Renard Eric

H Mosbah

M C Vantyghem

E Nobécourt

F Andreelli

F Archambeaud

E Bismuth

C Briet

M Cartigny

B Donadille

A Daguenel

M Fichet

J F Gautier

S Janmaat

I Jéru

C Legagneur

L Leguier

J Maitre

E Mongeois

C Poitou

E Renard

Y Reznik

A Spiteri

F Travert

B Vergès

J Zammouri

C Vigouroux

C Vatier

Therapeutic indications and metabolic effects of metreleptin in patients with lipodystrophy syndromes: Real-life experience from a national reference network

Aims: To describe baseline characteristics and follow-up data in patients with lipodystrophy syndromes treated with metreleptin in a national reference network, in a real-life setting.

Patients and Methods: Clinical and metabolic data from patients receiving metreleptin in France were retrospectively collected, at baseline, one year and at the latest follow-up during treatment.

Results: Forty-seven patients with lipodystrophy including generalized lipodystrophy (GLD, n=28) and partial lipodystrophy (PLD, n=19) received metreleptin over the last decade. At baseline, age was 29.3 [16.6-47.6] (years, median [interquartile range]); BMI 23.8 kg/m 2 [21.2-25.7]; serum leptin 3.2 [1.0-4.9] ng/mL); 94% of patients had diabetes (66% insulin-treated), 53% hypertension, and 87% dyslipidemia. Metreleptin therapy, administered during 31.7 [14.2-76.0] months, was ongoing in 77% of patients at the latest follow-up. In patients with GLD, HbA1c (%) and fasting triglycerides (mmol/L) significantly decreased from baseline to one year-metreleptin treatment, from 8.4 [6.5-9.9] to 6.8 [5.6-7.4], and 3.6 [1.7-8.5] to 2.2 [1.1-3.7] respectively (p<0.001), with a sustained efficacy thereafter. In patients with PLD, HbA1c (%) was not significantly modified from 7.7 [7.1-9.1] at baseline vs 7.7 [7.4-9.5] at one-year), and the decrease in fasting triglycerides (mmol/L), from 3.3 [1.9-9.9] to 2.5 [1.6-5.3], p<0.01) was not confirmed at the latest assessment (5.2 [2.2-11.3]). However, among PLD patients, at one-year, 61 % were responders regarding glucose homeostasis, with lower baseline leptin levels compared to non-responders, and 61% were responders regarding triglyceridemia. Liver enzymes significantly decreased only in the GLD group.

Conclusions: In this real-life setting study, metabolic outcomes are improved under metreleptin therapy in patients with GLD. The therapeutic indication of metreleptin needs to be clarified in patients with PLD.

Introduction

Lipodystrophy syndromes (LD) are rare diseases of acquired or genetic origin characterized by a generalized or partial loss of adipose tissue and subsequent risk of severe metabolic complications associated with insulin resistance, i.e. glucose tolerance abnormalities, hypertriglyceridemia, liver steatosis, atherosclerotic events, and ovarian hyperandrogenism in females [START_REF] Brown | The Diagnosis and Management of Lipodystrophy Syndromes: A Multi-Society Practice Guideline[END_REF]. LD are probably largely underdiagnosed, as illustrated by their estimated prevalence, initially reported at 1.3 to 4.7 cases per million [START_REF] Fernández-Pombo | Familial partial lipodystrophy syndromes[END_REF], but recently re-evaluated at rather 1 in 20 000 [START_REF] Chiquette | Estimating the prevalence of generalized and partial lipodystrophy: findings and challenges[END_REF][START_REF] Gonzaga-Jauregui | Clinical and Molecular Prevalence of Lipodystrophy in an Unascertained Large Clinical Care Cohort[END_REF].

Leptin deficiency has been shown to contribute to ectopic fat storage that drives the metabolic complications in LD [START_REF] Ebihara | Transgenic overexpression of leptin rescues insulin resistance and diabetes in a mouse model of lipoatrophic diabetes[END_REF][START_REF] Colombo | Transplantation of adipose tissue lacking leptin is unable to reverse the metabolic abnormalities associated with lipoatrophy[END_REF][START_REF] Chevalier | Metreleptin treatment of non-HIV lipodystrophy syndromes[END_REF][START_REF] Chan | Clinical effects of long-term metreleptin treatment in patients with lipodystrophy[END_REF]. Several open-label prospective studies have demonstrated that replacement therapy with metreleptin, a recombinant leptin analog, is efficient in reducing hyperphagia and improving insulin sensitivity, HbA1c, triglycerides, and hepatic steatosis, in patients with generalized lipodystrophy (GLD) [START_REF] Chan | Clinical effects of long-term metreleptin treatment in patients with lipodystrophy[END_REF][START_REF] Brown | Long-term effectiveness and safety of metreleptin in the treatment of patients with generalized lipodystrophy[END_REF][START_REF] Araujo-Vilar | Recombinant human leptin treatment in genetic lipodystrophic syndromes: the long-term Spanish experience[END_REF][START_REF] Petersen | Leptin reverses insulin resistance and hepatic steatosis in patients with severe lipodystrophy[END_REF][START_REF] Oral | Leptin-replacement therapy for lipodystrophy[END_REF][START_REF] Adamski | Effects of metreleptin in patients with lipodystrophy with and without baseline concomitant medication use[END_REF]. Improvement in health self-perception, quality of life, and morphotypeassociated stigmatization, were also reported on leptin-replacement therapy [START_REF] Cook | Effects of Metreleptin on Patient Outcomes and Quality of Life in Generalized and Partial Lipodystrophy[END_REF][START_REF] Vatier | Adherence with metreleptin therapy and health self-perception in patients with lipodystrophic syndromes[END_REF][START_REF] Simsir | Metreleptin replacement treatment improves quality of life and psychological well-being in congenital generalized lipodystrophy[END_REF]. However, to date, the therapeutic efficacy of metreleptin has not been studied in prospective placebo-controlled trials [START_REF] Chevalier | Metreleptin treatment of non-HIV lipodystrophy syndromes[END_REF]. In addition, concerning patients with partial forms of lipodystrophy (PLD), the metabolic effects of metreleptin seem variable and could depend on the initial severity of leptin deficiency and/or metabolic complications [START_REF] Adamski | Effects of metreleptin in patients with lipodystrophy with and without baseline concomitant medication use[END_REF][START_REF] Diker-Cohen | Partial and generalized lipodystrophy: comparison of baseline characteristics and response to metreleptin[END_REF][START_REF] Ajluni | Efficacy and Safety of Metreleptin in Patients with Partial Lipodystrophy: Lessons from an Expanded Access Program[END_REF][START_REF] Chong | Efficacy of leptin therapy in the different forms of human lipodystrophy[END_REF][START_REF] Park | Long-term efficacy of leptin replacement in patients with Dunnigan-type familial partial lipodystrophy[END_REF][START_REF] Oral | Long-term effectiveness and safety of metreleptin in the treatment of patients with partial lipodystrophy[END_REF].

Metreleptin is approved as an orphan drug in Japan and USA, and obtained a European Marketing Authorization (EMA) in 2018 for the treatment of metabolic complications associated with leptin deficiency in patients with lipodystrophy. Metreleptin treatment is authorized in Europe, as an adjunct to diet, in adults and children with GLD, from 2 years onwards, and in patients with PLD from 12 years onwards, for whom conventional treatments failed in achieving adequate metabolic control [START_REF]Authorization for Myalepta from the European Medicines Agency[END_REF]. The French National Health Authority (HAS) followed the favorable opinion of the EMA, but recommended a multi-disciplinary decision from the French Reference Center for Rare Diseases of Insulin Secretion and Insulin Sensitivity (PRISIS) network for the validation of the metreleptin therapy option [START_REF]Recommendations from the Transparence Committee of the French National Authority for Health (HAS) on Metreleptin Therapy[END_REF].

The aim of this observational study is to describe the characteristics of patients with lipodystrophy treated with metreleptin therapy in France, and to evaluate treatment efficacy, in a real-life setting.

Patients and Methods

Study Design

This multicenter retrospective observational cohort study included all patients with lipodystrophy who started metreleptin therapy in France between 2009 and 2020. Data collection was coordinated by the PRISIS National Reference Center, Endocrinology Department, Saint-Antoine Hospital, Paris. Patients' files were included in the CEMARA National Rare Disease Database (French data protection agency CNIL # 909474). The study followed the principles of the Declaration of Helsinki and all patients gave their informed consent for data collection.

Patients

Forty-seven patients with genetic or acquired, partial or generalized lipodystrophy, in the absence of Human Immunodeficiency Virus (HIV) infection, were included. Twenty-seven patients (14 with GLD and 13 with PLD) entered a compassionate program of metreleptin therapy approved by HAS from 2009 to 2017 [START_REF] Vatier | One-year metreleptin improves insulin secretion in patients with diabetes linked to genetic lipodystrophic syndromes[END_REF]. The 20 remaining patients (14 with GLD and 6 with PLD) received metreleptin therapy following EMA authorization in 2018. The proportion of patients with GLD or PLD did not significantly differ according to the time of metreleptin initiation, i.e. before or after 2018 (p=0.24).

Metreleptin was administered as a daily subcutaneous injection and dose adjustments were based on patients' response and tolerance as recommended [START_REF] Brown | The Diagnosis and Management of Lipodystrophy Syndromes: A Multi-Society Practice Guideline[END_REF].

Methods

Patients' records were reviewed using structured data collection forms. The following parameters were collected at baseline (before metreleptin treatment), after 12±3 months of metreleptin therapy (short-term response) and at the latest visit on metreleptin therapy, ≥15 months of treatment (longterm response):

-Sex, weight, Body Mass Index (BMI), ongoing treatments (lipid lowering and antidiabetic drugs), daily insulin dose and daily metreleptin dose -HbA1c, liver enzymes (aspartate aminotransferase (ASAT), alanine aminotransferase (ALAT), gammaglutamyl transferase (GGT), total, High Density Lipoprotein (HDL)-, and Low Density Lipoprotein (LDL)-cholesterol, fasting triglycerides, creatinine, and albuminuria At baseline, additional parameters were recorded including age at diagnosis of lipodystrophy (LD), subtype of LD (generalized GLD or partial PLD, genetic or acquired), gene pathogenic variant when applicable, presence of hypertension, dyslipidemia and/or diabetes, age at diagnosis of diabetes when applicable, and pretreatment serum leptin level, measured by ELISA (Quantikine, R&D Systems). Fat mass percentage was evaluated with Dual energy X-ray absorptiometry (DEXA).

Diabetes was defined by a glycated hemoglobin (HbA1c) ≥ 6.5 % or at least one antidiabetic treatment.

In adults, hypertension was defined by systolic blood pressure (SBP) ≥ 140 mmHg and/or diastolic blood pressure (DBP) ≥ 90 mmHg or at least one anti-hypertensive treatment. In children below the age of 16, hypertension was defined as SBP and/or DBP persistently superior to the 95 th percentile for sex, age and height measured on at least three separate occasions [START_REF]National High Blood Pressure Education Program Working Group on High Blood Pressure in Children and Adolescents. The fourth report on the diagnosis, evaluation, and treatment of high blood pressure in children and adolescents[END_REF]. Dyslipidemia was defined by serum fasting triglycerides ≥ 1.7 mmol/L, LDL-cholesterol ≥ 4.88 mmol/L, HDL-cholesterol ≤ 1.03 mmol/L (men) or ≤ 1.28 mmol/L (women) or at least one lipid-lowering therapy.

We also collected and reviewed serious adverse events that occurred under metreleptin treatment.

A favorable effect of metreleptin on glucose control was defined, in patients with diabetes at baseline, as a ≥ 0.5-point decrease in HbA1c, or HbA1c stability with a decrease of more than 50% in total daily insulin, or discontinuation of at least one antidiabetic class between baseline and short-term metreleptin therapy [START_REF] Vatier | One-year metreleptin improves insulin secretion in patients with diabetes linked to genetic lipodystrophic syndromes[END_REF]. A favorable effect of metreleptin on triglycerides was defined, in patients with serum fasting triglycerides ≥ 1.7 mmol/L at baseline, as serum triglyceride levels < 1 .7 mmol/L, or as a decrease of more than 30% in serum triglycerides between baseline and short-term metreleptin therapy [START_REF] Chan | Clinical effects of long-term metreleptin treatment in patients with lipodystrophy[END_REF][START_REF] Brown | Long-term effectiveness and safety of metreleptin in the treatment of patients with generalized lipodystrophy[END_REF][START_REF] Diker-Cohen | Partial and generalized lipodystrophy: comparison of baseline characteristics and response to metreleptin[END_REF][START_REF] Chong | Efficacy of leptin therapy in the different forms of human lipodystrophy[END_REF][START_REF] Beltrand | Resistance to leptinreplacement therapy in Berardinelli-Seip congenital lipodystrophy: an immunological origin[END_REF][START_REF]Long-Term Efficacy of Leptin Replacement in Treatment of Lipodystrophy[END_REF][START_REF] Meral | Endogenous Leptin Concentrations Poorly Predict Metreleptin Response in Patients with Partial Lipodystrophy[END_REF].

During follow-up, seven patients were switched from metreleptin to GLP-1R (Glucagon-like peptide-1 receptor) agonists. In these patients, we compared the metabolic results of the latest visit while on metreleptin therapy to those on GLP-1 analog therapy thereafter.

Statistical analysis

Results are reported as median and interquartile ranges [25 e percentile-75 e percentile] for quantitative variables, and as number and percentage for qualitative variables. In patients under the age of 18, weight-standard deviation scores are expressed as Z-scores. Comparisons between two groups were conducted by unpaired Mann-Whitney U-test, and between more than two groups by Wilcoxon signed-rank test. Chi-square test of independence was used for qualitative variables. Correlation analyses used Spearman non-parametric test. Descriptive and comparative statistical analyses were performed using GraphPad Prism (Windows version 9.0, GraphPad Software, San Diego, CA, USA). Two-sided P values ≤ 0.05 are considered statistically significant.

Results

Baseline characteristics

General characteristics of patients with LD initiating a metreleptin therapy in France

Patients' baseline characteristics are shown in Table 1. Among the forty-seven patients initiating a metreleptin therapy in France, 28 have GLD and 19 PLD. Women represent 75% of the whole cohort, and 95% of the PLD group. Patients with GLD are diagnosed with LD earlier, and are younger at metreleptin initiation, than patients with PLD.

Most patients were diagnosed with a genetic form of LD (83%). Genes and variants involved are provided in Supplemental Table 1, including AGPAT2 biallelic pathogenic variants in 8 patients (Congenital Generalized Lipodystrophy type 1), and LMNA p.Arg482 heterozygous substitutions in 12 patients (Dunnigan Familial Partial Lipodystrophy). Three patients present with autoimmune GLD. In five patients, the etiology of LD is unknown.

Metabolic parameters at baseline (Table 1)

As expected, leptin level and fat mass are correlated (r=0.64, p<0.001), and patients with GLD have lower leptin levels and lower fat mass than those with PLD. There is no significant difference between the GLD and PLD groups concerning the prevalence of diabetes, hypertension and dyslipidemia at metreleptin initiation. However, diabetes is diagnosed earlier in patients with GLD than in those with PLD (median age 13.0 vs 21.0 years, p<0.001). Seventy-nine percent of patients with diabetes are treated with metformin and 66% with insulin, with a high median daily insulin dose (median 2.0 U/kg/day), in keeping with LD-associated insulin resistance. Median HbA1c is 8.1 %, not different in the GLD and PLD groups (p=0.78). Most patients have increased albuminuria (median value 21.3 mg/L), with higher levels associated with GLD vs PLD (median 72.5 vs 11.5 mg/L respectively, p=0.05).

Transaminases are frequently high, especially ALAT in patients with GLD, without any significant differences between groups. A majority of patients are on lipid-lowering drugs at metreleptin initiation (60%), especially patients with PLD (95%). Serum fasting triglycerides are increased in both GLD and PLD groups (median 3.6 mmol/L) and HDL-cholesterol is low (median 0.8 mmol/L), but LDL cholesterol is not increased (median 2.3 mmol/L).

Response to metreleptin treatment

The median metreleptin treatment duration is 31.7 [IQR: 14.2-76.0] months. Patients with GLD are treated with a lower metreleptin dose (adjusted to weight) than patients with PLD (Supplemental Table 2). In five metreleptin-treated patients, all with GLD, the last metabolic profile has been measured before the one year-treatment data point: three patients chose to interrupt their treatment, considered as burdensome, after 2.5 (n=2) or 4.1 months. Data are missing for one patient after 6 months. In one patient, metreleptin was withdrawn after one month following an allergic reaction.

Since evolution of their HbA1c and triglyceride levels at the latest assessment under metreleptin therapy is not significantly different from those collected in patients with GLD after 12±3 months of metreleptin treatment, data from these five patients are included in the "short-term response" group.

We first analyzed the metreleptin response in all patients with LD (Supplemental Table 3). At shortterm under metreleptin therapy, LD patients loose weight, and improve fasting triglyceride and HbA1c levels while their insulin requirements decrease. These effects are maintained in the long term. Total cholesterol and liver enzymes also decrease rapidly following metreleptin therapy, but are no longer different from pretreatment levels on the long term. Finally, albuminuria does not significantly change at long-term versus baseline (median values 33.5 vs 21.3mg/L, mean values 354 vs 345 mg/L, p=0.94).

Response to metreleptin in patients with GLD

Metreleptin response is different in GLD patients as compared to PLD patients. Weight and BMI significantly decrease after short-term metreleptin therapy in adults and children with GLD.

Thereafter, BMI is not significantly modified in adult patients, whereas children showed a slight, but significant, improvement of weight Z score during long-term metreleptin therapy, which however remained significantly lower than baseline levels (Table 2).

Glucose homeostasis improves in GLD patients under metreleptin therapy. Indeed, median HbA1c significantly decreases from baseline to short-term (median 8.4 to 6.8%, p<0.0001), allowing five patients to stop insulin therapy, while others significantly decrease their daily insulin doses (Table 2, Figure 1A and1B). At long-term, we did not observe a significant rebound in the median HbA1c. Fasting triglycerides improve significantly, from a median value of 3.6 mmol/L at baseline to 2.2 and 1.9 mmol/L after short-term and long-term respectively, p<0.001 and <0.01 vs baseline (Table 2, Figure 1C and 1D). Total cholesterol follows a similar trajectory during metreleptin treatment, towards a significant decrease at short-term, with stable values therafter. The short-term effect of metreleptin on liver transaminases and albuminuria is not maintained in the long term. The use of lipid-lowering therapy and antidiabetics other than insulin is not significantly modified during the treatment period in patients with GLD (Table 2). Only one patient, with GLD, was able to discontinue its antidiabetic treatment (metformin) after 6 months on metreleptin, with HbA1c remaining <6% at the last followup (at 30 months of metreleptin treatment). Regarding dyslipidemia, only one patient, also with GLD, stopped its lipid-lowering therapy after 5 months of metreleptin, and maintained normal lipid levels after 40 months of metreleptin therapy. No patient discontinued any antihypertensive treatment.

Response to metreleptin in patients with PLD

Patients with PLD (n=19) also display a sustainable weight loss under metreleptin therapy (median BMI of 24.8 at baseline, 23.9 at short-term (p=0.03), then 23.5 at long-term (p=0.02 compared to baseline) (Table 3). However, in patients with PLD, median HbA1c values (7.7% All patients with PLD had diabetes at metreleptin initiation. Regarding glucose homeostasis, previously used criteria to define a favorable effect of metreleptin [START_REF] Vatier | One-year metreleptin improves insulin secretion in patients with diabetes linked to genetic lipodystrophic syndromes[END_REF] allow us to identify 11 patients (61%) as responders following short-term metreleptin treatment. Responders and non-responders do not differ in terms of age at treatment initiation, age at diagnosis of lipodystrophy and age at diagnosis of diabetes, nor in daily insulin dose, fat mass percentage or changes in weight/BMI upon metreleptin therapy. However, responders have lower leptin levels compared to non-responders (median [IQR]: 3.9 [3.3-4.4] vs 8.0 [3.4-8.7], p=0.05). Among patients with PLD treated with insulin (n=11), only 4 (36%) are responders when evaluated in the short-term, whereas the proportion of responders is 100% in patients with PLD-associated diabetes not treated with insulin therapy (n=7) (p=0.01). Regarding the effect of metreleptin therapy on triglycerides levels, 11 patients with PLD (61%) can be defined as responders (Supplemental Figure 1). Comparison of the responder and non-responder groups do not reveal any differences in any of the factors studied above.

When combining the study of both glucose and triglycerides levels, only seven patients with PLD (39%) are classified as responders to metreleptin (Supplemental Figure 1).

Diker-Cohen et al have previously proposed that baseline HbA1c > 8%, triglycerides > 5.6 mmol/L and serum leptin < 4 ng/mL could be predictive factors for improvement of glucose and triglycerides under metreleptin in patients with PLD [START_REF] Diker-Cohen | Partial and generalized lipodystrophy: comparison of baseline characteristics and response to metreleptin[END_REF]. Nine patients with PLD met those criteria, including four patients with FPLD3 due to PPARG pathogenic variants, three patients with FPLD2 due to p.(Arg482) LMNA variant, one patient with FPLD4 due to PLIN1 pathogenic variant, and one patient with an unknown genetic cause. Among them, the patient with FPLD4 is the only one that does not respond to metreleptin treatment, neither regarding glucose homeostasis nor triglycerides levels. All the other patients are classified as responders to metreleptin for either glucose and/or triglyceride control.

Adverse events

Four patients with LD died during metreleptin therapy, from cardiovascular events (n=2, cardiomyopathy with heart failure or stroke), hepatic insufficiency (n=1), or metastatic pancreatic cancer (n=1) (Supplemental Table 2). One patient, with an autoimmune form of GLD, developed neutralizing anti-metreleptin autoantibodies and concomitantly worsened her metabolic parameters after 45 months of treatment. One patient developed skin allergy to metreleptin leading to treatment interruption after one month of treatment.

Metabolic changes in patients switched from metreleptin to GLP-1R agonists therapy during followup

Seven lipodystrophy patients (3 with GLD and 4 with PLD) were switched from metreleptin to GLP-1R agonist therapy. They received metreleptin for 77 months (median [interquartile range]), then GLP-1 analogs, i.e. dulaglutide for five patients, and liraglutide for two patients, for 20 [START_REF] Colombo | Transplantation of adipose tissue lacking leptin is unable to reverse the metabolic abnormalities associated with lipoatrophy[END_REF][START_REF] Chevalier | Metreleptin treatment of non-HIV lipodystrophy syndromes[END_REF][START_REF] Chan | Clinical effects of long-term metreleptin treatment in patients with lipodystrophy[END_REF][START_REF] Brown | Long-term effectiveness and safety of metreleptin in the treatment of patients with generalized lipodystrophy[END_REF][START_REF] Araujo-Vilar | Recombinant human leptin treatment in genetic lipodystrophic syndromes: the long-term Spanish experience[END_REF][START_REF] Petersen | Leptin reverses insulin resistance and hepatic steatosis in patients with severe lipodystrophy[END_REF][START_REF] Oral | Leptin-replacement therapy for lipodystrophy[END_REF][START_REF] Adamski | Effects of metreleptin in patients with lipodystrophy with and without baseline concomitant medication use[END_REF][START_REF] Cook | Effects of Metreleptin on Patient Outcomes and Quality of Life in Generalized and Partial Lipodystrophy[END_REF][START_REF] Vatier | Adherence with metreleptin therapy and health self-perception in patients with lipodystrophic syndromes[END_REF][START_REF] Simsir | Metreleptin replacement treatment improves quality of life and psychological well-being in congenital generalized lipodystrophy[END_REF][START_REF] Diker-Cohen | Partial and generalized lipodystrophy: comparison of baseline characteristics and response to metreleptin[END_REF][START_REF] Ajluni | Efficacy and Safety of Metreleptin in Patients with Partial Lipodystrophy: Lessons from an Expanded Access Program[END_REF][START_REF] Chong | Efficacy of leptin therapy in the different forms of human lipodystrophy[END_REF][START_REF] Park | Long-term efficacy of leptin replacement in patients with Dunnigan-type familial partial lipodystrophy[END_REF][START_REF] Oral | Long-term effectiveness and safety of metreleptin in the treatment of patients with partial lipodystrophy[END_REF][START_REF]Authorization for Myalepta from the European Medicines Agency[END_REF][START_REF]Recommendations from the Transparence Committee of the French National Authority for Health (HAS) on Metreleptin Therapy[END_REF] months.

GLP-1 analogs were used with antidiabetic doses (1.5mg/week for dulaglutide and 1.8 mg/day for liraglutide). We do not identify any significant difference between BMI (p=0.44), HbA1c (p=0.30), fasting triglycerides (p=0.81) and albuminuria (p=0.16) after metreleptin or GLP-1R agonist therapy in those patients (Supplemental Table 4). However, considering only the three patients with GLD, all were treated only transiently with GLP-1 analogs, for 3, 6 and 20 months, and all were restarted with metreleptin thereafter. Indeed, HbA1c increases in all of them (by 0.8, 0.

Discussion

This study included the largest cohort of patients with LD initiating a metreleptin therapy in a real-life setting. It provides an overview of i) characteristics of patients with lipodystrophy treated with metreleptin therapy in France ii) the metabolic effects of short and long-term metreleptin therapy in patients with GLD or PLD.

Importantly, it confirms that metreleptin treatment is efficient, in a sustainable manner, in reducing hyperglycemia and hypertriglyceridemia in patients with GLD. Conversely, metabolic efficacy of metreleptin in patients with PLD was highly variable, and did not reach statistical significance when considering the whole PLD group.

Lipodystrophy syndromes are rare diseases whose metabolic complications are difficult to treat due to the severe insulin resistance linked to adipose tissue failure. Although the orphan drug metreleptin does not aim to replace adipose tissue, it was shown to alleviate ectopic lipid storage by central and peripheral effects improving satiety and insulin sensitivity [START_REF] Petersen | Leptin reverses insulin resistance and hepatic steatosis in patients with severe lipodystrophy[END_REF][START_REF] Aotani | Functional magnetic resonance imaging analysis of food-related brain activity in patients with lipodystrophy undergoing leptin replacement therapy[END_REF][START_REF] Grover | Leptin Decreases Energy Expenditure Despite Increased Thyroid Hormone in Patients With Lipodystrophy[END_REF][START_REF] Brown | Metreleptinmediated improvements in insulin sensitivity are independent of food intake in humans with lipodystrophy[END_REF][START_REF] Schlögl | Leptin Substitution in Patients With Lipodystrophy: Neural Correlates for Long-term Success in the Normalization of Eating Behavior[END_REF]. Following the international guidelines published in 2016 [START_REF] Brown | The Diagnosis and Management of Lipodystrophy Syndromes: A Multi-Society Practice Guideline[END_REF], and the recommendations of the French National Health Authority [START_REF]Recommendations from the Transparence Committee of the French National Authority for Health (HAS) on Metreleptin Therapy[END_REF], serum leptin<4 ng/mL, HbA1c>8% and/or triglycerides> 5.6 mmol/L are widely used prerequisites for metreleptin prescription in France [START_REF] Brown | The Diagnosis and Management of Lipodystrophy Syndromes: A Multi-Society Practice Guideline[END_REF][START_REF] Diker-Cohen | Partial and generalized lipodystrophy: comparison of baseline characteristics and response to metreleptin[END_REF]. However, metreleptin was previously used in France through compassionate programs before marketing authorization, which could explain why patients with PLD had lower median values of HbA1c (7.7% [7.1-9.1]) and triglycerides (3.3 mmol/L [1.9-9.9]) at metreleptin initiation.

Not surprisingly, in comparison to patients with PLD, whose disease manifests predominantly during late childhood or puberty, patients with GLD, mainly affected by congenital forms of the disease, were younger at metreleptin initiation. However, both groups had comparable metabolic markers at baseline, even though patients with PLD had higher BMI, fat mass and endogenous leptin levels than those with GLD.

Metreleptin was previously reported to be less efficient in patients with PLD compared to those with GLD [START_REF] Araujo-Vilar | Recombinant human leptin treatment in genetic lipodystrophic syndromes: the long-term Spanish experience[END_REF][START_REF] Adamski | Effects of metreleptin in patients with lipodystrophy with and without baseline concomitant medication use[END_REF][START_REF] Diker-Cohen | Partial and generalized lipodystrophy: comparison of baseline characteristics and response to metreleptin[END_REF][START_REF] Lee | Effects of Metreleptin on Proteinuria in Patients With Lipodystrophy[END_REF]. Herein, in patients with PLD, BMI and fasting triglycerides, but not HbA1c, albuminuria or liver enzymes, significantly improved at short-term under metreleptin therapy, in line with previous observations [START_REF] Ajluni | Efficacy and Safety of Metreleptin in Patients with Partial Lipodystrophy: Lessons from an Expanded Access Program[END_REF][START_REF] Park | Long-term efficacy of leptin replacement in patients with Dunnigan-type familial partial lipodystrophy[END_REF][START_REF] Lee | Effects of Metreleptin on Proteinuria in Patients With Lipodystrophy[END_REF][START_REF] Simha | Comparison of efficacy and safety of leptin replacement therapy in moderately and severely hypoleptinemic patients with familial partial lipodystrophy of the Dunnigan variety[END_REF]. The decrease in triglycerides was in the same range than that recently reported in 36 patients with PLD included in clinical trials (-28.7%) [START_REF] Adamski | Effects of metreleptin in patients with lipodystrophy with and without baseline concomitant medication use[END_REF]. In that same study, HbA1c decreased by a mean -0.61% in patients with PLD, compared to -2.16% in patients with GLD (n=59). In other studies, HbA1c did not improve during metreleptin therapy in patients with PLD [START_REF] Park | Long-term efficacy of leptin replacement in patients with Dunnigan-type familial partial lipodystrophy[END_REF][START_REF] Simha | Comparison of efficacy and safety of leptin replacement therapy in moderately and severely hypoleptinemic patients with familial partial lipodystrophy of the Dunnigan variety[END_REF]. Heterogeneous causes of PLD, and small numbers of patients included, might explain these discrepancies. Moreover, in patients with chronic metabolic complications, treatments are generally less effective in real-life studies than in clinical trials [START_REF] Araujo-Vilar | Recombinant human leptin treatment in genetic lipodystrophic syndromes: the long-term Spanish experience[END_REF][START_REF] Thompson | Replication of Randomized, Controlled Trials Using Real-World Data: What Could Go Wrong? Value[END_REF]. Interestingly, PLD patients with inadequate metabolic control (diabetes and/or triglycerides) were mostly responders to metreleptin treatment, as previously reported [START_REF] Diker-Cohen | Partial and generalized lipodystrophy: comparison of baseline characteristics and response to metreleptin[END_REF][START_REF] Lambadiari | Case Report: Metreleptin Treatment in a Patient With a Novel Mutation for Familial Partial Lipodystrophy Type 3, Presenting With Uncontrolled Diabetes and Insulin Resistance[END_REF][START_REF] Melzer | Rapid response to leptin therapy in a FPLD patient with a novel PPARG missense variant[END_REF].

In our study, insulin treatment at baseline was predictive of a poorer response to metreleptin therapy on HbA1c in patients with PLD. In line, in clinical trials including patients with PLD, insulin therapy at baseline was associated with a smaller decrease in HbA1c at one-year metreleptin therapy, although HbA1c was higher at metreleptin initiation [START_REF] Adamski | Effects of metreleptin in patients with lipodystrophy with and without baseline concomitant medication use[END_REF]. However, in patients with GLD, responders to metreleptin were more frequently insulin users [START_REF] Brown | Long-term effectiveness and safety of metreleptin in the treatment of patients with generalized lipodystrophy[END_REF]. As proposed by Adamski et al [START_REF] Adamski | Effects of metreleptin in patients with lipodystrophy with and without baseline concomitant medication use[END_REF], insulin use may be an indicator of beta-cell failure in PLD, whereas it may be an indicator of more severe insulin resistance in GLD. A hypothesis could be that metreleptin might improve HbA1c in patients with preserved beta cell function only [START_REF] Adamski | Effects of metreleptin in patients with lipodystrophy with and without baseline concomitant medication use[END_REF]. Besides, patients with PLD were older than patients with GLD, which can be associated with more beta-cell dysfunction.

In our study, patients with PLD showing a favorable response to metreleptin on glucose homeostasis had lower serum leptin levels than non-responders. This was reported in some, but not all previous studies [START_REF] Diker-Cohen | Partial and generalized lipodystrophy: comparison of baseline characteristics and response to metreleptin[END_REF][START_REF] Simha | Comparison of efficacy and safety of leptin replacement therapy in moderately and severely hypoleptinemic patients with familial partial lipodystrophy of the Dunnigan variety[END_REF]. The endogenous serum leptin cut-off point to predict metreleptin response in patients with LD is still debated, partly due to the different sensitivity of the leptin assays currently available [START_REF] Meral | Endogenous Leptin Concentrations Poorly Predict Metreleptin Response in Patients with Partial Lipodystrophy[END_REF]. Nevertheless, the baseline serum leptin level, assessed by the same assay, could be an important criterion for the indication of metreleptin therapy, as observed here. We failed to identify other predictive factors for metreleptin efficacy. Larger effectives of patients, as reached through international rare diseases registries (38) could help to understand the role of factors such as LD etiologies, genotypes involved, or specificities of body composition, on metreleptin response.

Interestingly, patients with GLD had a better response to metreleptin than those with PLD, whereas their metreleptin doses, corrected for weight, were lower. As patients with GLD had lower baseline leptin levels, we could have expected higher doses of metreleptin to get a favorable response. The need to increase the metreleptin dose in patients with PLD is an indirect sign of lower efficacy, which might be related to a state of relative "leptin-resistance".

Adverse events were rare during metreleptin therapy. As previously described, we report the development of anti-metreleptin autoantibodies in a patient with auto-immune GLD, leading to secondary inefficacy of metreleptin, and allergy to metreleptin in one patient [START_REF] Brown | The Diagnosis and Management of Lipodystrophy Syndromes: A Multi-Society Practice Guideline[END_REF][START_REF] Sollier | Lipodystrophic syndromes: From diagnosis to treatment[END_REF]. Deaths observed during follow-up were not considered to be related to metreleptin treatment, but rather to disease progression and severity. Three patients stopped metreleptin because of excessive constraints. This lack of compliance may result, at least in part, from the need to reconstitute the product from powder extemporaneously and to perform daily subcutaneous injections without pre-filled devices [START_REF] Vatier | Adherence with metreleptin therapy and health self-perception in patients with lipodystrophic syndromes[END_REF]. Seven patients were switched to GLP-1R agonists after cessation of metreleptin therapy. Due to the effects of GLP1-R agonists on fat mass and insulin resistance [START_REF] Knudsen | The Discovery and Development of Liraglutide and Semaglutide[END_REF][START_REF] Nauck | GLP-1 receptor agonists in the treatment of type 2 diabetes -state-of-the-art[END_REF], metabolic improvements of diabetes, serum triglycerides and/or hepatic steatosis, could have been expected, as reported in a few case reports [START_REF] Banning | Insulin secretory defect in familial partial lipodystrophy Type 2 and successful long-term treatment with a glucagon-like peptide 1 receptor agonist[END_REF][START_REF] Oliveira | Glucagon-like peptide-1 analogues -an efficient therapeutic option for the severe insulin resistance of lipodystrophic syndromes: two case reports[END_REF]. Nevertheless, we did not observe any additional metabolic benefit of GLP-1R agonists following metreleptin therapy in a real-life setting. Importantly, metreleptin was restarted in all three patients with GLD who were switched on GLP-1R agonists, due to deterioration of glucose and triglycerides values. These data have to be considered with caution in this very low number of patients, who were not controlled for compliance. It will certainly be useful to compare metreleptin and GLP-1R agonist treatments in a larger cohort of patients. Similarly, efficacy of SGLT2-inhibitors (iSGLT2) on metabolic complications associated with LD has been described in two case reports only [START_REF] Nagayama | Case Report: Metreleptin and SGLT2 Inhibitor Combination Therapy Is Effective for Acquired Incomplete Lipodystrophy[END_REF]45), and more data are needed regarding the use of this therapeutic drug class in lipodystrophy syndromes.

Our study has some limitations, such as its retrospective setting. As it was conducted during an extended period (2009-2020) over which antidiabetic drugs have evolved, this could have biased some outcomes. However, in France, no new antidiabetic drugs were marketed between 2009 and 2020 (iSGLT2 were approved in April 2020). We did not have access to data about treatment's compliance, which represents a potentially confounding factor. Heterogeneity in the medical management of patients could have occurred, but this risk should be minor due to the close interactions of clinical centers within our National Rare Disease Reference network.

The results of our observational study point out that metreleptin treatment is efficient in patients with GLD. However, as highlighted by published guidelines [START_REF] Brown | The Diagnosis and Management of Lipodystrophy Syndromes: A Multi-Society Practice Guideline[END_REF][START_REF]Recommendations from the Transparence Committee of the French National Authority for Health (HAS) on Metreleptin Therapy[END_REF], it should be given only to selected patients with PLD. Our study suggests that leptin levels, measured with the same assay, should be taken into account for this therapeutic decision. Metreleptin treatment should be regularly reevaluated and stopped if inefficient. Patients with LD also may require stronger educational support.

Studies on larger effectives and double-blind randomized controlled trials are needed to better evaluate the metabolic efficacy of metreleptin treatment in PLD, and to define predictive factors of response.

1 )

 1 [7.1-9.1]) do not significantly improve under metreleptin therapy, neither in the short-term (7.7 [7.4-9.5]), nor in the long-term (8.0 [7.6-8.5]) (Table 3, Figure 2A and 2B). Insulin therapy and other antidiabetic treatments are not significantly modified. No patient discontinued antidiabetic treatment during metreleptin therapy. Serum fasting triglycerides significantly decrease after short-term metreleptin treatment (from 3.3 [1.9-9.9] mmol/L to 2.5[1.6-5.3], p<0.01) but return to baseline levels afterwards (5.2 [2.2-11.3] mmol/L, p=0.94 compared to baseline) (Table 3, Figure 2C and 2D). A large majority of patients with PLD is treated with lipid-lowering drugs during metreleptin therapy. No patient discontinued lipidlowering or antihypertensive treatments during metreleptin therapy. We do not observe any significant changes in liver enzymes nor in albuminuria values during metreleptin therapy. Predictive factors of metreleptin therapy efficacy in PLD patients (Supplemental Figure Since patients with PLD demonstrate a highly variable metabolic response to metreleptin, we sought to determine which factor(s) could predict treatment efficacy. Values regarding glucose and triglycerides control are available for 18 PLD patients (follow-up data missing for one patient).

  5 and 0.1 %, respectively), and triglycerides increase in 2 out of 3 patients during GLP-1 analog treatment. Concerning the four patients with PLD, they were treated with metreleptin between 18.5 and 85.0 months, then switched to GLP-1 analogs, and none of them restarted metreleptin thereafter. Their median HbA1c values were 7.7% on metreleptin then 7.9% on GLP-1 analogs. Their median values of serum triglycerides were 6.3 mmol/L on metreleptin then 2.2 mmol/L on GLP-1 analogs.

  re expressed a s median [25% percentile-75% percentile] for quantitative va riables a nd a s number (n) a nd percentage (%) for qualitative va riables. Results a re from all patients unless s pecified (n). Diabetes is defined by HbA1c ≥ 6.5 % or a ntidiabetic treatment. Hypertension is defined by s ystolic blood pressure ≥ 140 mmHg and/or diastolic blood pressure ≥ 90 mmHg or antihypertensive treatment. Dyslipidemia i s defined by s erum tri glycerides ≥ 1.7 mmol /L, LDL cholesterol ≥ 4.88 mmol /L, HDL cholesterol ≤ 1.03 mmol/L (men) or ≤ 1.28 mmol /L (women) or lipidl owering therapy. N: normal values. p va lues are obtained from Wilcoxon tests for quantitative va riables a nd from chi-square tests for qualitative variables.

  re expressed a s median [25% percentile-75% percentile] for quantitative va riables a nd a s number (n) a nd percentage (%) for qualitative va riables. Results a re from all patients unless s pecified (n). Diabetes is defined by HbA1c ≥ 6.5 % or a ntidiabetic treatment. Hypertension is defined by s ystolic blood pressure ≥ 140 mmHg and/or diastolic blood pressure ≥ 90 mmHg or antihypertensive treatment. Dyslipidemia is defined by s erum tri glycerides ≥ 1.7 mmol /L, LDL cholesterol ≥ 4.88 mmol /L, HDL cholesterol ≤ 1.03 mmol /L (men) or ≤ 1.28 mmol /L (women) or lipidl owering therapy. N: normal values. p va lues are obtained from Wilcoxon tests for quantitative va riables a nd from chi-square tests for qualitative variables.

Table 2 : Anthropometric and metabolic parameters during metreleptin therapy in patients with generalized lipodystrophy (n=28)

 2

		At baseline	Short-term	Long-term	p baseline	p baseline	p short-
		(before metreleptin initiation)	response	response	vs short-	vs long-	term vs
		(n=28)	(n=28)	(n=20)	term	term	long-term
					response	response	response
	Anthropometrics
	Weight (Zscore)
	< 18 yea rs (n=15)	0.8 [-0.4;1.1]	0.4 [-1.7;1.2]	0.5 [-1.2;1.1] (n=9)	<0.01	0.02	0.03
	Body Mas s Index (kg/m 2 )
	> 18 yea rs (n=13)	21.1 [19.3-23.1]	19.1 [18.5-22.2]	19.0 [18.0-22.2] (n=8)	<0.01	0.05	0.78
	Glucose homeostasis
	HbA1c (%)	8.4 [6.5-9.9]	6.8 [5.6-7.4] (n=23)	6.9 [5.5-8.7]	<0.0001	<0.01	0.33
	Pa ti ents on i nsulin: n (% of patients with diabetes)	17/25 (68%)	12/25 (46%)	10/20 (50%)	0.41	0.56	>0.99
	Da i ly i nsulin dose according to weight (IU/kg)	1.3 [0.8-3.5] (n=16)	1.0 [0.4-2.4] (n=11)	1.1 [0.5-3.7] (n=9)	<0.01	0.69	0.50
	Number of antidiabetic therapeutic classes used	1.0 [0.0-1.0]	1.0 [0.0-1.0]	1.0 [1.0-1.0]	>0.99	0.34	0.18
	(except i nsulin)
	Lipids
	Serum tri glycerides (mmol/L)	3.6 [1.7-8.5]	2.2 [1.1-3.7] (n=23)	1.9 [1.3-3.3] (n=19)	<0.001	<0.01	0.57
	Tota l cholesterol (mmol/L)	4.6 [3.9-4.9]	3.5 [3.6-4.3] (n=23)	3.8 [3.5-4.5] (n=15)	0.02	<0.01	0.68
	LDL-chol esterol (mmol/L)	2.3 [1.5-3.1] (n=17)	2.2 [1.9-2.4] (n=18)	2.2 [1.7-2.6] (n=18)	0.80	0.68	0.71
	HDL-cholesterol (mmol/L)	0.8 [0.5-1.0]	0.8 [0.6-0.9] (n=23)	0.8 [0.7-1.1] (n=17)	0.47	0.06	0.95
	Pa ti ents on lipid-lowering drugs: n (%)	10/28 (36%)	8/20 (44%)	7/17 (41%)	0.77	0.76	>0.99
	Liver enzymes
	ASAT (IU/L) (N 6-35)	31.0 [24.5-49.5] (n=25)	27.0 [22.0-33.0] (n=19)	32.5 [24.7-65.7] (n=18)	<0.01	0.12	0.23
	ALAT (IU/L) (N 8-43)	57.0 [24.0-81.0] (n=27)	27.0 [20.0-57.0] (n=19)	51.5 [23.5-82.5] (n=18)	<0.01	0.26	0.44
	GGT (IU/L) (N 6-45)	42.5 [27.5-65.8] (n=24)	32.5 [17.2-62.0] (n=16)	42.0 [22.0-65.7]	0.15	0.29	0.11

Table 3 : Anthropometric and metabolic parameters during metreleptin therapy in patients with partial lipodystrophy (n=19) At baseline (before metreleptin initiation) (n=19) Short-term response (n=19) Long-term response (n=16) p baseline vs short term response p baseline vs long term response p short term vs long term response Anthropometrics

 3

	Body Mas s Index (kg/m 2 ) > 18 yea rs	24.8 [23.1-26.0]	23.9 [22.1-25.7] (n=17)	23.5 [21.5-25.9]	0.03	0.02	0.37
	Glucose homeostasis
	HbA1c (%)	7.7 [7.1-9.1]	7.7 [7.4-9.5] (n=18)	8.0 [7.6-8.5]	0.45	0.73	0.29
	Pa ti ents treated with i nsulin: n (% of patients	12/19 (63%)	12/19 (63%)	11/16 (48%)	>0.99	>0.99	>0.99
	wi th diabetes)
	Da i ly i nsulin dose according to weight (IU/kg)	2.7 [1.1-3.7] (n=12)	1.0 [0.8-3.1] (n=9)	1.8 [0.8-4.0] (n=11)	0.13	0.38	0.38
	Number of antidiabetic therapeutic classes used	2.0 [1.0-2.0]	1.0 [1.0-2.0]	1.5 [1.0-2.7]	0.12	>0.99	0.19
	(except i nsulin)
	Lipids
	Serum tri glycerides (mmol/L)	3.3 [1.9-9.9]	2.5 [1.6-5.3]	5.2 [2.2-11.3]	<0.01	0.94	0.08
	Tota l cholesterol (mmol/L)	4.6 [3.9-6.5] (n=18)	3.8 [3.4-5.5] (n=14)	4.9 [3.9-5.5] (n=12)	0.51	0.85	0.30
	LDL-chol esterol (mmol/L)	2.3 [1.8-2.8] (n=12)	2.0 [1.6-2.6] (n=11)	1.9 [1.5-3.8] (n=10)	0.76	>0.99	0.03
	HDL-cholesterol (mmol/L)	0.8 [0.5-0.8] (n=18)	0.9 [0.8-0.9] (n=14)	0.8 [0.4-1.0] (n=15)	0.05	0.80	>0.99
	Pa ti ents treated with lipid-lowering drugs: n (%)	18/19 (95%)	13/14 (93%)	14/14 (100%)	>0.99	>0.99	>0.99

Acknowledgements

We thank the patients who participated in this study, the nurses and all members of PRISIS network, and Amylin/Bristol-Myers Squibb/AstraZeneca and Aegerion Pharmaceuticals for generously providing metreleptin during the French compassionate program of metreleptin therapy. We thank Prof. Jean-Philippe Bastard and Dr Soraya Fellahi, AP-HP, Henri-Mondor Hospital, Department of Biochemistry-Pharmacology-Molecular Biology, Paris Est Créteil University, France, for leptin measurements, and Dr Alice Guilleux, La Réunion University Hospital, Clinical Investigation Center and Department of Clinical Epidemiology, Inserm U1410, France, for statistical support. This work was supported by the French Ministry of Solidarity and Health, Assistance-Publique Hôpitaux de Paris and Sorbonne University, France.

Conflicts of interest

Camille Vatier, Corinne Vigouroux and Marie-Christine Vantyghem report meeting fees from Aegerion Pharmaceuticals. Héléna Mosbah reports educational fees from Amryt.

Legends to figures :


Alexandre Veremme

Daniel Dupont

Éric Lefevre

David Mercier

Belief assignment on compound hypotheses within the framework of the Transferable Belief Model

Keywords: mass function construction, compound hypotheses, belief theory, evidence theory, Jenks' method, information criteria, forensic entomology

Within the framework of evidence theory, the construction of mass functions remains an important problem which can considerably influence final results. Several propositions of construction are available in literature but generally, these methods do not consider the belief masses on compound hypotheses and do not exploit all the strengths of the theory. In this article, we propose two methods of grouping hypotheses allowing to reduce the number of focal elements during the construction of mass functions. The first one is based on a well-known approach in cartography and geoinformation fields: Jenks' method. The second one is based on information criteria. We present and confront both methods. Results in a forensic entomology application are then presented.

Introduction

Representing and managing knowledge in complex systems, where quantity and disparity of information are important, generally requires the system designer to set up a particular formalism of information processing. The goal is to better reason with imperfect information (imprecision, uncertainty, incompleteness etc.) and to handle these imperfections to make a final decision. Among the existing theories, the belief function theory [START_REF] Shafer | A Mathematical Theory of Evidence[END_REF] could be well adapted to take into account these different characteristics.

The transferable belief model (TBM) of Philippe Smets [START_REF] Smets | The transferable belief model[END_REF] can be seen as a subjective and non probabilistic interpretation of evidence theory. It is a flexible and efficient tool to be applied in many areas such as decision support, diagnosis etc. The TBM is all the more interesting as it has a range of tools to handle belief functions (discounting, conditioning operations etc.). Despite the importance of the model foundations, a significant problem remains: the construction (or estimation) of basic belief assignments.

Construction is a difficult step that is not universal and which generally depends on the application and the vision we have on it. We usually distinguish the following methods of estimation: those based on the calculations of distances [START_REF] Schubert | On nonspecific evidence[END_REF][START_REF] Denoeux | A k-nearest neighbour classification rule based on dempster-shafer theory[END_REF] and those based on likelihood inspirations [START_REF] Appriou | Multisensor signal processing in the framework of the theory of evidence[END_REF][START_REF] Smets | Belief functions: The disjonctive rule of combination and the generalized bayesian theorem[END_REF]. However, these two types of estimation present various inconveniences: the first one can supply very engaged mass functions but a solution can always be obtained even if the frame of discernment is large. The second method respects more the Minimal Commitment Principle and seems better to make a careful decision. But we are then confronted with a computing problem when the frame of discernment increases. This last method becomes thus unusable because of the number of focal elements, close to two to the power of the cardinality of the frame of discernment. Different works deal with the assignment of belief masses on the disjunctions (compound hypotheses) to better use the power of evidence theory and so to reduce the space complexity. Affectations on the disjunctions can be classified in two axes: those where a knowledge on the hypotheses is a priori known [START_REF] Bloch | Some aspects of dempster-shafer evidence theory for classification of multi-modality medical images taking partial volume effect into account[END_REF] and those consisting in a learning of the compound focal elements [START_REF] Milisavljevic | A two-level approach for modeling and fusion of humanitarian mine detection sensors within the belief function framework[END_REF]. More recently, [START_REF] Bloch | Defining belief functions using mathematical morphology application to image fusion under imprecision[END_REF] also introduces imprecision on mass functions by using dual fuzzy morphological operators. Interested readers can find a good range of methods of estimation in [START_REF] Bloch | Fusion en traitement d'images : spécificités et approches numériques[END_REF].

Within the framework of our works, consisting in realizing a decision support system intended for forensic medicine, we are confronted with this problem of construction. According to experts' observations, we try to determine the time of death of a person with a possible large frame of discernment (several hundred hours). In this paper, we present two methods allowing to reduce the number of focal elements and to affect our belief masses on disjunctions.

After a display of the transferable belief model (section 2), we propose two methods allowing to estimate the belief masses on disjunctions of hypotheses (section 3). The first one is based on a well-known method in cartography and geoinformation fields: Jenks' natural breaks classification [START_REF] Slocum | Thematic Cartography and Geographic Visualization[END_REF]. The second one is based on an approach of information criteria and originally comes from the study of histograms [START_REF] Colot | Apprentissage et détection automatique de changements de modèles -Application aux signaux électro-encéphalographiques[END_REF]. We are going to detail these two methods of estimation, present and compare their advantages and inconveniences in the framework of belief functions. We apply them to our forensic medicine problem and end on the interests of such methods in the framework of the TBM.

Transferable belief model

The evidence theory was initially introduced by Dempster [START_REF] Dempster | Upper and lower probabilities induced by multivalued mapping[END_REF] in relation to his work on lower and upper bounds of a distribution probability family. Using this mathematical formalism, Shafer [START_REF] Shafer | A Mathematical Theory of Evidence[END_REF] shows the benefits of belief functions to model uncertain knowledge. The usefulness of belief functions, as an alternative to subjective probabilities, was subsequently proved in an axiomatic way by Smets [START_REF] Smets | The transferable belief model[END_REF] in the transferable belief model thus giving a clear and coherent interpretation of the subjacent concept of this theory. In this section, the essential mathematical elements of the TBM will be introduced.

Basic concepts

The transferable belief model is based on the assumption that, from the beginning, a set Ω called the frame of discernment is known and which is defined as follows:

Ω = {ω 1 , ω 2 , . . . ω N } (1)
This set is composed of N exclusive hypotheses. From this frame of discernment, a power set noted 2 Ω can be built, including the 2 N proposals A of Ω:

2 Ω = {A|A ⊆ Ω} (2)
A belief function can be mathematically defined by a mass function (or allocation 1 ), noted m defined by 2 Ω in [0, 1], and that verifies:

A⊆Ω m(A) = 1 (3)
Each subset A ⊆ Ω such that m(A) = 0 is called a focal element of Ω. Thus, mass m(A) represents the degree of belief allocated to the proposal A and that cannot, in the present state of knowledge, be attributed to a more specific subset than A. The belief function for which m(∅) = 0 is called normal. In the transferable belief model, the condition ∅ =A⊆Ω m(A) = 1 is not imposed and it is possible to have m(∅) = 0. This can introduce the notion of open world while assuming that the belief cannot be attributed to a subset of Ω. In this case, ∅ can be interpreted as a proposal which is not in the frame of discernment Ω and that it is likely to be the solution to the problem as opposed to the closed world where the set Ω is assumed to be exhaustive. 1 The term bba for basic belief assignment is often found in literature

Discounting

When the information that results in the belief function is not totally reliable, it may be useful to discount this belief. In order to do that, a coefficient α is used, which represents knowledge of the source reliability. This coefficient will allow the transfer of the belief to the set Ω when the information is not totally reliable. The discounting belief function m α , defined by a reliability coefficient α can then be deduced from m by means of the following expression:

m α (A) = αm(A) m α (Ω) = 1 -α + αm(Ω). (4)
In literature several methods have been developed to compute the discounting factor [START_REF] Guo | Evaluating sensor reliability in classification problems based on Evidence Theory[END_REF][START_REF] Mercier | Refined modeling of sensor reliability in the belief function framework using contextuel discounting[END_REF].

Fusion of belief functions

In the transferable belief model, data from distinct sources are fused using the conjunctive rule of combination. This rule, which is commutative and associative, is defined by:

∀ A ∈ 2 Ω m(A) = m 1 (A) ∩ . . . ∩ m Q (A) (5)
where ∩ represents the combination operator. In cases of two sources noted S i and S j , giving respectively belief functions noted m i et m j , the combination can be written as follows:

m(C) = A∩B=C m i (A).m j (B) (6)
The value m(∅) = m 1 ∩ m 2 (∅) reflects the existing conflict between two sources S i and S j . When this factor equals 1, the sources are in total conflict. Various works deal anyway with the problem of the appearance of conflict with this combination rule [START_REF] Smets | Analyzing the combination of conflicting belief functions[END_REF]. In a more general way, within the framework of the evidence theory, different combination rules, not defined in the TBM, were proposed. Interested readers could refer to [START_REF] Dubois | A set-theoric view of belief functions: Logical operations and approximations by fuzzy sets[END_REF][START_REF] Florea | Robust combination rules for evidence theory[END_REF][START_REF] Lefevre | Belief function combination and conflict management[END_REF] for more details about some of these rules.

Pignistic transformation

Within the framework of the TBM, Smets [START_REF] Smets | The transferable belief model[END_REF] recommends to construct a probability function in order to take an optimal decision allowing to make decision on individual hypotheses. The transformation from a mass function to a probability function is achieved by the pignistic transformation and is given by:

∀ ω n ∈ Ω BetP (ω n ) = 1 1 -m(∅) A ωn m(A) |A| (7)
where |A| represents the cardinality of A ⊆ Ω.

Once the pignistic probability obtained, it is possible to use classic tools of statistical decision theory.

Readers could find justifications and details of this transformation in [START_REF] Smets | Decision making in a context where uncertainty is represented by the belief functions[END_REF].

When comparing it to the probability theory, the transferable belief model shows some advantages, the most important of which is the possibility of expressing the degree of uncertainty. In this model, the mass assignment on a subset does not mean that the remainder automatically goes to the complement. After this introduction to the basic concepts of the TBM, the next section deals with the construction of mass functions.

Construction of mass functions

As mentioned in the introduction, the wealth and flexibility of the TBM constitute an interesting frame to represent and manage incomplete information. However, the problem of the construction of basic belief assignments persists and the modelling choice can have important consequences on the final results and decisions. In the next sections (3.2 and 3.3), we will present two methods of automatic grouping of hypotheses to construct mass functions (section 3.4).

Context and objectives

Let S = {S 1 , . . . , S j , . . . , S J } be a set of sources expressing themselves on the validity of the hypotheses {ω 1 , . . . , ω n , . . . , ω N } = Ω with regard to a given question Q. In a general way, each source S j can answer: "Is the hypothesis ω n a possible solution of the question Q?". The validity degree δ Q j,n associated to this answer varies between the values 0 and 1. It takes the value 0 when the hypothesis is not a solution of the question Q and the value 1 when the hypothesis is exactly a possible solution. So a source S j supplies a distribution of validity degrees defined by ∆ Q j,Ω : Ω → [0, 1]. The final objective in general is to find the hypothesis ω n (or the set of hypotheses Z λ ) according to the validity degree(s) (function of a given criterion relating to the application and the objectives for example).

When the number N is important (N > 15), it is generally difficult to work by means of belief functions because of the number of focal elements. And it is not easy to estimate correctly the bbas with a good report commitment/time calculations. So, the purpose is to obtain an estimation ∆ Q j,Z of the distribution ∆ Q j,Ω to reduce these focal elements afterwards. This estimate distribution is defined by ∆ Q j,Z : Z → [0, 1] where Z corresponds to the set of the disjunctions of hypotheses having been grouped together in an "intelligent" way according to the validity degrees: Z = {z 1 , ..., z λ , ..., z Λ } ⊆ Ω, with Λ < N .

In this article, two methods allowing to get an estimation of the distribution ∆ Q j,Ω are presented. This estimation will then be at the basis of the assignment of belief masses. The first method is based on a well-known method in cartography and geoinformation fields: Jenks' natural breaks classification [START_REF] Jenks | Optimal data classification for choropleth maps[END_REF][START_REF] Slocum | Thematic Cartography and Geographic Visualization[END_REF]. The last one is based on an approach of information criterion and originaly comes from the optimization of histograms. Both methods also allow to reduce complexity by keeping a satisfactory expressiveness to make decisions. In the next sections, we present these two methods and go on the comparison in the framework of the forensic application.

Grouping of hypotheses based on Jenks' approach

Jenks' algorithm is a much used empirical approach within the geographical information systems (GIS) [START_REF] Fisher | On grouping for maximum homogeneity[END_REF][START_REF] Jenks | Optimal data classification for choropleth maps[END_REF]. It allows an optimal classification [START_REF] Slocum | Thematic Cartography and Geographic Visualization[END_REF] of a population in a certain number of classes (or partitions) and minimizes the sum of absolute deviations about class means. It starts with an arbitrary set of classes, calculates an error (also named GV F for Goodness of Variance Fit index), and attempts to maximise it by moving observations between adjacent classes. The optimization so minimizes within-class variance and maximize between-class variance in an iterative series of calculations.

In short, Jenks' algorithm can be broken down into separate sequences: 1). the determination of the class number, 2). an iterative heuristics allowing to maximize the GVF index.

Jenks' optimized breaks algorithm

We consider the source S j , the set of hypotheses Ω and the relative ditribution of validity degrees ∆ Q j,Ω with regard to a certain question Q (see the previous section).

For a fixed number Λ of classes and a number N of hypotheses, there is a number Γ, equaled to N ! Λ!.(N -Λ)! , of possible combinations. Let us denote Z Λ,N = {Z 1 , . . . , Z γ , . . . , Z Γ } the set of these possible combinations. We try to get the best combination Z which maximizes the GV F index. The GVF index is given by the following equation:

GV F (Z γ ) = SDAM -SDCM (Z γ ) SDAM (8)
where the SDAM index is the Squared Deviations from the Array Mean. It is obtained by:

SDAM = ωn∈Ω (δ Q j,n -M ean(Ω)) 2 (9)
with M ean(Ω), the mean of the validity degrees δ Q j,n . In the equation ( 8), the SDCM value is the Squared Deviations from the Class Means:

SDCM (Z γ ) = z λ ∈Zγ ωn∈z λ (δ Q j,n -M ean(z λ )) 2 (10)
In [START_REF] Bloch | Fusion en traitement d'images : spécificités et approches numériques[END_REF], M ean(z λ ) is the mean of the validity degrees of the class z λ with for the combination Z γ .

We notice that the GVF index so varies between 0 and 1 (its value is 1 when Λ = N ). This index can be viewed as an entropic index that we maximize for a fixed number of classes. A heuristic has to be developed to cross the set of Z Λ,N possible combinations. We do not detail it in this article.

Grouping of hypotheses based on the approach of information criteria

The main idea is to sum up the information given by each source S j by means of an optimum histogram in the sense of the maximum likelihood and of a mean square cost. As previously, sources could supply degrees of validity δ Q j,n on hypotheses ω n ∈ Ω. First, we get the distribution of validity degrees given by each source and then make the optimum histogram building which is led by the use of an information criterion. We will see that different information criteria initially designed for model selection can be used.

Histogram approximation

We consider the source S j , the set Ω of N hypotheses and the relative distribution of validity degrees ∆ Q j,Ω with regard to a question Q of interest (see the section 3.1). The aim is to approximate ∆ Q j,Ω with a histogram ∆ Q j,Z built on a subpartition Z = {z 1 , ..., z λ , ..., z Λ } ⊆ Ω with Λ bins such as Λ < N . The distribution ∆ Q j,Z built with Z is an optimum estimation of ∆ Q j,Ω according to a cost function to define. Z results from an information criterion called IC issued from the basic Akaike's information criterion (AIC) [START_REF] Colot | Apprentissage et détection automatique de changements de modèles -Application aux signaux électro-encéphalographiques[END_REF], AIC * or φ * [START_REF] Colot | Information criteria and abrupt changes in probability laws[END_REF] which are respectively Hannan-Quinn's criterion and Rissanen's criterion. These criteria have the following form:

IC(Λ) = g(Λ) -2 z λ ∈Z ∆ Q j,Z (z λ ) ln ∆ Q j,Z (z λ ) Card(z λ ) ( 11
)
where g(Λ) is a penalty term which differs from one criterion to another one. Card(z λ ) is the number of hypotheses ω k ∈ z λ . ∆ Q j,Z (z λ ) is the maximum likelihood estimator for the partition z λ which is given by the following equation:

∆ Q j,Z (z λ ) = 1 ωn∈Ω δ Q j,n ω k ∈z λ δ Q j,k (12)
An optimum histogram ∆ Q j,Z to approximate the distribution ∆ Q j,Ω is obtained in two steps. The first one consists in merging two contiguous bins in a histogram with Λ bins among the (Λ -1) possible fusions of two bins. This is made by minimizing the IC criterion. The second one consists in finding the "best" histogram with Λ bins. The optimum histogram with Λ = Λ opt bins is the one which minimizes IC.

Selection of the bin number of a histogram

Obtaining the optimum histogram is based on the use of an information criterion IC which gives the optimal number of bins thanks to a cost function based on Kullback's contrast or Hellinger's distance. We do not detail these cost functions but if the cost function is expressed according to KullBack's contrast [START_REF] El-Matouat | Sélection du nombre de classe d'un histogramme et contraste de kullback[END_REF], we obtain the criterion such as :

AIC * (Λ) = Λ (1 + ln P ) P -2 z λ ∈Z ∆ Q j,Z (z λ ) ln ∆ Q j,Z (z λ ) Card(z λ ) (13) where P = ωn∈Ω δ Q j,n .
This criterion can be used to select the optimum histogram with Λ bins to approximate the distribution of validity degrees. Other criteria exist (e.g. AIC or φ * criteria) but we do not detail them in this article, simply because the criterion AIC * gave us better results in our application in term of time of calculations and final decision obtained. Detailed demonstrations for this section are available in [START_REF] Colot | Apprentissage et détection automatique de changements de modèles -Application aux signaux électro-encéphalographiques[END_REF][START_REF] Colot | Information criteria and abrupt changes in probability laws[END_REF].

Optimum histogram building process

At first, an initial histogram with N bins is built thanks to the distribution ∆ Q j,Ω . Then, a partition with (N -1) bins is considered. For each possible fusion of two contiguous bins among (N -1) the criterion IC(N -1) is computed. The choice of the best fusion is made according to the minimization of IC(N -1). When it is done, we look for the best partition with (N -2) bins according to the same rule. Finally, the histogram with Λ bins that minimize IC(Λ) for Λ ∈ {1, . . . , N } is retained.

Mass function construction by affecting masses on disjunctions

Having detailed the methods of grouping hypotheses, we present in this part the method of mass assignments. Whatever the previous method used, we obtain the set Z = {z 1 , ..., z λ , ..., z Λ } ⊆ Ω of compound hypotheses. For a source S j , we propose to make the average of the validity degrees of each z λ -class to affect the masses. We create as many basic belief assignments as disjunctions z λ :

       m j,λ (z λ ) = 0 m j,λ (z λ ) = α j (1 - M ean(z λ ) z λ ∈Z M ean(z λ ) ) m j,λ (Ω) = 1 -α j + α j . M ean(z λ ) z λ ∈Z M ean(z λ ) (14)
In ( 14), M ean(z λ ) is the mean of validity degrees of the class z λ and α j is the reliability coefficient of the source S j . This method of construction means that the more the source S j believes in a disjunction z λ , the less it will allocate a belief mass to its opposite z λ .

When we have the Λ created bbas m j,λ , we can combine them by means of the Dempster combination. The bba m j from the source S j is obtained by:

m j = ∩ Λ λ=1 m j,λ (15)
As in the method presented in [START_REF] Appriou | Multisensor signal processing in the framework of the theory of evidence[END_REF], it is a careful way to generate mass functions. We are going to study its results in the following sections.

Applications

Confrontation of both methods

In this section, we confront mass functions given by both methods. Figure 1 presents the validity degree distribution given by a source S j for a frame of discernment Ω = {ω 0 , ..., ω 150 }. As described in the section 3.2, in order to use Jenks' method, it is necessary to know a priori the number of possible disjunctions. Within the framework of this pratical case, we subjectively consider that an expert would have classed hypotheses in three classes. So, the belief function creating process is detailed for Jenks' method in tables 1 and 2. Figure 2 allows to compare the results obtained by both methods. We can see that with Jenks' method (with only three classes), the two principal peaks are eliminated. These peaks are always present with the AIC * -criterion based method.

Mass function construction in the forensic entomology project

During a criminal investigation, it is essential to obtain a maximum of information on the conditions of the manslaughter. Many methods to exploit the indications on the murder scene are known but, for large post-mortem intervals (PMI), only one of these techniques is useful in practice: forensic entomology. It consists in studying the insects found on a cadaver to estimate the time of its death. The objective is then to date the first layings by calculating the insects' age. Modern PMI estimation methods are based on insect development models. These models consider that insect development speed is temperature-dependent and Table 1: The three mass functions obtained with the grouping of hypotheses based on Jenks' method from data presented in figure 1. m j,1 m j,1 ({ω 0 , . . . , ω 80 }) 0, 997 m j,1 (Ω) 0, 003 m j,2 m j,2 ({ω 81 , . . . , ω 114 }) 0, 013 m j,2 (Ω) 0, 987 m j,3 m j,3 ({ω 115 , . . . , ω 150 }) 0, 990 m j,3 (Ω) 0, 010 has the next general form:

∆a = t2 t1 f (T (t))dt (16)
where T (t) represents the temperature T felt by an insect at the time t, f is a development model and ∆a is the accumulated rate of development of the insect. The interested reader can refer to [START_REF] Stinner | An algorithm for temperature-dependant growth rate simulation[END_REF] for more details.

There are a lot of models in literature but they do not take into account the entire ecosystemic context (thermic inertia of the body, thermic behavior of the larvae etc.). No current method takes into account these parameters, reported in numerous articles as an important source of error in the estimating of the PMI [START_REF] Marchenko | Medicolegal relevance of cadaver entomofauna for the determination of the time of death[END_REF]. Moreover, these models are based on different original biological data that were not still verified in all the contexts, but are used by the experts because of the lack of better information. Thus, estimations performed using these methods are often overestimated and not as precise as they could be. To improve the decision-making Table 2: The final mass function m j obtained from the combination (eq. 15) of the mass functions m j,1 , m j,2 and m j,3 .

m j m j ({ω 0 , . . . , ω 80 }) 0.010 m j ({ω 0 , . . . , ω 114 })

1.4 × 10 -4 m j ({ω 81 , . . . , ω 114 }) 3.2 × 10 and assist the forensic scientists, a decision support system has been developed. This project is based on a predictive multiagent model of insect development and cadaver decomposition in a complex ecosystem. It is used to determine if a hypothesis -a possible time of death -is coherent with the observations available on the ecosystem of the crime scene and the entomofauna found on the victim. More information about this model and the validation process can be found in [START_REF] Morvan | A decision support system for forensic entomology[END_REF].

To sum up the notations of the section 3.1, there are several sources (entomological models) S j ∈ S supplying distributions of validity degrees ∆ Q j,Ω : Ω → [0, 1], Ω is the set of the possible hours. Each distribution indicates how the model thinks that the hypotheses of the time interval are in accordance with the reality Q observed by the forensic scientists.

In the figure 3, we present the distributions of validity degrees supplied by four entomological models on an interval of 150 hours. For Jenks' based construction, the entomological expert recommends to use 3 disjunctions/classes. This value corresponds to a "normal" expected shape of a distribution supplied by the models.

For the method based on information criteria, we used the AIC * criterion allowing to obtain a good number of disjunctions.

We notice in figures 4 and 5 that the method based on the AIC * -criterion retranscribes in a better way the reality of the information than Jenks' method. An important loss of information is indeed observed during the use of this last method: mass functions of S 2 and S 4 contain no more information of the second observable peak in figure 3 around the hypothesis ω 90 . This loss of information related to Jenks' method can be annoying during the final decision-making but could also be handled by means of the value of the Jenks index (see section 3.2). For the models S 2 and S 4 , the Jenks' calculated index is very low and a treatment a posteriori could be made for, either to discount the mass functions proportionally in the index value, or to make a treatment by considering a bigger number of classes.

However, fixing the number of classes to a weaker value for Jenks' method allows us to reduce the calculation time during the mass function combinations even if we lose in precision, as we can see in figure 6. tion of likelihoods. They allow a compromise between "being less engaged" and "being calculable". Both methods can be used in two different contexts: Jenks' method can be used when the modeller knows a priori the shape of the distributions ∆ Q j,Ω . It is also possible to use the value of the GVF index to verify that the number of classes was correct and to modify it if necessary to generate other bbas with more disjunctions. The method based on information criteria can lead to a higher number of classes and thus obtain a bba closer to raw information, while being less involved with methods based on the calculation of distances. It seems interesting, but can lead to too important a number of disjunctions and too expensive a computation time.

Discussion

Conclusion

When the number of hypotheses of the frame of discernment is too large, it is generally difficult to manage belief functions because of the number of potential focal elements. In this article, we propose two automatic methods of grouping hypotheses to reason on compound hypotheses. Jenks' method is particularly useful if the modeller has the a priori knowledge of the information provided by the sources to combine. The second one, based on information criteria, is interesting to group the closest hypotheses in a totally automatic way. Both methods allow to reduce the calculation time and to improve the quality of our decisions in comparison with the other existing construction methods. Afterwards, we shall work on the complexity of the presented methods and possibly analyse previous works from possibility distribution construction.

The results obtained within the framework of the forensic medicine application are promising and allow us to obtain less committed results while being explicit enough to choose one or several hours of death. In the future, we shall work on an iterative and interactive method allowing to approach gradually hypotheses supported by the majority of the models.

Figure 1 :

 1 Figure 1: Validity degree distribution given by a source S j . Between the hypotheses ω 0 and ω 70 and between the hypotheses ω 130 and ω 150 , the validity degrees are null.

Figure 2 :

 2 Figure 2: Pignistic probability distributions for both methods. The Jenks' curve is obtained from the pignistic transformation of the mass function m j detailed in table 2.

4 Figure 3 :

 43 Figure3: Validity degree distributions given by four entomological models. To make it simpler, we do not print hypotheses between ω 120 and ω 150 where validity degrees are null.

2 + 4 Figure 4 : 2 + 4 Figure 5 :

 244245 Figure 4: The four pignistic probability distributions obtained with Jenks' method.

Figure 6 :

 6 Figure 6: Final pignistic probability distributions obtained from the combination of the four models for both methods.

Acknowledgments.

This work is financed by the Norbert Ségard foundation. The authors thank Daniel Jolly, Gildas Morvan and Gilles Goncalves (LGI2A), Philippe Kubiak (LAGIS-École Centrale of Lille), Benoît Bourel and Damien Charabidze (Institute of forensic medicine of Lille) for their support.