M Delplanque

J Pouchot

S Ducharme-Bénard

B J Fautrel

A Benyamine

L Daniel

T Gendre

G Grateau

MD Sophie Georgin-Lavialle

AA amyloidosis secondary to adult onset Still's disease: about 19 cases Authors

Keywords: Adult onset Still's disease, AA amyloidosis, kidney disease, systematic literature review

Objective: Adult onset Still's disease (AOSD) is an inflammatory disorder characterized by high spiking fever, evanescent rash, polyarthritis, and many other systemic manifestations. Recurrent or persistent disease can lead to AA amyloidosis (AAA). Our objectives were to present 3 French cases and perform a systematic review of the literature, in order to determine the prevalence, characteristics, predisposing factors, and therapeutic response of AOSD-related AAA.

Methods: A systematic literature review was performed by searching MEDLINE from 1971 to 2018. Two independent investigators selected reports of AAA complicating AOSD. New French cases were identified with the help of the Reference Center for rare Auto-Inflammatory Diseases and Amyloidosis (CEREMAIA). Patients with juvenile idiopathic arthritis were excluded.

Results:

The prevalence of AAA in AOSD was 0.88% (95%CI [0.49-1.28]) based on 45 articles.

In addition to 3 new cases from the CEREMAIA, 16 patients were assessed for clinical presentation, risk factors, and therapeutic response of AOSD-related AAA. Mean age at AOSD onset was 29.612.6 years, with a mean delay before AAA diagnosis of 16.755.8 years. Renal involvement was the most common manifestation of AAA. The majority of patients presented active AOSD at AAA diagnosis. Various treatments of AOSD-related AAA were attempted including corticosteroids and biotherapies.

Conclusion:

AAA is a rare and severe complication that may occur during the course of uncontrolled active AOSD. It could be prevented by early diagnosis and better control of AOSD, with more frequent use of biotherapies.

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Materials and Methods

. No previously available protocol was used. We searched MEDLINE from 1971, when the first AOSD case was published, to 2018, in order to identify all studies pertaining to AAA in AOSD. The search was limited to articles published in French or English. Patients with sJIA, defined as onset of symptoms or diagnosis before 16 years old, were excluded.

We searched MEDLINE through PubMed using MESH terms. Separate searches were performed for prevalence and description of AOSD-related AAA. Two independent readers (SGL and MD) first screened titles and abstracts to exclude irrelevant articles and duplicates. Only original cases satisfying the Yamaguchi or Fautrel classification criteria were selected.

Keywords used for the prevalence assessment study were: "Still's Disease, Adult-Onset/complications"[Majr] OR "Still's Disease, Adult-Onset/diagnosis"[Majr] OR "Still's Disease, Adult-Onset/epidemiology"[Majr]. Both prospective and retrospective studies were eligible, with no limit regarding patient number and length of follow-up. We decided to only include articles reporting complications of AOSD (joint destruction, death, amyloidosis…), in order to avoid underestimating the prevalence of AAA by selecting articles that focused on diagnostic features of AOSD. Articles restricted to subgroups of AOSD patients based on age or a specific feature of the disease were excluded, in order to reflect AAA prevalence in the general AOSD population. Trials of second line therapy, which involve patients with more severe disease at higher risk of AAA, were also excluded to avoid selection bias.

Keywords used for the descriptive study were "Still's Disease, Adult-Onset"[Majr] AND "Amyloidosis" [Majr] and "adult onset still disease and amyloidosis". Clinical features of patients with both AOSD and AAA were assessed through case reports. An investigator (MD) extracted

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Introduction

Adult onset Still's disease (AOSD) is a systemic inflammatory disorder of unknown etiology characterized by high spiking fever, evanescent rash, polyarthritis, and many other manifestations including odynophagia, lymphadenopathy, splenomegaly, hepatic and pulmonary involvement, and serositis. It was first described in 1971 by Bywaters [START_REF] Bywaters | Still's disease in the adult[END_REF] in a case series of 14 women presenting with clinical features similar to those of systemic juvenile idiopathic arthritis (sJIA).

AOSD is a rare disease with an annual incidence of 0.16 to 0.4 cases per 100 000 persons [START_REF] Kadavath | Adult-onset Still's disease-pathogenesis, clinical manifestations, and new treatment options[END_REF]. In the absence of pathognomonic clinical features or biomarkers, the diagnosis of AOSD relies on the Yamaguchi or Fautrel classification criteria [START_REF] Yamaguchi | Preliminary criteria for classification of adult Still's disease[END_REF][START_REF] Fautrel | Proposal for a new set of classification criteria for adult-onset still disease[END_REF] after exclusion of a wide differential diagnosis. Patients were classically treated with corticosteroids and synthetic disease-modifying antirheumatic drugs (DMARDS), most commonly methotrexate. Recent studies have revealed a pivotal role of proinflammatory cytokines, namely tumor necrosis factor  (TNF), interleukin (IL)-1, and IL-6, thus paving the way for novel targeted therapies that may potentially allow to control refractory disease [START_REF] Castañeda | Adult-onset Still's disease: Advances in the treatment[END_REF].

AA amyloidosis (AAA) is a multisystemic disease related to the deposition in tissues of serum amyloid A (SAA) protein secondary to chronic inflammation. Causes are multiple and include chronic rheumatic and inflammatory bowel diseases, monogenic autoinflammatory diseases, chronic infections, and less frequently cancers and immune deficiencies. Recurrent or persistent AOSD due to suboptimal control of the disease is associated with multiple complications, such as chronic destructive arthritis, increased long-term morbidity and mortality, and AAA [START_REF] Stojanovic | AA amyloidosis][END_REF].

Our objectives were to present new French cases and perform a systematic review of the literature in order to determine the prevalence, characteristics, predisposing factors, and therapeutic response of AOSD-related AAA.

[Tapez ici] data from selected articles according to a predefined form. Descriptive statistics were performed using the excel software and R software. Characteristics of AOSD with and without AAA were compared using Fisher's exact test and chi-square test. To represent AOSD without AAA, we selected the case series from the literature that included the largest number of patients while providing detailed information regarding clinical presentation, biotherapy, follow-up, and complications. A 2-sided p-value < 0.05 was considered statistically significant.

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Results

French cases identified via the CEREMAIA Case 1: A 17-year-old man developed the first manifestations of AOSD in 1984. Disease course was characterized by severe joint involvement leading to left hip replacement, as well as recurrent febrile episodes with myalgia, pharyngitis, splenomegaly, and lung involvement.

Biological manifestations included leukocytosis with more than 80% neutrophils. Given the suspicion of an autoinflammatory disease, analysis of TNFRSF1A and NLRP3 genes by Sanger sequencing respectively excluded TNF receptor-associated periodic fever syndrome (TRAPS) and cryopyrin-associated periodic syndrome (CAPS). The patient was diagnosed with AOSD at the age of 23 years. He was then sequentially treated with colchicine, corticosteroids, nonsteroidal anti-inflammatory drugs (NSAIDs), hydroxychloroquine, methotrexate, leflunomide, and etanercept, with suboptimal response to all agents. Anakinra was started at the age of 48 years. At that moment, the patient presented with a nephrotic syndrome. Kidney biopsy confirmed the diagnosis of AAA. A few months later, declining renal function required initiation of hemodialysis.

Case 2 (previously reported by Serratrice et al [START_REF] Serratrice | Resolution with etanercept of nephrotic syndrome due to renal AA amyloidosis in adult Still's disease[END_REF]): In 1996, a woman had been followed since the age of 37 years for recurrent attacks of fever, migratory maculopapular erythema, pharyngitis, and arthritis, with concomitant neutrophilia. She also presented an episode of pericarditis and an episode of idiopathic thrombocytopenic purpura. She was finally diagnosed with AOSD 6 years later and treated with corticosteroids. Disease course was marked by several relapses, partly due to poor compliance. At the age of 51 years, diagnosis of renal AAA was made upon development of a nephrotic syndrome. The introduction of etanercept and colchicine resulted in stabilization of renal function and a marked decrease in proteinuria (0.1 g/L). There was no mutation identified in the TNFRSF1A gene. with renal involvement was made at the age of 29 years, based on a renal biopsy revealing diffuse glomerular and vascular lesions with chronic tubulointerstitial involvement. The patient was unsuccessfully treated with corticosteroids, cyclophosphamide, chlorambucil, and infliximab until he received a renal transplant at the age of 33 years. Persistent active AOSD despite treatment with anakinra lead to a second renal transplant 13 years later for recurrent amyloidosis in the graft. Of note, the patient also suffered from destructive polyarthritis requiring synovectomy and joint replacements.

Literature search

Fifty-seven articles were selected following our 2 searches (figure A). Forty-five were used to assess AAA prevalence [START_REF] Bywaters | Still's disease in the adult[END_REF]. Sixteen [START_REF] Vigneron | Amyloidosis in adult Still's disease. Apropos of 2 cases[END_REF][START_REF] Cush | Adult-onset Still's disease. Clinical course and outcome[END_REF][START_REF] Bambery | Adult onset Still's disease: clinical experience with 18 patients over 15 years in northern India[END_REF][START_REF] Abdelmoula | Adult onset Still's disease: about 11 cases[END_REF][START_REF] Harrington | Amyloidosis in adult onset Still's disease[END_REF][START_REF] Takahashi | Improvement of endoscopic and histologic findings of AA-type gastrointestinal amyloidosis by treatment with dimethyl sulfoxide and prednisolone[END_REF][START_REF] Horlait | Secondary amylosis in Still's disease in an adult. Value of biopsy of the subcutaneous abdominal fatty tissue[END_REF][START_REF] Wendling | Adult onset Still's disease and related renal amyloidosis[END_REF][START_REF] Ishii | Systemic amyloidosis in a patient with adult onset Still's disease[END_REF][START_REF] Rivera | Vascular renal AA amyloidosis in adult Still's disease[END_REF][START_REF] Fautrel | Corticosteroid sparing effect of low dose methotrexate treatment in adult Still's disease[END_REF][START_REF] Oh | Secondary renal amyloidosis in adult onset Still's disease: case report and review of the literature[END_REF][START_REF] Ghorbel | Amyloidosis in adult onset Still's disease[END_REF][START_REF] Amemori | Oral dimethyl sulfoxide for systemic amyloid A amyloidosis complication in chronic inflammatory disease: a retrospective patient chart review[END_REF][START_REF] Kishida | Successful tocilizumab treatment in a patient with adult-onset Still's disease complicated by chronic active hepatitis B and amyloid A amyloidosis[END_REF][START_REF] Benito | Secondary bladder amyloidosis: a new case report[END_REF] were used for the descriptive analysis of patients with AOSD and AAA, including 4 articles selected for the prevalence study and 12 individual case reports.

Prevalence study

The analysis included prospective and retrospective studies published between 1971 and 2016, representing a total of 2157 AOSD patients from different countries. AAA was reported in 19 patients, with a prevalence of 0.88 (95%CI [0.49-1.28]). The mean duration of follow-up was 5.44.5 years (Table A). Fifty-two deaths were reported during follow-up, including at least 3 patients suffering from AAA. None of the case series published since 2012 reported AAA as a complication of AOSD.

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Description of patients with AOSD-related AAA

Overall, 19 patients, including our 3 French cases, were reported with sufficient information to be analyzed. One of the French cases had previously been reported by Serratrice et al [START_REF] Serratrice | Resolution with etanercept of nephrotic syndrome due to renal AA amyloidosis in adult Still's disease[END_REF]. Eight patients were female. The sex ratio was 1.4. The mean age at diagnosis of AOSD was 30.711.8 years, with a delay from onset of symptoms of 3.32.8 years. There was no familial aggregation.

Clinical and laboratory features when available (n=17) are detailed in Table B. Fever and arthralgia/arthritis were present in all patients, and skin rash in 12 of them (64.7%). Neutrophilic leukocytosis and elevation of ESR and/or CRP were reported in 15 patients (82.3%). Thirteen patients had a chronic articular course with joint destruction. Three developed uncomplicated pericarditis. None evolved into a macrophage activation syndrome (MAS). (Table C).

Before the diagnosis of amyloidosis, all patients received corticosteroids. Six were also treated with methotrexate, and three with colchicine. Recent studies reported 2 patients treated with biologic therapies, which consisted in TNF inhibitors. Other treatments administered included NSAIDs (n=6), hydroxychloroquine (n=3), cyclophosphamide (n=2), azathioprine (n=2), gold salts (n=1), penicillamine (n=1) and isoprinosine (n = 1). Only five patients received biologic therapies. When AOSD response was described, Tocilizumab lead to remission (n=1) meanwhile TNF inhibitors and anti-IL1 therapy were ineffective (French case 1 and 3).

The mean delay between the first symptoms of AOSD and the diagnosis of AAA was 16.85.8 years. The most common presentation of AAA was renal involvement with nephrotic syndrome (n=8), sub-nephrotic proteinuria (n=5), and isolated renal insufficiency (n=3). Diagnosis of amyloidosis was confirmed on renal (n=14), intestinal (n = 5), bladder (n=1), abdominal fat (n=1) and salivary gland (n=1) biopsies. According to pathology reports, amyloid deposition was predominantly vascular and perivascular. After diagnosis of AAA, patients were treated with corticosteroids (n=10), cyclophosphamide (n=4), colchicine (n=5), chlorambucil (n=3), dimethylsulfoxide (n=2), TNF inhibitors (n=2), anakinra (n=2), tocilizumab (n=2), NSAIDs (n=1), and methotrexate (n=1). Apart from the 3 French cases, there was only scarce information regarding therapeutic outcome. Tocilizumab lead to renal remission with complete disappearance

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of proteinuria and a decrease in creatinine [START_REF] Kishida | Successful tocilizumab treatment in a patient with adult-onset Still's disease complicated by chronic active hepatitis B and amyloid A amyloidosis[END_REF]. Meanwhile, anti-IL1 therapy proved ineffective (French case 1 and 3), whereas TNF inhibitors lead to mixed outcomes: a partial response with etanercept (French case 2), and no response with disease progression with etanercept (French case 1) and infliximab (French case 3). Among the 19 patients, 4 deaths were reported. Three were attributed to AAA. The highest number of reported cases of AAA in AOSD was between 1990 and 1999. None were reported after 2012 (figure B) [START_REF] Tamesis | Etanercept in recalcitrant Adult Onset Still's Disease[END_REF][START_REF] Rudinskaya | Successful treatment of a patient with refractory adult-onset still disease with anakinra[END_REF][START_REF] Nakahara | A case report of a patient with refractory adult-onset Still's disease who was successfully treated with tocilizumab over 6 years[END_REF].

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Comparison of AOSD with and without AAA

Ruscitti's cohort study [START_REF] Ruscitti | Adult-onset Still's disease: evaluation of prognostic tools and validation of the systemic score by analysis of 100 cases from three centers[END_REF] was the study that best met our predefined criteria, namely a large number of patients with detailed information regarding clinical presentation, biologic therapy, complications, and duration of follow-up. None of the patients from this Italian study suffered from AAA. They were compared to the 19 patients with AOSD and AAA. No sex predilection was observed. Patients with AAA tended to be younger at diagnosis of AOSD. They also had more frequent weight loss. On the other hand, odynophagia, myalgia, hepatomegaly, splenomegaly, and lymphadenopathy were less common. No laboratory feature was positively associated with the development of AAA. Joint destruction was not specifically reported in Ruscitti's cohort study. However, 30% of patients presented a chronic disease course, which tends to be complicated by significant articular damage (table D).

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Discussion

AAA is an uncommon complication of AOSD. We estimated its overall prevalence at 0.88 % (95%CI [0.49-1.28]). Based on our study, a typical case of AOSD-related AAA would be a patient between 35 and 40 years old with active AOSD despite treatment, who develops a nephrotic syndrome 17 years after disease onset. This is the first systematic literature review regarding AAA as a complication of AOSD. We intentionally chose to include only patients with adult onset Still disease, because development of AAA in this population has never been specifically assessed. Furthermore, even though AAA in sJIA continuing into adulthood has previously been reported, case series and reports on this topic have insufficient information to allow detailed descriptive analysis as was done in our review.

As a result, AAA prevalence in AOSD has never been evaluated. Estimation of prevalence through case series carries limitations, but rarity of both diseases prevents us from conducting an epidemiological study. In JIA, the prevalence varies between 1.8% and 15% [START_REF] Dhillon | Amyloidosis in the rheumatic diseases[END_REF][START_REF] Filipowicz-Sosnowska | The amyloidosis of juvenile rheumatoid arthritis--comparative studies in Polish and American children. I. Levels of serum SAA protein[END_REF][START_REF] David | Amyloidosis in juvenile chronic arthritis: a morbidity and mortality study[END_REF][START_REF] Ozdogan | Juvenile chronic arthritis in a Turkish population[END_REF] based on old studies. More recently, a Finnish study showed an important decline in the occurrence of AAA in JIA, with no new case reported between 1990 and 2005 [START_REF] Immonen | Why can we no longer find juvenile idiopathic arthritisassociated amyloidosis in childhood or in adolescence in Finland?[END_REF]. Similarly, prevalence of end-stage renal disease due to AAA in rheumatoid arthritis, ankylosing spondylitis, and JIA has been decreasing, concurrent to a decline in the incidence of AAA over the previous decade. This has been attributed to improvement in the therapeutic management of rheumatic diseases [START_REF] Immonen | Longterm outcome of amyloidosis associated with juvenile idiopathic arthritis[END_REF][START_REF] Immonen | A marked decline in the incidence of renal replacement therapy for amyloidosis associated with inflammatory rheumatic diseases -data from nationwide registries in Finland[END_REF][START_REF] Vasala | More evidence of declining incidence of amyloidosis associated with inflammatory rheumatic diseases[END_REF]. AOSD is no exception, especially since the introduction of biotherapies around the year 2000. Approximately a quarter of patients with AOSD are currently treated with biotherapies [START_REF] Castañeda | Adult-onset Still's disease: Advances in the treatment[END_REF], allowing to control refractory forms of the disease [START_REF] Franchini | Efficacy of traditional and biologic agents in different clinical phenotypes of adult-onset Still's disease[END_REF][START_REF] Gerfaud-Valentin | Adult-onset Still's disease[END_REF][START_REF] Ia | Biologic treatments for adult-onset Still's disease[END_REF]. Thus, the low prevalence of AOSD-related AAA observed in our review is probably due to a combination of factors, namely better awareness of the disease, shorter diagnostic delay, and improved therapeutic management.

In our review, none of the cases reporting AAA as a complication of AOSD was published after 2012 and publication bias might be a possibility. Indeed, without novelty, submission of new AAA cases usually be rejected by journals. However, there is no new case of such association in the prospective and retrospective studies used to assess prevalence either, and for these articles

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editor's selection did not depend on the novelty of AAA as a complication of AOSD. The absence of new case since 2012 could also be justified by the very low incidence of AAA secondary to AOSD but in the past forty years there has never been such a long period without any published case of AAA secondary to AOSD.

The average delay between the first symptoms of AOSD and diagnosis of AAA was 16.8 years, similar to that observed by Lachman et al. [START_REF] Lachmann | Natural history and outcome in systemic AA amyloidosis[END_REF] for inflammatory diseases associated with AAA.

In our review, patients experienced significant diagnostic delay and/or multiple therapeutic failures, leading to chronic inflammation. As in rheumatoid arthritis and autoinflammatory diseases, longer disease activity is probably a key element for AAA development [START_REF] Koivuniemi | Amyloidosis is frequently undetected in patients with rheumatoid arthritis[END_REF]. Renal involvement, particularly nephrotic syndrome, was the most common presentation of AAA in AOSD. In a previous case series describing natural history and outcome in AAA, renal dysfunction was also the main manifestation of the disease, affecting 97% of patients [START_REF] Lachmann | Natural history and outcome in systemic AA amyloidosis[END_REF].

The first step in the management of AAA is to identify and treat its underlying etiology, in order to control the chronic inflammation causing AAA [START_REF] Lebrazi | Mécanismes de l'amylose et protéines impliquées[END_REF]. SAA has been evaluated as a prognostic marker for AAA, with favorable outcome associated with serum concentration inferior to 10 mg/L [START_REF] Gillmore | Amyloid load and clinical outcome in AA amyloidosis in relation to circulating concentration of serum amyloid A protein[END_REF]. However, SAA measurement is not widely available in current practice. Nevertheless, except for the minority of patients in whom SAA and CRP are not correlated, CRP monitoring is usually sufficient to follow and guide treatment of AOSD-related AAA.

Therapeutic management of AAA remains controversial. In AAA from any cause, early studies suggested some efficacy of azathioprine [START_REF] Shapiro | Regression of the nephrotic syndrome in rheumatoid arthritis and amyloidosis treated with azathioprine. A case report[END_REF], methotrexate associated with corticosteroids [START_REF] Komatsuda | Remission of the nephrotic syndrome in a patient with renal amyloidosis due to rheumatoid arthritis treated with prednisolone and methotrexate[END_REF], chlorambucil [START_REF] Ortiz-Santamaría | Treatment of AA amyloid with chlorambucil[END_REF], colchicine [START_REF] Zemer | Colchicine in the prevention and treatment of the amyloidosis of familial Mediterranean fever[END_REF], and cyclophosphamide [START_REF] Chevrel | Renal type AA amyloidosis associated with rheumatoid arthritis: a cohort study showing improved survival on treatment with pulse cyclophosphamide[END_REF]. However, others showed only a partial regression of renal disease with chlorambucil, corticosteroids, cyclophosphamide, and colchicine [START_REF] Harrington | Amyloidosis in adult onset Still's disease[END_REF][START_REF] Horlait | Secondary amylosis in Still's disease in an adult. Value of biopsy of the subcutaneous abdominal fatty tissue[END_REF][START_REF] Ishii | Systemic amyloidosis in a patient with adult onset Still's disease[END_REF][START_REF] Oh | Secondary renal amyloidosis in adult onset Still's disease: case report and review of the literature[END_REF][START_REF] Ghorbel | Amyloidosis in adult onset Still's disease[END_REF]. Therefore, despite these initial glimmers of hope, AAA remained until recently a difficult-to-treat complication with a dismal prognosis. This was well illustrated in a study of 374 patients with AAA, of whom 44% died within a median of 86 months [START_REF] Lachmann | Natural history and outcome in systemic AA amyloidosis[END_REF].

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Fortunately, biologic therapy may reverse this trend. In our review, tocilizumab lead to a renal remission, meanwhile anti-IL1 therapy was ineffective and TNF inhibitors lead to mixed outcomes: partial response but also 2 cases with disease progression.. However, those patients whose AAA responded poorly to biotherapies also presented active AOSD despite treatment. In 7 cases of sJIA with renal amyloidosis from the literature [START_REF] Immonen | Longterm outcome of amyloidosis associated with juvenile idiopathic arthritis[END_REF][START_REF] Duarte | Renal amyloidosis: an uncommon complication of juvenile idiopathic arthritis[END_REF][START_REF] La | Rapid and sustained response to tocilizumab, anti-interleukin-6 receptor antibody, in a patient with nephrotic syndrome secondary to systemic juvenile idiopathic arthritis-related amyloidosis[END_REF][START_REF] Sharma | Systemic Juvenile Idiopathic Arthritis with Amyloidosis: An Uncommon Complication with a Favourable Outcome[END_REF][START_REF] Topaloglu | Anti-interleukin 1 treatment in secondary amyloidosis associated with autoinflammatory diseases[END_REF][START_REF] Chantarogh | Clinical improvement of renal amyloidosis in a patient with systemic-onset juvenile idiopathic arthritis who received tocilizumab treatment: a case report and literature review[END_REF][START_REF] Gottenberg | Anti-tumor necrosis factor alpha therapy in fifteen patients with AA amyloidosis secondary to inflammatory arthritides: a followup report of tolerability and efficacy[END_REF], complete biological remission was described in 2 patients using tocilizumab, while partial remission was achieved in 5 patients using anakinra, tocilizumab, or etanercept. These results concur with studies pertaining to AAA in general. TNF inhibitors have demonstrated some efficacy in AAA secondary to rheumatic diseases [START_REF] Gottenberg | Anti-tumor necrosis factor alpha therapy in fifteen patients with AA amyloidosis secondary to inflammatory arthritides: a followup report of tolerability and efficacy[END_REF][START_REF] Fernández-Nebro | Long-term TNF-alpha blockade in patients with amyloid A amyloidosis complicating rheumatic diseases[END_REF], and in AAA irrespective of its cause [START_REF] Esatoglu | Long-term follow-up of secondary amyloidosis patients treated with tumor necrosis factor inhibitor therapy: A STROBEcompliant observational study[END_REF]. IL6-antagonists have also shown promising results [START_REF] Courties | AA amyloidosis treated with tocilizumab: case series and updated literature review[END_REF]. Similarly, efficacy of IL-1 antagonists has been reported in both AAA of undetermined etiology [START_REF] Lane | Safety and efficacy of empirical interleukin-1 inhibition using anakinra in AA amyloidosis of uncertain aetiology[END_REF] and in association with familial Mediterranean fever (FMF) [START_REF] Varan | Efficacy and safety of interleukin-1 inhibitors in familial Mediterranean fever patients complicated with amyloidosis[END_REF]. Thus, despite the scarcity of published cases, the current data pertaining to biologic therapy is promising. Several questions remain, namely regarding the effectiveness and long-term safety of biotherapies, as well as the role of the underlying inflammatory disease in selecting the optimal treatment for AAA.

AAA may lead to end-stage renal failure and require a renal transplant [START_REF] Sherif | Long-term outcome of live donor kidney transplantation for renal amyloidosis[END_REF]. Thanks to the CEREMAIA, we reported the first case of renal transplantation for AAA secondary to AOSD.

Recurrence of AAA in the renal graft has previously been reported [START_REF] Hartmann | Fifteen years' experience with renal transplantation in systemic amyloidosis[END_REF]100]. Canaud et al. [101] noted a 14% recurrence rate, which was associated with a significantly increased risk of mortality. Unfortunately, AAA developed in the graft of our patient, thus highlighting the importance of sustained AOSD control and regular screening for its complications.

Our review highlights the poor outcome of AOSD complicated by AAA. Four out of 19 patients died. These results concur with those of Smith and al., who followed 389 AOSD and JIA patients for a mean of 11 years. Sixteen of them developed AAA. Among those, 7 (43%) died, compared to 14 (3.7%) patients without AAA [102]. AOSD is heterogeneous in terms of clinical presentation, evolution, and severity, such that AAA is not its only prognostic factor. Other severe complications of AOSD can lead to death, including severe organ failure, MAS, and adverse treatment effects [103]. Other than AAA, life-threatening complications were rare in our

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patients. None of them presented MAS, which is one of the most common and serious complications of AOSD with an incidence of 12-15% [START_REF] Gerfaud-Valentin | Adult-onset Still's disease[END_REF]103]. Given that MAS tends to be an early complication of AOSD [104-106] with severe symptoms that rarely go unnoticed, its presence may expedite the diagnosis and treatment of AOSD, thus avoiding prolonged inflammation.

Finally, comparison of characteristics of AOSD with and without AAA has several limitations, including the small number of patients with AAA and the questionable comparability of the 2 groups. Indeed, the AOSD with AAA group is not a true cohort, but rather a compilation of case reports. Nevertheless, our work revealed some trends, namely increased risk of AAA in patients developing AOSD at a young age; this will need to be confirmed in future studies. We also observed a high prevalence of joint destruction in AOSD patients suffering from AAA (76%), which represents a more chronic form of the disease. Prevalence of joint destruction was twice as high as its previously reported prevalence in the chronic form of AOSD [107], thus suggesting an association between joint destruction and AAA. In FMF, joint involvement has also been identified as a risk factor for AAA [108,109]. This association may be explained by the

Case 3 :

 3 AOSD was diagnosed in a 17-year-old man with arthritis, odynophagia, and fever with polynuclear leukocytosis. He was initially treated with corticosteroids alone, achieving disease control for 5 years. Because of a recrudescence in disease activity, methotrexate, hydroxychloroquine, and immunoglobulins were consecutively administered. Diagnosis of AAA [Tapez ici]

  production of an amyloid precursor by the synovial membrane, as was shown by O'Hara et al. in patients with active rheumatoid arthritis [110]. [Tapez ici] Conclusion AA amyloidosis is a rare complication of AOSD, caused by persistent or recurrent inflammation due to suboptimal disease control. Early diagnosis and treatment of AOSD could prevent this complication. The growing use of biologic therapy is now allowing to control previously refractory disease. It may therefore contribute to a decline in AAA development. Finally, renal involvement is a frequent and serious early manifestation of AAA. As a result, closer surveillance of renal function and proteinuria should be considered in uncontrolled AOSD. [Tapez ici]

Figure A .

 A Figure A. Flow-chart diagram depicting selection process 57 articles selected : 45 were used to assess AAA prevalence and sixteen for the descriptive analysis of patients with AOSD and AAA.

Figure B .

 B Figure B. Number of new cases of AOSD-related AAA published per decade (colors should be used) Decline in the incidence of AAA and mortality due to AAA in AOSD over decades

Table A . Case series used for assessment of AAA prevalence in AOSD

 A

		Country of study	Number of patients	Women	Men	Follow-up (years)	Number of amyloidosis cases	Death (with amyloidosis /without amyloidosis)
	BYWATERS et al. 1971 [1]	ENGLAND	14	14	0	NS	0	0
	ESDAILE et al 1980 [8]	CANADA	6	NS	NS	NS	0	0/0
	ELKON et al 1982 [9]	ENGLAND	11	11	0	20.2	1	0/1
	LARSON et al 1984 [10]	USA	14	6	8	NS	0	0
	VIGNERON et al 1986 [11]	FRANCE	42	NS	NS	NS	2	NS
	REGINATO et al 1987 [12]	USA	23	11	12	NS	0	0/1
	WOUTERS et al [13]	NETHERLANDS	25	7	18	NS	0	0
	CUSH et al 1987 [14]	USA	21	13	8	NS	1	0/2
	FLIPO et al 1989 [15]	FRANCE	11	6	5	3	0	0
	CABANE et al 1990 [16]	FRANCE	8	NS	NS	14.5	3	2/0
	OHTA et al 1990 [17]	JAPAN	90	NS	NS	NS	1	1
	POUCHOT et al 1991 [18]	CANADA	60	NS	NS	6	0	NS
	SANCHEZ LORIA et al 1992 [19]	ARGENTINA	15	10	5	3.7	0	0
	SINGH YN. et al 1992 [20]	INDIA	27	7	20	2.2	1	0/1
	BAMBERY et al 1992 [21]	INDIA	18	10	8	15	1	1/1
	MASSON et al 1996 [22]	FRANCE	65	NS	NS	NS	0	1
	MOK et al 1998 [23]	CHINA	16	NS	NS	8	0	0
	LOUTHRENOO et al 2001 [24]	THAILAND	16	13	3	NS	0	0
	AL-ARFAJ et al 2001 [25]	SAUDI ARABIA	14	6	14	2.43	0	0
	APPENZELLER et al 2005 [26]	BRAZIL	17	9	7	4.8	0	0
	PAY et al 2006 [27]	TURKEY	95	50	45	1.1	0	0
	AKRITIDIS et al 2006 [28]	GREECE	11	6	5	8.2	0	0
	EVENSEN et al 2006 [29]	NORWAY	13	3	10	6.3	0	0/1
	MARZOUK et al 2006 [30]	TUNISIA	19	14	5	NS	3	0
	CHEIKHROUHOU et al 2007 [31]	TUNISIA	11	7	4	NS	1	0
	UPPAL et al 2007 [32]	KUWAIT	28	22	6	3.7	0	0
	SINGH S. et al 2008 [33]	INDIA	14	5	9	1.6	0	0
	MEHRPOOR et al 2008 [34]	IRAN	28	21	7	NS	0	0
	CAGATAY et al 2009 [35]	TURKEY	84	59	27	3.7	3	0
	ZHU et al 2009 [36]	CHINA	77	54	23	NS	0	0/1
	ZENG et al 2009 [37]	CHINA	61	45	16	3.6	0	0/6
	LEE et al 2009 [38]	SOUTH KOREA	71	63	8	3.2	0	0/9
	PRIORI et al 2010 [39]	ITALY	41	23	18	NS	0	0/1
	KONG et al 2010 [40]	CHINA	104	NS	NS	3.5	0	0
	RIERA et al 2011 [41]	SPAIN	41	25	16	9.4	1	0
	JIANG et al 2012 [42]	CHINA	70	44	26	2.5	0	0/3
	FRANCHINI et al 2012 [43]	ITALY	66	38	28	2.2	1	0/1
	KIM HA et al 2012 [44]	SOUTH KOREA	54	39	15	2.2	0	0
	ILIOU et al 2013 [45]	GREECE	44	23	21	7	0	0
	GERFAUD-VALENTIN et al 2014 [46]	FRANCE	57	30	27	8.4	0	0/3
	KIM YJ. et al 2014 [47]	SOUTH KOREA	82	60	22	3	0	0/2
	LIU et al 2015 [48]	CHINA	75	44	31	1	0	0
	KALYONCU et al 2016 [49]	TURKEY	356	210	146	1.83	0	0
	BALCI et al 2016 [40]	TURKEY	42	32	10	6.25	0	0/1
	RUSCITTI et al 2016 [51]	ITALY	100	34	66	3.5	0	0/16
			2157	1074	699	mean: 5.4±4.5	19	55
	[Tapez ici]

Table B . Summary of clinical and biological features of 17 patients with AOSD and AAA Patients with AOSD and AAA n=17 (%)

 B

	Clinical signs
	Fever	17 (100)
	Arthritis / Arthralgia	17 (100)
	Maculopapular erythema / Rash	11 (64.7)
	Pericarditis	3 (17.6)
	Splenomegaly	7 (41.2)
	Hepatomegaly	5 (29.4)
	Lymphadenopathy	5 (29.4)
	Myalgia	5 (29.4)
	Pharyngitis / Odynophagia	4 (23.5)
	Deterioration of general state	5 (29.4)
	Biological signs
	Leukocytosis	14 (82.3)
	with neutrophilia	14 (82.3)
	Elevation of ESR and/or CRP	14 (82.3)
	Negative ANA and RF	10 (58.9)
	Abnormal liver function tests	2 (11.7)


Deborah C Navarro Morales

Olga Kuldavletova

Gaëlle Quarck

Pierre Denise

Gilles Clément
email: gilles.clement@unicaen.fr

Time perception in astronauts on board the International Space Station

We perceive the environment through an elaborate mental representation based on a constant integration of sensory inputs, knowledge, and expectations. Previous studies of astronauts on board the International Space Station have shown that the mental representation of space, such as the perception of object size, distance, and depth, is altered in orbit. Because the mental representations of space and time have some overlap in neural networks, we hypothesized that perception of time would also be affected by spaceflight. Ten astronauts were tested before, during, and after a 6-8-month spaceflight. Temporal tasks included judging when one minute had passed and how long it had been since the start of the workday, lunch, docking of a vehicle, and a spacewalk. Compared to pre-flight estimates, there is a relative overestimation for the 1-min interval during the flight and a relative underestimation of intervals of hours in duration. However, the astronauts quite accurately estimated the number of days since vehicle dockings and spacewalks. Prolonged isolation in confined areas, stress related to workload, and high-performance expectations are potential factors contributing to altered time perception of daily events. However, reduced vestibular stimulations and slower motions in weightlessness, as well as constant references to their timeline and work schedule could also account for the change in the estimation of time by the astronauts in space.

INTRODUCTION

To construct a mental representation of our world, we perceive our environment by constantly processing sensory inputs from the visual, vestibular, and somatosensory systems. This representation is also influenced by our expectations and our experience, which are derived from our knowledge of the costs and consequences of acting in this environment. The central neurovestibular system naturally takes gravity into account during spatial orientation, balance, and motor control. This system is also indispensable for constructing our mental representation of the world. It continuously processes data from the visual, vestibular, and somatosensory channels to update our spatial maps. Previous research suggests that distances are underestimated when subjects are in weightlessness during parabolic [START_REF] Clément | Perception of egocentric distance during gravitational changes in parabolic flight[END_REF][START_REF] Clément | Horizontal and vertical distance perception in altered gravity[END_REF] or orbital [START_REF] Clément | Distance and size perception in astronauts during long-duration spaceflight[END_REF] flight. This distance underestimation is thought to be due to adaptive changes in the processing of gravitational information by the neurovestibular system that alter the construction of spatial maps [START_REF] Clément | Geometric illusions in astronauts during long-duration spaceflight[END_REF][START_REF] Clément | Long-duration spaceflight increases depth ambiguity of reversible perspective figures[END_REF] .

Exposure to weightlessness during spaceflight is known to elicit changes in vestibular responses, i.e., orientation illusions, errors in sensory localization, changes in vestibulo-ocular reflexes, and space motion sickness [START_REF] Clément | Neuroscience in Space[END_REF] . Cognitive tasks involving the neurovestibular system, such as mental rotation [START_REF] Frederici | Spatial reference in weightlessness: Perceptual factors and mental representation[END_REF][START_REF] Léone | The effect of gravity on human recognition of disoriented objects[END_REF][START_REF] Bigelow | Vestibular involvement in cognition: Visuospatial ability, attention, executive function, and memory[END_REF] , perceived orientation [START_REF] Dyde | The effect of altered gravity states on the perception of orientation[END_REF] and judgments of distance [START_REF] Clément | Geometric illusions in astronauts during long-duration spaceflight[END_REF] are also affected during spaceflight. Some astronauts and cosmonauts have reported a 'time compression syndrome' in orbit, whereby they perceive time as compressed relative to the perceptions gained during training and simulation [START_REF] Albery | Time and mass perception in non-terrestrial environments[END_REF] . In weightlessness, routine tasks require different cognitive demands than they would on Earth. Astronauts in orbit also report that they require more time than normal to execute standard mental activity [START_REF] Casler | Cognitive performance in space and analogous environments[END_REF][START_REF] Manzey | Mental performance during short-term and long-term spaceflight[END_REF] . 'Space fog' is another reported syndrome that affects cognitive performance during the first weeks of a mission [START_REF] Welch | Cognitive performance during prismatic displacement as a partial analogue of 'space fog[END_REF] . After astronauts have adapted to weightlessness and they re-enter the atmosphere, they also encounter a condition called 'entry motion sickness', which slows the speed of decision making and alters their ability to control the vehicle and their movements [START_REF] Ortega | Space and entry motion sickness[END_REF] . The neurovestibular challenges that occur when the crewmember returns to normal gravity include alterations in manual control [START_REF] Moore | Long-duration spaceflight adversely affects post-landing operator proficiency[END_REF] , inability to egress the vehicle [START_REF] Reschke | Verbal reports of neurovestibular symptoms in astronauts after short-duration space flight[END_REF] , postural imbalance [START_REF] Wood | Assessing sensorimotor function following ISS with computerized dynamic posturography[END_REF] , and impaired locomotion [START_REF] Mulavara | Physiological and functional alterations after spaceflight and bed rest[END_REF] .

Einstein [START_REF] Einstein | Über das Relativitätsprinzip und die aus demselben gezogenen Folgerungen[END_REF] revolutionized physics a century ago with his theory that space and time are intertwined [START_REF] Morrone | Keeping vision stable: rapid updating of spatiotopic receptive fields may cause relativistic-like effects[END_REF] . Based on this theory, we hypothesized that the absence of gravitational reference alters the construction of the mental representation of both space and time. Because astronauts underestimate distances and because the pace of their motion is slower in weightlessness, we hypothesized that astronauts would also underestimate the relative time between events. Prior to the present study, subjective perception of time during long-duration spaceflight had not been investigated. The results of this study have operational implications: altered perception of time during spaceflight might impact operations that require critical timing, such as docking operations and piloted landings.

Experimental protocol

In the flight study, the tests were performed before, during, and after 6-8-month spaceflights (M = 202, SD = 28 days). The pre-flight test sessions occurred at launch minus (L-) 205 ± 51 days, L-149 ± 55 days, and L-116 ± 45 days. In-flight test sessions were conducted approximately every month: i.e., on flight day (FD) FD17 ± 6 (M ± SD), FD46 ± 8, FD71 ± 6, FD99 ± 7, FD134 ± 8, and FD164 ± 7. After the astronauts returned to Earth, tests were performed at return plus (R + ) 1 day, R + 5 ± 1 day, and R + 9 ± 1 day.

A psychophysics test was administered, during which the subjects were asked to judge when one minute had passed. During this test, subjects wore a head-mounted display (Oculus Rift, Oculus VR, Menlo Park, CA), and used a finger trackball connected to a laptop to report their responses. Subjects were wearing earphones for listening to the instructions and for attenuating/masking noises from the spacecraft. Using the finger mouse, they pressed a 'go' button and waited one minute before pressing on a 'stop' button. Only the 'go' and'stop' buttons were displayed during the test. Subjects were not allowed to count the seconds passing by.

On the ground, this test was performed in the seated upright position; on the ISS, astronauts were in the free-floating conditions (Fig. 1). During the free-floating conditions there are no proprioceptive, tactile, or static vestibular cues that participate in spatial orientation. Previous studies have demonstrated that the perception of distance and the depth of objects are altered when free-floating in orbit [START_REF] Clément | Distance and size perception in astronauts during long-duration spaceflight[END_REF][START_REF] Clément | Geometric illusions in astronauts during long-duration spaceflight[END_REF] . To investigate the relationship between these changes in spatial perception and changes in time perception, the psychophysics test was performed in the same conditions as the previous spatial perception tests.

The second part of the experiment included a series of questions to document the potential changes in the astronauts' perception of longer periods of time, i.e. hours and days, while in orbit. The subjects doffed the head-mounted display and used the laptop keyboard to answer the following 5 questions: In the flight study, the start of the workday was defined as the termination of the morning daily planning conference (DPC), which occurred about one hour after the crew awoke. The time of lunch was determined by the astronauts' daily schedule. The dates of docking and EVA was determined from the Flight Program Integration Panel document provided by NASA Mission Integrations and Operations Office. Because this information was not available before and after the flight, the questionnaire was only administered during the flight.

In the ground-based study, the subjects performed the 'How Long is a Minute?' test in the laboratory while sitting upright and using identical computer hardware and software as on board the ISS. The subjects were also asked the perceived number of days since the last test session and how many minutes since they woke up and had breakfast (which they noted in a diary). The number of days between test sessions of the control subjects in the laboratory (M = 44.1, SD = 10.2) was similar to the number of days separating the 3 pre-flight sessions with the astronauts (M = 45.2, SD = 28.4), and so were the durations since wake up and lunch.

Statistical analysis

The errors between the perceived durations and the actual durations were calculated and time errors were computed in terms of percentage or days.

First, linear mixed models (LMM) were used to compare the ground-based responses of the 2 subject groups (astronauts and controls) and to establish whether they differed, and whether the results of the 3 test sessions differed (dependent variable: time error; fixed effects: tests sessions; group: astronauts or controls; random effects: subjects). A second set of LMM was used to compare measurements from different sessions within the flight phases (pre, in, post) in astronauts (dependent variable: time error; fixed effect: test sessions; random effects: subjects). A third set of LMM was used to compare the time errors during the pre-, in-, and post-flight sessions (depended variable: time error; fixed effects: flight phase; random effects: subjects). Post-hoc pairwise comparisons were then conducted using Bonferroni adjustment for multiple comparisons. Fixed effects estimates, confidence limits, and random effects standard deviations of these LMMs are provided in Supplementary Table 1.

When preflight measurements were not available in the astronauts (perceived durations since beginning of workday and lunch), an independent Sample Mann-Whitney test was used to compare their inflight responses with those of the control subjects on the ground. When these responses were not available in the control subjects neither (perceived durations since last docking and last EVA) a one-sample t-test was used to determine if the time errors during the flight were different from zero. Statistical analysis was performed with JASP 0.16.1.0 and IBM SPSS Statistics 27.0 software.

Reporting summary

Further information on research design is available in the Nature Research Reporting Summary linked to this article.

RESULTS

How long is a minute? On the ground, on average subjects indicated the duration of one minute to be 74.1 ± 19.5 s (mean ± SD) (Fig. 2). The LMM indicated no significant differences between the 3 test sessions in the control subjects and the astronauts [F(2,46) = 0.58, p = 0.56], which suggests that there was no learning or adaptive effect due to the repetition of the test. No significant differences were detected between the 2 groups of subjects [F(1,23) = 0.008, p = 0.928].

A LMM on the perceived durations of 1 min during the test sessions performed within each flight phase indicated that there were no significant differences within the sessions pre

Duration between test sessions

The time interval between the test sessions was approximately the same before flight (34-56 days) and for sessions FD45 to R + 1 (31.0 ± 10 d). The time interval between the last pre-flight session (L-116) and FD17 was much longer (133.8 ± 43 d). When comparing the astronaut's perceived duration between the test sessions, the largest time errors occurred on FD17 and R + 1, i.e., shortly after the transition in gravity levels (Fig. 3). The mean time error for these two sessions was -26.0% (SD = 24.3).

The LMM indicated no significant differences between the 2 test sessions in the control subjects and the astronauts [F(1,23) = 0.023, p = 0.881], which suggests that there was no learning or adaptive effect due to the repetition of the test. No significant differences were detected between the 2 groups of subjects [F(1,23) = 0.005, p = 0.942].

We compared whether the perceived durations were different within each flight phase. A LMM indicated that these differences were only significant within the post-flight measurements [F(2,27) = 4.913, p = 0.009), and particularly between R + 1 and R + 4 (p = 0.018) and between R + 1 and R + 8 (p = 0.027). We then pooled measurements within each flight phases and we conducted another LMM, which indicated no significant main effect of flight phases [F(2,98) = 2.390, p = 0.097].

Duration since start of workday and lunch

On the ISS, a typical workday begins with the morning DPC between the astronauts and the mission control center. On average, this experiment took place 4.4 ± 1.0 h (mean ± SD) after the end of the DPC. The same interval was used when testing the control subjects in the laboratory (4.5 ± 1.1 h). Two separated LMMs indicated that there were no significant differences in the perceived duration since the beginning of the workday between the 3 pre-flight sessions in the control subjects [F(2,42) = 0.286, p = 0.880], and between the 6 pre-flight sessions in the astronauts [F(5,54) = 0.469, p = 0.797). However, there were significant differences between the perceived durations of the astronauts in-flight and the control subjects on the ground (Mann-Whitney, p = 0.001) (Fig. 4a). Overall, the astronauts underestimated the duration since the beginning of their workday by -14.2% (SD = 24.2%). On average, the experiment occurred 2.7 ± 0.7 h (mean ± SD) after lunch in orbit and 2.5 ± 0.5 h after lunch in the laboratory. Two separated LMMs indicated that there were no significant differences between the 3 pre-flight sessions in the control subjects [F(2,42) = 0.014, p = 0.986], and between the 6 in-flight sessions in the astronauts [F(5,45) = 0.591, p = 0.707]. However, there were significant differences between the perceived durations of the astronauts in-flight and the control subjects on the ground (Mann-Whitney, p = 0.037) (Fig. 4a,b). Overall, the astronauts underestimated the duration since their lunch by -19.2% (SD = 36.1%).

It is possible, however, that the astronauts could have underestimated the duration after the start of their workday and/or lunch before the flight. Unfortunately, we could not measure these durations because we did not have access to the astronaut's schedule during their training.

Duration since last docking and extra-vehicular activity (EVA)

The docking of a vehicle to the ISS occurred an average of 26.0 ± 12.4 days (mean ± SD) before administering the questionnaire. Astronauts were quite accurate in their estimations of this duration: their errors were less than one day (+2.2%) (Fig. 5a). A LMM indicated that there were no significant differences in the perceived duration since the last docking between the 6 in-flight sessions [F(5,45) = 0.695, p = 0.630]. However, this time error was not significantly different from zero (t-test, p = 0.797).

Similarly, the EVAs occurred an average of 49.4 ± 15.9 days (mean ± SD) before administering the questionnaire. The time error for the estimations of this duration was 2.8 ± 11.9 days (+5.6%) (Fig. 5b). A LMM indicated that there were no significant differences in the perceived duration since the last EVA between the 6 in-flight sessions [F(5,45) = 0.312, p = 0.903]. However, this time error was not significantly different from zero (t-test, p = 0.167).

DISCUSSION

This study indicates that astronauts' perceived the duration of one minute to be less during spaceflight (-20.0%) than before flight. In addition, they underestimated durations ranging from 2 h (since lunch) and 5 h (since start of their workday) by -36.1% and -24.2%, respectively. They also underestimated the time elapsed since the last test session when there was a change in gravity level between these sessions (-26.5% on R + 1). However, they were essentially correct in estimating the durations in days elapsed since the last docking of a vehicle to the ISS (+2.2%) and since the last EVA (+5.6%).

The method for assessing time perception of a 1-min time period is the method of production, i.e. indicating a 1-min interval by pressing a button. Before the flight, a clock time of on average 74.1 s was judged as 60 s on average by the subjects. In other words, at a clock time of 60 s the subjects still thought it was perhaps 50 s, and waited longer before pressing the button. Similarly, the relative underproduction during the flight by the astronauts (59.6 s) as compared to before (74.5 s) refers to a relative overestimation of duration. In other words, astronauts in space feel that more time (the 60 s) has gone by after 60 s (they then pressed the button) than the astronauts on the ground who at 60 s felt a relatively shorter duration (say, 50 s) and waiting a little longer to press the button (on average at 74.5 s). This is a classic dissociation in the interpretation of a time production and a time estimation task. In the latter the observer waits through the designated time and then verbally reports clock time ('about 60 s).

The fact that there is a relative overestimation for the 1-min interval is not at odds with the relative underestimation of intervals in the hours range. There is ample evidence that different time intervals are governed by different mechanisms. The 1-min interval could still be just within the working memory span (and a clock mechanism could apply) but the hour time range is way beyond and memory processes apply [START_REF] Jokic | Individual differences in self-rated impulsivity modulate the estimation of time in a real waiting situation[END_REF][START_REF] Baldauf | Time perception as a workload measure in simulated car driving[END_REF] .

Why are the astronauts actually more accurate in their estimate of minutes during spaceflight? One possible interpretation is related to the way the astronauts' work is organized on board the ISS. By every ISS workstation, there is a daily minute-by-minute schedule displayed on a computer (the Onboard Short-Term Plan Viewer, or OSTPV). Superimposed on this schedule is a vertical red bar that moves from left to right and symbolizes the passing of time. Each day, the astronauts perform different activities (e.g. experiments, equipment maintenance, taking photographs, etc.) scheduled at a given time for a given duration. Consequently, crewmembers may have lunch at different time from one day to another. The large variability in the astronauts' time errors for estimating the time elapsed since the beginning of the workday and lunch is presumably due to the different type and schedule of activities they perform each day. The red bar displayed on the OSTPV indicates when crewmembers must start and end each activity. The speed of the red bar movement, and the comparison between the perceived time for completing an activity and the actual time when crewmember looks at the OSTPV presumably contribute to them becoming more accurate in their estimates of minutes during spaceflight.

According to Einstein's theory of relativity [START_REF] Einstein | Über das Relativitätsprinzip und die aus demselben gezogenen Folgerungen[END_REF] , an observer traveling at high speed will experience time passing more slowly than an observer at rest. At the speed of the International Space Station (28,160 km/h), this difference is in the order to 4 ×10 -11 s. This effect is negligible in our results, which show changes in the order of minutes or days.

A confounding factor for accurate time perception during space operations is the fact that different time scales are used on the ISS, which can be confusing. The official time is Greenwich Mean Time (GMT), but the astronauts and cosmonauts often use Houston time (GMT-5 h) or Moscow time (GMT + 2 h) to communicate with mission control centers and their families. Also, critical operations such as vehicle docking and EVA are documented in mission elapsed time, i.e. the time elapsed since the launch or entry into the airlock.

The conditions during spaceflight, including weightlessness, prolonged isolation in confined areas, stress related to workload, and high-performance expectations, are known to affect human physiological and psychological responses [START_REF] Clément | Neuroscience in Space[END_REF][START_REF] Casler | Cognitive performance in space and analogous environments[END_REF] . These conditions could also alter the perceived temporal relationships between events. During his 8-min EVA, Alexey Leonov [START_REF] Leonov | Perception of Space and Time in Outer Space[END_REF] clearly experienced an underestimation of time: 'I was disappointed,' he wrote in his post-flight report, 'that the time assigned for working outside the craft flew by very quickly. The entire period I remained in outer space seemed to be only 1 or 2 min.' Only 2 studies have been performed previously to assess time perception during spaceflight. The first experiment took place after the historical one-orbit flight of Yuri Gagarin, when Gherman Titov flew on the Vostok-2 for a full day (17 Earth orbits). The objective was to assess his ability to evaluate time intervals. After starting a stopwatch, he began to count 20 s in his head; when he estimated subjectively that 20 s had passed, he stopped the stopwatch and looked at the actual elapsed time. The results were recorded in his onboard diary. The average time estimates during the 4 in-flight sessions were 20.3, 20.2, 20.1, and 20.1 s. These estimates were not significantly different from those measured during training, but they were biased by the fact that he was counting in his head and he had continuous feedbacks on his performance [START_REF] Leonov | Perception of Space and Time in Outer Space[END_REF] .

The second experiment on time perception during spaceflight was performed on 4 astronauts during a 4-day Space Shuttle mission. The test was a classic time reproduction (non-counting) task. Subjects viewed a visual target traversing a display and, while it was obscured, estimated the time of its arrival at a predetermined point by any means other than counting. The target moved at various speeds, so that the duration of the task ranged from 2 to 16 s. The 4 astronauts were tested the day before their flight, each day during the flight, 3 h after landing, and again 3 days later [START_REF] Ratino | Quantification of reaction time and time perception during Space Shuttle operations[END_REF] . As the time duration of the task increased, the subjects tended to underestimate duration and these errors in duration estimates increased each day as the flight progressed. Three hours after landing the duration estimates were also significantly larger than on FD4 [START_REF] Ratino | Quantification of reaction time and time perception during Space Shuttle operations[END_REF] .

The results of the above experiment and the present study suggest that the ability to estimate brief intervals of time deteriorates during a space mission and shortly after landing. Similar effects were also observed in subjects exposed to hypergravity in a centrifuge [START_REF] Albery | Time and mass perception in non-terrestrial environments[END_REF] . A potential consequence of these effects is that crewmembers who need to make quick decisions and perform critical tasks during flight and re-entry may exhibit some delays in their responses, which could compromise safety.

Gibson [START_REF] Gibson | Events are perceivable but time is not[END_REF] pointed out that some measures of time are intrinsic, i.e., they are physical phenomena 'out there' in the world. These include the year (the Earth's rotation around the sun which, due to the tilt of the Earth's axis, yields a sequence of seasons), the month (the Moon's rotation around the Earth, with its visible phases), and the day (the Earth's rotation around its own axis, yielding dawn, Noon, dusk, and so on). These intrinsic measures of time contrast with the extrinsic measures of time, such as seconds, minutes, and hours. Whereas the duration of the year, the month, and the day are fixed by physical facts, the duration of the second is arbitrary; it is a convention that works solely because many people have agreed to it. The duration of the second can be changed. We could, if we wished, have only 10 h in the day, with minutes and seconds that were much longer. By contrast, we cannot change the physical duration of the day. The day is an event, in Gibson's sense, and is perceivable as any physical event. The second is not an event-it does not exist 'out there' to be perceived, but exists only in the mind, as a social convention. In addition, circadian and circalunar rhythms follow 24-h and 30-day cycles, respectively.

The results of our study indicate that astronauts are quite accurate when estimating intrinsic measures of time (days, months), but are inaccurate when estimating extrinsic measures of time (seconds, minutes, hours). However, the accuracy when estimating intrinsic measures of time (days) is altered following transitions between gravity levels (0 g to 1 g).

The results of psychophysical experiments on Earth indicate that subjects have distorted time perception when they are in stressful situations [START_REF] Nijhawan | Space and Time in Perception and Action[END_REF] . For example, during dangerous events such as plane and car accidents, many people report an overestimation to time: fractions of seconds can be perceived as minutes [START_REF] Arstila | Time slows down during accidents[END_REF] . During parachute jumps, subjects tend to overestimate the time intervals and delay the opening of their parachute [START_REF] Leonov | Perception of Space and Time in Outer Space[END_REF] . By contrast, the results of our study indicate that subjects either underestimate periods of time, or report accurate time perception. It is therefore unlikely that the results of this study are due to the stress of spaceflight.

Depending on the situation, time is perceived as either passing slowly or flying by. When we are bored, we feel that time passes more slowly than when we are entertained. Impulsive people feel that time is excruciatingly slow when nothing is happening [START_REF] Jokic | Individual differences in self-rated impulsivity modulate the estimation of time in a real waiting situation[END_REF] . Time experience and time judgments are also altered in depressive and manic patients [START_REF] Bschor | Time experience and time judgment in major depression, mania and healthy subjects. A controlled study of 93 subjects[END_REF] . In general, institutionalized individuals, such as individuals in homes for the elderly, whose days are highly regulated and monotonous, experience time as passing slowly, i.e. they overestimate time durations [START_REF] Baum | Time perception and psychological well-being in the elderly[END_REF][START_REF] Zulley | The influence of isolation on psychological and physiological variables[END_REF] . But the same individuals may also experience time as speeding up when enjoyable and memorable events occur, such as a visit from family members or social events [START_REF] Locsin | Time experience of selected institutionalized adult clients[END_REF][START_REF] Wittmann | The experience of time spent in hospital and its relation to psychological distress[END_REF] . This effect is also likely to happen during spaceflight. As indicated above, astronauts' working days are filled with various activities in a timeline that changes from one day to another. During this busy schedule, time is perceived as going faster, and durations in minutes or hours are therefore underestimated [START_REF] Gable | Time flies when you're having approach-motivated fun: effects of motivational intensity on time perception[END_REF] . Whereas, when unique events take place, such as vehicle docking and EVA, these days are memorized more accurately.

Vicario et al. [START_REF] Vicario | Optokinetic stimulation affects temporal estimation in healthy humans[END_REF] . showed that optokinetic stimulation influences time perception. Subjects overestimated time intervals after optokinetic stimulation compared with their estimations before optokinetic stimulation. Because optokinetic stimulation interacts with the vestibular system, for example by generating nystagmus or illusion of self-motion (vection), this result suggests that a stimulation of the vestibular system affects the perception of time intervals. In agreement with this hypothesis, Binetti et al. [START_REF] Binetti | Adaptive tuning of perceptual timing to whole body motion[END_REF] . found that subjects overestimated time duration during vestibular stimulation produced by whole-body rotations.

The role of the vestibular system in cognitive processes such as spatial orientation, navigation, object recognition, and memory, has been demonstrated in many studies [START_REF] Hitier | Vestibular pathways involved in cognition[END_REF] . For example, patients with central vestibular lesions exhibit difficulties with counting tasks, immediate and short-term memory [START_REF] Jacob | Psychiatric consequences of vestibular dysfunction[END_REF][START_REF] Smith | Does vestibular damage cause cognitive dysfunction in humans?[END_REF][START_REF] Hanes | Cognitive-vestibular interactions: a review of patient difficulties and possible mechanisms[END_REF] , and alterations in the perception of size and distance of objects [START_REF] Clément | Mental representation of space in vestibular patients with otolithic or rotatory vertigo[END_REF] . The central neurovestibular system naturally takes gravity into account for spatial orientation, balance, and motor control. This system continuously processes data from the visual, vestibular, and somatosensory channels to update our spatial maps. Previous research suggests that astronauts underestimate distances in weightlessness [START_REF] Clément | Distance and size perception in astronauts during long-duration spaceflight[END_REF][START_REF] Clément | Geometric illusions in astronauts during long-duration spaceflight[END_REF] . The interpretation for this distance underestimation is that the adaptive changes in the processing of gravitational information by the neurovestibular system alter the construction of spatial maps. One plausible interpretation for the results of the present study is that the absence of gravitational reference alters the construction of the mental representation of temporal events, and this results in underestimation of the relative time between events. Due to the absence of static otolith inputs at rest, the vestibular system is less stimulated in weightlessness than in normal gravity. In addition, motions are slower in weightlessness: it takes about 50% more time in orbit to execute the same experimental procedures as on Earth. Less stimulation of the vestibular system could lead to an underestimation of the duration of time in weightlessness. In agreement with this hypothesis, perception of both distances 1,2 and durations [START_REF] Clément | Perception of time in microgravity and hypergravity during parabolic flight[END_REF] are altered during transient exposure to weightlessness in parabolic flight.

Our previous studies indicated that astronauts' subjective perception of their body motion and position and the size and distance of objects were altered during spaceflight [START_REF] Clément | Distance and size perception in astronauts during long-duration spaceflight[END_REF][START_REF] Clément | Geometric illusions in astronauts during long-duration spaceflight[END_REF] . The present study shows that astronauts' perception of durations ranging from one minute to several hours (which are human conventions) is also altered during long-duration spaceflight, and strikingly so in about the same percentage as for perceived distances [START_REF] Clément | Distance and size perception in astronauts during long-duration spaceflight[END_REF] . No significant differences were seen across flight days, which indicate that these alterations occurred within 2 weeks in orbit and that no adaptation took place during long-duration spaceflight. This observation raises operational concerns regarding the ability for crewmembers to manually perform docking and landing maneuvers after two weeks in orbit without any assistance.

These results also support the existence of an overlapping perception of time and space. It has been proposed that representations of space and time both share the same metrics and cortical network, presumably located in the right parietal cortex (for stimuli shorter than one-second duration) [START_REF] Walsh | A theory of magnitude: common cortical metrics of time, space and quantity[END_REF] . A potential neuronal basis for the interaction between these representations comes from neurophysiological recordings in rodents hippocampus and enthorhinal cortex (which have connections with the parietal cortex) showing that place and grid cells can simultaneously code for space and time [START_REF] Whitlock | Navigating from hippocampus to parietal cortex[END_REF][START_REF] Riemer | On the (a)symmetry between the perception of time and space in large-scale environments[END_REF] . In agreement with this interpretation, recent brain imaging studies have shown that the spontaneous or evoked activity in the right parietal cortex was modified in astronauts after spaceflight 46-50 .

  METHODSParticipants 10 healthy crewmembers (9 male, 1 female; age M = 44.1, SD = 4.6) who flew on the International Space Station (ISS) participated in this study. All crewmembers passed a United States Air Force Class III medical examination and had no known history of vestibular or oculomotor abnormalities. 15 healthy subjects (6 females, 9 males; age M = 43.2, SD = 18.8) participated in a control study in the laboratory. The test procedures were approved by the European Space Agency Medical Board and the NASA Johnson Space Center Institutional Review Board and were performed in accordance with the ethical standards laid down in the 1964 Declaration of Helsinki. All subjects gave a written informed consent before participating in the study. Informed consent was obtained from the subject for publication of identifying information/images in an online open-access publication.

  (a) How long has it been since the last time you performed this test?; (b) How long has it been since you started your work day?; (c) How long has it been since you had your lunch?; (d) How long has it been since the last vehicle docked at the ISS?; and (e) How long has it been since the last extra-vehicular activity (EVA)?

Fig. 1

 1 Fig. 1 Astronaut performing the experiment while free-floating on board the ISS. Photo credit NASA.

  -flight [F(2,18) = 0.939, p = 0.409], in-flight [F(5,45) = 0.593, p = 0.705] and post-flight [F(2,18) = 1.212, p = 0.321]. When measurements within each flight phases were pooled, the LMM indicated a significant main effect of flight phases [F(2,108) = 15.050, p < 0.001]. Post-hoc tests indicated that this difference was significant between pre-flight and in-flight (p < 0.001) and between post-flight and in-flight (p = 0.002), but not significant between pre-flight and post-flight (p = 0.484). During the flight, the averaged perceived duration of one minute was 59.6 ± 9.1 s (mean ± SD), which corresponds to a 20.0% decrease from before flight (74.5 ± 20.2 s). Interestingly, although the perceived duration of one minute was less during flight relative to before flight, the perception during flight was essentially accurate throughout the 6 in-flight sessions.

Fig. 2

 2 Fig. 2 How long is a minute. Box and whisker plots of 10 astronauts' perceived duration of one minute before (L-), during (FD), and after (R+) spaceflight. Filled symbol represents the mean, center line represents the median, bounds of box represent the first and third quartiles, and whiskers represent the minimum and maximum values in the set. The dotted line represents the average of all pre-flight measurements in the astronaut group. The gray area represents the mean ± standard deviation of measurements taken from 15 control subjects in the laboratory during 3 test sessions separated by approximately one month. L-Days before launch, FD Flight days, R+ Days after return. *p < 0.05 (linear mixed model).

Fig. 3

 3 Fig. 3 How long since last test session. Box and whisker plots of the error in 10 astronauts' perceived duration since the last test session before (L-), during (FD), and after (R+) spaceflight. Filled symbol represents the mean, center line represents the median, bounds of box represent the first and third quartiles, and whiskers represent the minimum and maximum values in the set. The dotted line represents the average of all pre-flight measurements in the astronaut group. The gray area represents the mean ± standard deviation of measurements taken from 15 control subjects in the laboratory. *p < 0.05 (linear mixed model).

Fig. 4

 4 Fig. 4 Time error for durations in hours. Box and whisker plots of the error in perceived duration since the start of the work day (a) and since lunch (b) in 10 astronauts during spaceflight and 15 subjects during 3 sessions (S1, S2, S3) in the laboratory. Filled symbol represents the mean, center line represents the median, bounds of box represent the first and third quartiles, and whiskers represent the minimum and maximum values in the set. *p < 0.05 (Sample Mann-Whitney test).

Fig. 5

 5 Fig. 5 Time error for durations in days. Box and whisker plots of the error in 10 astronauts' perceived duration since the last vehicle docking to the ISS (a) and the last spacewalk (Extra-Vehicular Activity, EVA) (b). Filled symbol represents the mean, center line represents the median, bounds of box represent the first and third quartiles, and whiskers represent the minimum and maximum values in the set. *p < 0.05 (one-sample t-test).

npj Microgravity (2023) 6 Published in cooperation with the Biodesign Institute at Arizona State University, with the support of NASA 1234567890():,;

Published in cooperation with the Biodesign Institute at Arizona State University, with the support of NASA npj Microgravity (2023) 6

npj Microgravity (2023) 6Published in cooperation with the Biodesign Institute at Arizona State University, with the support of NASA

ACKNOWLEDGEMENTS

The author thank Jean-François Clervoy for sharing his thoughts on time perception during space travel; Robert Macgregor, François Derache, and Maurice Marnat for their help in implementing this experiment; and Kerry George for editing the manuscript. European Space Agency provided access to the astronauts and the International Space Station (no grant number) and Centre National d'Etudes Spatiales provided the equipment and travel support for the investigators (grant # 2021/480001115).

DATA AVAILABILITY

Data will be made available on reasonable request to Dr. Gilles Clement (gilles.clement@unicaen.fr).

AUTHOR CONTRIBUTIONS

D.C.N.M., O.K., G.Q., and P.D. contributed to data analysis and manuscript; G.C. contributed to study design, data collection, data analysis, and manuscript.

COMPETING INTERESTS

The authors declare no competing interests.

ADDITIONAL INFORMATION Supplementary information

The online version contains supplementary material available at https://doi.org/10.1038/s41526-023-00250-x.

Correspondence and requests for materials should be addressed to Gilles Clément.

Reprints and permission information is available at http://www.nature.com/ reprints Publisher's note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.


C Almarcha

B Radisson

E Al Sarraf

E Villermaux

B Denet

J Quinard

Rapid Communications Interface dynamics, pole trajectories, and cell size statistics

We demonstrate the possibility to reproduce the experimental evolution of an interface, here a flame front, through the trajectory of a few poles whose position in the complex plane expresses the interface shape. These poles are analytical solutions of the Sivashinsky equation and they evolve according to an ordinary differential equation. The direct comparison with experimental flame fronts propagating in a quasi-two-dimensional configuration is made at the nonlinear but deterministic stages of the front dynamics, reproducing a cell creation and fusion process. At later times, when the front is sensitive to noise as in the Kardar-Parisi-Zhang equation, we demonstrate that the cell size distribution is still ruled by the pole attractive nature. DOI: 10.1103/PhysRevE.98.030202

An interface is a thin out-of-equilibrium region separating a metastable bulk phase from a stable one. At this layer, phase transitions or reactions can occur [1][2][3], involving diverse fluxes with diverse lengths and timescales. A paradigmatic example is a flame front, a submillimeter interface that propagates into a mixture of reacting gas and transforms it into products. Through this thin layer, density, normal velocity, and temperature change by an order of magnitude due to the energy released by hundreds of competing exothermic chemical reactions. When this interface remains flat, its normal propagation velocity is constant and can be computed in a onedimensional model taking into account the chemical reaction rates and the diffusivities of the species and temperature. However, interfaces suffer various instabilities and in general do not remain flat. Their evolution, sometimes modeled by partial differential equations (PDEs), is routinely studied by direct numerical simulations.

The corrugations of a flame can be induced by an external forcing, when it is propagating in turbulent flow [4]. In that case the flame is wrinkled by the underlying flow where it propagates. But flame wrinkling can also develop in the absence of forcing, due to the intrinsic Darrieus-Landau (DL) instability [5]. The increase of volume and normal velocity through the flame leads to this nonlocal hydrodynamical instability, which is not only encountered in the domain of combustion but also in ionizing waves in inertial confinement fusion or in thermonuclear process in white dwarfs to name a few [START_REF] Liberman | Introduction to Physics and Chemistry of Combustion[END_REF][START_REF] Clavin | Combustion Waves and Fronts in Flows[END_REF]. A nonlinear PDE, the Michelson Sivashinsky (MS) equation [START_REF] Sivashinsky | [END_REF], is supposed to reproduce the DL instability in flames and their evolution but the detailed comparison with real flame experiments is tricky and has been limited up to now to the saturation of a single wavelength [9]. When a moderate external forcing is acting on the flame as a source of noise, the MS equation can be extended to a forced MS equation (FMS) [10] whose structure is close to the Kardar-Parisi-Zhang (KPZ) equation [11], the simplest nonlinear Langevin equation for a local growth, extensively used to describe the interface growth by additive white noise [12]. They only differ by nonlocal instability term in FMS. Such extensions of the KPZ equation (sometimes named HKPZ equation [13]) with the addition of nonlocal instability terms [14] can be obtained in the context of molecular beam epitaxy leading to Asaro-Tiller-Grinfeld elastic instability [15], for the displacement of fluids with different viscosities leading to a Saffman-Taylor instability, for solidification fronts under the Mullins-Sekerka instability, or for reactive infiltration in porous media [13]. In contrast, growing interfaces in nematic liquid crystals [16] and flameless paper combustion [17], do not possess a nonlocal instability term and satisfy KPZ equation scalings, at least at sufficiently large scale [18].

Flame fronts are classically described by fractal dimensions [19,20], wrinkling factors [21], and various scalings in experiments [4] or numerical simulations [22][23][24][25]. Here we show that the deterministic nonlinear dynamics of a whole flame can be recovered using the MS equation with a very good accuracy. Moreover, since pole solutions to the MS equation exist [26], we demonstrate that experimental flame shapes can be described analytically by the trajectory of a few poles evolving in the complex plane according to a few coupled ordinary differential equations (ODEs). In addition, we show that in the late time evolution, when the importance of noise cannot be neglected anymore and the direct comparison with deterministic pole trajectories fails, the geometrical statistical description of the flame is still ruled by the pole attraction natural dynamics.

The experimental facility consists of a Hele-Shaw cell [27][START_REF] Sharif | Premixed-gas flame propagation in Hele-Shaw cells[END_REF][START_REF] Wongwiwat | Flame propagation in narrow channels at varying Lewis number[END_REF]: two vertically oriented glass-ceramic plates, 0.5 m wide and 1.5 m high, separated by a 5-mm gap width, closed on the sides, opened at the top are continuously filled with a premixed propane-air mixture at the desired equivalence ratio until the flame is ignited at the top (Fig. 1). The evolution of the luminous flame front location is recorded using a high-speed camera (500 fps). Qualitative results in this twodimensional geometry are presented in [START_REF] Almarcha | [END_REF], and emphasize its interest compared to tube propagation [31]. An example of evolution in the course of time is reported in Fig. 2(a) for lean propane air flame at equivalence ratio 0.9. The field of view is 203 mm large, with 0.006 s between each frame showing an initial linear stage with growth of corrugations from an initially flat flame. Soon, cusps, local crests pointing toward burnt gases, grow and merge, thus coarsening the pattern. Eventually new cusps are formed in locally flat zones.

Our first aim is to compare quantitatively this cusp dynamics to the integration of the MS equation. The latter accounts for the evolution of the position φ(x, t ) of the flame front location along the (horizontal) axis x defined by its mean position. This equation was originally derived in the limit of long wavelengths and unburnt to burnt gas expansion ratios θ = ρ u /ρ b close to unity, and later extended to larger gas expansions [32] and nonpotential flows [33][34][35] while keeping the same functional structure. It writes as

φ t + u A 2 φ 2 x = 4 σ M k c φ xx k c + I (φ, x) , ( 1
)
where I (•, x) is a nonlocal operator corresponding to the multiplication by |k| in Fourier space. It stands for the Darrieus-Landau instability coming from Euler's equations in a potential flow. The velocity u A depends on the nature of the reactants and on θ while σ M and k c are, respectively, the maximum growth rate and the cutoff wavelength of the associated dispersion relation

σ = 4 σ M k c |k| - k 2 k c . ( 2
)
These two quantities are measured experimentally in the linear stage of the front destabilization. Starting from an initially flat flame front anchored on the upper border of the Hele-Shaw cell, we analyze by way of Fourier transform the successive records of the flame contour when it penetrates the Hele-Shaw cell, as soon as the fresh gas mixture intake flow is stopped. A plate with periodic ripples placed a few millimeters above the burner allows one to excite a specific mode. This original procedure gives access to the linear growth rate. An example for propane air mixture with equivalence ratio 0.9 is reported in Fig. 3 and is compared to the best fit of the dispersion relation (2). The growth rates obtained compare favorably with the few measured previously in tubes [36], where, however, only the growth rate of the most unstable mode was accessible, thus demonstrating the interest of the Hele-Shaw burner for the study of flame dynamics [27,37]. The MS equation (1) admits exact "pole-decomposable" periodic solutions [26] of the form

φ = -A 2N n=1 ln sin π (x -z n ) , ( 3
)
where is the width of the domain and A = 8σ M /u A k 2 c . The poles come in complex conjugate pairs, N being the number of poles with a positive imaginary part. Each pole z n evolves according to the following coupled ODEs:

żn = 4σ M k c ⎧ ⎨ ⎩ 2N p =n 2π k c cot π z p -z n -i sgn[Im(z n )] ⎫ ⎬ ⎭ , ( 4
) where Im(z n ) is the imaginary part of z n .

In order to assess the validity of this system of equations, we used as an initial condition a frame (in bold) extracted from the experimental sequence reported in Fig. 2(a). By way of the Levenberg-Marquardt least-squares algorithm, we interpolated this experimental front presenting 16 cusps as the composition of 32 conjugate pole pairs. Their locations are represented by points over the frame in Fig. 2(b). Poles with a small imaginary part (vertical coordinate) induce visible cusps while the others are flattening the cells between the cusps. The pole trajectories (4) were integrated numerically in time [Fig. 2(c)]. The evolution of the simulated front exhibits cusp merging, and compares favorably with the evolution of the experimental one at the same instants of time. The remaining differences have their origin in the boundary conditions of the experimental front, the limited resolution of the camera, the flame front detection procedure, and the choice in the initial number of poles which, when integrating the ODEs (4), does not change in time. The creation of cells seen in Fig. 2(c) is thus entirely caused by poles present in the initial condition that get closer to the real axis, leading to visible new crests. At later times, the integration of (4) leads to the merging of all the cusps producing a unique cell, indicating that all the poles [and p(d ) in the inset] between successive cusps for propane-air flames with equivalence ratios between 0.9 and 1.4, in linear scale (a) and semilogarithmic scale (b). The result of the numerical integration of the MS equation is also reported. In the plots of the distribution of normalized distances ξ , the solid line is the gamma distribution with order ν = 2. are at the same abscissa. This is not what is happening in the experiment: new cusps separating new cells are appearing in the course of time [START_REF] Almarcha | [END_REF] showing that the comparison with pole trajectories is limited in time if there is no addition of new poles. Sources of physical noise, for instance fluctuations in the upstream flow, are capable of creating these new poles [38], leading to a statistical steady state of the flame. To take into account the physical noise effects in numerical simulations, it is then possible to periodically introduce poles in the numerical simulation as proposed in [39], or to proceed to integration of the MS equation with the addition of a noise term introduced in the same way as in the KPZ equation. Another limitation of both the MS equation and of the pole description is that the front is single valued, which is not always the case in the experiment, as seen on the late frames of Fig. 2(a), an effect even more pronounced at a higher equivalence ratio.

E.R = 0.9 E.R = 1.1 E.R = 1.3 E.R = 1.4 Sivashinsky (a) (b) FIG. 5. Distribution of distances p(ξ = d/ d )
However, we use the MS model and its pole solution to investigate important general aspects of the flame dynamics and of its geometrical features. In particular, we focus on the distribution of cell sizes (i.e., the distance d between two cusps). Considering two pairs of conjugate poles separated by a distance d sufficiently small for neglecting the interactions with other pairs of poles and in the limit of small imaginary parts, Eq. ( 4) reduces to ḋ ≈ -2D π cot πd , (5) where D = 16σ M /k 2 c , whose solution at a short distance is singular as

d 2 (t ) ≈ d 0 2 -4Dt, (6)
showing that the squared distance between two cusps initially separated by d 0 decreases linearly in time until fusion at d 0 2 /4D. This is verified experimentally in Fig. 4. Cusp fusion means adjacent cells sizes addition [41], and if these additions are made at random, irrespective of the initial cell sizes at merging, then the distribution of their sizes p(d ) must be stable by self-convolution [40], and therefore of the gamma type [40,41]:

p(ξ = d/ d ) = ν ν (ν) ξ ν-1 e -νξ , (7)
where ν depends on the microscopic dynamics producing the sizes d in Eq. ( 7). The aggregation dynamics in Eq. ( 6) is such that all cells initially smaller than the current average size d have a merging time smaller than the mean. The incomplete convolution theory [40] explains that this should occur for a critical size given by 2 d /ν, thus selecting the order ν = 2. Experimental distributions are reported in Fig. 5(a) for propane-air mixtures at equivalence ratios between 0.9 and 1.4. After rescaling by the mean size d , all distributions collapse, showing that the creation-fusion dynamics of the cells is universal, and is reasonably close to that obtained by numerical integration of the MS equation. Here for 200 unstable modes in the numerical domain, the round-off and truncation errors [42] create a numerical noise [38] that induces cusp creations and leads to a stationary distribution. When increasing the level of noise, the cusp creations increase [38] leading to smaller mean cell size, but we found that the normalized distribution remains the same, extremely close to the expected gamma of order ν = 2. The experimental distributions present a more rapid falloff for very large cells. The source of this discrepancy may be in gravity (not present in the numerics) which limits the height and the width of the larger cells. This will be discussed in a forthcoming paper.

We have thus demonstrated that the rich dynamics of an interface can be represented by a single equation; in the present case, the MS equation which is a form of the KPZ equation where a nonlocal instability term is added or replaces the source of noise term. In that framework, the dynamics of the interface in the absence of noise amounts to computing the trajectories of only a few poles in the complex plane. This method, which can be extended to three-dimensional flames [43], provides insight into the local dynamics of the front exhibiting cells fusion, and into its statistical features in the presence of noise, namely, the cell size distribution. It is a major simplification effort whose interest is likely to expand beyond the domain of flame fronts, and may concern any physical phenomenon which presents a developing interface.

(

  Received 27 February 2018; published 14 September 2018)

FIG. 1 .

 1 FIG. 1. Left: Sketch of the vertical Hele-Shaw cell. Right: Zoom on a lean propane-air flame front at different heights (inverted colors). The flame is initially anchored on the upper edge. When the upstream flow of reacting gas is stopped the flame moves downward and is destabilized.

FIG. 2 .

 2 FIG. 2. (a) Downward propagation of an experimental propaneair flame front (equivalence ratio is 0.9). Red dots are placed on the cusps in order to track their trajectory. (b) Poles evaluated from the initial condition are represented by blue points. (c) Evolution of the pole solutions according to the system of ODEs (4).

1 FIG. 3 .

 13 FIG.3. Measurements of growth rate σ for lean propane air flame at equivalence ratio 0.9. The solid line is the least-square parabolic fit in accordance with the dispersion relation (2) exhibiting the maximum σ M and cutting the horizontal axis at k c .

FIG. 4 .

 4 FIG.4. Detail of a two-cusp fusion process in experiments. Field of view is 22 mm large. Inset: evolution of the squared distance between cusps in time (normalized by initial distance).

We thank Guy Joulin for many stimulating discussions and the "Agence Nationale de la Recherche" for funding of the ANR "PDF" ANR-14-CE05-0006 and ANR "FISICS" ANR-15-CE30-0015-03, the Excellence Initiative of Aix-Marseille University-A*MIDEX, and Labex MEC, for funding.