MD Soizic Leguy

PhD Mathilde Lefort

MD Lucile Lescot

Audrey Michaud

MD, PhD Sandra Vukusic

MD Emmanuelle Le Page

MD, PhD Anne Kerbrat

MD Christine Lebrun-Frenay

; Audrey Michaud

MD ; Gilles Edan

MD, PhD Jérôme De Sèze

MD, PhD David-Axel Laplaud

MD Sandrine Wiertlewski

PhD Emmanuelle Leray

MD, PhD Laure Michel
email: laure.michel@chu-rennes.fr

COPP-MS: COrticosteroids during the Post-Partum in relapsing Multiple Sclerosis patients

Keywords: multiple sclerosis, post-partum, relapses, disability, preventive corticosteroids

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

INTRODUCTION

Multiple sclerosis (MS) is a common inflammatory demyelinating disease of the central nervous system which affects preferentially young women.

Pregnancy influences the clinical course with a decrease of the annualized relapse rate (ARR) during pregnancy, especially during the third trimester, and an increase during the post-partum (PP) period, with 13% to 30% of women facing a relapse [START_REF] Confavreux | Rate of pregnancy-related relapse in multiple sclerosis. Pregnancy in Multiple Sclerosis Group[END_REF][START_REF] Vukusic | Pregnancy and multiple sclerosis (the PRIMS study): clinical predictors of postpartum relapse[END_REF][START_REF] Vukusic | Oral nomegestrol acetate and transdermal 17-beta-estradiol for preventing post-partum relapses in multiple sclerosis: The POPARTMUS study[END_REF][START_REF] Hughes | Predictors and dynamics of postpartum relapses in women with multiple sclerosis[END_REF][START_REF] Fragoso | The effects of long-term exposure to disease-modifying drugs during pregnancy in multiple sclerosis[END_REF][START_REF] Hellwig | Multiple sclerosis and pregnancy: experience from a nationwide database in Germany[END_REF][START_REF] Alroughani | Relapse occurrence in women with multiple sclerosis during pregnancy in the new treatment era[END_REF][START_REF] Bsteh | Pregnancy and multiple sclerosis in the DMT era: A cohort study in Western Austria[END_REF][START_REF] Berenguer-Ruiz | Relapses and obstetric outcomes in women with multiple sclerosis planning pregnancy[END_REF][START_REF] Finkelsztejn | The Brazilian database on pregnancy in multiple sclerosis[END_REF][START_REF] Correia | Pregnancy in Multiple Sclerosis: A Portuguese cohort study[END_REF][START_REF] Langer-Gould | Pregnancy-related relapses and breastfeeding in a contemporary multiple sclerosis cohort[END_REF].

Recent studies suggest that the proportion of patients having a relapse in PP is currently decreasing, probably due to a better control of the disease activity before and during pregnancy with disease modifying drugs (DMDs) [START_REF] Hughes | Predictors and dynamics of postpartum relapses in women with multiple sclerosis[END_REF][START_REF] Fragoso | The effects of long-term exposure to disease-modifying drugs during pregnancy in multiple sclerosis[END_REF][START_REF] Hellwig | Multiple sclerosis and pregnancy: experience from a nationwide database in Germany[END_REF][START_REF] Alroughani | Relapse occurrence in women with multiple sclerosis during pregnancy in the new treatment era[END_REF][START_REF] Bsteh | Pregnancy and multiple sclerosis in the DMT era: A cohort study in Western Austria[END_REF][START_REF] Berenguer-Ruiz | Relapses and obstetric outcomes in women with multiple sclerosis planning pregnancy[END_REF][START_REF] Finkelsztejn | The Brazilian database on pregnancy in multiple sclerosis[END_REF][START_REF] Correia | Pregnancy in Multiple Sclerosis: A Portuguese cohort study[END_REF][START_REF] Langer-Gould | Pregnancy-related relapses and breastfeeding in a contemporary multiple sclerosis cohort[END_REF][START_REF] Vukusic | Natalizumab for the prevention of post-partum relapses in women with multiple sclerosis[END_REF][START_REF] Portaccio | Pregnancy decision-making in women with multiple sclerosis treated with natalizumab: II: Maternal risks[END_REF][START_REF] Canibaño | Pregnancy-related issues in women with multiple sclerosis: an evidence-based review with practical recommendations[END_REF]. However, this risk of disease reactivation still exists and the therapeutic management of MS patients in PP is a matter of debate. Several strategies have already been assessed (corticosteroids, intravenous immunoglobulin, hormonal treatment or early DMDs restart [START_REF] Vukusic | Oral nomegestrol acetate and transdermal 17-beta-estradiol for preventing post-partum relapses in multiple sclerosis: The POPARTMUS study[END_REF][START_REF] Fragoso | The effects of long-term exposure to disease-modifying drugs during pregnancy in multiple sclerosis[END_REF][START_REF] Bsteh | Pregnancy and multiple sclerosis in the DMT era: A cohort study in Western Austria[END_REF][START_REF] Berenguer-Ruiz | Relapses and obstetric outcomes in women with multiple sclerosis planning pregnancy[END_REF][START_REF] Vukusic | Natalizumab for the prevention of post-partum relapses in women with multiple sclerosis[END_REF][START_REF] Portaccio | Pregnancy decision-making in women with multiple sclerosis treated with natalizumab: II: Maternal risks[END_REF][START_REF] De Seze | Intravenous corticosteroids in the postpartum period for reduction of acute exacerbations in multiple sclerosis[END_REF][START_REF] Avila-Ornelas | The Role of Postpartum Intravenous Corticosteroids in the Prevention of Relapses in Multiple Sclerosis[END_REF][START_REF] Hellwig | Immunomodulation and postpartum relapses in patients with multiple sclerosis[END_REF][START_REF] Brandt-Wouters | The effect of postpartum intravenous immunoglobulins on the relapse rate among patients with multiple sclerosis[END_REF][START_REF] Rosa | There is no benefit in the use of postnatal intravenous immunoglobulin for the prevention of relapses of multiple sclerosis: findings from a systematic review and meta-analysis[END_REF][START_REF] Ledinek | Intravenous immunoglobulins for the prevention of postpartum relapses in multiple sclerosis[END_REF][START_REF] Winkelmann | Intravenous immunoglobulin treatment in multiple sclerosis: A prospective, rater-blinded analysis of relapse rates during pregnancy and the postnatal period[END_REF][START_REF] Portaccio | Postpartum relapses increase the risk of disability progression in multiple sclerosis: the role of disease modifying drugs[END_REF][START_REF] Canibaño | Pregnancy-related issues in women with multiple sclerosis: an evidence-based review with practical recommendations[END_REF]). Among them, the effectiveness of preventive high-dose corticosteroids (CTCs) has been evaluated in two retrospective clinical studies suggesting a beneficial effect [START_REF] De Seze | Intravenous corticosteroids in the postpartum period for reduction of acute exacerbations in multiple sclerosis[END_REF][START_REF] Avila-Ornelas | The Role of Postpartum Intravenous Corticosteroids in the Prevention of Relapses in Multiple Sclerosis[END_REF]. Limited level of evidence leads to heterogenous practices between MS centers and between patients.

In this context, using the French MS Registry (Observatoire Français de la Sclérose En Plaques-OFSEP) [START_REF] Vukusic | Observatoire Français de la Sclérose en Plaques (OFSEP): A unique multimodal nationwide MS registry in France[END_REF], we aimed to assess the real-life effectiveness of high-dose corticosteroids in prevention of PP relapses in relapsing-remitting MS (RRMS) patients, comparing two different strategies: [START_REF] Confavreux | Rate of pregnancy-related relapse in multiple sclerosis. Pregnancy in Multiple Sclerosis Group[END_REF] systematic preventive treatment by high-dose CTCs [START_REF] Vukusic | Pregnancy and multiple sclerosis (the PRIMS study): clinical predictors of postpartum relapse[END_REF] versus no preventive treatment by high-dose CTCs.

Patients and Methods

The OFSEP cohort and participating centers OFSEP is a French multimodal nationwide MS registry [START_REF] Vukusic | Observatoire Français de la Sclérose en Plaques (OFSEP): A unique multimodal nationwide MS registry in France[END_REF]. Data are prospectively collected during follow-up in the European Database for MS (EDMUS) software [24, ACCEPTED MANUSCRIPT/ CLEAN COPY 25]. For the current study, five French MS centers were selected according to their systematic strategies in PP. Two of them, defined as "treating centers" (TC), propose a systematic standardized intravenous corticotherapy after delivery contrary to the three others, defined as "non-treating centers" (NTC).

Standard protocol approvals, registrations, and patient consents

All patients signed an informed consent to allow the registration of their medical data in the OFSEP database. The present study received approval from the Comité de Protection des Personnes on April 16th 2019 (registration number SI: 19.03.05.63609) and was declared to the Agence Nationale de Sécurité du Médicament et des produits de santé (ID RCB:2019-A00567-50). The study was also registered in clinicaltrials.gov (NCT03917589). All patients received an information letter and a non-consent form.

Study cohort: the COPP-MS cohort

The inclusion criteria were: (1) women with RRMS defined by the 2010 revised MacDonald criteria [START_REF] Polman | Diagnostic criteria for multiple sclerosis: 2010 revisions to the McDonald criteria[END_REF], non-opposed to participation, (2) aged from 15 to 49 years at pregnancy onset, (3) having at least one pregnancy with live birth after MS onset from January 1, 2007 to January 1, 2017, (4) having at least one neurological visit per year within two years preceding the pregnancy and within two years after the delivery. In case of several pregnancies in the period, only the last one was analyzed.

Preventive treatment in PP

The systematic preventive treatment protocol was one gram of intravenous methylprednisolone (IVMP) per month, started the day after delivery at maternity and given for 3 to 6 months.

Data collection

ACCEPTED MANUSCRIPT/ CLEAN COPY

Following variables were collected using extraction from EDMUS database (from 2 years before pregnancy onset to 2 years after delivery): age and MS duration at delivery, clinical activity (ARR, relapses, EDSS), radiological activity and DMDs exposure (considering first-line therapies (interferon, glatiramer acetate, teriflunomide, dimethyl-fumarate), second-line therapies (natalizumab, fingolimod), third-line therapy (mitoxantrone, alemtuzumab) and off-label therapies (azathioprine, cyclophosphamide, laquinimod, methotrexate, mycophenolate mofetil, rituximab)).

Missing data were collected directly on site by two neurologists (SL, LL) and data (as relapses, EDSS, MRI parameters) were checked using MS patients' files in each center.

A questionnaire about breastfeeding (exclusivity/duration), vitamin D supplementation and smoking habits before/during/after pregnancy was sent to patients.

Outcomes

Primary outcomes were the (1) time to first relapse, (2) time to first EDSS progression and (3) time to first new MRI activity, from delivery over two years. EDSS progression was defined as follows: +1.5 point if baseline EDSS=0, +1 if 1≤EDSS <5.5, +0.5 if EDSS³5.5. New MRI activity was defined as either Gadolinium enhancing lesion or a new T2-lesion. Delays were censored at the earliest of: secondary progressive MS date, end of follow-up date or end of study date (i.e. delivery +2 years).

Secondary binary outcomes were the risk of occurrence of at least one relapse in the first six months, the first year and the first two years PP, and at least one EDSS progression and one MRI activity after delivery and until two years.

In a first descriptive step, before propensity-score weighting, we described characteristics at baseline and calculated the ARR before, during and after pregnancy.

Statistical analysis

ACCEPTED MANUSCRIPT/ CLEAN COPY Demographics and clinical characteristics were described and compared using Student test for continuous variables and Fisher test for categorical ones (level of significance: 5%). ARR were defined as the total number of relapses divided by the follow-up duration in each period of interest.

In the main analysis, we defined two cohorts, patients from TC and patients from NTC, independently of IVMP status, to compare centers practicing or not preventive corticosteroids strategy. A secondary analysis was performed with a per protocol method (exact treatment received), defining a cohort of patients actually treated and a cohort of patients who did not, independently of their origin center. Both analyses followed a similar modeling strategy.

Our study was not randomized, so in order to improve group balancing, a propensity score (PS) was used. Using PS, we want to balance between the both groups to make them comparable at therapeutic decision. This therapeutic decision should reflect the decision of the neurologist based on the risk to reach the outcome, for instance, the risk of relapse. Thus, PS was obtained from a multivariate logistic regression including variables statistically associated with each specific outcome (Cox model, level of significance: 20%) [START_REF] Brookhart | Variable selection for propensity score models[END_REF][START_REF] Karim | The use and quality of reporting of propensity score methods in multiple sclerosis literature: A review[END_REF][START_REF] Simoneau | Recommendations for the use of propensity score methods in multiple sclerosis ACCEPTED MANUSCRIPT/ CLEAN COPY research[END_REF] For each outcome, following variables were considered potentially associated with: age at delivery, MS duration at delivery, ARR over the two years before pregnancy, number of relapse during pregnancy, EDSS at pregnancy onset, MRI within the two years before pregnancy, breastfeeding, DMD within the year before pregnancy and DMD during pregnancy. The association of each variable with each outcome were studied using Cox model (level of significance: 20%). Then, outcomes were studied using Kaplan-Meier methods weighting by the stabilized inverse probability of treatment weights (IPTW), a weight obtained from PS and the corresponding log-rank test. In each balanced group, we reported and compared the ACCEPTED MANUSCRIPT/ CLEAN COPY weighted restricted mean time (RMST) before reaching outcomes [START_REF] Zhao | On the restricted mean survival time curve in survival analysis[END_REF] (significance level: 5%). RMST were restricted at six months, one year and two years of PP. Binary outcomes were assessed using IPTW weighted logistic regression. Confidence interval were obtained by bootstrap (k=1000).

Analysis was performed using R software (R4.0.3), the weighted Kaplan-Meier estimators were obtained using the package RISCA [START_REF]RISCA: Causal Inference and Prediction in Cohort-Based Analyses version 0.9 from CRAN[END_REF].

Results

Characteristics at baseline and at delivery

Database extraction identified 390 eligible patients. Inclusion process led to 350 patients retained for the analysis (Figure 1). 1.A for the main analysis and available in supplementary for the secondary analysis (eTable 1.A). Overall, 116 (33.1%) patients were followed in TC and 234 (66.9%) in NTC, and 71 (20.3%) patients were actually treated with IVMP in the PP period (69 from TC, 2 from NTC). Clinical data (ARR, EDSS), radiological activity and DMDs use were similar between patients of TC and NTC within the two years before pregnancy. Overall, 150 patients were exposed to DMDs during pregnancy (72.7% with first line, 25.3% with second line and 2% with off-label therapy), with a mean duration of 2.4 (±3) months and no difference between TC and NTC. Comparison of treated (N=69) and untreated patients (N=47) in TC showed no statistical difference for characteristics at baseline (data not shown).

Characteristics at baseline are presented in Table

Characteristics at delivery and in PP are presented in Table 1.B for the main analysis (in supplementary for the secondary analysis, eTable 1.B). Breastfeeding was more frequent in TC (66.2%) compared to NTC (42.6%) (p<0.001). Among breastfeeding mothers, exclusive breastfeeding (only breast-milk used for at least two months) ACCEPTED MANUSCRIPT/ CLEAN COPY included 32% of patients from TC and 22% of patients from NTC (p=0.088). Mean breastfeeding duration was 4.8 months (±4.3) in TC and 6.4 months (±9.2) in NTC. On average, treated patients received monthly IVMP during 4.56 (±1.8) months after delivery. Median delay before DMDs resumption was not different between TC and NTC (p=0.700). Indeed, 41% of patients from TC and 46% from NTC have resumed their DMDs in the first three months of PP. Considering treated patients, during the preventive corticotherapy, 71.7% breastfed and 68% resumed a DMD before the end of the month following the last infusion (eTable 1.B).

ARR before pregnancy, during pregnancy and over two years of PP every 3 months for patients of TC and NTC.

As shown in Figure 2, ARR were similar in the two groups before pregnancy (p=0.841).

Regarding the whole cohort, ARR decreased during pregnancy compared to prepregnancy rates and increased during the first trimester of PP compared to pregnancy rates. At one year, ARR remain slightly lower than pre-pregnancy rate.

Comparison of ARR along time using linear regression between TC and NTC showed no difference between curves, in main analysis (p=0.404). Overall, 11% of patients from TC and 14% from NTC faced at least one relapse during the first PP trimester (p=0.505).

Time to first relapse in PP: No difference in both analyses

Regarding this outcome, PS model included: ARR over the two years before pregnancy, number of relapses during pregnancy and DMDs during pregnancy (Table 2). No statistical difference was found comparing the Kaplan-Meier estimates for the time to first relapse from delivery (p=0.691, Figure 3), as well as the restricted mean time (RMST) before reaching the first relapse in PP at six months, one year and two years between groups (Figure 4). Odds-Ratio (OR) associated with the proportion of patients ACCEPTED MANUSCRIPT/ CLEAN COPY facing at least one relapse in each group was 0.99 [0.53; 1.63] in the first six months after delivery, 0.87 [0.41; 1.53] in the first year and 0.85 [0.44; 1.42] in the first two years. In the TC group, PP relapse rates were not different between treated and untreated patients (p>0.05). Results were confirmed in the secondary analysis without statistical difference between the two curves (p=0.537) (Figure 3), and no difference between RMST at the three time-points (Figure 4).

Time to first EDSS progression in PP: No difference in both analyses

PS model for this outcome included: year of pregnancy (Table 2). No statistical difference was found comparing the curves representing time to first EDSS progression (p=0.840, Figure 3) as well as RMST after delivery at six months, one year and two years (Figure 4). OR associated to the proportion of patients facing at least one EDSS progression was 0.93 [0.52; 1.54]. The secondary analysis was consistent with these results, with no difference between Kaplan-Meier curves (p=0.453, Figure 3) and no difference between RMST at the three timepoints (Figure 4).

Time to first MRI activity in PP: No difference in main analysis

PS model for this outcome included age and MS duration at delivery, number of relapses during pregnancy, MRI activity within the year before pregnancy, breastfeeding at delivery, DMDs over the two years before pregnancy and DMDs during pregnancy (Table 2). MRI was available for 284 patients, 105 (37%) were done within the 6-months PP, 39 (14%) between 6-and 12-months PP and 140 (49.2%) after 12-months PP.

No difference was found between the two groups comparing Kaplan-Meier estimates (p=0.093) and RMST at six months, one year and two years (Figure 4). OR associated to the proportion of patients with a new MRI activity was 1.40 [0.67; 2.45]. In the secondary analysis, a difference between treated and untreated groups was observed, in ACCEPTED MANUSCRIPT/ CLEAN COPY favor of untreated group (p=0.042) (Figure 3). Although, differences of RMST were not different at six months, a difference at one year and two years of PP was observed, in favor of untreated group (-0.012 [-0.21; -0.01] at one year and -0.26 [-0.48; -0.03] at two years) (Figure 4).

Discussion

Our main objective was to assess, in a large cohort of MS patients, the effectiveness of preventive IVMP in the PP period on disease activity by comparing two strategies: (1) Systematic preventive treatment and (2) No preventive treatment.

Using two complementary statistical analyses, time to first relapse and to first EDSS progression were not statistically different between patients of TC and NTC and between treated and untreated patients.

Until now, only two retrospective observational studies of small size [START_REF] De Seze | Intravenous corticosteroids in the postpartum period for reduction of acute exacerbations in multiple sclerosis[END_REF][START_REF] Avila-Ornelas | The Role of Postpartum Intravenous Corticosteroids in the Prevention of Relapses in Multiple Sclerosis[END_REF] have assessed this question, showing a decreased risk of PP relapses with preventive corticotherapy, and motivating some teams to apply a standardized protocol. In the first one, the ARR of 22 untreated MS women appeared higher in the first PP trimester (2±0.66) than the ARR of 20 MS women treated by one gram of IVMP monthly during six months in PP (0.8±0.41) [START_REF] De Seze | Intravenous corticosteroids in the postpartum period for reduction of acute exacerbations in multiple sclerosis[END_REF]. In the second study, 17.9% of 39 MS patients treated directly after delivery with one gram of IVMP faced a relapse in PP compared to 46.2% of 13 untreated MS patients [START_REF] Avila-Ornelas | The Role of Postpartum Intravenous Corticosteroids in the Prevention of Relapses in Multiple Sclerosis[END_REF]. In these two uncontrolled studies, the groups were small, and the untreated groups were particularly active in the 3 months PP with ARR of 2 in the French study (1.2 in PRIMS), and 46.2% of patients facing a relapse in the American one (28% in PRIMS) [START_REF] Confavreux | Rate of pregnancy-related relapse in multiple sclerosis. Pregnancy in Multiple Sclerosis Group[END_REF][START_REF] Vukusic | Pregnancy and multiple sclerosis (the PRIMS study): clinical predictors of postpartum relapse[END_REF]. Moreover, critical variables could have influenced disease activity in the PP: DMDs exposure before pregnancy was not ACCEPTED MANUSCRIPT/ CLEAN COPY described in the French study and appeared substantially different, although not statistically, in the American one (89.7% and 69.2% patients on DMDs respectively in treated and untreated group). Considering breastfeeding, no difference between the two groups was reported in the American study, but data were not available for the French one. Considering the statistical analysis, the authors did not balance their treated/not treated groups that were not comparable and they did not consider the indication bias especially in the American study.

In our study, we retrospectively included 350 patients of five French MS centers with a prospective follow-up. Our findings were consistent with recent literature, confirming an increase of ARR in PP but lower than the ARR firstly described in historical cohorts, with 13% of patients presenting a relapse during the 3 months PP [4, 7, 8, 10-12, 15, 32]. These data confirmed a decrease of the risk of inflammatory disease reactivation in PP. Unlike the two previous studies, we found no benefit for a preventive corticotherapy in our cohort on clinical parameters [START_REF] De Seze | Intravenous corticosteroids in the postpartum period for reduction of acute exacerbations in multiple sclerosis[END_REF][START_REF] Avila-Ornelas | The Role of Postpartum Intravenous Corticosteroids in the Prevention of Relapses in Multiple Sclerosis[END_REF], maybe partly explained by the difference in term of disease activity before pregnancy between our contemporary cohort and these two older cohorts (ARR of 0.5 in the year before pregnancy in our cohort compared to 0.6 and 1 in the two groups of the French study), but also during the PP (with 11% and 14% of patients relapsing in the 3 months PP in our cohort compared to 17.9 and 46.2% in the two groups of the American study). Indeed, changes in the therapeutic management of MS have led to a significant reduction of the disease activity over the last decade and especially during the PP [START_REF] Hughes | Predictors and dynamics of postpartum relapses in women with multiple sclerosis[END_REF][START_REF] Fragoso | The effects of long-term exposure to disease-modifying drugs during pregnancy in multiple sclerosis[END_REF][START_REF] Hellwig | Multiple sclerosis and pregnancy: experience from a nationwide database in Germany[END_REF][START_REF] Alroughani | Relapse occurrence in women with multiple sclerosis during pregnancy in the new treatment era[END_REF][START_REF] Bsteh | Pregnancy and multiple sclerosis in the DMT era: A cohort study in Western Austria[END_REF][START_REF] Berenguer-Ruiz | Relapses and obstetric outcomes in women with multiple sclerosis planning pregnancy[END_REF][START_REF] Finkelsztejn | The Brazilian database on pregnancy in multiple sclerosis[END_REF][START_REF] Correia | Pregnancy in Multiple Sclerosis: A Portuguese cohort study[END_REF][START_REF] Langer-Gould | Pregnancy-related relapses and breastfeeding in a contemporary multiple sclerosis cohort[END_REF][START_REF] Vukusic | Natalizumab for the prevention of post-partum relapses in women with multiple sclerosis[END_REF][START_REF] Portaccio | Pregnancy decision-making in women with multiple sclerosis treated with natalizumab: II: Maternal risks[END_REF][START_REF] Canibaño | Pregnancy-related issues in women with multiple sclerosis: an evidence-based review with practical recommendations[END_REF].

Concerning the time to first MRI activity, no statistical difference was found with the main analysis, but the secondary analysis found a difference at one year and two years after delivery with an earlier occurrence of MRI activity for treated patients compared ACCEPTED MANUSCRIPT/ CLEAN COPY to untreated patients. It was not explained by the time before resuming DMDs in PP which was not delayed for treated patients (p=0.600). However, comparison of MRI activity has to be interpreted cautiously due to missing data, and to a substantial variability of delay between delivery and MRI acquisition between patients with probably more frequent control MRIs for patients with active disease.

In our cohort, breastfeeding was more frequent in TC (66.2%) compared to NTC (42.6%) (p<0.001), but not statistically different considering exclusive breastfeeding (p=0.088). Some studies showed a potential protective effect of breastfeeding that could influence the risk of disease reactivation in this period [START_REF] Hellwig | Multiple sclerosis and pregnancy: experience from a nationwide database in Germany[END_REF][START_REF] Langer-Gould | Pregnancy-related relapses and breastfeeding in a contemporary multiple sclerosis cohort[END_REF][START_REF] Canibaño | Pregnancy-related issues in women with multiple sclerosis: an evidence-based review with practical recommendations[END_REF][START_REF] Hellwig | Exclusive Breastfeeding and the Effect on Postpartum Multiple Sclerosis Relapses[END_REF][START_REF] Krysko | Association Between Breastfeeding and Postpartum Multiple Sclerosis Relapses: A Systematic Review and Meta-analysis[END_REF]. This variable, considered at baseline, was not included in the PS because not associated with the outcome in the model. Characteristics of breastfeeding such as exclusivity or duration, assessed after baseline, could thus be influenced by that occurs after baseline, such as disease activity, preventing their inclusion in the PS.

Early DMD resumption after delivery could reduce the risk of disease reactivation [START_REF] Bsteh | Pregnancy and multiple sclerosis in the DMT era: A cohort study in Western Austria[END_REF][START_REF] Berenguer-Ruiz | Relapses and obstetric outcomes in women with multiple sclerosis planning pregnancy[END_REF][START_REF] Vukusic | Natalizumab for the prevention of post-partum relapses in women with multiple sclerosis[END_REF][START_REF] Portaccio | Pregnancy decision-making in women with multiple sclerosis treated with natalizumab: II: Maternal risks[END_REF][START_REF] Portaccio | Postpartum relapses increase the risk of disability progression in multiple sclerosis: the role of disease modifying drugs[END_REF]. In our study, no difference for delay of DMD resumption was found between TC and NTC and did not influence our results.

One of the main strengths of our study is the large sample of MS patients with a prospective follow up, composing a representative cohort. There were very few missing data concerning our clinical endpoints, thanks to the use of EDMUS database, on-site data control and additional data collected with questionnaires. Regarding methodological choices, all women of TC did not receive IVMP (40% of patients from TC remained untreated). We cannot exclude that this high proportion of patients could be explained by unobserved or unknown confounders. A consequence of such unobserved data is a potential selection bias if we perform a direct comparison between treated patients and untreated patients [START_REF] Couvrat-Desvergnes | Comparison of graft and patient survival according to the transplantation centre policy for 1-year screening biopsy among stable kidney recipients: a propensity scorebased study[END_REF]. Similar to randomized clinical trial ACCEPTED MANUSCRIPT/ CLEAN COPY analyses, the intention-to-treat (ITT) principle was applied whereby the results were based on the practice of the center, and not on the treatment actually given. That point led us to perform two complementary statistical methods. The main analysis, equivalent to an ITT analysis, is a conservative method considering all patients as having received the treatment's strategy of their own center. The secondary analysis, equivalent to a per-protocol analysis, investigates the real exposure considering each patient according to the treatment actually received. To ensure comparability of our two populations, already very closed, we used a PS. Our conclusions are strengthened by concordance of both analyses and comforted by data of literature as the monthly high dose methylprednisolone benefit for controlling disease activity has never been clearly documented in MS, especially in very active MS [START_REF] Edan | Therapeutic effect of mitoxantrone combined with methylprednisolone in multiple sclerosis: a randomised multicentre study of active disease using MRI and clinical criteria[END_REF].

Currently, no preventive treatment has demonstrated its benefit to prevent PP relapses. Studies assessing intravenous immunoglobulins effect showed conflicting results [START_REF] Hellwig | Immunomodulation and postpartum relapses in patients with multiple sclerosis[END_REF][START_REF] Brandt-Wouters | The effect of postpartum intravenous immunoglobulins on the relapse rate among patients with multiple sclerosis[END_REF][START_REF] Rosa | There is no benefit in the use of postnatal intravenous immunoglobulin for the prevention of relapses of multiple sclerosis: findings from a systematic review and meta-analysis[END_REF][START_REF] Ledinek | Intravenous immunoglobulins for the prevention of postpartum relapses in multiple sclerosis[END_REF]. No protective effect was reported with hormonal treatments analyzed in a large randomized controlled study [START_REF] Vukusic | Oral nomegestrol acetate and transdermal 17-beta-estradiol for preventing post-partum relapses in multiple sclerosis: The POPARTMUS study[END_REF]. An interesting lead could be an early reintroduction of DMDs in the PP, which could reduce the risk of rebound [START_REF] Bsteh | Pregnancy and multiple sclerosis in the DMT era: A cohort study in Western Austria[END_REF][START_REF] Berenguer-Ruiz | Relapses and obstetric outcomes in women with multiple sclerosis planning pregnancy[END_REF][START_REF] Vukusic | Natalizumab for the prevention of post-partum relapses in women with multiple sclerosis[END_REF][START_REF] Portaccio | Pregnancy decision-making in women with multiple sclerosis treated with natalizumab: II: Maternal risks[END_REF][START_REF] Portaccio | Postpartum relapses increase the risk of disability progression in multiple sclerosis: the role of disease modifying drugs[END_REF], that will be analyzed in a second study on the same cohort of patients.

Main limitations of our study are classical ones in such retrospective studies, linked to the challenge of data collection. We ensured quality of our collected data using a national database, on sites files' checking and questionnaires. However, a selection bias could exist as we included patients followed exclusively in MS expert centers whose representativeness may be questionable. Only 71 patients of the entire cohort were actually treated, among which 69 patients of TC. The quite low proportion of treated patients in TC could be explained by different management strategies in PP according to neurologist. Moreover, even in a systematic approach, several patients ACCEPTED MANUSCRIPT/ CLEAN COPY may have declined the proposal of a preventive corticotherapy administered monthly. Despite these limitations, collected data remains of great interest as no randomized study would be easily achievable on this purpose on a cohort of MS patients. Thanks to our large cohort and robust statistical methods, we assess in this study the absence of benefit of high dose corticosteroids in PP period, providing class III evidence that IVMP do not prevent clinical activity during the PP period in women with MS.

Statements and Declarations

Competing interests: LM received honoraria as speaker from Biogen, Merck Serono, BMS Celgene, Sanofi Genzyme, Roche, and Novartis. SV reports consulting and lecture fees, travel grants and research support from Biogen, Celgène, Novartis, Merck, Roche, Sanofi Genzyme and Teva Pharm. DAL has received consulting and lecturing fees, travel grants and unconditional research support from Biogen, Genzyme, Novartis, Merck, Roche, Sanofi, Medday, Teva Pharma and BMS. SW received honoraria as speaker or consultant from Alexion, Biogen, Merck Serono, Roche, and Novartis. ELP received honoraria for lectures or consulting from Biogen Idec, Merck Serono, Novartis, Sanofi-Aventis, TEVA, Roche, Alexion. SL, ML, LL, AM, GE, AK, JDS, CLF, EL have no no conflict of interest regarding this work. For all of the subfigures, the blue line represents the curve for the patients from the TC or the treated patients; the gray line represents the curve for the patients from the NTC of the untreated patients. The dotted lines represent the 95% confidence interval in each group according to the same color legend. The weighted number of at risk patients is indicated in blue for the patients from the TC/treated patients and in gray for the patients from the NTC/untreated patients. The p-value was computed using the propensity score weighted log-rank statistical test. ).

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Figure 1 :

 1 Figure 1: Flowchart of the study population.

Figure 2 :

 2 Figure 2: Mean annual relapse rate (main analysis)

Figure 3 :

 3 Figure 3: Propensity score weighted Kaplan-Meier estimates of the time before

Figure 4 :

 4 Figure 4: Restricted mean time before reaching outcomes (main and secondary

2 A

 2 : first relapse, B: first EDSS progression, C: first new MRI activity from delivery over six months, one year and two years of post-partum.(1) Main analysis comparing treating centers (TC) to non-treating centers (NTC). (2) Secondary analysis comparing treated patients to untreated patients, For all of the subfigures, the blue bar represents RMST for patients from TC (main analysis, left column) or treated patients (secondary analysis, right column); the gray line represents the RMST for patients from NTC or the untreated patients. The difference of RMST is indicated in the table for each corresponding pair of bars.ACCEPTED MANUSCRIPT/ CLEAN COPY

Figure 1 :

 1 Figure 1:

Figure 2 :

 2 Figure 2:

Figure 3 :

 3 Figure 3:

Figure 4 :

 4 Figure 4:

Table 1 : Patients' characteristics (A) before pregnancy, (B) at delivery and during the post-partum (main analysis).Table 1 : Characteristics in each cohort

 11

	Total Cohort	Treating Centers	Non-Treating Centers	p-value
	N = 350	N = 116	N = 234

Table 2 : Variables included in the propensity score models for each outcome of the COPP-MS study

 2

Table 2 . Variables included in the propensity score models for each outcome of the COPP-MS study Variables Time to first relapse from delivery over two years Time to first EDSS progression from delivery over two years Time to first new MRI activity from delivery over two years Year of

 2 Cox model was built included the pool of variables potentially associated with the treatment decision. The table presents the p value relating to this association. A level of 20% was considered to decide if the variables was includedin the propensity score model corresponding to the outcome.

	pregnancy	0.293	0.176	0.404
	Age at delivery	0.998	0.448	0.360
	MS duration at delivery	0.283	0.317	0.009
	ARR over the two years before pregnancy	0.005	0.611	0.755
	Number of relapses during pregnancy	0.021	0.356	0.057
	EDSS at pregnancy onset (±3 months)	0.586	0.547	0.279
	MRI within the two years before pregnancy	0.393	0.507	0.010
	Breastfeeding	0.569	0.636	0.274
	DMD within the year before pregnancy	0.243	0.787	0.001
	DMD exposure during pregnancy	0.025	0.602	0.007
	*bold p means <20% i.e. included in the propensity score corresponding to the outcome.
	Note: for each outcome, a

Acknowledgments:

The present study is supported by a grant from the ARSEP. This work has been done with data from the OFSEP (http://www.ofsep.org/fr/), which is supported by a grant provided by the French State and handled by the Agence Nationale de la Recherche within the framework of the Investments for the Future program under the reference ANR-10-COHO-002 OFSEP and by the Eugene Devic Foundation Against Multiple Sclerosis. The authors thank the technicians of participating centers for their help (Mr Leport from Rennes, Mrs Moyon from Nantes, Mrs Callier from Nice, Mrs Berthe from Strasbourg). The authors also thank Merck Serono for partly supporting this study. Acknowledgments: The present study is supported by a grant from the ARSEP. This work has been done with data from the OFSEP (http://www.ofsep.org/fr/), which is supported by a grant provided by the French State and handled by the Agence Nationale de la Recherche within the framework of the Investments for the Future program under the reference ANR-10-COHO-002 OFSEP and by the Eugene Devic Foundation Against Multiple Sclerosis. The authors thank the technicians of participating centers for their help (Mr Leport from Rennes, Mrs Moyon from Nantes, Mrs Callier from Nice, Mrs Berthe from Strasbourg). The authors also thank Merck Serono for partly supporting this study.

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research support from Biogen, Celgène, Novartis, Merck, Roche, Sanofi Genzyme and Teva Pharm. DAL has received consulting and lecturing fees, travel grants and unconditional research support from Biogen, Genzyme, Novartis, Merck, Roche, Sanofi, Medday, Teva Pharma and BMS. SW received honoraria as speaker or consultant from Alexion, Biogen, Merck Serono, Roche, and Novartis. ELP received honoraria for lectures or consulting from Biogen Idec, Merck Serono, Novartis, Sanofi-Aventis, TEVA, Roche, Alexion. SL, ML, LL, AM, GE, AK, JDS, CLF, EL have no no conflict of interest regarding this work. Study funding: This work has been granted by the Eugene Devic EDMUS Foundation against Multiple Sclerosis and by the Aide à la Recherche sur la Sclérose en Plaques (ARSEP) foundation.

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Jean-Baptiste Welcomme

Marie-Pierre Gleizes

Romaric Redon

Thierry Druot

Self-Regulating Multi-Agent System for Multi-Disciplinary Optimisation Process

This article presents a multi-agent method to tackle multidisciplinary optimisation, based on the notions of cooperation and self-regulation. It is focused on the preliminary aircraft design, which is a complex compromise. In our approach several cooperative agents collectively act to achieve a common goal, i.e. optimising a multi-objective function, even if the environment of the system (the user's requirements) changes during the solving process. In MASCODE, one agent encapsulates one discipline and is designed individually without considering the dependencies with the others. So the computation is conceptually distributed without central control. Experimental results including efficiency comparison with the classical FSQP method are presented, and show that the adaptive behaviour of MASCODE provides new capabilities to understand and manage the complexity of the preliminary aircraft design.

Introduction 1.Preliminary Aircraft Design

Preliminary aircraft design involves a lot of disciplines like weight, range, aerodynamic and operating cost estimations [START_REF] Kroo | Distributed Multidisciplinary Design and Collaborative Optimization[END_REF]. In addition to the multidisciplinary aspect, manufacturers and airlines have different objectives on the product. Most of the time, manufacturers search to design product families, whereas airlines are looking for aircraft satisfying their needs at best (number of passenger, range, depreciation...). So during this design process a lot of compromises are carried out in order to specify the high-level design achieving expected aircraft performances (number of seats, cruise range, takeoff distance, etc.). These compromises are difficult to achieve, because constraints are numerous and dependent. Preliminary aircraft design is organised in two steps. First, a simulation function is built. It is obtained from the complex assembling of disciplinary models that represent a physic as a mathematical function with a set of inputs and outputs. Then, when the technical requirements (product performances) are known, the simulation is used to calculate the performances (Max Take Off Weight, Range, Operating Weight Empty); this is the design direction. Unfortunately, a mathematical inverse problem must be solved iteratively, because computational models are only known in the analysis direction ; computing product performances from design parameters. In the particular case of aircraft design, a lot of parameters are shared between disciplines. So the parameters and performances are highly interdependent, and constrained by their mutual tradeoffs [START_REF] Fujita | Product Variety Optimization Simultaneously Designing Module Combination and Module Attributes[END_REF].

Multi-Objective Optimisation

As presented, preliminary aircraft is an optimisation problem. Mathematical tools using response surfaces allow dealing with it. Especially the Feasible Sequential Quadratic Programming (FSQP), a gold standard method, enables to define objectives on performances and on design parameters, and then to find solutions to the design problem [START_REF] Zhou | User's Guide for FFSQP Version 3.7: A FOR-TRAN Code for Solving Constrained Nonlinear (Minimax) Optimization Problems[END_REF]. However the number of degrees of freedom and the parameter interdependencies imply that the solution space is discontinuous. So, these traditional mathematical methods are not really adapted to the preliminary aircraft design, because the discontinuity makes difficult to find design points that satisfy all the constraints and then to optimise them. Genetic algorithms (GA) offer very interesting robustness to tackle this problem and to find admissible point, because they are independent of the discontinuity. However GAs optimise the design as a global problem, and provide a limited view on compromise solution, since it is obvious that the aircraft is never a mathematical optimum but an alchemic compromise [START_REF] Badufle | Automatic satisfaction of constraints set in aircraft sizing studies[END_REF]. Pareto front are computed to compensate for this lack by pointing the region of the best compromise rather than providing only one optimal design. However if they improve the solution quality by providing more information, they do not really offer a better understanding since they are difficult to compute and to visualise especially when the targeted solution is really multi-objectives.

Multi-Disciplinary Optimisation

Kroo defines MDO (Multi-Disciplinary Optimisation) as "a methodology for the design of complex engineering systems and subsystems that coherently exploits the synergism of mutually interacting phenomena" [START_REF] Kroo | Distributed Multidisciplinary Design and Collaborative Optimization[END_REF]. During the last three decades, various types of computational or computer-aided design systems have been developed in MDO domain. A lot of issues were addressed like interoperability, problem decomposition, design robustness analysis and uncertainty propagation. Several strategies were proposed for the global optimisation and the subsystems linkage, exploiting the synergy of interactions through Fixed Point Iteration (FPI) algorithms [START_REF] Allison | Complex System Optimization: A Review of Analytical Target Cascading, Collaborative Optimization, and Other Formulations[END_REF]. Many relations between mathematic analyser and optimiser were studied, in which an analyser defines an execution order for computing the different models, whereas an optimiser compares their results and adapts the design parameters to converge on target criteria, like in All at Once (AAO), Multi Disciplinary Feasible (MDF) and Individual Disciplinary Feasible (IDF). However these strategies are finally first decomposed in subsystems and then centralised within an optimiser. So the decomposition of the system becomes a key point and influences the resolution. More complete approaches such as Collaborative Optimisation (CO), Concurrent Sub-Space Optimisation (CSSO) offer multi-level architectures, where each disciplinary has its individual optimisation strategy [START_REF] Kroo | Distributed Multidisciplinary Design and Collaborative Optimization[END_REF]. Analytical Target Cascading (ATC) is another alternative, in which each component is itself an optimiser [START_REF] Allison | Complex System Optimization: A Review of Analytical Target Cascading, Collaborative Optimization, and Other Formulations[END_REF]. As a consequence, the system is hierarchical and each component tries to minimise its individual objectives and those of its neighbours. The MASCODE1 approach presented in this paper have some similitude with it, but it processes are adaptive and dynamic.

Self-Organising Multi-Agent Approach

Distributed Constraint Optimisation Problems (DCOP) are an important research area for multiagent systems. Its objective is to propose an optimal assignment to a set of variables spread over a number of agents. A number of powerful distributed algorithms such as SynchBB [START_REF] Hirayama | Distributed Partial Constraint Satisfaction Problem[END_REF], ADOPT [START_REF] Modi | ADOPT: Asynchronous Distributed Constraint Optimization with Quality Guarantees[END_REF], OptAPO [START_REF] Mailler | Solving Distributed Constraint Optimization Problems Using Cooperative Mediation[END_REF] have been developed, and provided solutions either optimally, or close to optimality. However as these approaches are inspired from non-distributed combinatorial optimisation, they present lacks to be well-used to solve a dynamics problem with continuous parameters. Self-organising multi-agent approach works on the apparition of a functional structure spontaneously maintained in a dynamic equilibrium by all the participating components [START_REF] Heylighen | The meaning of self-organization in computing[END_REF]. As described in [START_REF] Di Marzo Serugendo | Self-Organisation and Emergence in Multi-Agent Systems: An Overview[END_REF], self-organising MAS (Multi-Agent Systems) offer opportunities to simulate real complex systems, because of agents have ideally autonomous behaviours; adapt constantly their state relative to each other; learn from experience; and create dynamically group and organization. As described previously, the preliminary aircraft design is a complex process, because of multi-disciplinary aspects and of multi-objectives criteria. In addition, the interdependencies between the parameters imposed to make a lot of compromises that dynamically change the problem formulation. All these characteristics make self-organising multi-agent approach a promising solution to support the preliminary aircraft design. In this paper, we present a cooperative multi-agent approach based on the AMAS theory (Adaptive Multi-Agent Systems) to solve the preliminary aircraft problem [START_REF] Bernon | Engineering Self-Adaptive Multi-Agent Systems : the ADELFE Methodology[END_REF]. According to the "organisaction" principles [START_REF] Morin | La mthode : La vie de la vie[END_REF], a self-organising system is described as to be able to self-regulate, self-relate and self-product. The paper is focused on the description of the self-regulation process, which aims

TakeOffW

WeightPerformance

EmptyW Weight

Span Mission Range

Awing Geometry MainGeometry

Figure 1: A simplified example of relation between models at finding the optimised values of several parameters in a given system. The system properties make that some parameters are shared between several disciplines and/or strongly interdependent. The paper is structured as follows. First, the principle of using a MAS for enabling preliminary aircraft design through cooperative reasoning are detailed; then, MASCODE results are described and compared with FSQP results; finally we highlight the main long term opportunities associated to our approach compared to latest MDO research works.

2 System Design

Introduction

MASCODE uses the specific resolution strategy of AMAS theory. This is a cooperative strategy, which is focused on the identification of a set of local Non Cooperative Situations (NCS) for the agents. By now, a cooperative agent is assigned to a discipline and the aim of an agent is to cooperate with its neighbours to find the values of different parameters in a given system, as shown in figure 1. Agents are drawn in square and shared parameters in oval.

For the system, some parameters are inputs (TakeOffW, Span, Awing), outputs (TakeOffW) and intermediates (Range, EmptyW...). Any one of these parameters can be a user objective. However regarding the characteristics of the problem, the only freedom degrees are input parameters. Due to interdependencies between parameters, decomposition of the global problem by disciplinary and subtask, it seems possible to gain advantages to use a distributed resolution process that will take into account the shared constraints between entities as in ADOPT or DPOP algorithms, or in multi-agent approaches in general.

In MASCODE, one agent controls one discipline. Therefore, these agents are called Disciplinary Agents (DA). The multi-agent system is a network of DA corresponding to the model hierarchy commonly found in preliminary aircraft design. Each DA owns representation knowledge of the model and learned knowledge from experiences, which are used through a set of behaviours to communicate and to take decision according to environment perception.

DA's Knowledge

This knowledge is static or dynamic, and is twofold: knowledge on its relations (connection with neighbours) and knowledge on its model.

Knowledge on Relations

To interact, each DA knows its provider and user agents. For example, in figure 1, for the agent Weight the users are WeightPerformance and Mission, and the providers are Geometry, Mission and WeightPerformance. A user agent uses the computed value of one of its output parameter, and a provider sends the value of one of its input parameters. In addition to this static knowledge, DA learns experiences during the execution, and builds memories. Memory is a key element in the AMAS approach, because an agent adapts its behaviour, and takes decisions in function of its past experiences (see section 2.3.2).

Knowledge on Model

Each model possesses physical properties. They are explainable on each input of models and defined with validity domains:

• Lower and upper bounds of the design variables provide a physical validity interval (physical limits) for each input, in which the domain is computable.

• An objective validity interval (objective limits) describes a preferred interval. All the values inside this range fit the user constraints.

With these intervals, we defined a parametric2 piecewise continuous mathematical function, shown in figure 2, that enables for the agent to compute a satisfaction criteria. It indicates whether the agent respects its physical limits and its objective limits:

• when the input value is inside the objective validity interval the critical value is negative,

• when the input value is inside the physical validity interval but outside the objective validity interval, the critical value is positive and inferior to a critical threshold, predefined by the designer, which is the maximal critical value in the system,

• when the input value is outside the physical limits, its critical value is equal to the critical threshold.

Finally, the non-satisfaction degree of the agent is defined as the maximum of its input critical values from its input parameters.

DA's Behaviour

Each agent is able to receive and send messages. In a first phase, agents compute their modules and transmit, via a forward message, the value of their outputs to their user agents. This phase is completed once an agent received all its forward messages from its providers. Consequently to the reception of forward messages, agents may send backward messages to inform providers when the provided value is not relevant. This second phase is completed once the agent received all its backward messages from its users. Thus, according to the received information in backward messages and to its individual state, it sends a modification request to its predecessors.

Cooperative Reasoning

The Cooperative Reasoning is designed across Non Cooperative Situations (NCS) [START_REF] Bernon | Engineering Self-Adaptive Multi-Agent Systems : the ADELFE Methodology[END_REF], composed of a description (conditions, triggers) and a set of actions. The description can be viewed as a rule containing all the conditions necessary to recognise the NCS. The sets of actions describe how the agents can improve the cooperation of their neighbourhoods. When all NCS are identified, the main objectives and the high-level decision model of agents are known.

In our cooperative approach and due to a set of NCS, we define the aim of each agent, which consists to do the action that decreases the most critical situation in the system. By measuring a non-satisfaction degree (the maximum of all the critical input values) in function of its objectives and of its physical limits, each agent can compare its critical value with the critical values of its neighbours (received requests). Then, it takes local cooperative decisions according to the following main principles:

• When the agent is the most critical, it builds a modification request for itself.

• When the agent is less critical than a modification request, it acts for the modification request.

For that it computes its Jacobian matrix 3 and finds the local dependencies between the concerned output and its inputs. Thus with the modification request and with its local dependencies, the agent is able to send a new modification request to its neighbours, that would help the received one.

Learned Experiences and Adaptive Input Variation Steps

The reasoning can be cooperative only if the decision model takes into account the past experiences of the agent. Without any reasoning on the past experiences the system is open to oscillations and chaotic phenomenons. However in MASCODE, the memory is quite simple. While moving to a solution, if the modification direction of an input is successively the same, the agent considers it as a positive feedback and increases an input variation step. Conversely, if the modification direction is changing, agent considers it as a negative feedback, and decreases the variation step. The initial variation step is a percentage of the total interval of the objective limits given in the figure 2. This behaviour allows a dynamic equilibrium when the system converges to a global solution as shown in section 3.

Algorithm

To sum up the DA's behaviours, the backward message phase, is presented in figure 3. The modifications are propagated across the system. DA agent uses its cooperative reasoning to select the modification requests it wants to create or transmit on each input. For each output, an agent possesses several users, because one parameter is often shared between several disciplines. So in a first time, it receives the modification for each output parameter (step 1a). Then, it selects for each output the most critical request and uses its knowledge on intput/output dependencies to build the corresponding request on its inputs (step 1b). Then, it selects the modification to transmit to its provider. When all critical situations have disappeared, all agents are in a satisfied state and the system has converged.

Experiments and Results

MASCODE uses the framework JADE (Java Agent DEvelopment Framework). To validate the approach, some experiments have been done for a sample preliminary aircraft design case study with 10 models and 60 parameters (20 inputs, 17 outputs, 23 intermediates), in which 14 parameters are objectives (7 inputs design freedom degrees, 7 outputs performances). MASCODE solutions have been compared with FSQP constraints satisfaction mode. For MAS-CODE and FSQP, the same intervals are provided on the objective parameters. Then the system adapts its parameters until constraints would be satisfied. Experimental solutions are plotted on the figures 4 and 5. The input objectives of the problem are illustrated with figure 4 and output objectives with 5. Results show that the found solutions are similar. For the same problem, two different solutions called "MASCODE 1" and "MASCODE 2", are presented on the histograms. Solutions can be different at each resolution, but are equivalent because they respect the problem constraints. By contrast, FSQP provides always the same solution, because its optimisation process is deterministic.

Comparison with FSQP

A MASCODE Execution

Figure 6 shows the evolution of the objectives parameters4 during the solving process. X-axis represents the time and Y-axis the normalised parameter values. Thus, all parameters can be plotted on a same graph. Figure 7 shows the same evolution for the critical value of each objective parameter. X-axis represents the time and Y-axis the parameter critical values. The system finds a solution, when the critical values are all null. As shown in figure 7, the system finds four solutions during the computation. In fact at each solution, the user introduces some new constraints in the system, figure 8. These new constraints breaks the equilibrium by introducing new critical situations (new disorders). Then a new self-adaptive process is engaged, because the problem formulation has changed. An entire scenario is explained in the next section. Figure 7 shows that the critical values globally decrease. However the decrease is sometimes discontinuous, because parameters are more or less sensible to the modifications. So an agent can decide a modification without knowing it would not be really a cooperative choice. However each agent learns progressively this kind of non-cooperative situations and the system converges.

Adaptive Behaviour of MASCODE

MASCODE provides user interfaces that help to understand/manage the system. For example, it provides a view of the system with the repartition of the critical values, individual interfaces for each DA and various graphics to pay attention on the parameter evolutions. The scope of this article is not to present all these properties. However, we provide a general survey of MASCODE capabilities to provide a really dynamic and adaptive system. Figures 8 and9 detail some parameters during a resolution, described in 3.2. These figures illustrate the dependencies between the parameters RA (RAnge of the mission), M T OW (Max Take Off Weight) and M W E (Manufacturing Weight Empty). Some of the relations of these parameters are illustrated on the figure 1. To provide an example and to simplify, they could be expressed as follows:

• The range RA impacts the fuel weight and so M T OW .

• MTOW impacts the manufacturing weight empty. When MTOW increases, the aircraft structural constraints change.

• If the aircraft structure changes, geometry could evolve.

• If the geometry of the aircraft evolves, aerodynamic forces and the range could also be modified.

During the presented process, the constraints was changed by the user as follow (see figures 8 and 9):

1. At time t = 84s, the objective on RA (range) was increased of 1%. It immediately introduces a new critical value for RA. But this modification does not impact M T OW and M W E.

2. At t = 100s, user asks for a diminution of the M T OW . First the critical value of M T OW increases and then the new constraint is shared between RA, M T OW and M W E. Then system is unable to converge, because it is over-constrained.

3. At t = 146s, a modification of the M W E objective provides new freedom degrees. This modification is not important (see figure 8) but enough to decrease M T OW without changing the mission RA.

4. At t = 150s, the M T OW constraint is reinforced (a lowering of 2%).

5. At t = 190s, the mission performance RA is degraded and enables the system to converge, because of links between RA and f uel, and between f uel and M T OW . About the results and this scenario, it is quite clear that MASCODE helps the designer to understand and to manage the constraints in the system. Each time the problem formulation changes, the agents adapt their behaviour and search a new equilibrium. When a new stable state is not reachable 5 , agents self-regulate the critical values in the system and help the user to identify conflicting parts. Thanks to this information, he can alter the strongest constraints and let the system converge toward another relevant solution.

Discussion

In addition to the comparison to the FSQP method for the case study in the previous section, MASCODE can be compared to other MDO approaches. This comparison is done at a relative high-level, since existing methods are not agent-based, and do not tackle dynamic and changeable problems.

Solution quality

The quality of MASCODE solutions is equivalent to the solution found by FSQP. FSQP method is based on a gradient descent, recognised as finding good solutions in nonlinear optimisation problems. However we used FSQP only in its constrained solving mode. So we need to go further in a multi-objective approach for a full comparison.

Convergence speed

The time of convergence is the same as FSQP, but none systematic measure of convergence speed has been realised for large problems, because it is not our first intention. Nevertheless as described in [START_REF] Allison | Complex System Optimization: A Review of Analytical Target Cascading, Collaborative Optimization, and Other Formulations[END_REF], the convergence speed depends on the problem decomposition.

Our problem decomposition is close to AAO, which is considered as the faster in comparison with IDF and MDF.

Robustness and disciplinary knowledge integration

In MASCODE, validity intervals are local knowledge about physical models. Introducing this knowledge in the resolution process is a first key point for improving the result consistency. Thus, it will be possible to add other knowledge in the reasoning, and to include it in the agent decision model. As other Disciplinary openness MASCODE requires no global information and decision process. Consequently, adding or deleting physical models consists only in updating the MAS. In all other non-agent MDO approaches, the openness capability is never invoked. However, this incremental functionality could be very useful for designing complex systems, because as we saw the context is dynamic and the designers often change their requirements.

Parameter adjustment In MASCODE, users can adjust values and associated validity domains of parameters in real time, because agents will dynamically change their behaviours according to this new knowledge. There is no information about parameter modification in runtime for the other approaches. By now, this is the most relevant property of MASCODE, because it permits to understand the relation between disciplines by the negotiations and to manage/adapt dynamically the constraints of the problem. As described, to integrate this particularity, the system needs to be robust to change and have to be itself dynamic and self-adaptive.

Conclusion

This article has presented a multi-agent method to tackle multidisciplinary optimisation, based on the notions of cooperation and self-adaptation. In MASCODE, the physical models are encapsulated in cooperative agents which negotiate and cooperate to find an optimal (or near optimal) solution. This approach is efficient and provides relevant results, in comparison to the classical FSQP method. The method is non complete, but the main objective, in the preliminary aircraft design context, is to quickly provide approximated solutions, with some user guidances and interferences during the solving process. From learned lessons, DA approach for multidisciplinary design optimisation can be considered without doubt as relevant for many reasons: • Each disciplinary model can be design individually without considering the dependencies with its neighbours. This ability reduces greatly the complexity of the MDO framework.

• An agent can encapsulate the disciplinary model, but also all the associated knowledge such as critical values, execution time, precision and granularity. So the quality of the solution is potentially better.

• The MASCODE computation is conceptually distributed without central control. Thus, the running can be entirely concurrent leading to a time reduction.

The design process of an aircraft is multi-disciplinary, multi-objective and also multi-level. All these aspects increase the complexity of an aircraft design. By now, we consider some of the multi-disciplinary and multi-objective aspects. However to provide a self-organized system, we need to add other knowledge on the disciplinary models (granularity, precision, computation time, semantic...) and by consequence to consider new cooperative situations for fully re-organize the system. For example, a re-organization process could be to change a model inside a discipline for a concurrent one or to change the granularity of a model. This is the main focus of our research and developments by now.

Figure 2 :

 2 Figure 2: Evaluation functions of critical values (interval validity functions).

begin 1 . 2 .Figure 3 :

 123 Figure 3: Backward message phase procedure for a DA.

2 Figure 4 : 2 Figure 5 :

 2425 Figure 4: Comparison of input objectives obtained with FSQP and Mascode.

Figure 6 :

 6 Figure 6: Evolution of objective parameters during a Mascode execution

Figure 7 :

 7 Figure 7: Evolution of critical values in the system during a Mascode execution

Figure 9 :

 9 Figure 9: Critical values of parameters RA, M T OW and M W E

MASCODE : Multi-disciplinary Aircraft Simulation for COnceptual DEsign

The parameter alpha is used to indicate whether a constraint is hard or not (higher alpha is, harder the constraint is).

These values are normalised for visual representation.

The system is over-constrained