, especially with rare but extreme risks like earthquake or flooding. The present paper is concerned with situations of that kind.

Framework and motivations

Let us begin by recalling the model in its standard version. The reserves of the company, {R(t), t ≥ 0}, are given by

R(t) = u + ct -S(t), t ≥ 0, (1.1)
where u ≥ 0 is the amount of initial reserves, c > 0 is the premium income rate and S(t) is the cumulated claim amount up to time t. {S(t), t ≥ 0} is a compound Poisson process, i.e.

S(t) = N(t) i=1 X i , t ≥ 0, (1.2)
where {N(t), t ≥ 0} is a Poisson process, with parameter λ, which counts the claim occurrences until time t, and {X i , i = 1, 2, . . .} are the successive claim amounts which are represented by non-negative independent identically distributed random variables.

Let {V i , i = 1, 2, . . .} be the interarrival times of successive claims. By construction, these random variables are independent with a common exponential distribution. Moreover, the inter-occurrence times are assumed to be independent of the claim amounts.

A statistics of great interest is the probability of (non-)ruin over any fixed horizon of finite length. Denote by ψ(u, t) the probability of ruin before time t, t ≥ 0, for initial reserves u : ψ(u, t) = P(∃ s ∈ [0, t], R(s) < 0 | R(0) = u), u, t ≥ 0.

(1.3) Different methods have been proposed to evaluate ψ(u, t) ; see, e.g., [START_REF] Lefèvre | On finite-time ruin probabilities for classical risk models[END_REF].

Recently, much research has been devoted to the evaluation of ruin probabilities, over finite or infinite horizon, when some independence and stationarity assumptions of the model are relaxed. A number of references will be mentioned later in the section (the list being non-exhaustive, of course).

In practice, the independence assumptions on, and between, the sequences {X i } and {V i } may be too unrealistic. Typically, this arises in the case of natural disasters like earthquake (or flooding). The occurrence of an earthquake often increases the probability of by-claims in a near future. If the last earthquake occurred a long time ago, the next earthquake is likely to be more severe. If two earthquakes occur in a short time span, the second one may cause unusual damages like flooding.

This question has already been raised and discussed in the literature. [START_REF] Albrecher | A ruin model with dependence between claim sizes and claim intervals[END_REF] consider that the time between two claim occurrences depends on the previous claim amount. Exact expressions for the Laplace transform of the (ultimate) survival probability are derived. In [START_REF] Albrecher | Exponential behavior in the presence of dependence in risk theory[END_REF], the interclaim time and the subsequent claim size are dependent through an arbitrary copula structure. Asymptotic results for both the finite and infinite-time ruin probabilities are then derived. [START_REF] Boudreault | On a risk model with dependence between interclaim arrivals and claim sizes[END_REF] consider a particular form of dependence among the interclaim time and the subsequent claim size : if the current interclaim time exceeds a certain threshold, the distribution of the next claim is modified. The defective renewal equation satisfied by the expected Gerber-Shiu discounted penalty function is then obtained. In [START_REF] Meng | On a risk model with dependence between claim sizes and claim intervals[END_REF], the time between two claim occurrences determine the distribution of the next claim. Some exact and approximation results are derived for the survival probability. [START_REF] Ambagaspitiya | Ultimate ruin probability in the sparre andersen model with dependent claim sizes and claim occurrence times[END_REF] determines the ruin probability for two forms of dependence between claim size and occurrence in the Sparre Andersen model. See also the references given in these papers.

The present paper discusses several scenarios of dependence between claim amounts and interclaim arrivals. Such scenarios are motivated by earthquake or flooding-type risks. Our purpose is to provide approximations to the finite-time ruin probabilities, for large initial reserves, when the claim amounts have heavy-tailed distributions. This work can also be seen as a sequel to a recent paper by [START_REF] Biard | Impact of correlation crises in risk theory : Asymptotics of finite-time ruin probabilities for heavy-tailed claim amounts when some independence and stationarity assumptions are relaxed[END_REF]. Here the claim amount distributions are allowed to depend, to some extent, on the history of the claim arrival process.

Basic assumptions and implications

As in the classical model, claims occur according to a Poisson process {N(t), t ≥ 0}. Let U i = i j=1 V j , i ≥ 1, be the claim arrival times. Let us turn to the claim amounts {X i , i ≥ 1}. Firstly, the X i is allowed to be one of two different types of random variables Y i or Z i depending on the behavior of past interarrival times V j , j ≤ i. Three different models that describe this dependence will be examined.

Secondly, all the severities Y i have the same distribution function (d.f.) F, and each vector Y (j) = (Y 1 , ..., Y j ), j ≥ 1, is of multivariate regular variation of index -α with α > 0, i.e. there exists a θ ∈ S j-1 , where S j-1 is the unit sphere with respect to a norm

| • |, such that P |Y (j) | > tx, Y (j) /|Y (j) | ∈ • P (|Y (j) | > x) v - → t -α P S j-1 (θ ∈ •),
where v -→ denotes vague convergence on S j-1 (see e.g. [START_REF] Resnick | The extremal dependence measure and asymptotic independence[END_REF] and [START_REF] Basrak | A characterization of multivariate regular variation[END_REF]). Analogously, the amounts Z i too have a common distribution function G, and each vector Z (j) = (Z 1 , ..., Z j ), j ≥ 1, is again of multivariate regularly varying but of index -β with β > 0. The sequences {Y i , i ≥ 1}, {Z i , i ≥ 1} and {V i , i ≥ 1} are independent of each other.

A key tool in the analysis will be the following well-known result (see e.g. [START_REF] Barbe | On the tail behavior of sums of dependent risks[END_REF]). If (Y 1 , ..., Y j ) is regularly varying of index -α with common marginal d.f. F, then the right tail of the partial sum S j = Y 1 + . . . + Y j can be approximated as

P(Y 1 + ... + Y j > x) ∼ q j,α F(x) for large x, (1.4)
where ∼ means that the ratio tends to 1 as x → ∞. The parameter q j,α in (1.4) depends on j, α and on the dependence structure inside the vector (Y 1 , ..., Y j ).

Table 1 provides examples of q j,α for some classical multivariate copulas (see [START_REF] Nelsen | An Introduction to Copulas[END_REF] for a nice introduction to copulas). The independent copula corresponds to the special case where the Y i , 1 ≤ i ≤ j, are independent. The Fréchet upper bound represents the case where the Y i 's are comonotonic. The non-degenerate Gaussian copula is the case where the dependence between the Y i 's is drawn from a multivariate Gaussian distribution with a correlation matrix with coefficients strictly than ±1 outside of the diagonal. A copula introduced in [START_REF] Biard | Impact of correlation crises in risk theory : Asymptotics of finite-time ruin probabilities for heavy-tailed claim amounts when some independence and stationarity assumptions are relaxed[END_REF], denoted BBL in the sequel, is built as follows :

Y i = I i W 0 + (1 -I i )W i , 1 ≤ i ≤ j, (1.5)
where W i , 0 ≤ i ≤ j, are i.i.d. random variables, and I i , 1 ≤ i ≤ j, are i.i.d. Bernoulli random variables with parameter p, these two sequences being mutually independent.

Special copulas q j,α Independent j Fréchet upper bound

j α Non-degenerate Gaussian j BLL j i=0 j i p i (1 -p) j-i j -i + i α T. 1 -Values of q j,α in (1.4) for different copulas.
Let us mention that the study of the sum of dependent random variables has received much attention in actuarial sciences. See e.g. W üthrich (2003), [START_REF] Alink | Diversification of aggregate dependent risks[END_REF], [START_REF] Alink | Analysis of the expected shortfall of aggregate dependent risks[END_REF], [START_REF] Barbe | On the tail behavior of sums of dependent risks[END_REF], Albrecher et al. (2006), [START_REF] Biard | Impact of correlation crises in risk theory : Asymptotics of finite-time ruin probabilities for heavy-tailed claim amounts when some independence and stationarity assumptions are relaxed[END_REF] and [START_REF] Kortschak | Asymptotic results for the sum of dependent non-identically distributed random variables[END_REF], among many others.

The paper is organized as follows. In Section 2, we obtain the asymptotic finite-time ruin probabilities for regularly varying claim sizes in the risk model introduced by [START_REF] Boudreault | On a risk model with dependence between interclaim arrivals and claim sizes[END_REF]. In Section 3, we derive such ruin probabilities in two other risk models, for earthquake or flooding-type risks, that take consecutive gauge-loading effects into account. The methods of proof will rely on the approximation (1.4) and the analysis of spacings in a conditioned Poisson process. Finally, some numerical illustrations are presented in Section 4.

Direct effects of each interclaim arrival

Our starting point is the model of [START_REF] Boudreault | On a risk model with dependence between interclaim arrivals and claim sizes[END_REF] where claim amounts are of two different types depending on the length of the previous interclaim arrival. Hereafter, we are going to assume that the claim amounts may be dependent and they have heavy tailed distributions.

Specifically, if an interarrival period V i is larger than a fixed threshold τ, then the next claim amount X i is given by a random variable Y i , and if not, X i corresponds to another random variable Z i . As stipulated before, each vector Y (j) = (Y 1 , ..., Y j ) is of multivariate regular variation of index -α and common d.f. F, while each vector Z (j) = (Z 1 , ..., Z j ) is of multivariate regularly varying of index -β and common d.f. G.

For instance, for earthquake-type risks, one would expect that α < β : the longer a period without any earthquake, more serious will be the next earthquake. For flooding risks, the inverse situation where α > β seems to be quite plausible.

Consider the random variable M(t, τ) that gives the number of spacings of the Poisson process {N(s), 0 ≤ s ≤ t} which are larger than τ. Conditioning by the number of events N(t), define the following.

Let M(n, t, τ) be the random variable that counts the number, during (0, t), of Poisson spacings which are larger than τ, given that N(t) = n (≥ 1) and 0 < τ < t.

Proposition 2.1 If α < β, for t > 0 and large u,

ψ(u, t) ∼        ∞ n=1 P[N(t) = n] min(n, t/τ ) j=1 P[M(n, t, τ) = j] q j,α        F(u + ct),
(2.1)

while if α > β, for t > 0 and large u,

ψ(u, t) ∼          ∞ n=1 P[N(t) = n] n j=max(1,n-t/τ ) P[M(n, t, τ) = n -j] q j,β          G(u + ct), (2.2)
with, for 0 ≤ j ≤ n,

P[M(n, t, τ) = j] = n i=j (-1) i-j n j n -j i -j P(V * 1 > τ t , . . . , V * i > τ t ), (2.3)
where

P(V * 1 > v, . . . , V * i > v) =          1, v ≤ 0, (1 -iv) n , 0 < v < 1/i, 0, 1/i ≤ v < 1.
(2.4)

Proof. Consider S(t), the aggregate claim amount (1.2). Following [START_REF] Biard | Impact of correlation crises in risk theory : Asymptotics of finite-time ruin probabilities for heavy-tailed claim amounts when some independence and stationarity assumptions are relaxed[END_REF], we know that if x → P[S(t) > x] is regularly varying, then for large u,

ψ(u, t) ∼ P[S(t) > u + ct].
(2.5)

To begin with, we find that

P[S(t) > x] = ∞ n=1 P[N(t) = n]P (X 1 + ... + X n > x) = ∞ n=1 P[N(t) = n] n j=1 P[M(t, τ) = j|N(t) = n] P[X 1 + ... + X n > x|N(t) = n, M(t, τ) = j],
and by the model assumptions,

P[S(t) > x] = ∞ n=1 P[N(t) = n] n j=1 P[M(n, t, τ) = j] P(Y 1 + ... + Y j + Z 1 + ... + Z n-j > x).
(2.6)

Note that P[M(n, t, τ) = j] = 0 for j > t/τ, so that the summation over j in (2.7) goes from 1 to min (n, t/τ ). Now, suppose that α < β, so that

G(x) = o(F(x)) for large x. Putting S k = Y 1 + ... + Y k and T k = Z 1 + ... + Z k , k ≥
1, and using Proposition 1.1 in [START_REF] Cai | On max-sum equivalence and convolution closure of heavytailed distributions and their applications[END_REF] and (1.4), we obtain

P(S j + T n-j > x) ∼ P(S j > x) + P(T n-j > x) ∼ P(S j > x) ∼ q j,α F(x), 1 ≤ j ≤ n.
(2.7)

Inserting (2.7) in (2.6) then yields

P[S(t) > x] ∼        ∞ n=1 P[N(t) = n] min(n, t/τ ) j=1 P[M(n, t, τ) = j] q j,α        F(x),
that is, the function P[S(t) > x] is regularly varying with parameter α. Therefore, (2.5) is applicable and gives the announced result (2.1). The approximation (2.2) is derived in a similar way. Now, it remains to evaluate the distribution of the variable M(n, t, τ). As {N(t), t ≥ 0} is a Poisson process, it is well-known that given N(t) = n, the claim instants U 1 , . . . , U n are distributed as the order statistics of n uniform random variables on [0, t], and the interclaim arrivals V 1 , . . . , V n (i.e. the associated spacings) are exchangeable random variables such that the p.d.f. of the vector (V * 1 , . . . , V * i ), where [START_REF] David | Order statistics[END_REF], Sec. 6.4. Consequently, whenever v 1 , . . . , v j ≥ 0 with v 1 + . . .

V * i = V i /t, 1 ≤ i ≤ n, is given by f (v 1 , . . . , v i ) = n! (n -i)! (1 -v 1 -. . . -v i ) n-i , for v 1 , . . . , v i ≥ 0, v 1 + . . . + v i ≤ 1; see e.g.
+ v i ≤ 1, P(V * 1 > v 1 , . . . , V * i > v i ) = (1 -v 1 -. . . -v i ) n .
(2.8)

In particular, choosing v i = τ/t = v for all i in (2.8), we find that P(V 1 > τ, . . . , V i > τ) is provided by the formula (2.4) above. Let us consider the event that exactly j interclaim arrivals are larger than τ, given that N(t) = n, with 1 ≤ j ≤ n. Following e.g. David and Nagaraja (2003) (p. 129-130), we then get

P[M(n, t, τ) = j] = 1 j! n-j l=0 (-1) l l! ( j + l)! i 1 <•••<i j+l P(V i 1 > τ, . . . , V i j+l > τ|N(t) = n) = 1 j! n-j l=0 (-1) l l! ( j + l)! n j + l P(V 1 > τ, . . . , V j+l > τ|N(t) = n) = n i=j (-1) i-j n! j!(i -j)!(n -i)! P(V * 1 > v, . . . , V * i > v), (2.9)
with v = τ/t.

Consecutive gauge-loading effects

In this section, we want to incorporate in the risk model the observation that several consecutive claims with large, or small, interoccurrence times are more susceptible to be followed by severe damages. By comparison with the previous model, it will be necessary this time to take (part of) the history of the claim arrival process into account. So, a gauge will be used to register the large, or small, interclaim arrivals ; initially, the gauge is empty.

Earthquake-type phenomenon

Roughly speaking, for a risk of earthquake-type, the severity of a catastrophe is expected to be more important if the latest catastrophes occur a long time ago. In this sense, let us assume that if k consecutive interclaim arrivals are larger than τ, then a more dangerous catastrophe may arise. Moreover, just after, the gauge is put at level 0. Damages caused by standard earthquakes give multivariate regular varying vectors Z (j) of index -β, while the more severe form regular varying vectors Y (j) with index -α and common d.f. F.

Let M + (n, k, t, τ) be the random variable that counts the number of sequences, during (0, t), of k consecutive Poisson spacings which are larger than τ, given that N(t) = n (≥ 1). Proposition 3.1 If α < β, for t > 0 and large u :

ψ(u, t) ∼        ∞ n=1 P[N(t) = n] min( n/k , t/kτ ) j=1 P M + (n, k, t, τ) = j q j,α        F(u + ct), (3.1)
with, for 0 ≤ j ≤ min ( n/k , t/kτ ),

P M + (n, k, t, τ) = j = k-1 i=0 x 1 ,...,x k x 1 + • • • + x k + j x 1 , . . . , x k , j P[M(n, t, τ) = n -x 1 -• • • -x k ]/ n x 1 + • • • + x k , (3.2)
where the inner summation is over all nonnegative integers x 1 , . . . ,

x k such that x 1 + 2x 2 + • • • + kx k = n -i -k j and x 2 + 2x 3 + . . . + (k -1)x k ≤ t/τ -i -k j.
Proof. It suffices to substitute M + (n, k, t, τ) for M(n, t, τ) in the proof of Proposition 2.1. By adapting and combining (2.6) and (2.7), we then find that

P[S(t) > x] = ∞ n=1 P[N(t) = n] n j=1 P[M + (n, t, τ) = j] q j,α F(x),
which leads again to the desired approximation (3.1).

One has now to determine the p.m.f. of the random variable M + (n, k, t, τ). This is closely related to the evaluation of the so-called binomial distributions of order k (see e.g. the book by [START_REF] Balakrishnan | Runs and scans with applications[END_REF]). Remember that the p.m.f. of M(n, t, τ) is given in Proposition 2.2. Denote by S the event [a Poisson spacing until time t is larger than τ, given that N(t) = n], and let F be the complementary event. Following [START_REF] Philippou | Successes, runs and longest runs[END_REF], we observe that a typical element of the event [M + (n, k, t, τ) = j] consists in an arrangement of the form a 1 a 2 . . . a x 1 +...+x k +j SS . . . S i , with 0 ≤ i ≤ k -1, where x 1 of the a's are F, x 2 of the a's are SF, . . ., x k of the a's are SS . . . S k-1 F and j of the a's are SS . . . S k . Note that these x k are subject to the constraint

x 1 + 2x 2 + . . . + kx k + k j + i = n.
Moreover, by construction of the model, the number of S has to be smaller than t/τ , so that x 2 + 2x 3 + (k -1)x k + k j + i ≤ t/τ . Now, such arrangements are in number

x 1 + ... + x k + j x 1 , . . . , x k , j ,
and each of them has probability

P[M(n, t, τ) = n -x 1 -• • • -x k ]/ n x 1 + • • • + x k ,
hence the formula (3.2).

Flooding-type phenomenon

For a risk of flooding-type, a close succession of claims is expected to be followed by a more important catastrophe. So, we here assume that if k consecutive interclaim arrivals are smaller than τ, then the next claim may be more severe. Moreover, the gauge becomes empty once an interoccurrence time is larger than τ (it corresponds to a reconstruction time). In case of standard floods, damages form multivariate regular varying vectors Y (j) of index -α, while the more severe give regular varying vectors Z (j) with index -β and common d.f. G.

Let M -(n, k, t, τ) be the random variable that counts the number of sequences, during (0, t), of at least k consecutive Poisson spacings which are smaller than τ, given that N(t) = n (≥ 1). Proposition 3.2 If α > β, for t > 0 and large u :

ψ(u, t) ∼        ∞ n=1 P[N(t) = n] min( (n+1)/(k+1) , t/(τ-1) ) j=1 P M -(n, k, t, τ) = j q j,β        G(u + ct), (3.3) with, for 0 ≤ j ≤ min ( (n + 1)/(k + 1) , t/(τ -1) ), P[M -(n, k, t, τ) = j] = n i=0 x 1 ,...,x n x 1 + • • • + x n x 1 , . . . , x n P[M(n, t, τ) = x 1 + • • • + x n ]/ n x 1 + • • • + x n , (3.4)
where the inner summation is over all nonnegative integers x 1 , . . . , x n such that x 1 + 2x 2 + • • • + nx n = n -i, x k+1 + . . . + x n + 1 {i≥k} = j and x 1 + . . . + x n ≤ t/τ .

Proof. Substituting this time M -(n, k, t, τ) for M(n, t, τ) in the proof of Proposition 2.1 allows us to derive similarly the desired approximation (3.3).

To determine the p.m.f. of the random variable M -(n, k, t, τ), one still may argue as for M + (n, k, t, τ). Let S be here the event [a Poisson spacing until time t is smaller than τ, given that N(t) = n], and let F be the complementary event. An element of the event

[M -(n, k, t, τ) = j] is an arrangement of the form a 1 a 2 . . . a x 1 +...+x n SS . . . S i , with 0 ≤ i ≤ n,
where x 1 of the a's are F, x 2 of the a's are SF, . . ., x n of the a's are SS . . . S n-1 F, with the constraints x 1 + 2x 2 + . . . + nx n + i = n and x k+1 + . . . + x n + 1 {i≥k} = j. In addition, as the number of spacings larger than τ is smaller than t/τ , one has x 1 + . . . + x n ≤ t/τ . Now, the number of such arrangements is x 1 + ... + x n x 1 , . . . , x n , each of them having probability

P[M(n, t, τ) = x 1 + ... + x n ]/ n x 1 + • • • + x n ,
so that the formula (3.4) then follows.

Numerical analysis

A central step of the numerical analysis is the computation of the p.m.f. of the random variables M(n, t, τ), M + (n, k, t, τ) and M - (n, k, t, τ). For that, one can use the exact formulas obtained before. Another possible way is to proceed by recursion. A recursive method is provided in the Appendix for the variables M + and M -(it is not simpler for M). Both methods have their own advantages and drawbacks.

-The exact formulas are easier to implement but the computation is longer.

-The recursive method works faster but implementation is fastidious.

Hereafter, we have chosen to work with the exact formulas because we aim to see the impact of various dependence parameters on the asymptotic ruin probabilities. Someone who wants to investigate larger time horizons for example, might prefer to follow the recursive method.

Impact of a dependence between claim amounts

In this first part, we consider asymptotic ruin probabilities ψ(u, t) for different values of q k,α (see Table 1). In order to have a spectrum of dependence, the parameter p (called "dependence parameter") in the BLL copula model is allowed to vary from 0 to 1 with step 0.01. When the dependence parameter equals 0 we have the independence case, and when it equals 1 we obtain the Fréchet upper bound case. The other parameters are : u c λ Distribution of the riskier claim amount 1, 000, 000 10 0.1 Pareto law with parameter α

For the model of Section 2, we examine the case where α < β. For t = 10 and for both α = 0.5 and α = 3, two cases are investigated, one with τ = 1 and the other with τ = 2. Each case is plotted first separately and then in a same graph. When α = 0.5 (Figure 1) the asymptotic ruin probability is a decreasing function of the dependence parameter p. It is the opposite when α = 3 (Figure 2).

For the model of Subsection 3.1, with k = 3, the same phenomenon is observed (see Figure 3 for α = 0.5 and Figure 4 for α = 3). F. 1 -Asymptotic ruin probability for the model of Section 2 as a function of the dependence parameter p in the BLL copula when α = 0.5 and t = 10.

Impact of a dependence between interclaim arrivals and claim amounts

In this second part, we examine the effect of two parameters on the asymptotic ruin probabilities : τ for the model of Section 2 and τ and k for the model of Subsection 3.2. Let us choose α = 3, for instance. For each case, we consider two dependence parameters p = 0.4 and p = 0.8 ; these are plotted separately and the relative difference is then plotted in a third graph.

For the model of Section 2 with t = 10, we obtain that as expected, the asymptotic ruin probability is a decreasing function of τ (see Figure 5).

For the model of Subsection 3.1 with t = 20, the asymptotic ruin probability is a decreasing function of both τ and k (see Figures 6 and7).

Appendix

We present in this section a recursive method to compute the p.m.f. of M + (n, k, t, τ) and M -(n, k, t, τ). The method of proof is directly inspired from [START_REF] Makri | On binomial and circular binomial distributions of order k for l-overlapping success runs of length k[END_REF] (Theorem 4.1).

Proposition 5.1 (a) For n ≥ k + 1 and 1 ≤ j ≤ min( n/k , t/(kτ) ), (e) Obvious fact.

P M + (n, k, t, τ) = j = p n,n (t)δ j, n/k + 1 t p 0,n (t) B(1, n) τ 0 P M + (n -1, k, t -s, τ) = j 1 - s t n ds + 1 t k-1 l=1 p l,n (t) B(l + 1, n -l) t lτ s t l 1 - s t n-l-1 P M + (n -1 -l, k, t -s, τ) = j ds + 1 t n-1 l=k p l,n (t) B(l + 1, n -l) t lτ s t l 1 - s t n-l-1 P M + (n -1 -l, k, t -s, τ) = j -l/k ds, (b) for n ≥ k + 1, P [M + (n, k, t, τ) = 0] = 1 t
Proposition 5.2 (a) For n ≥ k + 1 and 1 ≤ j ≤ min( (n + 1)/(k + 1) , t/(τ -1) ), P M -(n, k, t, τ) = j = δ j,1 q n,n (t)

+ 1 t k-1 l=0
q l,n (t) B(l + 1, nl)

F. 5 -

 5 Asymptotic ruin probability for the model of Section 2 as a function of τ when α = 3 and t = 10. F. 6 -Asymptotic ruin probability for the model of Subsection 3.1 as a function of τ when α = 3 and t = 20. F. 7 -Asymptotic ruin probability for the model of Subsection 3.1 as a function of k when α = 3 and t = 20.

P

  [M + (n -1, k, ts, τ) = 0] 1 -+ (n -1l, k, ts, τ) = 0] ds, (c) for n ≥ k + 1 and j > min( n/k , t/(kτ) ), P M + (n, k, t, τ) = j = 0, (d) for n = k, P [M + (n, k, t, τ) = 0] = 1p k,k(t),P [M + (n, k, t, τ) = 1] = p k,k (t), (e) for 0 ≤ n ≤ k -1, P [M + (n, k, t, τ) = 0] = 1, n (t) = 1p n,n (t),and for1 ≤ l ≤ n -1, p l,n (t) = + 1)τ ≤ t, 1 -lτ t nif lτ ≤ t < (l + 1)τ.

From 0 P

 0 [START_REF] David | Order statistics[END_REF] p. 134-135, we have, for l = 1, . . . , ., .) is the Beta function. So, we deduce part (a) of the proposition.(b) When j = 0 and n ≥ k, we observe that[M + (n, k, t, τ) = 0] = + (n, k, t, τ) = 0] ∩ A l } , so that P [M + (n, k, t, τ) = 0] = + (n, k, t, τ) = 0|A l ] .For l = 0,P [M + (n, k, t, τ) = 0|A 0 ] = τ [M + (n -1, k, ts, τ) = 0] f U 1 (s)ds. For 1 ≤ l ≤ k -1, P [M + (n, k, t, τ) = 0|A l ] = t lτ P [M + (n -1l, k, ts, τ) = 0] f U l+1 (s)ds. For l ≥ k, P [M + (n, k, t, τ) = 0|A l ] = 0.Thus, we get part (b) of the proposition.(c) Obvious fact.(d) For n = k,P [M + (n, k, t, τ) = 1] = P (V 1 > τ, V 2 > τ, . . . , V k > τ) = 1 -kτ t k ,andP [M + (n, k, t, τ) = 0] = 1 -P [M + (n, k, t, τ) = 1] .

  n (t) B(l + 1, nl) -(n -1l, k, ts, τ) = j -1 ds,

Acknowledgment

C. Lefèvre thanks the members of the I.S.F.A for their warm welcome during his visit, and the Banque Nationale de Belgique for a partial support.

Proof.

(a) Let n ≥ k + 1 and 1 ≤ j ≤ min( n/k , t/(kτ) ). Define, for l = 0, . . . , n -1, the event A l = [l interarrival times larger than τ precede the first interarrival time smaller than τ in the sequence of n interarrival times] and the event B = [there is no interarrival time smaller than τ in the sequence of n interarrival times]. Then, we can write that

so that we have

For l = 0,

From (2.4), we get, for 1 ≤ l ≤ n -1,

(c) for n ≥ k + 1 and j > min( (n + 1)/(k + 1) , t/(τ -1) ),

where hereabove,

and for 1 ≤ l ≤ n -1,

Proof. We will follow an analogous argument.

(a) Let n ≥ k -1 and 1 ≤ j ≤ min( (n + 1)/(k + 1) , t/(τ -1) ). Define this time, for l = 0, . . . , n -1, A l = [l interarrival times smaller than τ precede the first interarrival time larger than τ in the sequence of n interarrival times] and B = [there is no interarrival time larger than τ in the sequence of n interarrival times]. Then,

From [START_REF] David | Order statistics[END_REF] p. 126, 134-135, we have, for 1 ≤ l ≤ n -1,

Using again the expression of f U l (u), l = 1, . . . , n, we then obtain part (e) Obvious fact.


L Maurin

P Ferdinand

G Laffont

N Roussel

J Boussoir

L Auger

S Rougeault

. In such a context, the Smart Monitoring In Train Systems (SMITS) European project [2] is devoted to control contact forces between pantographs and Overhead Contact Lines (OCL), enabling infrastructure managers to control the rolling stock and to optimize the predictive maintenance. Optical Fibre Bragg Gratings (FBGs) sensors were integrated into current collectors, turning pantographs into 3-point bending-based sensors in contact with the OCL. A semi-empirical model was introduced into the software of an innovative selfcalibrated dedicated measurement system, providing accurate data at 500 Hz over several FBG optical lines. Ontrack tests were performed on a "TGV Duplex" train at speeds up to 320 km/h. Contact forces, and temperatures inside the collectors were computed in real time, on 4 optical fibers. During these runs, the FBG measurement system has demonstrated its reliability and accuracy with temperature measurements, and coherence with contact force in comparison with reference records provided by the SNCF.

Fiber Bragg Grating-based optical sensing for Smart Monitoring In Train Systems

Introduction

Transport deregulation is currently in process in Europe, transforming the rolling and standing stock activities, traditionally hosted within the same train transportation companies until the recent years, in two shared businesses, introducing new monitoring and control needs. In this frame, the interface between the pantograph and its Overhead Contact Line (OCL) appears to be critical in many aspects: vertical force and temperature being the two main relevant parameters to be controlled in order to develop qualitative predictive maintenance parameters based on current collector's wear. SMITS FP5 European project, led by Siemens and involving several R&D actors [3] introduced a new monitoring approach, based on embedded FBGs controlling the pantograph's current collector bending when submitted to the OCL contact force.

Force and temperature measurements on smart pantographs

Current collection is performed by the pantograph, whilst the OCL line is powered in part of France at 25 kV (AC).

Traditional contact force measurements are obtained through electrical strain gages-based force cells [START_REF] Kießling | Contact lines for electric railways, planning, design, implementation[END_REF] and force and momentum equilibriums relationships. These electrical force cells are usually located inside the mechanical supports of the current collectors, and need both electrical insulation and compensation of inertial forces. The European SMITS project approach solves partially these two drawbacks thanks to the silica optical fibers which allow measurements at 25 kV without additional specific electrical insulation. In the same time, the inertial forces dependence is partially eliminated with a location of the FBG-based sensors inside the pantohead, as close as possible to the contact point, turning it into a 3-point bending force and temperature sensor (Fig. 1 and 2) [START_REF] Maurin | High speed real-time contact measurements between a smart train pantograph with embedded fibre Bragg grating sensors and its overhead contact line[END_REF]. Rolling stock operators, like SNCF, have stringent specifications in terms of measurements frequency (500 Hz, equivalent to 0.2 m spatial resolution along the OCL at TGV commercial speed), and vertical force resolution (1 N) which turns to be equivalent to 0.6 pm in terms of FBG wavelength shift resolution. A semi-empirical model, validated by Finite Element Modelling as well as several static laboratory tests performed at high voltages, was introduced into the signal processing of a dedicated highly accurate and auto-calibrated FBG measurement system (accuracy & drift < 0.6 pm @ 1550 nm, and 0.5 kHz measuring rate), providing simultaneous measurements over several FBG optical fiber lines [START_REF] Laffont | Wavelength tunable fiber ring laser for high-speed interrogation of fiber Bragg grating sensors[END_REF]. This optoelectronics measurement system has been developed and embedded into a 19" rack, and include a dedicated software able to deliver, in real time, the requested parameters (Fig. 3). Such optoelectronic acts as a "Master Wavelength Server" (MWS) sending wavelengths data over an Ethernet link to a "supervision client" computing force, position and temperature parameters, and displaying them in real-time onto the screen (Fig. 4). Simultaneously, this "Supervision Client" provides 0-10 V analog outputs to the end-user reference system.

High speed train (TGV) trials

On-line validation tests were performed on more than 6000 km during 5 full days on a very high speed train "TGV Duplex" between Paris and Vendôme (France) at speeds up to 320 km/h (Fig. 5). Contact forces between the pantograph and the OCL, as well as temperatures inside the current collectors were computed in real time at 500 Hz, on 4 optical lines in parallel. During these TGV tests, the innovative FBG measurement system developed by the CEA LIST has demonstrated its reliability and accuracy with pantohead temperature measurements (up to 160°C), and coherence with contact force measurements in comparison with SNCF reference records.

Fig. 5: 180° view of measurement systems during tests at 320 km/h Conclusion

Both SMITS FBG-based sensors and related spectral measurement systems have demonstrated their ability to be used in harsh environments (high voltages, vibrations, shocks, temperature gradients, humidity…) for real-time measurements, as encountered in railway industry. Nowadays, industrial projects devoted to structure monitoring based on Optical Fiber Sensors, such as FBGs, require more and more high scanning rates, better resolutions, and simultaneous measurements on a higher number of sensors and optical lines. Based on its 15 years experience in this domain, the CEA LIST has been able to develop and build such a system compliant to the SMITS project specifications, and now continues these developments in a second European funded railway project, named CATIEMON, for enhanced performances and new parameters to monitor.

Fig. 1 :

 1 Fig.1 : Smart pantohead equipped with FBG sensors Fig. 2 Smart TGV pantograph

Fig

  Fig. 3: Optoelectronic MWS Fig. 4: Supervision Client screen with real time display of requested parameters.

nd Intern. Conference on Optical Complex Systems. OCS'2008, 17 -20 March 2008, Cannes, France


Nicolas Belleville
email: nicolas.belleville@cea.fr

Accelerating exp-log based finite field multiplication

Finite field multiplication is widely used for masking countermeasures against side-channel attacks. The execution time of finite field multiplication implementation is critical as it greatly impacts the overhead of the countermeasure. In this context, the use of exp-log tables is popular for the implementation of finite field multiplication. Yet, its performance is affected by the need for particular code to handle the case where one of the operands equals zero, as log is undefined for zero. As noticed by two recent papers, the zero case can be managed without any extra code by extending the exp table and setting log[0] to a large-enough value. The multiplication of a and b then becomes as simple as:

In this paper, we compare this approach with other implementations of finite field multiplication and show that it provides a good trade-off between memory use and execution time.

Introduction

Side-channel attacks represent a major threat for the security of embedded systems [START_REF] Mangard | Power Analysis attacks: Revealing the secrets of smart cards. Power Analysis Attacks: Revealing the Secrets of Smart Cards[END_REF]. In particular, they are particularly effective for finding secret keys of block ciphers, such as AES.

The masking countermeasures can provide a strong protection against these attacks [START_REF] Goubin | DES and Differential Power Analysis (The "Duplication" Method)[END_REF]. The general principle is to split the secret data into several variables, called shares, in a way that any strict subset of the shares do not provide information about the secret data. In other words, masking forces an attacker to learn information about all the shares in order to get information about the secret data.

The masking countermeasures require modifying how computations are performed. Indeed, instead of manipulating the secrets directly, one needs to use the shares instead. All the complexity relies on the requirement to avoid revealing any information about the secret during intermediate computations. Depending on the masking scheme used and the computation to perform, this code transformation may be complex, and the resulting overhead can be high.

In this context, finite field multiplication implementation can be critical. Indeed, finite field multiplication is a key operation for various masking types:

• Affine masking [START_REF] Fumaroli | Affine Masking against Higher-Order Side Channel Analysis[END_REF] and inner-product masking [START_REF] Balasch | Inner product masking revisited[END_REF] involve finite field multiplications for masking the secret data.

• Boolean masking can involve many finite field multiplications when masking SBoxes. Multiplications can be used to compute finite field powers, like x 254 for AES SBox that involves 4 multiplications of masked variables. It can also be used to compute a polynomial that interpolates a given SBox in a finite field [START_REF] Coron | Fast evaluation of polynomials over binary finite fields and application to side-channel countermeasures[END_REF].

In addition, in the case of boolean masking, each secure multiplication on masked variables involves 4 finite field multiplications at first order, i.e. when only the secret is decomposed into 2 shares [START_REF] Rivain | Provably Secure Higher-Order Masking of AES[END_REF]. When increasing the number of shares, the number of finite field multiplication required to performed one multiplication on masked data gets higher and higher.

As such, the number of finite field multiplications required can grow quickly and represent a big part of the overhead of the countermeasure, and there is a strong need for fast constant-time finite field multiplication algorithms.

In [START_REF] Goudarzi | How fast can higherorder masking be in software?[END_REF], the authors do an overview of the different methods to implement the finite field multiplication. Among these methods, the exp-log method provides a nice trade-off between execution time and memory consumption. However, its main issue is that it requires particular code to handle the case where one of the operand equals zero, as log is undefined for zero. Until recently, no approach had emerged to implement finite field multiplication with exp-log tables without such particular code for zero.

In this paper, we present an approach that solves this problem. This approach was independently discovered and presented in both [BCH + 20] and [START_REF] Cardoso Dos Santos | Rivain-Prouff on Steroids: Faster and Stronger Masking of the AES[END_REF].

We start by recalling the exp-log method in section 2.1. Then, we remind the well-known optimization of the modulo in section 2.2. We explain the technique that avoids any extra code to handle the case where one of the operands equals zero in section 2.3. We compare extensively the exp-log method with this optimization to the other methods for implementing finite field multiplication that were detailed in [START_REF] Goudarzi | How fast can higherorder masking be in software?[END_REF], showing the impact on memory (section 3.2), and execution time (section 3.3).

Exp-log multiplication and its optimization 2.1 Naïve implementation

The exp-log method is based on the use of a generator element e of the multiplicative group F * 2 n . We have:

∀x ∈ F * 2 n , ∃k, x = e k .
The number k is called the log of x. We can use this generator element to multiply 2 numbers in F * 2 n . Let log t be a 2 n entry table that contains the logs in base e of all 2 n -1 non-zero values (log t[0] is usually set to 0), and exp t a 2 n -1 entries table that contains e raised to the power of all values in 0, 2 n -2 . Then the multiplication of two non-zero elements a and b is given by1 :

a.b = exp t[(log t[a] + log t[b])%(2 n -1)]
Some code is needed to handle the case where a or b equals zero. In [START_REF] Goudarzi | How fast can higherorder masking be in software?[END_REF], the authors propose a sequence of 4 ARM instructions to create a bitmask that equals zero if and only if one of the operands equals zero (asr 32 designates the arithmetic shift right by 32bits, that propagates the sign bit): bitmask = asr 32 (-(asr 32 (-a)&b))

The multiplication of 2 numbers of F 2 n is given by:

a.b =asr 32 (-(asr 32 (-a)&b)) & exp t[ (log t[a] + log t[b]) % (2 n -1) ]

Removal of the modulo

A first optimization to make the implementation faster consists in removing the modulo [START_REF] Goudarzi | How fast can higherorder masking be in software?[END_REF]. To achieve this goal, a wider exp t table with 2 * (2 n -1) entries is used. It is defined as follows:

∀x ≤ 2 n+1 -2, exp t[x] = e x
The multiplication of 2 numbers of F 2 n with this optimization is given by:

a.b = asr 32 (-(asr 32 (-a)&b)) & exp t[log t[a] + log t[b]]

Removal of the code that handles the zero case

We and Cardoso dos Santos et al. proposed an optimization to remove the code that handles the case where one operand equals zero, at the price of a wider exp t table [BCH + 20, CdSGGS22]. Note that this optimization cannot be deployed without the modulo optimization. The general idea is that the sum of 2 logs is bounded by a maximum value: setting log t[0] higher than this maximum allows to easily detect any multiplication that involves a zero operand.

First, we notice that:

∀a ∈ F * 2 n , log(a) ≤ 2 n -2 We deduce that: ∀a, b ∈ F * 2 n , log(a) + log(b) ≤ 2 n+1 -4 Setting log t[0] = 2 n+1 -3, we have: ∀a, b ∈ F 2 n ,((log t[a] + log t[b] ≥ 2 n+1 -3) ⇔(a = 0 or b = 0))
With such a value for log t[0], we need the exp t table to be extended as the sum of 2 logs can now be as high as 2 * (2 n+1 -3) = 2 n+2 -6. The exp t table becomes a 2 n+2 -5 entries table and is set as follows:

• ∀x, 0 ≤ x ≤ 2 n+1 -4, exp t[x] = e x • ∀x, 2 n+1 -3 ≤ x ≤ 2 n+2 -6, exp t[x] = 0
The multiplication of 2 numbers of F 2 n is given by:

a.b = exp t[log t[a] + log t[b]]
Note: log t[0] can be set to other (higher) values if needed, for instance to get a particular hamming weight, at the price of a larger exp t table.

lut size 2 n+1 -1 3 * 2 n -2 c * 2 n -5 3 * 2 n 2 3n 2 +1 2 2n (in ES) c = 5 or c = 6
3 Comparison to alternative table-based implementations

Alternative approaches considered

In order to evaluate the interest of the optimization, we compare in the next sections its memory impact and number of cycles with other table-based finite field multiplication implementation from [START_REF] Goudarzi | How fast can higherorder masking be in software?[END_REF].

We consider the following methods: (1) exp-log tables with optimizations previously presented, (2) karatsuba, (3) half-table, (4) full table. In this section we briefly describe the general ideas of each method. Full algorithms descriptions can be found in [START_REF] Goudarzi | How fast can higherorder masking be in software?[END_REF].

The karatsuba method involves 3 lookup tables. Its main principle is to split the multiplication into computations involving smaller numbers represented on half as many bits. For instance, to multiply to bytes, this methods uses 4-bits numbers. Each table access combines 2 of those numbers.

The half-table method involves 2 lookup tables. Similarly to karatsuba, it splits the multiplication into computations involving smaller numbers represented on half as many bits. Yet, instead of combining 2 numbers, each table access depends on 3 numbers.

The full-table method consists in fully tabulating the multiplication, for all possible values of both operands. The multiplication is then done in one table access.

Memory impact

In this section we compare the size of the lookup tables of considered methods, for a finite field of 2 n elements.

Table 1 presents the total memory size taken by the lookup table of the various methods. The sizes are reported as multiples of the size taken by a single element, i.e. as a number of bytes for n ≤ 8, as halfwords for 8 < n ≤ 16, and as words for 16 < n ≤ 32. Our method is reported as "exp-log w/o zero" in the table. It has a variable overhead depending on n: c = 5 except if n is equal to 8 or 16 or 32. In these cases, c = 6. Indeed, for these cases, we need to change the size of the elements in the log t To make the comparison easier, the actual total size of lookup tables in bytes are reported in Table 2 for various n. While our optimization make the lookup table size clearly bigger than exp-log without this optimization and karatsuba method, the memory impact remains way lower than the one of half-tables.

Instruction count and estimated number of clock cycles

We compare the number of ARM assembly instructions necessary for the different methods, following the implementations provided by [START_REF] Goudarzi | How fast can higherorder masking be in software?[END_REF]. Figure 1 shows the assembly of exp-log method with our optimization. In order to save a load instruction, the exp table is stored right after the load table in memory.

The address to the exp table is thus retrieved from the address of the log table, with an addition. Note that the offset used in this addition has to be adjusted depending on the field size, as logs may be stored as halfwords or words instead of bytes.

The measured numbers are presented in Table 3. For each implementation, we indicate the number of arithmetic instructions (alu), and the number of loads (ldr). We estimate the clock cycles of each implementation by counting 1 clock cycle per alu instruction and 3 clock cycles per ldr instruction, similarly as in [START_REF] Goudarzi | How fast can higherorder masking be in software?[END_REF]. We also estimate the number of clock cycles without initialization, i.e. the number of clock cycles the multiplication would take if memory addresses of the various tables were already in registers. This number makes sense when several multiplications are done in a row. Such case happens for instance when a multiplication is done on masked data, requiring several finite field multiplications under the hood.

Our method has an estimated number of clock cycles per multiplication that is largely improved compared to previous exp-log methods and to the karatsuba method. It is also very close to the estimated number of clock cycles of the halftable method, both with and without initialization: in both cases, our method is only one clock cycle slower than the half-table implementation. As such, our method provides a good trade-off between impact on memory and on execution time: while its execution time is almost on par with half-table, its memory impact is much lower.

Conclusion

This short paper presents an optimization of the finite field multiplication with exp-log tables that handles the case where an operand equals zero by expanding the exp table and by setting a particular value for 0 in the log table. Compared with other table-based methods to implement the finite field multiplication, this optimization makes the exp-log implementation very competitive. It becomes almost as fast as the half-table implementation while having a way lower impact on memory, whatever the size of the finite field.

Figure 1 :

 1 Figure 1: Implementation of exp-log multiplication with our optimization

Table 1 :

 1 Total size of lookup tables (lut) used in various multiplication algorithms. ES is the size necessary to store one element of the field, e.g. byte for integers up to 255.

		exp-log	exp-log	exp-log	kara-	half	full
	Method	naïve w/o mod	w/o zero tsuba	table table

  table due to the larger value of log t[0]. Karatsuba and all exp-log methods including ours feature a table size in O(2 n ). This is not the case for half-table and full-table: with these methods the table size evolve in O(2

3n 2 ) and O(2 2n ) respectively.

Table 2 :

 2 Total size of lookup tables used in various multiplication algorithms,

	in bytes, for various n
		exp-log	exp-log	exp-log	kara-	half	full
		naïve w/o mod w/o zero	tsuba	table	table
	n=1	3 B	4 B	5 B	6 B	6 B	4 B
	n=2	7 B	10 B	15 B	12 B	16 B	16 B
	n=3	15 B	22 B	35 B	24 B	45 B	64 B
	n=4	31 B	46 B	75 B	48 B	128 B	256 B
	n=5	63 B	94 B	155 B	96 B	362 B	1 KB
	n=6	127 B	190 B	315 B	192 B	1 KB	4 KB
	n=7	255 B	382 B	635 B	384 B	3 KB	16 KB
	n=8	511 B	766 B	1.5 KB	768 B	8 KB	64 KB
	n=9	2 KB	3 KB	5 KB	3 KB	45 KB	512 KB
	n=10	4 KB	6 KB	10 KB	6 KB 128 KB	2 MB
	n=11	8 KB	12 KB	20 KB	12 KB 362 KB	8 MB
	n=12	16 KB	24 KB	40 KB	24 KB	1 MB	32 MB
	n=13	32 KB	48 KB	80 KB	48 KB	3 MB 128 MB
	n=14	64 KB	96 KB	160 KB	96 KB	8 MB 512 MB
	n=15 128 KB	192 KB	320 KB 192 KB	23 MB	2 GB
	n=16 256 KB	384 KB	768 KB 384 KB	64 MB	8 GB

Table 3 :

 3 Number of ARM assembly instruction for each method. Loads and alu operations are separated as their performance impact is different. An estimated number of clock cycles is computed, considering 1 clock cycle per alu instruction and 3 clock cycles per load instruction.

			exp-log	exp-log	exp-log kara-	half	full
			naïve w/o mod w/o zero tsuba table table
	# ldr	init only w/o init total	1 3 4	1 3 4	1 3 4	1 3 4	1 2 3	1 1 2
	# alu	init only w/o init total	1 7 8	1 5 6	1 1 2	2 7 9	1 3 4	1 1 1
	cycles	full w/o init	20 16	18 14	14 10	21 16	13 9	7 4

% designates the modulo and & designates the AND operator

Acknowledgements

This work was funded as part of the SARMENTI project, which has received funding from the European Union's Horizon 2020 research and innovation programme under grant agreement No 825325, and by the French National Research Agency in the framework of the "Investissements d'avenir" program (ANR-10-AIRT-05).